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=210mm =165mmReIm [email protected] Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100190, ChinaSchool of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China =15pt We investigate the initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii (GP) equations with 4× 4 Lax pair on the half-line. The solution of this system can be obtained in terms of the solution of a 4× 4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. The relevant jump matrices of the RH problem can be explicitly found using the two spectral functions s(k) and S(k), which can be defined by the initial data, the Dirichlet-Neumann boundary data at x=0. The global relation is established between the two dependent spectral functions. The general mappings between Dirichlet and Neumann boundary values are analyzed in terms of the global relation. An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4× 4 Lax pair on the half-line Zhenya Yan=============================================================================================================================15ptIn 1967, Gardner, Greene, Kruskal, and Miura presented a powerful inverse scattering transformation (IST) to investigate solitons of the KdV equation with an initial value problem. After that this method was used to solve the initial value problems for many integrable nonlinear evolution partial differential equations (PDEs) with the Lax pairs. Moreover, the IST method was further extended such as the Fokas' unified transformation method. The Fokas unified method can be used to study the initial-boundary value problems for some integrable nonlinear integrable evolution PDEs with 2× 2 and 3× 3 Lax pairs on the half-line and the finite interval. To the best of our knowledge, so far there is no work on the IBV problems of integrable equations with 4× 4 Lax pairs on the half-line. In this paper, We investigate the initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii (GP) equations with 4× 4 Lax pair on the half-line. The solution of this system can be obtained in terms of the solution of a 4× 4 matrix Riemann-Hilbert problem formulated in the complex k-plane. The relevant jump matrices of the RH problem can be explicitly found using the two spectral functions s(k) and S(k), which can be defined by the initial data, the Dirichlet-Neumann boundary data at x=0. The global relation is established between the two dependent spectral functions. The general mappings between Dirichlet and Neumann boundary values are analyzed in terms of the global relation. § INTRODUCTIONThe initial value problems for many integrable nonlinear evolution partial differential equations (PDEs) with the Lax pairs can be solved in terms of the inverse scattering transform (IST) <cit.>. After that, there exist some important extensions of the IST such as the Deift-Zhou nonlinear steepest descent method <cit.> and the Fokas unified method <cit.>. Particularly, the Fokas unified method can be used to study the initial-boundary value problems for both linear and nonlinear integrable evolution PDEs with 2× 2 Lax pairs on the half-line and the finite interval, such as the nonlinear Schrödinger equation <cit.>, the sine-Gordon equation <cit.>, the KdV equation <cit.>, the mKdV equation <cit.>, the derivative nonlinear Schrödinger equation <cit.>, Ernst equations <cit.>, and etc. (see Refs. <cit.> and references therein). Recently, Lenells extended the Fokas method to study the initial-boundary value (IBV) problems for integrable nonlinear evolution equations with 3× 3 Lax pairs on the half-line <cit.>. After that, the idea was extended to study IBV problems of some integrable nonlinear evolution equations with 3× 3 Lax pairs on the half-line or the finite interval, such as the Degasperis-Procesi equation <cit.>, the Sasa-Satsuma equation <cit.>, the coupled nonlinear Schrödinger equations <cit.>, and the Ostrovsky-Vakhnenko equation <cit.>. To the best of our knowledge, so far there is no work on the IBV problems of integrable equations with 4× 4 Lax pairs on the half-line.The aim of this paper is to develop a methodology for analyzing the IBV problems for integrable nonlinear evolution equations with 4× 4 Lax pairs on the half-line by extending the method <cit.> for the integrable nonlinear PDEs with 2× 2 and 3× 3 Lax pairs. In this paper, we will study the IVB problem of the integrable spin-1 GP equations {[ ị q_1t+ q_1xx-2α(|q_1|^2+2|q_0|^2)q_1-2αβ q_0^2q̅_-1=0,;ị q_0t+ q_0xx-2α(|q_1|^2+|q_0|^2+|q_-1|^2)q_0-2αβ q_1q_-1q̅_0=0,; ị q_-1t+ q_-1xx-2α(2|q_0|^2+|q_-1|^2)q_-1-2αβ q_0^2q̅_1=0, α^2=β^2=1, ]. with the initial-boundary value conditions[Initial conditions:q_j(x, t=0)=q_0j(x)∈𝕊(ℝ^+), j=1, 0, -1, 0<x<∞,; Dirichletboundaryconditions: q_j(x=0, t)=u_0j(t),j=1, 0, -1,0<t<T,; Neumannboundaryconditions:q_jx(x=0, t)=u_1j(t), j=1, 0, -1, 0<t<T, ]where the complex-valued spinor condensate wave functions q_j=q_j(x,t),j=1,0,-1 are the sufficiently smooth functions defined in the finite region Ω={(x,t) |x∈ [0, ∞),t∈ [0, T]} with T>0 being the fixed finite time, the overbar denotes the complex conjugate, 𝕊(ℝ^+) denotes the space of Schwartz functions,the initial data q_0j(x), j=1,0,-1and boundary data u_0j(t), u_1j(t),j=1,0, -1 are sufficiently smooth and compatible at points (x,t)=(0, 0).The spin-1 GP system (<ref>) can describe soliton dynamics of an F=1 spinor Bose-Einstein condensates <cit.>. The four types of parameters: (α, β)={(1,1), (1, -1), (-1, 1), (-1, -1)} in the spin-1 GP system (<ref>)correspond to the four roles of the self-cross-phase modulation (nonlinearity) and spin-exchange modulation,respectively, that is,(attractive, attractive), (attractive, repulsive), (repulsive, attractive), and ((repulsive, repulsive).In particular, Eq. (<ref>) with the attractive meanCfield nonlinearity and ferromagnetic spin-exchange modulationwas shown to possess multi-bright soliton solutions <cit.>.Eq. (<ref>) with the repulsive mean-field nonlinearity and ferromagnetic spin-exchange modulation was shown to possess multi-dark soliton solutions <cit.>. Moreover, double-periodic wave solutions ofEq. (<ref>) were also found <cit.>. System (<ref>) is associated with a variational principle iq_jt(x,t)=δℰ_GP/δq̆_j(x,t),q̆_1(x,t)=q̅_1(x,t),q̆_0(x,t)=2q̅_0(x,t),q̆_-1(x,t)=q̅_-1(x,t), with the energy functional being of the form ℰ_GP=∫̣dx{∑_j=1,0,-1|q_jx|^2+α[|q_1|^4+|q_-1|^4+2|q_0|^4+4(|q_1|^2+|q_-1|^2)|q_0|^2] +2αβ(q_0^2q̅_1q̅_-1)}. The rest of this paper is organized as follows. In Sec. 2, we study thespectral analysis of the associated 4× 4 Lax pair of Eq. (<ref>). Sec. 3 presents the corresponding 4× 4 matrix RH problem in terms of the jump matrices found in Sec. 2. The global relation is used to establish the map between the Dirichlet and Neumann boundary values in Sec. 4. § THE SPECTRAL ANALYSIS OF THE LAX PAIRIn this subsection, we will simultaneously consider the spectral analysis of the Lax pair (<ref>) to present sectionally its analytic eigenfunctions in order to formulate a 4× 4 matrix RH problem defined in the complex k-plane. §.§ (a) The closed one-form for the Lax pairThe spin-1 GP equations (<ref>) admits the 4 × 4 Lax pair <cit.>{[ψ_x+ikσ_4ψ=U(x,t)ψ,; ψ_t+2ik^2σ_4ψ=V(x,t,k)ψ, ].where ψ=ψ(x,t,k) is a 4×4 matrix-valued or 4× 1 column vector-valued spectral function, k∈ℂ is an isospectral parameter, σ_4= diag(1,1,-1,-1).and the 4 × 4 matrix-valued functions U(x,t) and V(x,t,k) are defined byU(x,t)=([00q_1q_0;00β q_0 q_-1;αq̅_1 αβq̅_000;αq̅_0 αq̅_-100 ]),V(x,t,k)=2kU+V_0, V_0=iσ_4(U_x-U^2). A new eigenfunction μ=μ(x,t,k) is defined by the transformμ(x,t,k)=ψ(x,t,k)e^i(kx+2k^2t)σ_4, such that the Lax pair (<ref>) is changed into an equivalent form {[ μ_x+ikσ̂_4μ= U(x,t)μ,; μ_t+2ik^2σ̂_4μ=V(x,t,k)μ, ].where σ̂_4μ=[σ_4, μ],σ̂_4 denote the commutator with respect to σ_4 and the operator acting on a 4× 4 matrix X by σ̂_4X=[σ_4, X] such that e^xσ̂_4X=e^xσ_4Xe^-xσ_4. The Lax pair (<ref>) leads to a full derivative formd[e^i(kx+2k^2t)σ̂_4μ(x,t,k)]=W(x,t,k),where the closed one-form W(x,t,k) is W(x,t,k)=e^i(kx+2k^2t)σ̂_4[U(x,t)μ(x,t,k)dx+V(x,t,k)μ(x,t,k)dt]. §.§ (b) The basic eigenfunctions μ_j'sFor any point (x,t) in the considered region Ω={(x,t)| 0<x <∞,0< t< T} (see Fig. <ref>(a)),{γ_j}_1^3 denote the three contours in the domain Ω connecting (x_j, t_j) to (x,t), respectively, where (x_1, t_1)=(0, T),(x_2, t_2)=(0, 0),(x_3, t_3)=(∞, t) (see Figs. <ref>(b)-(d)). Thus for the point (ξ, τ) on the each contour, we have [ γ_1:x-ξ≥ 0,t-τ≤ 0,; γ_2:x-ξ≥ 0,t-τ≥ 0,; γ_3:x-ξ≤ 0, t-τ = 0, ] It follows from the one-form (<ref>) that we can use the Volterra integral equations to define its three eigenfunctions {μ_j}_1^3 on the above-mentioned three contours {γ_j}_1^3[ μ_j(x,t,k)=I+∫_(x_j, t_j)^(x,t)e^-i(kx+2k^2t)σ̂_4W_j(ξ,τ,k), j=1,2,3, (x,t)∈Ω, ] where 𝕀= diag(1,1,1,1), the integral is over a piecewise smooth curve from (x_j, t_j) to (x,t),W_j(x,t,k) is given by Eq. (<ref>) with μ(x,t,k) replaced by μ_j(x,t,k). Since the one-form W_j is closed, thus μ_j is independent of the path of integration. If we choose the paths of integration to be parallel to the x and t axes, then the integral Eq. (<ref>) becomes (j=12,3) [μ_j= I+∫_x_j^x e^-ik(x-ξ)σ̂_4(Uμ_j)(ξ,t,k)dξ+e^-ik(x-x_j)σ̂_4∫_t_j^te^-2ik^2(t-τ)σ̂_4 (Vμ_j)(x_j,τ,k)dτ, ] Eq. (<ref>) implies that the first, second, third, and fourth columns of the matrices μ_j(x,t,k)'s contain these exponentials [μ_j]_1:e^2ik(x-ξ)+4ik^2(t-τ), e^2ik(x-ξ)+4ik^2(t-τ), [μ_j]_2:e^2ik(x-ξ)+4ik^2(t-τ), e^2ik(x-ξ)+4ik^2(t-τ), [μ_j]_3:e^-2ik(x-ξ)-4ik^2(t-τ), e^-2ik(x-ξ)-4ik^2(t-τ),[μ_j]_4:e^-2ik(x-ξ)-4ik^2(t-τ), e^-2ik(x-ξ)-4ik^2(t-τ), To analyse the bounded domains of the eigenfunctions {μ_j}_1^3 in the complex k-plane, we need to use the curve 𝕂={k∈ℂ |f(k) · g(k)=0,f(k)=ik,g(k)=ik^2}, to separate the complex k-plane into four regions (see Fig. <ref>): [ D_1={k∈ℂ |f(k)=- k<0and g(k)=-2kk<0},; D_2={k∈ℂ |f(k)=- k<0and g(k)=-2kk>0},; D_3={k∈ℂ |f(k)=- k>0and g(k)=-2kk<0},;D_4={k∈ℂ |f(k)=- k>0 and g(k)=-2kk>0}, ] Thus it follows from Eqs. (<ref>), (<ref>) and (<ref>) that the domains, where the different columns of eigenfunctions {μ_j}_1^3 are bounded and analytic in the complex k-plane, are presented as follows: {[ μ_1: (f_-(k) ∩ g_+(k),f_-(k) ∩ g_+(k),f_+(k) ∩ g_-(k),f_+(k) ∩ g_-(k))=: (D_2, D_2, D_3, D_3),; μ_2: (f_-(k) ∩ g_-(k),f_-(k) ∩ g_-(k),f_+(k) ∩ g_+(k),f_+(k) ∩ g_+(k))=: (D_1, D_1, D_4, D_4),;μ_3: (f_+(k) ,f_+(k) ,f_-(k) ,f_-(k))=: (C^-, C^-, C^+, C^+), ].where C^-=D_3∪ D_4,C^+=D_1∪ D_2,f_+(k)=:f(k)=- k>0,f_-(k)=: f(k)=- k<0,g_+(k)=:g(k)=-2kk>0,and g_-(k)=: g(k)=-2kk<0. §.§ (c) Symmetries of eigenfunctionsFor the convenience, we write a 4× 4 matrix X=(X_ij)_4× 4 as [ X=([ X̃_11 X̃_12; X̃_21 X̃_22 ]),X̃_11=([ X_11 X_12; X_21 X_22 ]),X̃_12=([ X_13 X_14; X_23 X_24 ]),;X̃_21=([ X_31 X_32; X_41 X_42 ]),X̃_22=([ X_33 X_34; X_43 X_44 ]), ] Let 𝕌(x,t, k)=-ikσ_4+U(x,t),𝕍(x,t, k)=-2ik^2σ_4+V(x,t,k). Then the symmetry properties of 𝕌(x,t, k) and 𝕍(x,t, k) imply that the eigenfunction μ(x,t,k) have the symmetries(μ̃(x,t,k))_11=P^β (μ̃(x,t,k̅))_22P^β, (μ̃(x,t,k))_12=α (μ̃(x,t,k̅))_21^T,where P^β= diag(1, β), β^2=1.SinceP_±^α 𝕌(x,t, k̅)P_±^α=-𝕌(x,t,k)^T,P_±^α 𝕍(x,t, k̅)P_±^α=-𝕍(x,t,k)^T,where P_±^α= diag(±α, ±α,∓ 1, ∓ 1), α^2=1.According to Eq. (<ref>) (see the similar proof in Ref. <cit.>), we know that the eigenfunction ψ(x,t,k) of the Lax pair (<ref>) and μ(x,t,k) of the Lax pair (<ref>) areof the same symmetric relation [ ψ^-1(x,t,k)=P_±^α ψ(x,t,k̅)^TP_±^α,μ^-1(x,t,k)=P_±^α μ(x,t,k̅)^TP_±^α, ]Moreover, In the domains where μ is bounded, we have μ(x,t,k)=𝕀+O(1/k), k→∞,and det [μ(x,t,k)]=1 since tr (𝕌(x,t, k))= tr (𝕍(x,t,k))=0.§.§ (d)The minors of eigenfunctionsThe cofactor matrix X^A (or the transpose of the adjugate) of a 4× 4 matrix X is given byadj(X)^T=X^A=([m_11(X) -m_12(X)m_13(X) -m_14(X); -m_21(X)m_22(X) -m_23(X)m_24(X);m_31(X) -m_32(X)m_33(X) -m_34(X); -m_41(X)m_42(X) -m_43(X)m_44(X) ]),where m_ij(X) denote the (ij)th minor of X and (X^A)^TX = adj(X) X= X.It follows from Eq. (<ref>) that be shown that the matrix-valued functions μ_j^A's satisfy the Lax pair {[ μ_j,x^A-ikσ̂_4μ_j^A= -U^Tμ_j^A,; μ_j,t^A-2ik^2σ̂_4μ_j^A=-V^Tμ_j^A, ]. whose solutions can be expressed as [μ_j^A(x,t,k)= I-∫_x_j^x e^ik(x-ξ)σ̂_4(Uμ_j^A)(ξ,t,k)dξ-e^ik(x-x_j)σ̂_4∫_t_j^te^2ik^2(t-τ)σ̂_4 (Vμ_j^A)(x_j,τ,k)dτ, ] by using the Volterra integral equations, where U^T and V^T denote thetransposes of U and V, respectively.It is easy to check that the regions of boundedness of μ_j^A: {[ μ_1^A(x,t,k)isbounded for k∈ (D_3, D_3, D_2, D_2),; μ_2^A(x,t,k)isbounded for k∈ (D_4, D_4, D_1, D_1),; μ_3^A(x,t,k)isbounded for k∈ (C^+, C^+, C^-, C^-), ].which are symmetric ones of μ_j about the k-axis (cf. Eq. (<ref>)). §.§ (e) The spectral functions and the global relationLet us introduce the 4× 4 matrix-valued functions S(k),s(k), and 𝔖(k) by μ_j,j=1,2,3{[ μ_1(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4S(k),; μ_3(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4s(k),; μ_3(x,t,k)=μ_1(x,t,k)e^-i(kx+2k^2t)σ̂_4𝔖(k), ]. Evaluatingsystem (<ref>) at (x,t)=(0,0) and (x, t)=(0, T), respectively, we have {[ S(k)=μ_1(0,0,k)=e^2ik^2Tσ̂_4μ_2^-1(0,T,k),; s(k)=μ_3(0,0,k),; 𝔖(k)=μ_1^-1(0,0,k)μ_3(0,0,k)=S^-1(k)s(k)=e^2ik^2Tσ̂_4μ_3(0,T,k), ].These relations among μ_j are displayed in Fig. <ref>. Thus these three functions S(k),s(k), and 𝔖(k) are dependent such that we only consider two of them, e.g., S(k) and s(k).According to the definition (<ref>) of μ_j, Eq. (<ref>)implies that [ s(k)=I-∫_0^∞ e^ikξσ̂_4(Uμ_3)(ξ,0,k)dξ; S(k)= I-∫_0^T e^2ik^2τσ̂_4(Vμ_1)(0, τ,k)dξ =[𝕀+∫_0^T e^2ik^2τσ̂_4(Vμ_2)(0,τ,k)dτ]^-1, ] where μ_j(0,t,k), j=1,2 and μ_3(x,0,k),0<x<∞,0<t<T satisfy the Volterra integral equations [μ_3(x,0,k)=I-∫_x^∞e^-ik(x-ξ)σ̂_4 (Uμ_3)(ξ,0,k)dξ, 0<x<∞, k∈ (C^-, C^-, C^+, C^+),; μ_1(0,t,k)=I-∫_t^Te^-2ik^2(t-τ)σ̂_4 (Vμ_1)(0,τ,k)dτ, 0<t<T, k∈ (D_2∪ U_4, D_2∪ U_4, D_1∪ U_3, D_1∪ U_3),; μ_2(0,t,k)=I+∫_0^te^-2ik^2(t-τ)σ̂_4 (Vμ_2)(0,τ,k)dτ,0<t<T,k∈ (D_1∪ U_3, D_1∪ U_3, D_2∪ U_4, D_2∪ U_4), ] Thus, it follows from Eqs. (<ref>) and (<ref>) that s(k) and S(k) are determined by U(x,0,k) and V(0,t,k), i.e., by the initial data q_j(x, t=0) and the Dirichlet-Neumann boundary data q_j(x=0, t) and q_jx(x=0, t),j=1,0,-1, respectively. In fact, μ_3(x,0,k) and μ_1,2(0,t,k) satisfy the x-part and t-part of the Lax pair (<ref>) at t=0 and x=0, respectively, that is,x- part: {[ μ_x(x,0,k)+ik[σ_4, μ(x,0,k)]=U(x, t=0)μ(x,0,k),; ḷịṃ_x→∞μ(x,0,k)=𝕀, 0<x<∞, ].t- part: {[ μ_t(0,t,k)+2ik^2[σ_4, μ(0,t,k)]=V(x=0,t,k)μ(0,t,k),0<t<T,; μ(0,0,k)=𝕀, μ(0,T,k)=𝕀, ].Moreover, the functions{S(k),s(k)} and{S^A(k),s^A(k)} have the following boundedness:{[S(k) isbounded for k∈(D_2∪ D_4, D_2∪ D_4, D_1∪ D_3, D_1∪ D_3),;s(k) isbounded for k∈(C^-, C^-, C^+, C^+),; S^A(k) isbounded for k∈(D_1∪ D_3, D_1∪ D_3, D_2∪ D_4, D_21∪ D_4),;s^A(k) isbounded for k∈(C^+, C^+, C^-, C^-), ]. It follows from the third one inEq. (<ref>) that we have the so-called global relationc(T,k)=μ_3(0, T, k)=e^-2ik^2Tσ̂_4[S^-1(k)s(k)],where μ_3(0,t,k),0<t<T satisfies the Volterra integral equation μ_3(0,t,k)=I-∫_0^∞ e^ikξσ̂_4(Uμ_3)(ξ,t,k)dξ, 0<t<T, k∈ (C^-, C^-, C^+, C^+),§.§ (f)The definition of matrix-valued functions M_n'sIn each domain D_n, n=1,2,3,4 of the complex k-plane,the solution M_n(x,t,k) of Eq. (<ref>) is(M_n(x,t,k))_lj=δ_lj+∫_(γ^n)_sj(e^-i(kx+2k^2t)σ̂_4W_n(ξ,τ,k))_lj,k∈ D_n,l,j=1,2,3,4.via the Volterra integral equations, where W_n(x,t,k) is given by Eq. (<ref>) with μ(x,t,k) replaced with M_n(x,t,k), and the definition of the contours (γ^n)_lj's is given by(γ^n)_lj={[ γ_1,if f_l(k)<f_j(k)and g_l(k)≥ g_j(k),; γ_2,if f_l(k)<f_j(k)and g_l(k) <g_j(k),;γ_ 3,if f_l(k) ≥ f_j(k), ].for k∈ D_n, where f_1,2(k)=-f_3,4(k)=-ik,g_1,2(k)=-g_3,4(k)=-ik^2. The definition (<ref>) of (γ^n)_ljimplies that the matrices γ^n(n=1,2,3,4) are of the forms [ γ^1=( [ γ_3 γ_3 γ_3 γ_3; γ_3 γ_3 γ_3 γ_3; γ_2 γ_2 γ_3 γ_3; γ_2 γ_2 γ_3 γ_3 ]), γ^2=( [ γ_3 γ_3 γ_3 γ_3; γ_3 γ_3 γ_3 γ_3; γ_1 γ_1 γ_3 γ_3; γ_1 γ_1 γ_3 γ_3 ]), γ^3=( [ γ_3 γ_3 γ_1 γ_1; γ_3 γ_3 γ_1 γ_1; γ_3 γ_3 γ_3 γ_3; γ_3 γ_3 γ_3 γ_3 ]), γ^4=( [ γ_3 γ_3 γ_2 γ_2; γ_3 γ_3 γ_2 γ_2; γ_3 γ_3 γ_3 γ_3; γ_3 γ_3 γ_3 γ_3 ]), ]According to the similar proof for the 3× 3 Lax pair in <cit.> and the above-mentioned properties of μ(x,t,k), we have the bounedness and analyticity of M_n:Proposition 2.1. The matrix-valued functions M_n(x,t,k),n=1,2,3,4 are weill defined by Eq. (<ref>) for k∈D̅_n and (x,t)∈Ω̅. For any fixed point (x,t), M_n's are the bounded and analytic function of k∈ D_n away from a possible discrete set of singularity {k_j} at which the Fredholm determinants vanish. M_n(x,t,k) also admits the bounded and continuous extensions to D̅_n and M_n(x,t,k)=𝕀+O(1/k),k∈ D_n,k→∞,n=1,2,3,4.§.§ (g) The jump matricesThe new spectral functions S_n(k)(n=1,2,3, 4) are introduced by S_n(k)=M_n(0,0,k),k∈ D_n, n=1,2,3,4.Let M(x,t,k) stand for the sectionally analytic function on the Riemann k-spere which is equivalent to M_n(x,t,k) for k∈ D_n. Then M(x,t,k) solves the jump equations M_n(x,t,k)=M_m(x,t,k)J_mn(x,t,k), k∈D̅_n∩D̅_m,n,m=1,2,3,4, n≠ m,with the jump matrices J_mn(x,t,k) defined byJ_mn(x,t,k)=e^-i(kx+2k^2t)σ̂_4[S_m^-1(k)S_n(k)]. Proposition 2.2. The matrix-valued functions S_n(x,t,k)(n=1,2,3,4) defined by M_n(x,t,k)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4S_n(k),k∈ D_n, can be determined by the entries of s(k)=(s_ij)_4× 4,S(k)=(S_ij)_4× 4 (cf. Eq. (<ref>)) as follows: [ S_1(k)=([ m_22(s)/n_33,44(s) m_21(s)/n_33,44(s) s_13 s_14; m_12(s)/n_33,44(s) m_11(s)/n_33,44(s) s_23 s_24;00 s_33 s_34;00 s_43 s_44 ]),S_2(k)=([ S_2^(11) S_2^(12) s_13 s_14; S_2^(21) S_2^(22) s_23 s_24; S_2^(31) S_2^(32) s_33 s_34; S_2^(41) S_2^(42) s_43 s_44 ]),;S_3(k)=([ s_11 s_12 S_3^(13) S_3^(14); s_21 s_22 S_3^(23) S_3^(24); s_31 s_32 S_3^(33) S_3^(34); s_41 s_42 S_3^(43) S_3^(44) ]), S_4(k)=([ s_11 s_1200; s_21 s_2200; s_31 s_32 m_44(s)/n_11,22(s) m_43(s)/n_11,22(s); s_41 s_42 m_34(s)/n_11,22(s) m_33(s)/n_11,22(s) ]), ] where n_i_1j_1,i_2j_2(X) denotes the determinant of the sub-matrix generated by taking the cross elements of i_1,2th rows and j_1,2th columns of the 4× 4 matrix X and {[ S_2^(1j)=n_1j,2(3-j)(S)m_2(3-j)(s)+n_1j,3(3-j)(S)m_3(3-j)(s)+n_1j,4(3-j)(S)m_4(3-j)(s)𝒩([S]_1[S]_2[s]_3[s]_4),; S_2^(2j)=n_2j,1(3-j)(S)m_1(3-j)(s)+n_2j,3(3-j)(S)m_3(3-j)(s)+n_2j,4(3-j)(S)m_4(3-j)(s)𝒩([S]_1[S]_2[s]_3[s]_4),; S_2^(3j)=n_3j,1(3-j)(S)m_1(3-j)(s)+n_3j,2(3-j)(S)m_2(3-j)(s)+n_3j,4(3-j)(S)m_4(3-j)(s)𝒩([S]_1[S]_2[s]_3[s]_4),; S_2^(4j)=n_4j,1(3-j)(S)m_1(3-j)(s)+n_4j,2(3-j)(S)m_2(3-j)(s)+n_4j,3(3-j)(S)m_3(3-j)(s)𝒩([S]_1[S]_2[s]_3[s]_4), ]. j=1,2, {[ S_3^(1j)=n_1j,2(7-j)(S)m_2(7-j)(s)+n_1j,3(7-j)(S)m_3(7-j)(s)+n_1j,4(7-j)(S)m_4(7-j)(s)𝒩([s]_1[s]_2[S]_3[S]_4),; S_3^(2j)=n_2j,1(7-j)(S)m_1(7-j)(s)+n_2j,3(7-j)(S)m_3(7-j)(s)+n_2j,4(7-j)(S)m_4(7-j)(s)𝒩([s]_1[s]_2[S]_3[S]_4),; S_3^(3j)=n_3j,1(7-j)(S)m_1(7-j)(s)+n_3j,2(7-j)(S)m_2(7-j)(s)+n_3j,4(7-j)(S)m_4(7-j)(s)𝒩([s]_1[s]_2[S]_3[S]_4),; S_3^(4j)=n_4j,1(7-j)(S)m_1(7-j)(s)+n_4j,2(7-j)(S)m_2(7-j)(s)+n_4j,3(7-j)(S)m_3(7-j)(s)𝒩([s]_1[s]_2[S]_3[S]_4), ].j=3,4,where 𝒩([S]_1[S]_2[s]_3[s]_4)= det(n([S]_1, [S]_2, [s]_3, [s]_4)) denotes the determinant of the matrix generated by choosing the first and second columns of S(k) and the third and fourth columns of s(k), and 𝒩([s]_1[s]_2[S]_3[S]_4)= det(n([s]_1, [s]_2, [S]_3, [S]_4)). Proof.Let γ_3^x_0 with x_0>0 denote the contour (x_0, 0)→ (x,t) in the (x,t)-plane and μ_3(x,t,k; x_0) be determined by Eq. (<ref>) with j=3 and the contour γ_3 replaced by γ_3^x_0. M_n(x,t,k; x_0) is defined by Eq. (<ref>) withthe contour γ_3 replaced by γ_3^x_0.We introduce the functions R_n(k; x_0), S_n(k; x_0), and T_n(k; x_0)in the form {[M_n(x,t,k; x_0)=μ_1(x,t,k)e^-i(kx+2k^2t)σ̂_4R_n(k; x_0),;M_n(x,t,k; x_0)=μ_2(x,t,k)e^-i(kx+2k^2t)σ̂_4S_n(k; x_0),; M_n(x,t,k; x_0)=μ_3(x,t,k; x_0)e^-i(kx+2k^2t)σ̂_4T_n(k; x_0), ]. It follows from Eq. (<ref>) that we have the relations {[ R_n(k; x_0)=e^2ik^2Tσ̂_4M_n(0,T,k; x_0),; S_n(k; x_0)=M_n(0,0,k; x_0),; T_n(k; x_0)=e^ikx_0σ̂_4[μ_3^-1(x_0,0,k;x_0)M_n(x_0,0,k; x_0)], ].and {[ S(k)=μ_1(0,0,k)=S_n(k; x_0)R_n^-1(k; x_0),; s(k; x_0)=μ_3(0,0,k; x_0)=S_n(k; x_0)T_n^-1(k; x_0), ].which can in general deduce the functions {R_n(k; x_0), S_n(k; x_0), T_n(k; x_0)} for the given spectral functions {s(k), S(k)}.Moreover, we can also determine some entries of {R_n(k; x_0), S_n(k; x_0), T_n(k; x_0)} as {[ (R_n(k; x_0))_ij=0, if(γ^n)_ij=γ_1,; (S_n(k; x_0))_ij=0,if (γ^n)_ij=γ_2,; (T_n(k; x_0))_ij=δ_ij, if (γ^n)_ij=γ_3, ]. by using Eqs. (<ref>) and (<ref>). System (<ref>) contains 32 scalar equations for 32 unknowns. Thus it follows from system (<ref>) that we have S_n(k; x_0). Then taking the limit x_0→∞ of S_n(k; x_0) yields Eq. (<ref>). □ §.§ (h) The residue conditionsSince μ_2(x,t,k) is an entire function, it follows from Eq. (<ref>) that M(x,t,k) only has the singularities at the points where the S_n(k)'s have the singularities. The S_n(k)'s given byEq. (<ref>) imply that the possible singularities of M(x,t,k) are as follows: * [M]_j, j=1,2 could have poles in D_1 at the zeros of n_33,44(s)(k);* [M]_j,j=1,2 could have poles in D_2 at the zeros of 𝒩([S]_1[S]_2[s]_3[s]_4)(k);* [M]_j,j=3,4 could have poles in D_3 at the zeros of 𝒩([s]_1[s]_2[S]_3[S]_4)(k);* [M]_j,j=3,4 could have poles in D_4 at the zeros of n_11,22(s)(k). We use {k_j}_1^N to denote the above-mentioned possible zeros and suppose that they satisfy the following assumption.Assumption 2.3. We supposethat* n_33,44(s)(k) admits n_1 possible simple zeros in D_1 denoted by {k_j}_1^n_1; * 𝒩([S]_1[S]_2[s]_3[s]_4)(k) admits n_2-n_1 possible simple zeros in D_2 denoted by {k_j}_n_1+1^n_2; * 𝒩([s]_1[s]_2[S]_3[S]_4)(k) admits n_3-n_2 possible simple zeros in D_3 denoted by {k_j}_n_2+1^n_3; * n_11,22(s)(k) admits N-n_3 possible simple zeros in D_4 denoted by {k_j}_n_3+1^N;and that none of these simple zeros coincide. Moreover, none of these functions are assumed to have zeros on the boundaries of the D_n's (n=1,2,3,4).Proposition 2.4.Let {M_n(x,t,k)}_1^4 be the eigenfunctions given by Eq. (<ref>) and suppose that the set {k_j}_1^N of singularities are as the above-mentioned Assumption 2.3. Then we have the following residue conditions: [Ṛẹṣ_k=k_j[M_1(x,t,k)]_l= m_2(3-l)(s)(k_j)s_24(k_j)-m_1(3-l)(s)(k_j)s_14(k_j)ṅ_33,44(s)(k_j)n_13,24(s)(k_j)[M_1(x,t,k_j)]_3e^-2θ(k_j); +m_1(3-l)(s)(k_j)s_13(k_j)-m_2(3-l)(s)(k_j)s_23(k_j)ṅ_33,44(s)(k_j)n_13,24(s)(k_j)[M_1(x,t,k_j)]_4e^-2θ(k_j),; for 1≤ j≤ n_1, k∈ D_1, l=1,2, ][Res_k=k_j[M_2(x,t,k)]_l= S_2^(1l)(k_j)s_24(k_j)-S_2^(2l)(k_j)s_14(k_j)𝒩̇([S]_1[S]_2[s]_3[s]_4)(k_j)n_13,24(s)(k_j)[M_2(x,t,k_j)]_3e^-2θ(k_j); +S_2^(2l)(k_j)s_13(k_j)-S_2^(1l)(k_j)s_23(k_j)𝒩̇([S]_1[S]_2[s]_3[s]_4)(k_j)n_13,24(s)(k_j)[M_2(x,t,k_j)]_4e^-2θ(k_j),;for n_1+1≤ j≤ n_2, k∈ D_2,l=1,2, ][ Res_k=k_j[M_3(x,t,k)]_l=S_3^(1l)(k_j)s_22(k_j)-S_3^(2l)(k_j)s_12(k_j)𝒩̇([s]_1[s]_2[S]_3[S]_4)(k_j)n_11,22(s)(k_j)M_3(x,t,k_j)]_1e^2θ(k_j);+S_3^(2l)(k_j)s_11(k_j)-S_3^(1l)(k_j)s_21(k_j)𝒩̇([s]_1[s]_2[S]_3[S]_4)(k_j)n_11,22(s)(k_j)[M_3(x,t,k_j)]_2e^2θ(k_j),;for n_2+1≤ j≤ n_3, k∈ D_3,l=3,4, ][ Res_k=k_j[M_4(x,t,k)]_l= m_4(7-l)(s)(k_j)s_42(k_j)-m_3(7-l)(s)(k_j)s_32(k_j)ṅ_11,22(s)(k_j)n_31,42(s)(k_j)[M_4(x,t,k_j)]_1e^2θ(k_j);+m_3(7-l)(s)(k_j)s_31(k_j)-m_4(7-l)(s)(k_j)s_41(k_j)ṅ_11,22(s)(k_j)n_31,42(s)(k_j)[M_4(x,t,k_j)]_2e^2θ(k_j),; for n_3+1≤ j≤ N, k∈ D_4,l=3,4 ] where the overdot denotes the derivative with resect to the parameter k and θ=θ(k)=-i(kx+2k^2t).Proof.It follows from Eqs. (<ref>) and (<ref>) that we find the four columns of M_1(x,t,k) as[M_1]_1=[μ_2]_1 m_22(s)n_33,44(s)+[μ_2]_2m_12(s)n_33,44(s), [M_1]_2=[μ_2]_1 m_21(s)n_33,44(s)+[μ_2]_2m_11(s)n_33,44(s),[M_1]_3=[μ_2]_1 s_13e^2θ +[μ_2]_2s_23e^2θ+[μ_2]_3s_33+[μ_2]_4s_43,[M_1]_4=[μ_2]_1 s_14e^2θ +[μ_2]_2s_24e^2θ+[μ_2]_3s_34+[μ_2]_4s_44, For the case that k_j∈ D_1 is a simple zero of n_33,44(s)(k), it follows from Eqs. (<ref>) and (<ref>) that we obtain [μ_2]_1 and [μ_2]_2 and then substitute them into Eqs. (<ref>) and (<ref>) to yield [M_1]_1=m_22(s)s_24-m_12(s)s_14n_33,44(s)n_13,24(s)[M_1]_3e^-2θ +m_12(s)s_13-m_22(s)s_23n_33,44(s)n_13,24(s)[M_1]_4e^-2θ +m_42(s)[μ_2]_3+m_32(s)[μ_2]_4n_13,24(s)e^-2θ,[M_1]_2=m_21(s)s_24-m_11(s)s_14n_33,44(s)n_13,24(s)[M_1]_3e^-2θ +m_11(s)s_13-m_21(s)s_23n_33,44(s)n_13,24(s)[M_1]_4e^-2θ +m_41(s)[μ_2]_3+m_31(s)[μ_2]_4n_13,24(s)e^-2θ,whose residues at k=k_j,k_j∈ D_1 yield Eq. (<ref>).Similarly, we can showEq. (<ref>) for k_j∈ D_2,Eq. (<ref>) for k_j∈ D_3, and Eq. (<ref>) for k_j∈ D_4 by studyingEqs. (<ref>) and (<ref>) for n=2,3,4.□§ THE 4× 4 MATRIX RIEMANN-HILBERT PROBLEMBy using the district contours γ_j(j=1,2,3,4), the integral solutions of the revised Lax pair (<ref>), and S_n due to {S(k), s(k)}, we have defined the sectionally analytic function M_n(x,t,k),n=1,2,3,4, which solves a 4× 4 matrix Riemann-Hilbert (RH) problem. This RH problem can be formulated on basis of the initial conditions of the Schwartz class q_j(x, t=0) and Dirichlet-Neumann boundary data q_j(x=0,t) and q_jx(x=0,t),j=1,0,-1. Thus the solution of Eq. (<ref>) for all values of x,t can be refound by solving the RH problem.Theorem 3.1.Suppose that (q_1(x,t),q_0(x,t),q_-1(x,t)) is a solution of Eq. (<ref>) in the domainΩ={(x,t) | 0<x<∞,t∈ [0, T]} with sufficient smoothness and decay as x→∞. Then it can be reconstructed from the initial data defined by q_j(x, t=0)=q_0j(x), j=1,0,-1 and Dirichlet and Neumann boundary values defined by q_j(x=0, t)=u_0j(t) and q_jx(x=0, t)=u_1j(t),j=1,0, -1.We use the initial and boundary data to define the jump matrices J_mn(x, t, k),n, m = 1,..., 4, by Eq. (<ref>) as well as the spectral functions S(k), s(k)given by Eq. (<ref>). Assume that the possible zeros {k_j}^N_1 of the functions n_33,44(s)(k), 𝒩([S]_1[S]_2[s]_3[s]_4)(k), 𝒩([s]_1[s]_2[S]_3[S]_4)(k) and n_11,22(s)(k) are as in Assumption 2.4. Then the solution (q_1(x,t),q_0(x,t), q_-1(x,t)) of Eq. (<ref>) is given by M(x,t,k) in the form {[q_1(x,t)=2̣ilim_k→∞(kM(x,t,k))_13,; q_0(x,t)=2̣ilim_k→∞(kM(x,t,k))_14=2iβlim_k→∞(kM(x,t,k))_23,; q_-1(x,t)=2̣ilim_k→∞(kM(x,t,k))_24, ].where M(x,t,k) satisfies the following 4× 4 matrix Riemann-Hilbert problem: * M(x,t,k) is sectionally meromorphic on the Riemann k-sphere with jumpsacross the contours D̅_n∪D̅_m, (n, m = 1,2,3,4) (see Fig. <ref>a). *Across the contours D̅_n∪D̅_m(n, m = 1,2,3,4), M(x, t, k) satisfies thejump condition (<ref>). *The residue conditions of M(x,t,k) are satisfied in Proposition 2.4. * M(x, t, k) = I+O(1/k) as k→∞. Proof.System (<ref>) can be deduced from the large k asymptotics of the eigenfunctions. We can follow the similar one in Refs. <cit.>to show the rest proof of the Theorem. □§ NONLINEARIZABLE BOUNDARY CONDITIONSThe main difficulty of the initial-boundary value problems is to find the boundary values for a well-posed problem.All boundary conditions are required for the definition of S(k), and hence for the formulate the 4× 4 matrixRH problem. Our main conclusion exhibits the unknown boundary condition on basis of the prescribed boundary condition and the initial conditionin terms of the solution of a system of nonlinear integral equations.§.§ (a) The time evolution of the global relation By evaluating Eq. (<ref>) at (x,t)=(0, t) and considering the global relation (<ref>), we have c(t,k)=μ_2(0,t,k)e^-2ik^2tσ̂_4s(k), 0<t<T,k∈ (C^-, C^-, C^+, C^+), which can be written as[̣c(t,k)]_l=∑_j=1^2[μ_2(0,t,k)]_js_jl(k)+∑_j=3^4[μ_2(0,t,k)]_js_jl(k)e^-4ik^2t, l=1,2,[̣c(t,k)]_l=∑_j=1^2[μ_2(0,t,k)]_js_jl(k)e^4ik^2t+∑_j=3^4[μ_2(0,t,k)]_js_jl(k),l=3,4,Thus, the column vectors [c(t,k)]_l,l=1,2 are analytic and bounded in C^- away from the possible zeros of n_11,22(s)(k) and of order O(1/k) as k→∞, and the column vectors [c(t,k)]_l,l=3,4 are analytic and bounded in C^+ away from the possible zeros of n_33,44(s)(k) and of order O(1/k) as k→∞. §.§ (b)Asymptotic behaviors of eigenfunctionsIt follows from Eq. (<ref>) that we have the asymptotics of eigenfunctions {μ_j}_1^3 as k→∞[ μ_j(x,t,k)= I+∑_s=1^2 1/k^s([ μ_j,11^(s) μ_j,12^(s) μ_j,13^(s) μ_j,14^(s); μ_j,21^(s) μ_j,22^(s) μ_j,23^(s) μ_j,24^(s); μ_j,31^(s) μ_j,32^(s) μ_j,33^(s) μ_j,34^(s); μ_j,41^(s) μ_j,42^(s) μ_j,43^(s) μ_j,44^(s) ]) +O(1/k^3); = I+1/k([ ∫_(x_j, t_j)^(x,t)Δ_11^(1) ∫_(x_j, t_j)^(x,t)Δ_12^(1) -̣i/2q_1 -̣i/2q_0; ∫_(x_j, t_j)^(x,t)Δ_21^(1) ∫_(x_j, t_j)^(x,t)Δ_22^(1)-̣iβ/2q_0-̣i/2q_-1;%̣ṣ/̣%̣ṣiα2q̅_1 %̣ṣ/̣%̣ṣiαβ2q̅_0 ∫_(x_j, t_j)^(x,t)Δ_33^(1) ∫_(x_j, t_j)^(x,t)Δ_34^(1);%̣ṣ/̣%̣ṣiα2q̅_0 %̣ṣ/̣%̣ṣiα2q̅_-1 ∫_(x_j, t_j)^(x,t)Δ_43^(1) ∫_(x_j, t_j)^(x,t)Δ_44^(1) ]); +̣1/k^2([ ∫_(x_j, t_j)^(x,t)Δ_11^(2) ∫_(x_j, t_j)^(x,t)Δ_12^(2) μ_j,13^(2) μ_j,14^(2); ∫_(x_j, t_j)^(x,t)Δ_21^(2) ∫_(x_j, t_j)^(x,t)Δ_22^(2) μ_j,23^(2) μ_j,24^(2); μ_j,31^(2) μ_j,32^(2) ∫_(x_j, t_j)^(x,t)Δ_33^(2) ∫_(x_j, t_j)^(x,t)Δ_34^(2); μ_j,41^(2) μ_j,42^(2) ∫_(x_j, t_j)^(x,t)Δ_43^(2) ∫_(x_j, t_j)^(x,t)Δ_44^(2) ])+O(1/k^3), ] where we have introduced the following functions {[ Δ_11^(1)=-Δ_33^(1) =%̣ṣ/̣%̣ṣiα2(|q_1|^2+|q_0|^2)dx+α/2∑_j=0,1(q_jq̅_jx-q_jxq̅_j)dt,; Δ_22^(1)= -Δ_44^(1)=%̣ṣ/̣%̣ṣiα2(|q_-1|^2+|q_0|^2)dx+α/2∑_j=-1,0(q_jq̅_jx-q_jxq̅_j)dt,; Δ_12^(1)= -Δ̅_21^(1)=-Δ_34^(1)=Δ̅_43^(1)=iα/2(β q_1q̅_0+q_0q̅_-1)dx+α/2(β q_1q̅_0x-β q_1xq̅_0+q_0q̅_-1x-q_0xq̅_-1)dt, ]. {[ μ_j,13^(2)=%̣ṣ/̣%̣ṣ14q_1x+1/2i(q_1μ_j,33^(1)+q_0μ_j,43^(1)) =1/4q_1x+1/2i[q_1∫_(x_j,t_j)^(x,t)Δ_33^(1)+q_0∫_(x_j,t_j)^(x,t)Δ_43^(1)],; μ_j,14^(2)=%̣ṣ/̣%̣ṣ14q_0x+1/2i(q_1μ_j,34^(1)+q_0μ_j,44^(1)) =1/4q_0x+1/2i[q_1∫_(x_j,t_j)^(x,t)Δ_34^(1)+q_0∫_(x_j,t_j)^(x,t)Δ_44^(1)],; μ_j,23^(2)= %̣ṣ/̣%̣ṣβ4q_0x+1/2i(β q_0μ_j,33^(1)+q_-1μ_j,43^(1))=β/4q_0x+1/2i[β q_0∫_(x_j,t_j)^(x,t)Δ_33^(1)+q_-1∫_(x_j,t_j)^(x,t)Δ_43^(1)],; μ_j,24^(2)= %̣ṣ/̣%̣ṣ14q_-1x+1/2i(β q_0μ_j,34^(1)+q_-1μ_j,44^(1))=1/4q_-1x+1/2i[β q_0∫_(x_j,t_j)^(x,t)Δ_34^(1)+q_-1∫_(x_j,t_j)^(x,t)Δ_44^(1)], ]. {[ μ_j,31^(2)= %̣ṣ/̣%̣ṣα4q̅_1x+iα/2(q̅_1μ_j,11^(1)+βq̅_0μ_j,21^(1)) =α/4q̅_1x+iα/2[q̅_1∫_(x_j,t_j)^(x,t)Δ_11^(1)+βq̅_0 ∫_(x_j,t_j)^(x,t)Δ_21^(1)],; μ_j,32^(2)= %̣ṣ/̣%̣ṣαβ4q̅_0x+iα/2(q̅_1μ_j,12^(1)+βq̅_0μ_j,22^(1)) =αβ/4q̅_0x+iα/2[q̅_1∫_(x_j,t_j)^(x,t)Δ_12^(1)+βq̅_0 ∫_(x_j,t_j)^(x,t)Δ_22^(1)],; μ_j,41^(2)= %̣ṣ/̣%̣ṣα4q̅_0x+iα/2(q̅_0μ_j,11^(1)+q̅_-1μ_j,21^(1)) =α/4q̅_0x+iα/2[q̅_0∫_(x_j,t_j)^(x,t)Δ_11^(1)+βq̅_-1∫_(x_j,t_j)^(x,t)Δ_21^(1)],; μ_j,42^(2)= %̣ṣ/̣%̣ṣα4q̅_-1x+iα/2(q̅_0μ_j,12^(1)+q̅_-1μ_j,22^(1)) =α/4q̅_-1x+iα/2[q̅_0∫_(x_j,t_j)^(x,t)Δ_12^(1)+βq̅_-1∫_(x_j,t_j)^(x,t)Δ_22^(1)], ]. [Δ_11^(2)={α/4(q_1q̅_1x+q_0q̅_0x)+iα/2[(|q_1|^2+|q_0|^2)μ_j,11^(1)+(β q_1q̅_0+q_0q̅_-1)μ_j,21^(1)]}dx;+̣{α/4(q_1q̅_1t+q_0q̅_0t)+iα/4(q_1xq̅_1x+q_0xq̅_0x).-i/4[(|q_1|^2+|q_0|^2)^2;+̣ (β q_1q̅_0+q_0q̅_-1)(β q_0q̅_1+q_-1q̅_0)] +α/2(q_1q̅_1x-q_1xq̅_1+q_0q̅_0x-q_0xq̅_0)μ_j,11^(1); .+α/2(β q_1q̅_0x-β q_1xq̅_0+q_0q̅_-1x-q_0xq̅_-1)μ_j,21^(1)} dt, ] [ Δ_12^(2)= {α/4(β q_1q̅_0x+q_0q̅_-1x) +iα/2[(|q_1|^2+|q_0|^2)μ_j,12^(1)+(β q_1q̅_0+q_0q̅_-1)μ_j,22^(1)]}dx; +̣{α/4(β q_1q̅_0t+q_0q̅_-1t)-i/4(β q_1q̅_0+q_0q̅_-1)(|q_1|^2+2|q_0|^2+|q_-1|^2) .; +̣iα/4(β q_1xq̅_0x+q_0xq̅_-1x)+α/2(q_1q̅_1x-q_1xq̅_1+q_0q̅_0x-q_0xq̅_0)μ_j,12^(1); .+α/2(β q_1q̅_0x-β q_1xq̅_0+q_0q̅_-1x-q_0xq̅_-1)μ_j,22^(1)} dt, ] [ Δ_21^(2)= {α/4(β q_0q̅_1x+q_-1q̅_0x)+iα/2[(β q_0q̅_1+q_-1q̅_0)μ_j,11^(1)+(|q_-1|^2+|q_0|^2)μ_j,21^(1)]}dx; +̣{α/4(β q_0q̅_1t+q_-1q̅_0t) -i/4(β q_0q̅_1+q_-1q̅_0)(|q_1|^2+2|q_0|^2+|q_-1|^2).;+̣iα/4(β q_0xq̅_1x+q_-1xq̅_0x) +α/2(q_-1q̅_-1x-q_-1xq̅_-1+q_0q̅_0x-q_0xq̅_0)μ_j,21^(1); .+α/2(β q_0q̅_1x-β q_0xq̅_1+q_-1q̅_0x-q_-1xq̅_0)μ_j,11^(1)} dt, ] [Δ_22^(2)= {α/4(q_-1q̅_-1x+q_0q̅_0x)+iα/2[(β q_0q̅_1+q_-1q̅_0)μ_j,12^(1)+(|q_-1|^2+|q_0|^2)μ_j,22^(1)]}dx;+̣{α/4(q_-1q̅_-1t+q_0q̅_0t)+iα/4(q_-1xq̅_-1x+q_0xq̅_0x)-i/4[|q_0|^2+|q_-1|^2)^2.;+̣(β q_0q̅_1+q_-1q̅_0)(β q_1q̅_0+q_0q̅_-1)] +α/2(q_1q̅_-1x-q_-1xq̅_-1+q_0q̅_0x-q_0xq̅_0)μ_j,22^(1); .+α/2(β q_0q̅_1x-β q_0xq̅_1+q_-1q̅_0x-q_-1xq̅_0)μ_j,12^(1)} dt, ][Δ_33^(2)={α/4(q_1xq̅_1+q_0xq̅_0)-iα/2[(|q_1|^2+|q_0|^2)μ_j,33^(1)+(β q_-1q̅_0+q_0q̅_1)μ_j,43^(1)]}dx; +̣{α/4(q_1tq̅_1+q_0tq̅_0)-iα/4(q_1xq̅_1x+q_0xq̅_0x) +i/4[(|q_1|^2+|q_0|^2)^2.;+̣ (β q_-1q̅_0+q_0q̅_1)(β q_0q̅_-1+q_1q̅_0)] +α/2(q_1xq̅_1-q_1q̅_1x+q_0xq̅_x-q_0q̅_0x)μ_j,33^(1); .+α/2(β q_-1xq̅_0-β q_-1q̅_0x+q_0xq̅_1-q_0q̅_1x)μ_j,43^(1)} dt, ] [ Δ_34^(2)= {α/4(β q_-1xq̅_0+q_0q̅_1x) -iα/2[(|q_1|^2+|q_0|^2)μ_j,34^(1)+(β q_-1q̅_0+q_0q̅_1)μ_j,44^(1)]}dx; +̣{α/4(β q_-1tq̅_0+q_0tq̅_1) +i/4(β q_-1q̅_0+q_0q̅_1)(|q_1|^2+2|q_0|^2+|q_-1|^2).;-̣iα/4(q_-1xq̅_0x+q_0xq̅_1x) +α/2(q_1xq̅_1-q_1q̅_1x+q_0xq̅_x-q_0q̅_0x)μ_j,34^(1); .+α/2(β q_-1xq̅_0-β q_-1q̅_0x+q_0xq̅_1-q_0q̅_1x)μ_j,44^(1)} dt, ] [Δ_43^(2)= {α/4(β q_0xq̅_1+q_1xq̅_0)-iα/2[(β q_0q̅_-1+q_1q̅_0)μ_j,33^(1)+(|q_-1|^2+|q_0|^2)μ_j,43^(1)]}dx; +̣{α/4(β q_0tq̅_-1+q_1tq̅_0)+i/4(β q_0q̅_-1+q_1q̅_0)(|q_1|^2+2|q_0|^2+|q_-1|^2) .; -̣iα/4(β q_0xq̅_-1x+q_1xq̅_0x)+α/2(q_-1xq̅_-1-q_-1q̅_-1x+q_0xq̅_x-q_0q̅_0x)μ_j,43^(1); .+α/2(β q_0xq̅_-1-β q_0q̅_-1x+q_1xq̅_0-q_1q̅_0x)μ_j,33^(1)} dt, ] [Δ_44^(2)= {α/4(q_-1xq̅_-1+q_0xq̅_0)-iα/2[(β q_0q̅_-1+q_1q̅_0)μ_j,34^(1)+(|q_-1|^2+|q_0|^2)μ_j,44^(1)]}dx;+̣{α/4(q_-1tq̅_-1+q_0tq̅_0) +i/4[(β q_-1q̅_0+q_0q̅_1)(β q_0q̅_-1+q_1q̅_0)+(|q_0|^2+|q_-1|^2)^2].;-̣iα/4(q_-1xq̅_-1x+q_0xq̅_0x)+α/2(q_-1xq̅_-1-q_-1q̅_-1x+q_0xq̅_x-q_0q̅_0x)μ_j,44^(1); .+α/2(β q_0xq̅_-1-β q_0q̅_-1x+q_1xq̅_0-q_1q̅_0x)μ_j,34^(1)} dt, ] where the functions {μ^(i)_jl=μ^(i)_jl(x,t)}_1^3,i=1, 2 are independent of k.We define the matrix-valued function Ψ(t,k)=(Ψ_ij(t,k))_4× 4 as [μ_2(0, t,k)=Ψ(t, k)= I+∑_s=1^21/k^s([ Ψ_11^(s)(t) Ψ_12^(s)(t) Ψ_13^(s)(t) Ψ_14^(s)(t); Ψ_21^(s)(t) Ψ_22^(s)(t) Ψ_23^(s)(t) Ψ_24^(s)(t); Ψ_31^(s)(t) Ψ_32^(s)(t) Ψ_33^(s)(t) Ψ_34^(s)(t); Ψ_41^(s)(t) Ψ_42^(s)(t) Ψ_43^(s)(t) Ψ_44^(s)(t) ]) +O(1/k^3), ]By using the asymptotic of Eq. (<ref>) and the boundary data at x=0, we find {[ Ψ_13^(1)(t)=-%̣ṣ/̣%̣ṣi2u_01(t), Ψ_14^(1)(t)=βΨ_23^(1)(t)=-i/2u_00(t),Ψ_24^(1)(t)=-%̣ṣ/̣%̣ṣi2u_0-1(t),; Ψ_13^(2)(t)=%̣ṣ/̣%̣ṣ14u_11(t)-i/2[u_01(t)Ψ_33^(1)+u_00(t)Ψ_43^(1)],; Ψ_14^(2)(t)=%̣ṣ/̣%̣ṣ14u_10(t)-i/2[u_01(t)Ψ_34^(1)+u_00(t)Ψ_44^(1)],;Ψ_23^(2)(t)=%̣ṣ/̣%̣ṣβ4u_10(t)-i/2[β u_00(t)Ψ_33^(1)+u_0-1(t)Ψ_43^(1)],; Ψ_24^(2)(t)=%̣ṣ/̣%̣ṣ14u_1-1(t)-i/2[β u_00(t)Ψ_34^(1)+u_0-1(t)Ψ_44^(1)],;Ψ_33^(1)(t)=%̣ṣ/̣%̣ṣα2∫^t_0∑_j=0,1[u̅_0j(t)u_1j(t)-u_0j(t)u̅_1j(t)]dt,;Ψ_44^(1)(t)=%̣ṣ/̣%̣ṣα2∫^t_0∑_j=-1, 0[u̅_0j(t)u_1j(t)-u_0j(t)u̅_1j(t)]dt,;Ψ_34^(1)(t)=α/2∫^t_0[β u_11(t)u̅_00(t)-β u_01(t)u̅_10(t)+u_10(t)u̅_0-1(t)-u_00(t)u̅_1-1(t)]dt,;Ψ_43^(1)(t)=α/2∫^t_0[β u_10(t)u̅_01(t)-β u_00(t)u̅_11(t)+u_1-1(t)u̅_00(t)-u_0-1(t)u̅_10(t)]dt,; ]. Thus we have the the Dirichlet-Neumann boundary data at x=0: {[u_01(t)= 2iΨ_13^(1)(t),u_00(t)=2iΨ_14^(1)(t)=2iβΨ_23^(1)(t),u_0-1(t)= 2iΨ_24^(1)(t),;u_11(t)=4Ψ_13^(2)(t)+2i[u_01(t)Ψ_33^(1)(t)+u_00(t)Ψ_43^(1)(t)]; u_1-1(t)=4Ψ_24^(2)(t)+2i[β u_00(t)Ψ_34^(1)(t)+u_0-1(t)Ψ_44^(1)(t)],;u_10(t)=4Ψ_14^(2)(t)+2i[u_01(t)Ψ_34^(1)(t)+u_00(t)Ψ_44^(1)(t)]; =4βΨ_23^(2)(t)+2iβ[β u_00(t)Ψ_33^(1)(t)+u_0-1(t)Ψ_43^(1)(t)], ].For the vanishing initial values, it follows from Eqs. (<ref>) and (<ref>) that we have the following asymptotic of c_24(t,k) and c_1j(t,k), j=3,4.Proposition 4.1. The global relation (<ref>) implies that the large k behavior ofc_1j(t,k),j=3,4 and c_24(t,k) is of the form c_13(t,k)=Ψ_13^(1)/k+Ψ_13^(2)/k^2+O(1/k^3), c_14(t,k)=Ψ_14^(1)/k+Ψ_14^(2)/k^2+O(1/k^3), c_24(t,k)=Ψ_24^(1)/k+Ψ_24^(2)/k^2+O(1/k^3), Proof.The global relation (<ref>) can be written asc_13(t,k)=[Ψ_11(t,k)s_13+Ψ_12(t,k)s_23]e^-4ik^2t+Ψ_13(t,k)s_33+Ψ_14(t,k)s_43,c_14(t,k)=[Ψ_11(t,k)s_14+Ψ_12(t,k)s_24]e^-4ik^2t+Ψ_13(t,k)s_34+Ψ_14(t,k)s_44,c_24(t,k)=[Ψ_21(t,k)s_14+Ψ_22(t,k)s_24]e^-4ik^2t+Ψ_23(t,k)s_34+Ψ_24(t,k)s_44, According to the asymptotics (<ref>), we have ([ s_13; s_23; s_33; s_43 ])=([ 0; 0; 1; 0 ])+1/2ik([q_1(0,0); β q_0(0,0,); 2i∫_(∞, 0)^(0,0)Δ_33^(1)(0,0); 2i∫_(∞, 0)^(0,0)Δ_34^(1)(0,0) ]) +O(1/k^2),and ([ s_14; s_24; s_34; s_44 ])=([ 0; 0; 0; 1 ])+1/2ik([q_1(0,0); β q_0(0,0,); 2i∫_(∞, 0)^(0,0)Δ_34^(1)(0,0); 2i∫_(∞, 0)^(0,0)Δ_44^(1)(0,0) ]) +O(1/k^2), Recalling the time-part of the Lax pair (<ref>) μ_t+2ik^2[σ_4, μ]=V(x,t,k)μ, It follows from the first column of Eq. (<ref>) with μ=μ_2(0,t,k)=Ψ(t,k) that we have {[ Ψ_11,t(t,k)= 2k(u_01Ψ_31+u_00Ψ_41)+i(u_11Ψ_31+u_10Ψ_41);-iα[(|u_01|^2+|u_00|^2)Ψ_11+(β u_01u̅_02+u_00u̅_0-1)Ψ_21],; Ψ_21,t(t,k)= 2k(β u_00Ψ_31+u_0-1Ψ_41)+i(β u_10Ψ_31+u_1-1Ψ_41); -iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_11+(|u_0-1|^2+|u_00|^2)Ψ_21],; Ψ_31,t(t,k)= 4ik^2Ψ_31+2α k(u̅_01Ψ_11+βu̅_00Ψ_21) -iα(u̅_11Ψ_11+βu̅_10Ψ_21); +iα[(|u_01|^2+|u_00|^2)Ψ_31+(β u_0-1u̅_00+u_00u̅_01)Ψ_41]; Ψ_41,t(t,k)=4ik^2Ψ_41+2α k(u̅_00Ψ_11+u̅_0-1Ψ_21)-iα(u̅_10Ψ_11+u̅_1-1Ψ_21); +iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_31+(|u_0-1|^2+|u_00|^2)Ψ_41], ]. The second column of Eq. (<ref>) with μ=μ_2(0,t,k)=Ψ(t,k) yields {[ Ψ_12,t(t,k)= 2k(u_01Ψ_32+u_00Ψ_42)+i(u_11Ψ_32+u_10Ψ_42);-iα[(|u_01|^2+|u_00|^2)Ψ_12+(β u_01u̅_02+u_00u̅_0-1)Ψ_22],; Ψ_22,t(t,k)= 2k(β u_00Ψ_32+u_0-1Ψ_42)+i(β u_10Ψ_32+u_1-1Ψ_42); -iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_12+(|u_0-1|^2+|u_00|^2)Ψ_22],; Ψ_32,t(t,k)= 4ik^2Ψ_32+2α k(u̅_01Ψ_12+βu̅_00Ψ_22) -iα(u̅_11Ψ_12+βu̅_10Ψ_22); +iα[(|u_01|^2+|u_00|^2)Ψ_32+(β u_0-1u̅_00+u_00u̅_01)Ψ_42]; Ψ_42,t(t,k)=4ik^2Ψ_42+2α k(u̅_00Ψ_12+u̅_0-1Ψ_22)-iα(u̅_10Ψ_12+u̅_1-1Ψ_22); +iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_32+(|u_0-1|^2+|u_00|^2)Ψ_42], ]. The third column of Eq. (<ref>) with μ=μ_2(0,t,k)=Ψ(t,k) yields {[Ψ_13,t(t,k)= -4ik^2Ψ_13+2k(u_01Ψ_33+u_00Ψ_43)+i(u_11Ψ_33+u_10Ψ_43);-iα[(|u_01|^2+|u_00|^2)Ψ_13+(β u_01u̅_02+u_00u̅_0-1)Ψ_23],;Ψ_23,t(t,k)= -4ik^2Ψ_23+2k(β u_00Ψ_33+u_0-1Ψ_43)+i(β u_10Ψ_33+u_1-1Ψ_43); -iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_13+(|u_0-1|^2+|u_00|^2)Ψ_23],;Ψ_33,t(t,k)=2α k(u̅_01Ψ_13+βu̅_00Ψ_23) -iα(u̅_11Ψ_13+βu̅_10Ψ_23); +iα[(|u_01|^2+|u_00|^2)Ψ_33+(β u_0-1u̅_00+u_00u̅_01)Ψ_43];Ψ_43,t(t,k)= 2α k(u̅_00Ψ_13+u̅_0-1Ψ_23)-iα(u̅_10Ψ_13+u̅_1-1Ψ_23); +iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_33+(|u_0-1|^2+|u_00|^2)Ψ_43], ].and the fourth column of Eq. (<ref>) with μ=μ_2(0,t,k)=Ψ(t,k) yields {[Ψ_14,t(t,k)= -4ik^2Ψ_14+2k(u_01Ψ_34+u_00Ψ_44)+i(u_11Ψ_34+u_10Ψ_44);-iα[(|u_01|^2+|u_00|^2)Ψ_14+(β u_01u̅_02+u_00u̅_0-1)Ψ_24],;Ψ_24,t(t,k)= -4ik^2Ψ_24+2k(β u_00Ψ_34+u_0-1Ψ_44)+i(β u_10Ψ_34+u_1-1Ψ_44); -iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_14+(|u_0-1|^2+|u_00|^2)Ψ_24],;Ψ_34,t(t,k)=2α k(u̅_01Ψ_14+βu̅_00Ψ_24) -iα(u̅_11Ψ_14+βu̅_10Ψ_24); +iα[(|u_01|^2+|u_00|^2)Ψ_34+(β u_0-1u̅_00+u_00u̅_01)Ψ_44];Ψ_44,t(t,k)= 2α k(u̅_00Ψ_14+u̅_0-1Ψ_24)-iα(u̅_10Ψ_14+u̅_1-1Ψ_24); +iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_34+(|u_0-1|^2+|u_00|^2)Ψ_44], ]. Suppose that Ψ_j1's, j=1,2,3,4 are of the form ([ Ψ_11; Ψ_21; Ψ_31; Ψ_41 ]) =(a_10(t)+a_11(t)/k+a_12(t)/k^2+⋯)+(b_10(t)+b_11(t)/k+b_12(t)/k^2+⋯)e^4ik^2t,where the 4× 1 column vector functions a_1j(t),b_1j(t)(j=0,1,...,) are independent of k.By substituting Eq. (<ref>) into Eq.(<ref>) and using the initial conditions a_10(0)+b_10(0)=(1, 0,0,0)^T, a_11(0)+b_11(0)=(0, 0, 0, 0)^T, we have ([ Ψ_11; Ψ_21; Ψ_31; Ψ_41 ]) =([ 1; 0; 0; 0 ]) +1/k([ Ψ_11^(1); Ψ_21^(1); Ψ_31^(1); Ψ_41^(1) ]) +1/k^2([ Ψ_11^(2); Ψ_21^(2); Ψ_31^(2); Ψ_41^(2) ]) +O(1/k^3) +[1/k([ 0; 0; -iα/2u̅_01(0); -iα/2u̅_00(0) ])+O(1/k^2)]e^4ik^2t,Similarly, it follows from Eqs. (<ref>)-(<ref>) that we have the asymptotic formulae for Ψ_ij,i=1,2,3,4; j=2,3,4 in the form ([ Ψ_12; Ψ_22; Ψ_32; Ψ_42 ]) =([ 0; 1; 0; 0 ]) +1/k([ Ψ_12^(1); Ψ_22^(1); Ψ_32^(1); Ψ_42^(1) ]) +1/k^2([ Ψ_12^(2); Ψ_22^(2); Ψ_32^(2); Ψ_42^(2) ]) +O(1/k^3) +[1/k([0;0; -iαβ/2u̅_00(0); -iα/2u̅_0-1(0) ])+O(1/k^2)]e^4ik^2t, ([ Ψ_13; Ψ_23; Ψ_33; Ψ_43 ]) =([ 0; 0; 1; 0 ]) +1/k([ Ψ_13^(1); Ψ_23^(1); Ψ_33^(1); Ψ_43^(1) ]) +1/k^2([ Ψ_13^(2); Ψ_23^(2); Ψ_33^(2); Ψ_43^(2) ]) +O(1/k^3) +[1/k([i/2u_01(0); iβ/2u_00(0); 0; 0 ])+O(1/k^2)]e^-4ik^2t,and ([ Ψ_14; Ψ_24; Ψ_34; Ψ_44 ]) =([ 0; 0; 0; 1 ]) +1/k([ Ψ_14^(1); Ψ_24^(1); Ψ_34^(1); Ψ_44^(1) ]) +1/k^2([ Ψ_14^(2); Ψ_24^(2); Ψ_34^(2); Ψ_44^(2) ]) +O(1/k^3) +[1/k([i/2u_00(0); i/2u_0-1(0); 0; 0 ])+O(1/k^2) ]e^-4ik^2t, The substitution of Eqs. (<ref>) and (<ref>)-(<ref>) into Eq. (<ref>) yields Eq. (<ref>). Similarly, we can also get Eqs. (<ref>) and (<ref>). □ §.§ (c) The map between Dirichlet and Neumann problemsIn the following we mainly show that the spectral functions S(k) and S_L(k) can be expressed in terms of the prescribed Dirichlet and Neumann boundary data and the initial data using the solution of a system of integral equations.Define the new notations as F_± (t,k)=F(t,k)± F(t, -k), Σ_±(k)=e^2ikL± e^-2ikL.The sign ∂ D_j, j=1,...,4 stands for the boundary of the jth quadrant D_j, oriented so that D_j lies to the left of ∂ D_j. ∂ D_3^0 denotes the boundary contour which has not contain the zeros of Σ_-(k) and ∂ D_3^0=-∂ D_1^0. Theorem 4.2. Letthe initial data of Eq. (<ref>) q_j(x,t=0)=q_0j(x),j=1,0,-1 be the functions of Schwartz class on the domain x∈ [0, ∞) and0<t<T<∞. For the Dirichlet problem, the boundary data u_0j(t),(j=1,0,-1) on the interval t∈ [0, T) are sufficiently smooth and compatible with the initial data q_0j(x),(j=1,0,-1) at the point (x_2, t_2)=(0, 0), i.e., u_0j(0)=q_0j(0),j=1,0,-1. Similarly, for the Neumann problem, the boundary data u_1j(t),j=1,0,-1 on the interval t∈ [0, T) are sufficiently smooth and compatible with the initial data q_0j(x),j=1,0,-1 at the origin (x_2, t_2)=(0, 0). For simplicity, let n_33,44(s)(k) have no zeros in the domain D_1. Then the matrix-valued spectral function S(k) is defined by [ S(k)= ([ m_11(Ψ(T,k))-m_21(Ψ(T,k)) m_31(Ψ(T,k))e^4ik^2T-m_41(Ψ(T,k))e^4ik^2T;-m_12(Ψ(T,k)) m_22(Ψ(T,k))-m_32(Ψ(T,k))e^4ik^2T m_42(Ψ(T,k))e^4ik^2T;m_13(Ψ(T,k))e^-4ik^2T -m_23(Ψ(T,k))e^-4ik^2T m_33(Ψ(T,k))-m_43(Ψ(T,k)); -m_14(Ψ(T,k))e^-4ik^2Tm_24(Ψ(T,k))e^-4ik^2T-m_34(Ψ(T,k)) m_44(Ψ(T,k)) ]), ] and the complex-valued functions {Ψ_ij(t,k)}_i,j=1^4 have the following system of integral equations {[ Ψ_11(t,k)=1+∫_0^t {-iα[(|u_01|^2+|u_00|^2)Ψ_11+(β u_01u̅_02+u_00u̅_0-1)Ψ_21].; +. (2ku_01+iu_11)Ψ_31+(2ku_00+iu_10)Ψ_41}(t',k)dt',; Ψ_21(t,k)=∫_0^t{-iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_11+(|u_0-1|^2+|u_00|^2)Ψ_21].; +. (2k(β u_00+iβ u_10)Ψ_31+ (2ku_0-1+iu_1-1)Ψ_41}(t',k)dt',; Ψ_31(t,k)= ∫_0^te^4ik^2(t-t'){2α k(u̅_01Ψ_11+βu̅_00Ψ_21) -iα(u̅_11Ψ_11+βu̅_10Ψ_21).;.+iα[(|u_01|^2+|u_00|^2)Ψ_31+(β u_0-1u̅_00+u_00u̅_01)Ψ_41]} (t',k)dt',; Ψ_41(t,k)= ∫_0^te^4ik^2(t-t')[ 2α k(u̅_00Ψ_11+u̅_0-1Ψ_21)-iα(u̅_10Ψ_11+u̅_1-1Ψ_21).;.+iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_31+(|u_0-1|^2+|u_00|^2)Ψ_41]](t',k)dt', ]. {[Ψ_12(t,k)=∫_0^t{-iα[(|u_01|^2+|u_00|^2)Ψ_12+(β u_01u̅_02+u_00u̅_0-1)Ψ_22].; .+2k(u_01Ψ_32+u_00Ψ_42)+i(u_11Ψ_32+u_10Ψ_42)}(t',k)dt',;Ψ_22(t,k)= 1+∫_0^t{-iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_12+(|u_0-1|^2+|u_00|^2)Ψ_22].;. +2k(β u_00Ψ_32+u_0-1Ψ_42)+i(β u_10Ψ_32+u_1-1Ψ_42)}(t',k)dt',;Ψ_32(t,k)= ∫_0^t e^4ik^2(t-t'){2α k(u̅_01Ψ_12+βu̅_00Ψ_22) -iα(u̅_11Ψ_12+βu̅_10Ψ_22).;. +iα[(|u_01|^2+|u_00|^2)Ψ_32+(β u_0-1u̅_00+u_00u̅_01)Ψ_42]}(t',k)dt',;Ψ_42(t,k)=∫_0^t e^4ik^2(t-t'){2α k(u̅_00Ψ_12+u̅_0-1Ψ_22)-iα(u̅_10Ψ_12+u̅_1-1Ψ_22).;.+iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_32+(|u_0-1|^2+|u_00|^2)Ψ_42]}(t',k)dt', ]. {[Ψ_13(t,k)= ∫_0^te^-4ik^2(t-t'){-iα[(|u_01|^2+|u_00|^2)Ψ_13+(β u_01u̅_02+u_00u̅_0-1)Ψ_23].,; .+2k(u_01Ψ_33+u_00Ψ_43)+i(u_11Ψ_33+u_10Ψ_43)}(t',k)dt',;Ψ_23(t,k)= ∫_0^te^-4ik^2(t-t'){-iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_13+(|u_0-1|^2+|u_00|^2)Ψ_23].;. +2k(β u_00Ψ_33+u_0-1Ψ_43)+i(β u_10Ψ_33+u_1-1Ψ_43)}(t',k)dt',;Ψ_33(t,k)= 1+∫_0^t{2α k(u̅_01Ψ_13+βu̅_00Ψ_23) -iα(u̅_11Ψ_13+βu̅_10Ψ_23).; .+iα[(|u_01|^2+|u_00|^2)Ψ_33+(β u_0-1u̅_00+u_00u̅_01)Ψ_43]}(t',k)dt',;Ψ_43(t,k)=∫_0^t{2α k(u̅_00Ψ_13+u̅_0-1Ψ_23)-iα(u̅_10Ψ_13+u̅_1-1Ψ_23).;.+iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_33+(|u_0-1|^2+|u_00|^2)Ψ_43]}(t',k)dt', ].and {[ Ψ_14(t,k)= ∫_0^te^-4ik^2(t-t'){-iα[(|u_01|^2+|u_00|^2)Ψ_14+(β u_01u̅_02+u_00u̅_0-1)Ψ_24].; .+2k(u_01Ψ_34+u_00Ψ_44)+i(u_11Ψ_34+u_10Ψ_44)}(t',k)dt',; Ψ_24(t,k)=∫_0^te^-4ik^2(t-t'){-iα[(β u_00u̅_01+u_0-1u̅_00)Ψ_14+(|u_0-1|^2+|u_00|^2)Ψ_24].;.+ 2k(β u_00Ψ_34+u_0-1Ψ_44)+i(β u_10Ψ_34+u_1-1Ψ_44)}(t',k)dt',; Ψ_34(t,k)=∫_0^t[2α k(u̅_01Ψ_14+βu̅_00Ψ_24) -iα(u̅_11Ψ_14+βu̅_10Ψ_24).; .+iα[(|u_01|^2+|u_00|^2)Ψ_34+(β u_0-1u̅_00+u_00u̅_01)Ψ_44]}(t',k)dt',; Ψ_44(t,k)= 1+∫_0^t{2α k(u̅_00Ψ_14+u̅_0-1Ψ_24)-iα(u̅_10Ψ_14+u̅_1-1Ψ_24).; . +iα[(β u_00u̅_0-1+u_01u̅_00)Ψ_34+(|u_0-1|^2+|u_00|^2)Ψ_44]}(t',k)dt', ]. (i) For the known Dirichlet problem, the unknown Neumann boundary conditions u_1j(t),j=1, 0,-1,0<t<T can be found by[u_11(t)= ∫̣_∂ D_32/iπ[kΨ_13-(t,-k)-iu_01(t)+u_01(t)Ψ_33-(t,k)+u_00(t)Ψ_43-(t,k)]dk; +̣4i/π∫_∂ D_3 k{[Ψ_11(t,k)s_13+Ψ_12(t,k)s_23]e^-4ik^2t.+Ψ_13(t,k)(s_33-1)+Ψ_14(t,k)s_43}dk, ] [ u_10(t)=∫̣_∂ D_32/iπ[kΨ_14-(t,-k)-iu_00(t)+u_01(t)Ψ_34-(t,k)+β u_00(t)Ψ_44-(t,k)]dk;+̣4i/π∫_∂ D_3 k{[Ψ_11(t,k)s_14+Ψ_12(t,k)s_24]e^-4ik^2t. +Ψ_13(t,k)s_34+Ψ_14(t,k)(s_44-1)}dk, ] [ u_1-1(t)= ∫̣_∂ D_32/iπ[kΨ_24-(t,-k)-iu_0-1(t)+u_00(t)Ψ_34-(t,k)+β u_0-1(t)Ψ_44-(t,k)]dk; +̣4i/π∫_∂ D_3 k{[Ψ_21(t,k)s_14+Ψ_22(t,k)s_24]e^-4ik^2t.+Ψ_23(t,k)s_34+Ψ_24(t,k)(s_44-1)}dk, ] (ii) For the known Neumannproblem, the unknown Dirichlet boundary conditions u_0j(t),j=1, 0,-1,0<t<T can be found by[u_01(t)= %̣ṣ/̣%̣ṣ1π∫_∂ D_3^0Ψ_13+(t, -k)dk -2/π∫_∂ D_3{[Ψ_11(t,k)s_13+Ψ_12(t,k)s_23]e^-4ik^2t.; +̣Ψ_13(t,k)(s_33-1)+Ψ_14(t,k)s_43}dk, ] [u_00(t)= %̣ṣ/̣%̣ṣ1π∫_∂ D_3^0Ψ_14+(t, -k)dk -2/π∫_∂ D_3{[Ψ_11(t,k)s_14+Ψ_12(t,k)s_24]e^-4ik^2t.; +̣Ψ_13(t,k)s_34+Ψ_14(t,k)(s_44-1)}dk, ] [ u_1-1(t)= %̣ṣ/̣%̣ṣ1π∫_∂ D_3^0Ψ_24+(t, -k)dk -2/π∫_∂ D_3{[Ψ_21(t,k)s_14+Ψ_22(t,k)s_24]e^-4ik^2t.; +̣Ψ_23(t,k)s_34+Ψ_24(t,k)(s_44-1)}dk, ] where s_ij=s_ij(k),i,j=1,2,3,4.Proof. We can show Eq. (<ref>) by means of Eq. (<ref>), that is,S(k)=e^-2ik^2Tσ̂_4μ_2^-1(0,T,k)=e^-2ik^2Tσ̂_4(μ_2^A(0,T,k))^T=e^-2ik^2Tσ̂_4(Ψ^A(T,k))^T,Moreover, Eqs. (<ref>)-(<ref>) for Ψ_ij(t,k),i,j=1,2,3,4 can be obtained by using the Volteral integral equations of μ_2(0,t,k).(i) In the following we show Eqs. (<ref>)-(<ref>). Applying the Cauchy's theorem to Eq. (<ref>), we have [-iπ2Ψ_33^(1)(t)=∫_∂ D_2[Ψ_33(t,k)-1]dk=∫_∂ D_4[Ψ_33(t,k)-1]dk,;-iπ2Ψ_43^(1)(t)=∫_∂ D_2Ψ_43(t,k)dk=∫_∂ D_4Ψ_43(t,k)dk,; -iπ2Ψ_13^(2)(t)=∫_∂ D_2[kΨ_13(t,k)+i/2u_01(t)]dk=-∫_∂ D_4[kΨ_13(t,k)+i/2u_01(t)]dk, ]From Eq. (<ref>), we further find[iπΨ_33^(1)(t)= -̣(∫_∂ D_2+∫_∂ D_4)[Ψ_33(t,k)-1]dk = (∫_∂ D_1+∫_∂ D_3)[Ψ_33(t,k)-1]dk; = ∫̣_∂ D_3[Ψ_33(t,k)-1]dk-∫_∂ D_3[Ψ_33(t,-k)-1]dk = ∫̣_∂ D_3Ψ_33-(t,k)dk, ] iπΨ_43^(1)(t)=∫̣_∂ D_3Ψ_43-(t,k)dk, [iπΨ_13^(2)(t)= (∫_∂ D_1-∫_∂ D_3)[kΨ_13(t,k)+i/2u_01(t)]dk+C_1(t)=∫_∂ D_3[kΨ_13-(t,-k)-i u_01(t)]dk+C_1(t), ] where we have introduced the function C_1(t) in the formC_1(t)=2̣∫_∂ D_3[kΨ_13(t,k)+i/2u_01(t)]dk, We use the global relation (<ref>), the Cauchy's theorem and asymptotic (<ref>) to further reduce C_1(t) to be [ C_1(t)= 2̣∫_∂ D_3[kc_13(t,k)+i/2u_01(t)]dk-2̣∫_∂ D_3k[c_13(t,k)-Ψ_13(t,k)]dk,; = -̣iπΨ_13^(2)-2∫_∂ D_3k{[Ψ_11(t,k)s_13(k)+Ψ_12(t,k)s_23(k)]e^-4ik^2t. +Ψ_13(t,k)(s_33(k)-1)+Ψ_14(t,k)s_43(k)}dk, ]It follows from Eqs. (<ref>) and (<ref>) that we have [2iπΨ_13^(2)(t)= ∫̣_∂ D_3[kΨ_13-(t,-k)-i u_01(t)]dk-2∫_∂ D_3k{[Ψ_11(t,k)s_13(k)+Ψ_12(t,k)s_23(k)]e^-4ik^2t.; +̣Ψ_13(t,k)(s_33(k)-1)+Ψ_14(t,k)s_43(k)}dk, ] Thus substituting Eqs. (<ref>), (<ref>) and (<ref>) into the fourth one of system (<ref>), we can get Eq. (<ref>). Similarly, we can also show that Eqs. (<ref>) and (<ref>) hold.(ii) We now derive the Dirichlet boundary value conditions (<ref>)-(<ref>) at x=0 from the given Neumann boundary value problems. It follows from the first one of Eq. (<ref>) that u_01(t) can be expressed by means of Ψ_13^(1). Applying the Cauchy's theorem to Eq. (<ref>) yields [iπΨ_13^(1)(t)= (∫_∂ D_1+∫_∂ D_3)Ψ_13(t,k)dk= (∫_∂ D_1-∫_∂ D_3)Ψ_13(t,k)dk+C_2(t); =∫̣_∂ D_1Ψ_13+(t,k)dk+C_2(t)= ∫̣_∂ D_3Ψ_13+(t,-k)dk+C_2(t), ] where we have introduced the function C_2(t) in the formC_2(t)=2̣∫_∂ D_3Ψ_13(t,k)dk, We use the global relation (<ref>),the Cauchy's theorem and asymptotics (<ref>) to further reduce C_2(t) to be [ C_2(t)=2̣∫_∂ D_3c_13(t,k)dk-2̣∫_∂ D_3[c_13(t,k)-Ψ_13(t,k)]dk,; = -̣iπΨ_13^(1)-2∫_∂ D_3{[Ψ_11(t,k)s_13(k)+Ψ_12(t,k)s_23(k)]e^-4ik^2t.+Ψ_13(t,k)(s_33(k)-1)+Ψ_14(t,k)s_43(k)}dk, ]Eqs. (<ref>) and (<ref>) imply that [2iπΨ_13^(1)(t)= ∫̣_∂ D_3Ψ_13+(t,-k)dk-2∫_∂ D_3{[Ψ_11(t,k)s_13(k)+Ψ_12(t,k)s_23(k)]e^-4ik^2t.; +̣Ψ_13(t,k)(s_33(k)-1)+Ψ_14(t,k)s_43(k)}dk, ] Thus, the substitution of Eq. (<ref>) into the first one of Eq. (<ref>) yields Eq. (<ref>). Similarly, in terms of the global relation (<ref>) and (<ref>), we can also show Eqs. (<ref>) and (<ref>) by using the second and third ones of Eq. (<ref>) and Ψ_14^(1)(t) and Ψ_24^(1)(t).□ §.§ (d) Effective characterizations Substituting the perturbated expressions of the eigenfunction and initial-boundary data [ Ψ_ij(t,k)=Ψ_ij^[0]+ϵΨ_ij^[1]+ϵ^2Ψ_ij^[2]+⋯, i,j=1,2,3,4,; u_0j(t)=ϵ u_0j^[1](t)+ϵ^2 u_0j^[2](t)+⋯, j=1,0,-1,; u_1j(t)=ϵ u_1j^[1](t)+ϵ^2 u_1j^[2](t)+⋯, j=1,0,-1, ] where ϵ>0 is a small parameter,into Eqs. (<ref>)-(<ref>), we have these terms of O(1) and O(ϵ) asO(1): {[ Ψ_jj^[0](t,k)=1, j=1,2,3,4,; Ψ_ij^[0](t,k)=0, i,j=1,2,3,4,i≠j, ].O(ϵ): {[ Ψ_11^[1](t,k)=Ψ_12^[1]=Ψ_21^[1]=Ψ_22^[1]=Ψ_33^[1](t,k)=Ψ_34^[1]=Ψ_43^[1]=Ψ_44^[1]=0,; Ψ_13^[1](t,k)=∫̣_0^te^-4ik^2(t-t')(2ku_01^[1]+iu_11^[1])(t')dt',Ψ_14^[1](t,k)=∫̣_0^te^-4ik^2(t-t')(2ku_00^[1]+iu_10^[1])(t')dt',;Ψ_23^[1](t,k)=β̣∫_0^te^-4ik^2(t-t')(2ku_00^[1]+iu_10^[1])(t')dt',Ψ_24^[1](t,k)=∫̣_0^te^-4ik^2(t-t')(2ku_0-1^[1]+iu_1-1^[1])(t')dt',; Ψ_31^[1](t,k)=α̣∫_0^te^4ik^2(t-t')(2ku̅_01^[1]-iu̅_11^[1])(t')dt', Ψ_32^[1](t,k)=α̣β∫_0^te^4ik^2(t-t')(2ku̅_00^[1]-iu̅_10^[1])(t')dt',; Ψ_41^[1](t,k)=α̣∫_0^te^4ik^2(t-t')(2ku̅_00^[1]-iu̅_10^[1])(t')dt',Ψ_42^[1](t,k)=α̣∫_0^te^4ik^2(t-t')(2ku̅_0-1^[1]-iu̅_1-1^[1])(t')dt', ].If we assume that n_33,44(s) has no zeros, then we substitute the fourth one in Eq. (<ref>) into Eqs. (<ref>)-(<ref>) to find {[ u_11^[1](t)=∫̣_∂ D_3[2/iπ(kΨ_13-^[1](t, -k)-iu_01^[1]) +4ik/π s_13^[1]]dk,; u_10^[1](t)=∫̣_∂ D_3[2/iπ(kΨ_14-^[1](t, -k)-iu_00^[1]) +4ik/π s_14^[1]]dk,; u_1-1^[1](t)=∫̣_∂ D_3[2/iπ(kΨ_24-^[1](t, -k)-iu_0-1^[1]) +4ik/π s_24^[1]]dk, ].where s_13=ϵ s_13^[1](t)+ϵ^2 s_13^[2](t)+O(ϵ^3),s_14=ϵ s_14^[1](t)+ϵ^2 s_14^[2](t)+O(ϵ^3), ands_24=ϵ s_24^[1](t)+ϵ^2 s_24^[2](t)+O(ϵ^3). It further follows from Eq. (<ref>) that we have {[Ψ_13-^[1](t, -k)=-̣4k∫_0^te^-4ik^2(t-t')u_01^[1](t')dt',;Ψ_14-^[1](t, -k)=-̣4k∫_0^te^-4ik^2(t-t')u_00^[1](t')dt',; Ψ_24-^[1](t, -k)=-̣4k∫_0^te^-4ik^2(t-t')u_0-1^[1](t')dt', ]. Thus, the Dirichlet problem can now be solved perturbatively as follows: for n_33,44(s) having no zeros and given u_0j^[1], j=1,0,-1, we can obtain {Ψ_ij-^[1],i=12; j=3,4 from Eq. (<ref>) and further find u_1j^[1],j=1,0,-1 from Eq. (<ref>). Finally, we can have Ψ_ij^[1] from Eq. (<ref>). In fact, these arguments for Ψ_ij can be extended to all orders such that we can determine all orders of S(k). In fact, the above recursive formulae can be continued indefinitely. We assume that they hold for all 0≤ j≤ n-1, then for n>0, the substitution of Eq. (<ref>) into Eqs. (<ref>)-(<ref>) yields the terms of O(ϵ^n) asu_11^[n](t)=∫̣_∂ D_3[2/iπ(kΨ_13-^[n](t, -k)-iu_01^[n]) +4ik/π s_13^[n]]dk+lowerorderterms,u_10^[n](t)=∫̣_∂ D_3[2/iπ(kΨ_14-^[n](t, -k)-iu_00^[n]) +4ik/π s_14^[n]]dk+lowerorderterms, u_1-1^[n](t)=∫̣_∂ D_3[2/iπ(kΨ_24-^[n](t, -k)-iu_0-1^[n]) +4ik/π s_24^[n]]dk+lowerorderterms,where `lower order terms' stands for the result involving known terms of lower order.The terms of O(ϵ^n) in Eqs. (<ref>)-(<ref>) yield {[ Ψ_13^[n](t, k)=∫̣_0^te^-4ik^2(t-t')(2ku_01^[n]+iu_11^[n])(t')dt' +lowerorderterms,; Ψ_14^[n](t, k)=∫̣_0^te^-4ik^2(t-t')(2ku_00^[n]+iu_10^[n])(t')dt' +lowerorderterms,; Ψ_24^[n](t, k)=∫̣_0^te^-4ik^2(t-t')(2ku_0-1^[n]+iu_1-1^[n])(t')dt' +lowerorderterms, ].which leads to {[Ψ_13-^[n](t, -k)=-̣4k∫_0^te^-4ik^2(t-t')u_01^[n](t')dt'+lowerorderterms,;Ψ_14-^[n](t, -k)=-̣4k∫_0^te^-4ik^2(t-t')u_00^[n](t')dt'+lowerorderterms,; Ψ_24-^[n](t, -k)=-̣4k∫_0^te^-4ik^2(t-t')u_0-1^[n](t')dt'+lowerorderterms, ].It follows from system (<ref>) that Ψ_13-^[n](t, -k), Ψ_14-^[n](t, -k), and Ψ_24-^[n](t, -k) can be generated at each step from the known Dirichlet boundary data u_0j^[n](t),j=1,0,-1 such that we know that the Neumann boundary data u_1j^[n](t),j=1,0,-1 can be given by Eqs. (<ref>)-(<ref>) and then Ψ_13^[n](t, k), Ψ_14^[n](t, k), and Ψ_24^[n](t,k) can be determined by Eq. (<ref>) and other Ψ_ij^[n](t,k) can also be found.Similarly, it follows from Eqs. (<ref>)-(<ref>) that we have[u_01^[1](t)=%̣ṣ/̣%̣ṣ1π∫_∂ D_3[Ψ_13+^[1](t,-k)-2s_13^[1]]dk,;u_00^[1](t)=%̣ṣ/̣%̣ṣ1π∫_∂ D_3[Ψ_14+^[1](t,-k)-2s_14^[1]]dk,; u_0-1^[1](t)=%̣ṣ/̣%̣ṣ1π∫_∂ D_3[Ψ_24+^[1](t,-k)-2s_24^[1]]dk, ]It further follows from Eq. (<ref>) that we have {[Ψ_13+^[1](t,-k)=2̣i∫_0^te^-4ik^2(t-t')u_11^[1](t')dt',;Ψ_14+^[1](t,-k)=2̣i∫_0^te^-4ik^2(t-t')u_10^[1](t')dt',; Ψ_24+^[1](t,-k)=2̣i∫_0^te^-4ik^2(t-t')u_1-1^[1](t')dt', ].Thus, the Neumann problem can now be solved perturbatively as follows: for n_33,44(s) having no zeros and given u_1j^[1](t),j=1,0,-1, we can obtain {Ψ_13+^[1], Ψ_14+^[1], Ψ_24+^[1]} from Eq. (<ref>) and further find u_0j^[1],j=1,0,-1 from Eq. (<ref>). Finally, we can have Ψ_ij^[1] from Eq. (<ref>). In fact, these arguments for Ψ_ijcan be extended to all orders such that we can determine all orders of S(k). Similarly, the substitution of Eq. (<ref>) into Eqs. (<ref>)-(<ref>) yields the terms of O(ϵ^n) as u_01^[n](t)=∫̣_∂ D_3[Ψ_13+^[n](t,-k)-2s_13^[n]]dk+lowerorderterms, u_00^[n](t)=∫̣_∂ D_3[Ψ_14+^[n](t,-k)-2s_14^[n]]dk+lowerorderterms, u_0-1^[n](t)=∫̣_∂ D_3[Ψ_24+^[n](t,-k)-2s_24^[n]]dk+lowerorderterms, Eq. (<ref>) implies that {[Ψ_13+^[n](t,-k)=2̣i∫_0^te^-4ik^2(t-t')u_11^[n](t')dt'+lowerorderterms,;Ψ_14+^[n](t,-k)=2̣i∫_0^te^-4ik^2(t-t')u_10^[n](t')dt'+lowerorderterms,; Ψ_24+^[n](t,-k)=2̣i∫_0^te^-4ik^2(t-t')u_1-1^[n](t')dt'+lowerorderterms, ].It follows from system (<ref>) that Ψ_13+^[n], Ψ_14+^[n], Ψ_24+^[n]can be generated at each step from the known Neumann boundary data u_1j^[n],j=1,0,-1 such that we know that the Dirichlet boundary data u_0j^[n],j=1,0,-1 can then be given by Eqs. (<ref>)-(<ref>).Remark 4.3. We can also give the corresponding Gelfand-Levitan-Marchenko representations for the Dirichlet and Neumann boundary value problems and the IVB of spin-1 GP equation on the finite interval, which will be studied in another paper. The analogous analysis of the Fokas unified method can also be extended to study the IBV problems for other integrable nonlinear evolution PDEs with 4× 4 Lax pairs on the half-line or the finite interval. Acknowledgment This work was partially supported by the NSFC under Grant No.11571346 and the Youth Innovation Promotion Association, CAS.99ist C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Methods for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967) 1095-1097.soliton2M.J. Ablowitz, H. Segur, Solitons and the inverse scattering transform, SIAM, Philadelphia, 1981.soliton M.J. Ablowitz, P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press, Cambridge, 1991.rh P. Deift, Z. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems, Bull. Am. Math. Soc. 20 (1992) 119-123.f1 A.S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. Lond. Ser. A 453 (1997) 1411-1443. f3 A.S. Fokas, Integrable nonlinear evolution equations on the half-line, Comm. Math. Phys. 230 (2002) 1-39.f4 A.S. Fokas, A unified approach to boundary value problems, in: CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 2008.nls1 A.S. Fokas, A.R. Its, The linearization of the initial-boundary value problem of the nonlinear Schrödinger equation, SIAM J. Math. Anal. 27 (1996) 738-764.nls4 S. Kamvissis, Semiclassical nonlinear Schrödinger on the half line, J. Math. Phys. 44 (2003) 5849-5868.nls2 A.S. Fokas, A.R. Its, The nonlinear Schrödinger equation on the interval, J. Phys. A: Math. Theor. 37 (2004) 6091-6114.nls3 A.S. Fokas, A.R. Its, L.Y. Sung, The nonlinear Schrödinger equation on the half-line, Nonlinearity 18 (2005) 1771-1822.sg A.S. Fokas, A.R. Its, An initial-boundary value problem for the sine-Gordon equation, Theoret. Math. Phys. 92 (1992) 388-403.sg2 B. 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Math. 135 (2015) 310-346.cnls3 J. Xu, E. Fan, Initial-boundary value problem for integrable nonlinear evolution equation with3× 3lax pairs on the interval, Stud. Appl. Math. 136 (2016) 321-354.cnls4 S.F. Tian, Initial-boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method, J. Differential Equations 262 (2017) 506-558.os J. Xu, E. Fan, The initial-boundary value problem for the Ostrovsky-Vakhnenko equation on the half-line, Mathematical Physics, Analysis and Geometry, 19 (2016) 20. sgpJ. Ieda, T. Miyakawa, M. Wadati, Exact analysis of soliton dynamics in spinor Bose-Einstein condensates, Phys. Rev. Lett. 93 (2004) 194102.sgpb1 J. Ieda, T. Miyakawa, M. Wadati, Matter-wave solitons in an F=1 spinor Bose-Einstein condensate, J. Phys. Soc. Jpn. 73 (2004) 996.sgpd M. Uchiyama, J. Ieda, M. Wadati, Dark solitons in F=1 spinor Bose-Einstein condensate, J. Phys. Soc. Jpn.75 (2006) 064002.sgpp Z. Yan, K. W. Chow, B. A. Malomed, Exact stationary wave patterns in three coupled nonlinear Schrödinger/Gross-Pitaevskii equations, Chaos, Solitons & Fractals 42 (2009) 3013. | http://arxiv.org/abs/1704.08534v1 | {
"authors": [
"Zhenya Yan"
],
"categories": [
"nlin.SI",
"math-ph",
"math.AP",
"math.MP",
"quant-ph"
],
"primary_category": "nlin.SI",
"published": "20170427123212",
"title": "An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4x4 Lax pair on the half-line"
} |
Chemical Enhancements in Shock-accelerated Particles: Ab-initio Simulations Anatoly Spitkovsky December 30, 2023 ===========================================================================§ INTRODUCTIONObservations of cosmic-ray (CR) antiprotons are a sensitive probe of dark matter (DM) models with thermal annihilation cross sections <cit.>.In particular, with the very accurate recent measurement of the CR antiproton fluxby <cit.>,it is a timely moment to investigate this subject. A joint analysis of the CR fluxes of light nuclei and a potential DM contributionto the antiproton flux provides strong DM constraints <cit.>, as well as an hint for a DM signalcorresponding to a DM mass of about 80 GeV, and a thermal hadronic annihilation cross section, ⟨σ v ⟩≈ 3 × 10^-26 cm^3/s.These values have been derived in<cit.> in a novel analysis where CR propagation uncertainties have been marginalized away, taking into account possible degeneracies between CR uncertainties and DM. A similar result has been found in <cit.>, but using the boron over carbon ratio, also recently measured by AMS-02 <cit.>. In the present work, we shall extend the DM analysis of the CR antiproton flux presented in <cit.> to a comprehensive set of standard model (SM) annihilation channels,including gluons, bottom quarks, W, Z and Higgs bosons, as well as top quarks.Similarly, an excess in gamma-ray emission toward the center of our Galaxy has been reported by several analyses <cit.>. The spectrum and spatial distribution of this Galactic center excess (GCE) is consistent with a signal expected from DM annihilation, and consistent with the excess observed in antiprotons. The second goal which we will pursue in this work is to quantify more precisely the above statement, performing joint fits of the antiproton and gamma-ray signals for various individual DM annihilation channels. In performing this comparison we will also use the most recent results of gamma-ray observations from dwarf satellite galaxies of the Milky Way, which are a known sensitive probe of DM annihilation.Finally, while DM annihilation can be probed in a rather model-independent way by considering individual SM annihilation channels, it is interesting to also test specific models of DM. Such models typically predict CR and gamma-ray fluxesfrom a combination of various SM annihilation channels, and they can be confronted with direct and collider searches for DM. As an example, we shall thus consider a minimal Higgs portalDM model, which adds a real singlet scalar DM field S to the SM.We shall demonstrate that the scalar Higgs portal model can accommodate both the CR antiproton flux and the GCE, despite strong constraints from invisible Higgs decays and direct DM detection.The paper is organized as follows. In section <ref> we analyze the CR antiproton data for individual DM annihilation channels. The joint analysis of antiproton and gamma-ray fluxes, including both the GCE and dwarf galaxies, is presented in section <ref>. In section <ref> we consider the specific case of the scalar Higgs portal model, and present a global analysisincluding antiproton and gamma-ray fluxes, as well as constraintsfrom the DM relic density, invisible Higgs decays and direct DM searches. We conclude in section <ref>.§ COSMIC-RAY FITS FOR INDIVIDUAL DARK MATTER ANNIHILATION CHANNELSDM annihilation in the Galaxy results in a flux of antiprotons from the hadronization and decay of SM particles. The corresponding source term is given byq_p̅^(DM)(x, E_kin) =1/2( ρ(x)/m_DM)^2∑_f ⟨σ v ⟩_fN^f_p̅/ E_kin ,where m_DM istheDM mass and ρ(x) the DM density distribution.Thethermally averaged annihilation cross section for the SM final state f, DM+ DM→ f+f̅, is denoted by ⟨σ v ⟩_f, and N^f_p̅/ E_kin is the corresponding antiprotonenergy spectrum per DM annihilation. Note that the factor 1/2 corresponds to scalar or Majorana fermion DM.We use the NFW DM density profile <cit.>, ρ_NFW(r) = ρ_h r_h/r ( 1 + r/r_h )^-2,with a characteristic halo radius r_h=20kpc, and a characteristic halo density ρ_h, normalized to alocal DM density ρ_⊙ = 0.43GeV/cm^3 <cit.> at thesolar position r_ ⊙ = 8kpc. The choice of the DM profile has a negligible impact on our results,as demonstrated in <cit.>.The energy distribution and yield of antiprotons per DM annihilation, N^f_p̅/ E_kin, is determined by the DM mass andthe relevant SM annihilation channel. We usethe results presented in <cit.> for the annihilation into gluons, b b̅, tt̅ and hh. (The spectra for annihilation into light quarks are very similar to those for gluons.) For ZZ^* and WW^* final states we havegenerated the spectra with Madgraph5_aMC@NLO <cit.> and Pythia 8.215 <cit.>, adopting the default setting and scale choice, Q=m_DM. We note that the choice of the Pythia tune may introduce uncertainties up to about 15%, while varying the shower scale in a range between m_DM/6 and 2m_DM can result in uncertainties of up to 30%.[ Note that in Madgraph5_aMC@NLO for the default setting (dynamical scale) the scale is set to m_DM/6.] This difference is induced through the strength of the final state radiation. However, we have checked that the theoretical uncertainty in the prediction of the antiproton energy spectrumfrom this scale variation and different Pythia tunes does not affect our results.For the default Pythia settings, annihilation spectra into (on-shell) WW and ZZare in reasonable agreement with those of <cit.>. To analyze the impact of DM annihilation on the CR antiproton flux, we perform a joint analysis of the fluxes of protons, helium and antiprotons, including a potential contribution from DM annihilation, which would affect the antiproton to proton ratio. We solve the standard diffusion equation using Galprop <cit.>,assuming a cylindrical symmetry for our Galaxy, with a radial extension of 20kpc.In total, we analyze a parameter space with thirteen dimensions. Eleven parameters are related to the CR sources and the propagation of CRs, whilefor each individual SM annihilation channel,theDM component of the CR flux is specifiedby the DM mass and its annihilation cross section. The parameters describing the CR sources and propagation, as well as the DM contribution, are determined in a global fit of the AMS-02 proton and helium fluxes <cit.>, and the AMS-02 antiproton to proton ratio <cit.>, complementedby proton and helium data from CREAM <cit.> and VOYAGER <cit.>. We use MultiNest <cit.> to scan this parameter space and derive the corresponding profile likelihoods. For details of the propagation model and the numerical analysis we refer to <cit.>. We use as benchmark antiproton production cross section the default in Galprop, i.e.,the parameterization from <cit.>. In <cit.> we checked recent new updated models of the cross sectionfrom <cit.> and <cit.>, and we found that the results of the fit are substantially unchanged. The main effect is to slightly modify the region of parameter space preferred by DM at the level of 20–30%, leaving unchanged the values of the minimal χ^2. Adding a DM component significantly improves the global fit of the CR antiproton data. This is due to a sharp spectral feature in the antiproton flux at a rigidity of about 20 GV. Such a feature cannot be described by the smooth spectrum of secondary antiprotons produced by the interactions of primary protons and helium nuclei on the interstellar medium. The spectrum from DM annihilation, on the other hand, exhibits such a sharp feature from the kinematic cut-off set by the DM mass. Adding a DM component thus provides a significantly better description of the antiproton data. In figure <ref> we present the preferred range of DM masses and annihilation cross sections for the different SM annihilation channels.The regions are frequentist contour plots of the two-dimensional profile likelihood obtained minimizing the χ^2 with respect to the remaining eleven parameters in the fit. They, thus, include the uncertainties in the CR source spectra and CR propagation. All channels provide an improvement compared to a fit without DM: we find a χ^2/(number of degrees of freedom) of 71/165 for the fit without DM, which is reduced to46/163 (b b̅), 48/163 (hh), 50/163 (gluons and/or light quarks),50/163 (WW^*), 46/163 (ZZ^*),and 59/163 (t t̅), respectively, when adding a corresponding DM component (see also Table <ref>). Formally, Δχ^2 = 25 for the two extra parameters introduced by the DM component with annihilation into bb̅ corresponds to a significance of 4.5, although such an estimate does not account for possible systematic errors. Figure <ref> also shows that different annihilation channels would imply different preferred DM masses, ranging from m_ DM≈ 35 GeV for gluons and/or light quarks to m_ DM near the Higgs and top mass for annihilation into Higgs or top-quark pairs, respectively. For all the channels, the fit points to a thermal annihilation cross section ⟨σ v ⟩≈ 3 × 10^-26 cm^3/s.It can be noted that the values of the χ^2 are typically quite low for both the fits with and without DM. This is due to the fact that CR data error-bars are dominated by systematic errors rather than statistical errors. This is true in particular for the proton and helium data, while forthe antiproton to proton ratio the two errors have comparable weight. As a consequence, the use of χ^2 statistics to describe the data is not fully correct. A proper treatment would require a deeper knowledge of the systematic uncertainties so that to include them directly at the level of the likelihood rather than in the error-bars. This information is, however, not publicly available, and this approach is not possible at the moment. It would be desirable that such a more complete information is releasedfor future CR data publications and updates. § JOINT FIT OF ANTIPROTON AND GAMMA-RAY FLUXESDM annihilation would also result in a flux of gamma rays, predominantly from the decay of pions produced inthe fragmentation of SM particles. The gamma-ray flux per unit solid angle at a photon energy E_γ is dΦ/dΩdE = 1/2 m_ DM^2∑_fdN^f_γ/dE⟨σ v⟩_f/4 π∫_l.o.s.ds ρ^2(r(s,θ)) ,wheredN^f_γ/dE is the photon spectrum per annihilation for a given finalstate f, and ⟨σ v⟩_f is the correspondingannihilation cross section.The integral has to be evaluatedalong the line-of-sight (l.o.s.) at an observational angle θ towards the Galactic center. The l.o.s. integral of the DMdensity-squared, ρ^2, over the solid angle dΩ is called the J-factor.We adopt a generalized NFW profile <cit.> with an inner slope γ≃1.2for the DM density ρ.This is in contrast to the standard NFW profile applied in section <ref>.However, CRs and gamma-rays probe very different parts of the profiles.The l.o.s. integral in Eq. (<ref>) is very sensitive to the profile behavior close to the Galactic center,while CRs mostly probe the local DM distribution.Indeed, in the latter case we verified that even changing to the cored Burkert profile <cit.> does not affect the results of the CR fit <cit.>.From this point of view it is legitimate to use an NFW profile for CRs while adopting the generalized NFW profile for gamma-rays. An excess in the flux of gamma rays from the Galactic center has been reported by several groups <cit.> (but see also <cit.>). The GCE is peaked at photon energies of a few GeV, and consistent with a spherical morphology,extending up to at least 10^∘ away from the Galactic center,and a steep radial profile <cit.>.Various astrophysical processes have been proposed to explain the excess <cit.>. Also, studies based on photon-count statistic suggest that the excess ismore compatible with a population of unresolved point sources rather than with a pure diffuse emission <cit.>. Nonetheless, a DM interpretationis still viable. In particular, the excess is compatible with the signal expected from the annihilation of DM, with a cross section close to the thermalvalue and with a DM mass around 50 GeV. In our analysis of the GCE we will use the gamma-ray energy spectrum and error covariance matrix obtained in <cit.>.DM annihilation in gamma rays can also be sensitively tested by observations ofdwarf satellite galaxies of the Milky Way<cit.>. Here, we use the likelihood as a function of the flux for each dwarf provided byFermi-LAT <cit.>, and the gamma-ray spectra for the individual annihilation channels obtained in <cit.>. We consider a total of eleven dwarfs: the seven brightest confirmed dwarfs analyzed in <cit.>(Coma Berenices, Draco, Sculptor, Segue 1, Ursa Major II, Ursa Minor, Reticulum II) as well asWillman 1, Tucana III, Tucana IV and Indus II. Four of these dwarfs (Reticulum II, Tucana III, Tucana IV, Indus II) exhibit small excesses at the level of ∼2σ (local) each, which are compatible with a signal from DM annihilation with a thermal cross section <cit.>. The total likelihood is obtained as a product of likelihoods over each single dwarf as described in <cit.>.The likelihood of each dwarf contains a factor from the flux likelihood, and a log-normal factor from a deviation of the J-factor from its nominal value.For the seven brightest confirmed dwarfs we use the J-factors and corresponding uncertainties provided in <cit.>(which, in turn, drawsfrom <cit.>, except for Reticulum II, whose J-factor is taken from <cit.>), while for Willman we use the J-factor from <cit.>. ForTucana III, Tucana IV and Indus II we use the distance-based predictions providedin <cit.> with a medium estimated error of 0.6 dex. We marginalize over the J-factors of the individual dwarf galaxies in the fit. It should be noted that estimates of the J-factors present some differences depending on the analysis(compare for example <cit.>), with some analysis <cit.>finding somewhat lower values than the others. This, however, has onlya minor impact on our results, since, as we show below,the results of the fits are dominated by the GCE and CR signals. On the basis of the likelihoods obtained in the CR fit described in section <ref> we now perform a joint fit of CR antiprotons and of gamma-rays from the Galactic center and from dwarf galaxies.The gamma-ray fit follows the methodology described in <cit.>.The fit contains four input parameters, the model parameters,⟨σ v ⟩ and m_DM, as well as the J-factor for the Galactic center, log, and the local DM density ρ_⊙.The latter two parameters are, in principle, not independent. However, as already mentioned above, CRs and gamma-rays probes different parts of the DM distribution in the Galaxy and it is thus reasonable to explorethe uncertainties in these two parameters as independent. Forlogwe use a gaussian distribution (log-normal in ) with mean 53.54 and error 0.43, i.e., log (J/GeV^2cm^-5)=53.54± 0.43. This GC J-factorrefers to an integration region of 40^∘× 40^∘ around the GC and with a stripe of ± 2^∘ masked along the Galactic plane, in order to be compatible with the GCE data from Ref. <cit.> that we use. The details of the derivation ofthe distribution in log and the error are described in <cit.>. For the local DM density we also use Gaussian errors ρ_⊙=0.43±0.15 GeV/cm^3 <cit.>. We use the result of Ref. <cit.> in order to be conservative since ρ_⊙ has arelatively large error.A recent review on the status of the determination of ρ_⊙ is given in <cit.>. Figure <ref> shows the preferred range of DM masses and annihilation cross sections,where we have marginalized over log and ρ_⊙. We present 1, 2, and 3 σ contours for a fit to the GCE (blue), CR (red), CR+GCE (green) and CR+GCE+dwarfs (black) for the six annihilation channels gg, b b̅, WW^(*), ZZ^(*), hh and tt̅. Note that the fits to the CR fluxes in figure <ref> show a wider spread in ⟨σ v ⟩ than those displayed in figure <ref>, because in figure <ref> we marginalize over the local DM density, ρ_⊙=0.43±0.15 GeV/cm^3, while in figure <ref> a fixed value ρ_⊙=0.43 GeV/cm^3 is used.For most SM annihilation channels, we observe very good agreement between the DM interpretation of the CR antiprotons and the GCE gamma-ray flux. The preferred region in ⟨σ v ⟩ and m_DM is consistent when comparing the CR and GCE fits individually, and the combined CR+GCE fit. However, as can be seen in the upper left panel of figure <ref>, annihilation into gluons (or light quarks) is disfavored as an explanation of both the CR antiproton flux and the GCE, as both signals individually prefer different regions of DM mass.Annihilation into t quarks is also disfavored since it does not provide a good fit to either the GCE or antiprotons. Adding the constraints from dwarf galaxiesdisfavors large values for ⟨σ v ⟩, but hardly affectsthe combined CR+GCE fit.Numerical values of the best-fit χ^2 are reported in Table <ref>.From the figure we note also that CR prefer a somewhat largerthan the GCE and, hence,the joint fit pushes ρ_⊙ towards slightly larger values with respect to the assumed priorfrom <cit.>.More precisely, we find, with some variation depending on the DM channel, that the global fit gives a valueρ_⊙=0.55±0.15 GeV/cm^3, i.e., ∼ 0.1 GeV/cm^3 higher than the input prior. We find that ρ_⊙=0.3 GeV/cm^3 is at the lower edge of ∼ 3 σ range preferred by the fit.This means that if the true ρ_⊙ is significantly lower than 0.3 GeV/cm^3 it becomes difficult to reconcile the GCE with the CR data. Nonetheless, we also note that for z_h, the half-height of the CR propagation region in the Galaxy, we use the prior 2-7 kpc, but, since this parameter is unconstrained by the fit (see <cit.>), values up to 10 kpc or more areallowed. Thus, since the DM signal approximately scales linearly in z_h, higher z_h values would allow, in consequence, lower ρ_⊙ values, making possible a joint fit of the GCE and CRs down to aρ_⊙ value of 0.2 GeV/cm^3. This issue is also further discussed in the next section within the Higgs portal fit.§ INTERPRETATION WITHIN THE SINGLET SCALAR HIGGS PORTAL MODELWe now discuss a specific minimal model of DM, where we add a singlet scalar field S to theSM <cit.>.We will follow the analysis in <cit.>,[ An interpretation of the GCE within the singlet Higgs portal model has also been discussed in <cit.>.]with the main difference that now weincludeCR data.The scalar field interacts with the SM Higgs field H through the Higgs portal operatorS^2H^† H. Imposing an additional Z_2 symmetry, S → -S, the scalar particle is stable and thus a DM candidate. The Lagrangian of the scalar Higgs portal model reads L =L_SM + 1/2∂_μS∂^μS- 1/2 m_S,0^2S^2- 1/4λ_SS^4- 1/2λ_HSS^2 H^† H .After electroweak symmetry breaking, the last three terms of the above Lagrangian becomeL⊃- 1/2 m_S^2S^2- 1/4λ_S S^4 - 1/4λ_HSh^2 S^2 - 1/2λ_HSv h S^2 ,with H = (h+v, 0)/√(2), v = 246GeV, and where we introduced the physical mass of the singletfield, m_S^2= m_S,0^2 + λ_HS v^2 / 2. The phenomenology of the singlet Higgs portal model has been extensively studied in the literature, see e.g. the recent reviews <cit.> and references therein. While the scalar self-coupling, λ_S, is of importance for the stability of the electroweak vacuum, the DM phenomenology of the scalar Higgs portal model is fully specified by the mass of the scalar DM particle, m_S=m_ DM, and the strength of the coupling between the DM and Higgs particles, λ_HS. Even though the model is minimal, the S^2 H^† H interaction term implies a rich phenomenology, including invisible Higgs decays, h → SS, a DM-nucleon interaction through the exchange of a Higgs particle, and DM annihilation throughs-channel Higgs, t-channel scalar exchange, and the S^2 h^2 interactions.The region most relevant for the DM interpretation of the CR antiproton flux and the GCE is the regionm_S≲100 GeV. As this is below the Higgs-pair threshold, m_S < m_h, annihilation proceeds through s-channel Higgs exchange only, andthe relative weight of the different SM final states is determined by the SM Higgs branching ratios,independent of the Higgs-scalar coupling λ_HS.Above the Higgs-pair threshold, m_S ≥ m_h, the hh final state opens up. The strength of the annihilation into Higgs pairs, as compared to W,Z or top-quark pairs, depends on the size of the Higgs-scalar coupling λ_HS. However, as shown in <cit.>, within the scalar Higgs portal model the region above the Higgs-pair threshold that provides a good fit to the GCE (and to the CR) requires very large λ_HS which are excluded by direct detection limits. In the following, we will thus focus on DM masses m_ DM < m_h. We pursue two approaches. We first adopt a more model-independent point of view and consider a DM interpretation in terms of m_DM and . The only referenceto the Higgs portal model is through the relative weight of the different SM final states, which is determined by m_DM. Such an analysis probes whether a certaincombination of annihilation channels, considered individually in section<ref> and <ref>, can provide a fit of the observations.Note that this kind of analysis can, in general, not be performed based on the results presented for the individual channels.Instead, we perform a dedicated fit to the CR antiproton flux,constructing the injection spectra from the spectra of the individual channels according to their relative weights.The result is shown infigure <ref> (red contours), where we have marginalized overρ_⊙ and log.The preferred region of DM masses is around m_ DM≈ 60 GeV,where the Higgs portal model predicts annihilation pre-dominantly into bottom quarks,W-bosons and gluons with a weight of approximately 70, 20 and 10%, respectively.We find a χ^2/(number of degrees of freedom) of 47/163 for the Higgs portal model fit, compared to 71/165 for the fit without DM. Performing a joint fit of the CR antiproton flux with the GCE (green contours) as well as with the GCE and dwarf galaxies (black contours) shifts the preferred region to slightly smaller masses m_ DM≈ 55 GeV, with aχ^2/(number of degrees of freedom) of 49/163 for the CR and 20.8/22 for the GCE. Although the best-fit point for the GCE-only fit lies at smaller masses, around m_ DM≈ 45 GeV (cf. <cit.>), the χ^2/(number of degrees of freedom) for the GCE in the joint fit is almost as good as for the GCE-only fit (which yields 19.2/22). We can draw the quite general conclusion that DM models where the annihilation is pre-dominantly into bb̅, WW^(*) or ZZ^(*) final states, or any combination thereof, provide a very good fit of the CR antiproton flux, the GCE and gamma-rays from dwarf galaxies, and point to a DM mass in the vicinity of m_ DM≈ 60 GeV.We proceed with a more detailed analysis of the scalar Higgs portal model, where we take into account the various constraints on the parameter space from the Higgs invisible decay width, direct detection searches, searches for gamma-ray lines from the inner Galaxy and the DM relic density. Hence we consider the actual model parameters m_S=m_ DM and λ_HS defined in eq. (<ref>).We shall discuss the various constraints briefly in turn, and refer to <cit.> for more details.* For light DM below the Higgs threshold, m_ DM < m_h/2, the invisible Higgs decay h → SS is kinematically allowed. The LHC limits on the Higgs invisible branching ratio, BR_ inv≲ 0.23 <cit.>, thusimply an upper limit on the Higgs-scalar coupling λ_HS as a function of the DM mass.* The scalar Higgs portal model predicts a spin-independent DM-nucleon scattering cross section, σ_SI∝λ_HS^2/m_ DM^2, through the exchange of the SM Higgs boson. The model is therefore severely constrained by direct detection experiments. We use the recent direct detection limits from LUX <cit.> in our numerical analysis, updating the results presented in <cit.>. Furthermore, we introduce the local DM density ρ_⊙, relevant for the DM-nucleon scattering rate and the CR flux, as an additional nuisance parameter in the fit. * Searches for gamma-ray lines provide constraints on the cross section for the annihilation into mono-chromatic photons, ⟨σ v ⟩_γγ. We have calculated ⟨σ v ⟩_γγ using an Higgs effective Lagrangian as described in <cit.>, and constrain the model with data from the recent Fermi-LATsearch for spectral lines in the Milky Way halo <cit.>.* We require that the Higgs portal model provide the correct DM relic density as measuredby Planck, Ω h^2|_DM = 0.1198± 0.0015 <cit.>. We assume a standard cosmological history, but allow for the possibility that the dark sector is more complex thanassumed within our minimal model. Hence, the DM density provided by the scalar Higgs portal model is a certain fraction, R≤1, of the density of all gravitationally interacting DM, ρ_Higgs portal = Rρ_DM. The total DM density predicted by our model is then Ω h^2|_DM = Ω h^2|_Higgs portal /R. We will consider R as a free parameter in ourfit. Note that the annihilation signal today scales as ∝ R^2, while the direct detection limits scale ∝ R, thus implying a non-trivial interplay of the various constraints for R≠ 1.In figure <ref> we present a fit of the Higgs portal model to the CR antiproton flux and the GCE, including the constraints from dwarf galaxies and searches for gamma-ray lines, the invisible Higgs branching ratio, direct DM detection, and the relic density.[Compared to the analysis presented in <cit.> we have included the likelihood of the CR antiproton flux and updated the direct detection and dwarf galaxy limits.]Let us first consider the upper left panel, which shows the allowed region in the Higgs portal coupling, λ_HS, and the DM mass. The overall flux of antiprotons and photons scales with the annihilation cross section ⟨σ v ⟩∝λ_HS^2/[(m_h^2-4 m_ DM^2)^2+Γ_h^2 m_h^2], where Γ_h is the Higgs width. To accommodate the CR data and the GCE, either large couplings λ_HS or masses near the Higgs resonance, m_ DM≈ m_h/2, are required. However, large couplings are excluded by the invisible Higgs branching ratio for masses m_ DM≲ m_h/2, and by direct detection limits for masses m_ DM≳ m_h/2, leaving only the region near the Higgs threshold m_ DM≈ m_h/2, where the annihilation proceeds through resonant Higgs exchange. Upon closer inspection, we find two viable regions of parameter space, see the panel displaying the allowed region in the Higgs portal coupling and the scalar DM fraction R. In one region, λ_HS is of order O(10^-2) and R <1, so that an additional DM component is required. In the second region,the scalar particle constitutes a significant fraction or even all of DM, R≲ 1, but the Higgs portal coupling must be very small, of orderO(10^-3 - 10^-4). These two regions are a result of an interplay betweenthe strong velocity dependence of the annihilation cross section near the resonance and the non-trivial scaling of the CR and GCE signals and the relic density with the fraction R of scalar DM.The best-fit points as well as their χ^2 values are listed in table <ref> for the two regions described above. For comparison we also show the results for the fit where we leave out the CR likelihood (GCE+constraints) or the GCE likelihood (CR+constraints). Within the Higgs portal model the observations are very well compatible with each other.However, the CR signal prefers a flux corresponding to a slightly larger annihilationcross section. In the joint fit the nuisance parameters ρ_⊙ and log J/J_nom leave enough freedom to accommodate both signals. In fact, for ρ_⊙ and log J/J_nom the fit prefers somewhat larger and smaller values, respectively, than the nominal ones (cf. lower panels in figure <ref>). Note that ρ_⊙ also effects the direct detection rate ∝ρ_⊙Rλ_HS^2. As compared to the fit of the GCE presented in <cit.>, the improved LUX limits and, to a lesser extent, the larger DM density ρ_⊙ further constrain large values of R in the first region where λ_HS is of order O(10^-2).Another difference to the results of <cit.> arises from the fact that the recentresults from dwarf galaxies are less constraining and, in particular, are not in tension with the GCE anymore.This allows for largerand hence for a smaller value of log J/J_nom while still fitting the GCE signal.Note also that the ρ_⊙ range preferred by the Higgs portal model,ρ_⊙= 0.6±0.1 GeV/cm^3 (see figure <ref>), is different from the case of the single channel fits where, instead, ρ_⊙= 0.55±0.15 GeV/cm^3. The specific parameter region preferred by the Higgs portal fit, thus, further pushes ρ_⊙ toward higher values with respect to the single channel fit case.[!h] 3c |region 1 3cregion 2 log L contribution GCE+constr.CR+constr. GCE+CR+constr. GCE+constr.CR+constr. GCE+CR+constr.m_S [GeV] 62.58_-0.04^+0.76 62.60_-0.06^+0.2162.58_-0.03^+0.1862.541_-0.016^+0.003 62.532_-0.009^+0.01262.533_-0.011^+0.011 λ_HS0.017_-0.003^+0.015 0.015_-0.002^+0.006 0.015_-0.001^+0.004 0.0016_-0.0013^+0.0060 0.00032_-0.00012^+0.00815 0.00039_-0.00017^+0.00561 R0.019_-0.018^+0.204 0.041_-0.040^+0.124 0.020_-0.018^+0.100 0.021_-0.019^+0.979 0.39_-0.38^+0.61 0.29_-0.28^+0.71 log J/J_nom-0.065_-0.295^+0.341 -0.280_-0.793^+0.351 -0.303_-0.205^+0.304 -0.099_-0.275^+0.377 -0.415_-0.590^+0.468 -0.316_-0.201^+0.238 ρ_⊙ [GeVcm^-3] 0.43_-0.15^+0.15 0.56_-0.08^+0.090.59_-0.05^+0.1 0.43_-0.15^+0.15 0.56_-0.09^+0.090.59_-0.06^+0.09⟨σ v⟩ R^2 [10^-26 cm^3/s] 1.36_-0.44^+0.45 1.89_-0.53^+0.721.73_-0.47^+0.38 1.36_-0.45^+0.46 1.87_-0.51^+0.721.70_-0.32^+0.39 χ_GCE^2 26.2226.4926.69 26.4727.3526.88χ_CR^2 52.3248.0848.42 57.1448.0748.42 Fit parameters and ⟨σ v⟩ R^2 for the best fit points of region 1 and 2taking into account the log-likelihood contributions from GCE+constraints, CR+constraints and CR+GCE+constraints. Given errors are 1σ uncertainties. We also show the corresponding χ^2_GCE andthe χ^2_CR.§ CONCLUSION In this paperwe analyzeantiproton data from AMS-02 searching for a signature ofDM annihilation. Using the same methodology of <cit.>, we take into account CR propagation uncertainties by fitting at the same time DM and propagation parameters. With respect to <cit.> we explore a wider class of annihilation channels including gg, b b̅, WW^*, ZZ^*, hh and tt̅. We find that almost all the channels provide similar hints of a DM annihilation at about 4σ level (considering statistical uncertainties only) with masses ranging from 40 and 130 GeV depending on the annihilation channel. Annihilation into tt̅ provides a smaller fit improvement, at the 3σ level.We then investigate the compatibility of the antiproton DM hint with the GCE performing a joint gamma-ray and antiproton fit where we further introduce two nuisance parameters related to the distribution of DM in the vicinity of the Galactic center and in the Solar local neighborhood. We find that the two signals are well compatible for most of the channels, except for gg, wherethe two are somewhat in tension. Overall, we find that b b̅,ZZ^* and hh provides good fits to both the GCE and antiprotons, followed by WW^*, which fits only slightly worse.gg and tt̅are less favored, either because they do not fit well one of the two signals or because the two signals are found to be in tension. We also include in the fit the latest results from the analysis of dwarf galaxies in gamma raysand we find that dwarf constraints are compatible with the joint GCE and antiproton fit anddo not change significantly the conclusions.Finally, as an example, we perform the above joint fit for the specific case of the Higgs portal DM model, including, in this case, also constraints from direct detection and collider searches. We find that a surviving, although fine tuned, region corresponding to DM of mass equal to aboutm_h/2 annihilating via resonant Higgs exchange satisfies all constrains and provides a good fit to both antiprotons and gamma rays.§ ACKNOWLEDGEMENTSWe acknowledge support by the German Research Foundation DFG through theresearch unit “New physics at the LHC”.JHEP | http://arxiv.org/abs/1704.08258v2 | {
"authors": [
"Alessandro Cuoco",
"Jan Heisig",
"Michael Korsmeier",
"Michael Krämer"
],
"categories": [
"astro-ph.HE",
"hep-ph"
],
"primary_category": "astro-ph.HE",
"published": "20170426180001",
"title": "Probing dark matter annihilation in the Galaxy with antiprotons and gamma rays"
} |
Instituto Tecnológico de Aeronáutica, DCTA, 12228-900, S. José dos Campos, Brazil. Instituto de Estudos Avançados, Universidade de São Paulo C. P. 72012, 05508-970 São Paulo, SP, BrazilInstituto de Física, Universidade de São Paulo, C. P. 66318, 05314-970 São Paulo, SP, BrazilInstituto Tecnológico de Aeronáutica, DCTA, 12228-900, S. José dos Campos, Brazil.Instituto Tecnológico de Aeronáutica, DCTA, 12228-900, S. José dos Campos, Brazil.Instituto Tecnológico de Aeronáutica, DCTA, 12228-900, S. José dos Campos, Brazil. Instituto Tecnológico de Aeronáutica, DCTA, 12228-900, S. José dos Campos, Brazil. We present a recently developed theory for theinclusive breakupof three-fragment projectiles within a four-bodyspectator model <cit.>,for the treatment of the elastic andinclusive non-elastic break up reactions involving weakly bound three-cluster nuclei inA (a,b) X / a = x_1 + x_2 + b collisions. The four-body theory is an extension ofthe three-body approaches developed in the 80's byIchimura, Autern and Vincent (IAV) <cit.>, Udagawa and Tamura (UT) <cit.> and Hussein and McVoy (HM) <cit.>. We expect that experimentalists shall be encouraged to search for more information aboutthe x_1 + x_2 system in the elastic breakup cross section and that also further developmentsandextensions of the surrogate method will be pursued, based on the inclusive non-elastic breakup part of the b spectrum. Inclusive Breakup Theory of Three-Body Halos T. Frederico Received: date / Revised version: date ============================================ § INTRODUCTIONWe report a recently developed theory to treat the inclusivebreakup of three-fragment weakly bound nuclei<cit.>. The natural three fragmentcandidates for projectiles are Borromean, two-nucleon and unstable three-fragment halo nuclei. Our theory is an extension ofthe inclusive breakup models used forincomplete fusion reactions and in the surrogate method with two fragment projectiles. The three-body approach was developed in the 80's byIchimura, Autern and Vincent (IAV) <cit.>, Udagawa and Tamura (UT) <cit.> and Hussein and McVoy (HM) <cit.>. These three-body theories were extended to obtain the fragmentyield in the reaction A (a,b) X, where the projectile is a = x_1 + x_2 + b.The inclusive breakup cross section is a sum of the four-bodyelastic breakup cross section plus the inclusive non-elastic breakup cross section that involves theabsorption cross sections of the participant fragments, x_1 and x_2, and which generalizes the three-body formula reviewed in Austern, et al. <cit.>. The connection between IAV, UT and HM theories was shown in <cit.>.The newformula contains the four-body dynamicsboth in the elastic breakup cross section and in the inclusivenon-elastic breakup ones. It can be applied to treat reactions withstable/unstable projectiles composed of three-fragments, like weakly bound Borromean andtwo-nucleon halo nuclei, where fingerprints of Efimov physics <cit.>and universality <cit.>could be revealed through the appearance of long-range correlations between the three-fragments. In addition one can seek the generalization of the surrogate method applied to reactions like (d,p) and (d,n) to the reactions (t,d) and (^3He,d), among others. Of particular interest is alsothe two-fragment correlation contribution to the elastic breakup cross section and theinclusive non-elastic breakup cross section through a three-body absorption interaction, which appears naturally in the four-body formulation. One could examine, for example, the relation betweenthe pairing, in the case of two-neutron halos, and the three-body formulation of the projectile and its importance to the reaction mechanism underlying the inclusive breakup cross-sections.We expect that these developments could stimulate experimentalists to searchfor more information about the x_1 + x_2 system in the elastic breakup cross section,and that also further theoretical developmentsandextensions of the surrogate method will be pursued,based on the inclusive non-elastic breakup part of the b spectrum. § INCLUSIVE BREAK-UP THEORY The many-body Hamiltonian for the b + x_1 + x_2 + A system to be applied to derive the scattering dynamicsof the three-fragment projectile isH_(b, x_1, x_2, A) = T_b + T_x_1 + T_x_2 + V_b, x_1 + V_b, x_2 + V_x_1, x_2 +h_A + T_A +V_b,A +V_x_1,A + V_x_2,Awhere the kinetic energies are given by the T'sand the microscopic Hamiltonian of the target nucleus ish_A. In thespectator approximation themicroscopic potential V_b, A is associated by standard methods in reaction theoryto the optical potential U_b, and the target is considered infinitely massive, namelyT_A = 0. The inclusive breakup cross-section is an integral over the spectator fragment position r_b and a sum overthe x_1 + x_2 + A bound and scattering states. This leads to theb-spectrum and angular distributiond^2σ_b/dE_bdΩ_b = 2π/ħ vρ_b(E_b)∑_c δ(E - E_b -E_c) ×| ⟨χ_b^(-)Ψ^c_x_1x_2A|V_b,x_1 + V_b, x_2 + V_x_1, x_2|⟩|^2where (r_b, r_x_1, r_x_2, A) is the exact eigenstate of the A + a many bodyHamiltonian. The final state wave function is χ_b^(-)(k_b, r_b)Ψ^c_(x_1x_2A), where Ψ^c_(x_1x_2A) runs over bound and continuum states. Thedensity of b continuum states is ρ_b(E_b) ≡ [dk_b/(2π)^3]/[dE_bdΩ_b]= μ_bk_b/[(2π)^3ħ^3], withμ_b thereduced mass of the b + A system. The connection with the four-body (4B) scattering problem is obtained by eliminating the target internal degrees of freedom,and by using the product approximation, = Ψ_0^4B(+)Φ_A, whereΨ_0^4B (+) is the exact 4B scattering wave function in the incident channel, andΦ_A is the ground state wave function of the target nucleus. Together with the 4B approximation, theenergy conservation δ in Eq. (<ref>) is associated with theimaginary part of an optical model Green's function operator G^(+)_X=-π Ω^(-)_X δ(E_x - H_0)(Ω^(-)_X)^† -(G^(+)_X)^†W_X G^(+)_Xwhere the Möller operator is Ω^(-)_X = [1 + G^(-)_X(V_X)^†]. The imaginary part of the optical potential for particles x_1 and x_2 contains single fragment terms, W_x_i, and a three-body term. In addition, closure has to be usedto perform the sum over c andgeneral nuclear reaction theory to transform the microscopic interactionsV_x_1, A and V_x_2, A into complex optical potentials U_x_1 and U_x_2 (see e.g. <cit.>).§.§ Inclusive breakup cross-section Following the above steps, the inclusive breakup cross section is reduced to a sum of two distinct terms,the elastic breakup and the non-elastic breakup cross sectionsd^2σ_b/dE_bdΩ_b = d^2σ^EB_b/dE_bdΩ_b + d^2σ^INEB_b/dE_bdΩ_bwith the4B elastic breakup cross section contribution beingd^2σ^EB_b/dE_bdΩ_b =2π/ħ v_aρ_b(E_b)∫dk_x_1/(2π)^3dk_x_2/(2π)^3 × |⟨χ^3B(-)_x_1, x_2χ^(-)_b|V_bx_1 + V_bx_2 |Ψ_0^4B(+)⟩|^2×δ(E - E_b - E_(k_x_1, k_x_2))where χ^3B(-)_x_1, x_2is the full scattering wave function of the two unobserved fragments in the final channel.The inclusive cross-section for the inelastic breakup contains the optical potentials U_x_1, U_x_2 and the fragment-fragment interaction V_x_1, x_2 to all orders:d^2σ^INEB_b/dE_bdΩ_b =2/ħ v_aρ_b(E_b) ⟨ρ̂_x_1, x_2|W_x_1 + W_x_2 +W_3B|ρ̂_x_1, x_2⟩where the source functionρ̂_X(r_x_1,r_x_2) = (χ_b^(-)|Ψ_0^4B(+)⟩ = = ∫ dr_b[χ_b^(-)(r_b)]^†Ψ_0^4B(+)(r_b, r_x_1, r_x_2)carries the full 4B dynamics in the optical model description. Theimaginary parts of the optical potentials U_x_1 and U_x_2 are W_x_1 and W_x_2, respectively.Notice the presence of the imaginary part of a three-body optical potential (W_3B), associated with inelastic excitations that are irreducible to single fragment inelastic processes. This will be discussed in more detail in a later sections. We point out that the 4B inclusive non-elastic breakup cross section differs significantly from the 3B Austern formula <cit.>.The inelastic breakup cross-section is a sum of three terms, d^2σ^INEB_b/dE_bdΩ_b = ρ_b(E_b) k_a/E_a[E_x_1/k_x_1σ_R^x_1 + E_x_2/k_x_2σ_R^x_2 + E_CM(x_1, x_2)/(k_x_1+ k_x_2)σ_R^3B] where the approximation of a weakly bound projectile is used to write the kinetic energy:E_x_i,Lab = E_a, Lab(M_x_i/M_a) with M_a and M_x_i being themass numbers of the projectile and fragment, respectively. The single fragmentinclusive cross-sections are σ_R^x_1 = k_x_1/E_x_1⟨ρ̂_x_1, x_2|W_x_1|ρ̂_x_1, x_2⟩, σ_R^x_2 = k_x_2/E_x_2⟨ρ̂_x_1, x_2|W_x_2|ρ̂_x_1, x_2⟩, and the double fragment inclusive cross-section isσ_R^3B = (k_x_1+ k_x_2)/E_CM(x_1, x_2) ⟨ρ̂_x_1, x_2|W_3B|ρ̂_x_1, x_2⟩which represents the two-fragment irreducible inelastic processes. We elaborate here on the physical interpretation of the different contributions to the inclusive inelastic cross-section.The absorption of the fragment x_i by the target is given by σ_R^x_i (i=1,2), in the case in which th other fragmentx_j just scatters off the target through the optical potential U_x_j A. It is noteworthy tostress that σ_R^x_i is different from the one in the 3B theory of the b - x -A system. The cross-sections σ_R^x_1, σ_R^x_2 are related to the three-body IAV cross sectionthrough a convolution of the latter with the distorted wave densities|χ^(+)_x_2(r_x_2)|^2,and |χ^(+)_x_1(r_x_1)|^2 of the spectator fragments x_1 and x_2, respectively. If an eikonal-type approximation of the projectile distorted wave χ_a^(+)×Φ_a(r_b,r_x_1, r_x_2) (a≡ x_1+x_2+b) is used in the 4B theory, the differencewith the 3B formulation isclearly exposed. Thus, it is expected that theσ_R^x_1 in a (t,p) reaction will differ from the σ_R^x extracted in a (d,p) reaction.Furthermore, at low energiesσ_R^x_1, σ_R^x_2, and σ_R^3B are associated with theformation of compound nucleiA +x_1, A+x_2 and A + (x_1 +x_2). §.§ ExamplesThe first example to which we applythe 4B formulation of the inelastic breakup cross-sections is the case of the reaction ^9Be (= α + α + n) + ^208Pb, where 4B-CDCC (four-body Continuum Discretized Coupled Channels) calculations were performed byDescouvemont and collaborators <cit.> for the elastic scattering,elastic breakup, andtotal fusion cross-sections. However so far, CDCC cannot obtain the partial or incomplete fusioncross-sections. Despite this fact, we can point out that compound nuclei formationwith α detected, namely α + ^208Pb = ^210Po, n + ^208Pb = ^209Pb, and α + n + ^208Pb = ^211Po, would be very interesting to observe, in order to gather information about the spectrum of α particles in the analysis of the inclusive cross sections. It would be interesting to investigate the properties of these compound nuclei experimentally, as they are formed in such a hybrid reaction.A second example is a two-neutron Borromean nuclei. The inclusive α spectrum for the ^6He projectile accounts for the formation of n + A and 2n + A compound nuclei. However it would be experimentally difficult to distinguish these in CN decay. The third example is a two-protonBorromean nuclei.Onemight consider the inclusive proton spectra in the breakupof ^20Mg + ^208Pb with the formation of^209Bi, ^226U and^227Np at different excitation energies. However, ^20Mg presents a very short lifetime andlow intensity as a secondary beam.§IB CROSS SECTIONS: HM SOURCE FUNCTION The source function computed with theHussein and McVoy (HM) model <cit.> can be written as:⟨r_x_1, r_x_2|ρ̂^4B_HM⟩ = Ŝ_b(r_x_1, r_x_2)χ^(+)_x_1(r_x_1)χ^(+)_x_2(r_x_2)where the four-body scattering state is approximated by the product of the distorted waves of the three fragments and the projectile bound state wave function. The internal motion modifies the S-matrix of the b fragment asŜ_b(r_x_1, r_x_2) ≡∫ dr_bΦ_a(r_x_1, r_x_2, r_b)⟨χ^(-)_b|χ^(+)_b⟩(r_b),which should be compared to the 3BS-matrix elementof b given byS_k_b^', k_b=∫ dr_b⟨χ^(-)_b|χ^(+)_b⟩(r_b). In this approximation theinclusive cross-section in the 4B theory with a=x_1+x_2+b is E_x_1/k_x_1σ_R^x_1 = ∫ dr_x_1∫ dr_x_2 |Ŝ_b(r_x_1, r_x_2)|^2 × |χ^(+)_x_2(r_x_2)|^2 W(r_x_1)|χ^(+)_x_1(r_x_1)|^2 ,which should be contrasted with the cross-section in the 3B theory (a=x+b)E_x/k_xσ_R^x = ∫ dr_x |Ŝ_b(r_x)|^2 W(r_x)|χ^(+)_x(r_x)|^2 ,where Ŝ_b(r_x) ≡∫ dr_b⟨χ^(-)_b|χ^(+)_b⟩(r_b)Φ_a(r_b, r_x). Comparing the cross-sections in Eqs. (<ref>) and (<ref>), one singles out the distorted wave|χ^(+)_x_2(r_x_2)|^2 , which damps the 4B cross-section with respect to the 3B one.§ HM AND 3B GLAUBER THEORYWe begin with the Glauber phase <cit.> [ψ_p⃗_b^'^(-)(r⃗^⃗'⃗_b)]^⋆ψ_p⃗_b^(+)(r⃗_b) =exp[-im/ħ p_b^'∫_-∞^z_b^'dz^'V(√(b_b^'2+z^'2)) . . -im/ħ p_b∫_z_b^∞dz^'V(√(b_b^2+z^'2)) - q⃗·r⃗_b] , where ħq⃗= p⃗_b^ ' -p⃗_b, and p⃗_b (p⃗^ '_b)is the incoming(outgoing) momentum of the spectator. We take the initial value of the momentum to be p⃗_b=p_bk̂ and the final value to be p⃗_b^ '=p_b^'sinθ î+p_b^'cosθ k̂. The coordinates (b_b^', z_b^') can be written in terms of the coordinates (b_b, z_b) as b_b^'= b_bcosθ-z_bsinθz_b^'= b_bsinθ+z_bcosθ .Figure <ref> illustrates the coordinates (b_b, z_b) and (b_b^', z_b^'). In order to get insight into the structure of the source function, we approximate the potential by a square well V(r)=(V_0 - W_0)Θ(R-r) .The spectator distorteddensity then becomes [ψ_p⃗_b^'^(-)(r⃗^⃗'⃗_b)]^⋆ψ_p⃗_b^(+)(r⃗_b)=exp[-im/ħ^2(V_0+iW_0).. ×(1/p_b^'(√(R^2-b_b^'2)-z_b^')+1/p_b(z_b+√(R^2-b_b^2))) - q⃗·r⃗_b ] forb_b<R and exp[- q⃗·r⃗_b] for b_b≥ R.The formula simplifies for the case of strong absorption, where the spectator distorted wave density becomes[ψ_p⃗_b^'^(-)(r⃗^⃗'⃗_b)]^⋆ψ_p⃗_b^(+)(r⃗_b) =Θ(b_b-R) Θ(b_b^'-R)e^- q⃗·r⃗_b .Standard eikonal calculations ignore the difference (b_b, z_b) ≠ (b_b^', z_b^'), inwhich case the distorted wave density can be simply written as[ψ_p⃗_b^'^(-)(r⃗^⃗'⃗_b)]^⋆ψ_p⃗_b^(+)(r⃗_b) =Θ(b_b-R) e^- q⃗·r⃗_b The internal motion modified S-matrix of the b fragment within the approximations above is given byŜ_b(r_x_1, r_x_2) = ∫ dr_bΦ_a(r_x_1, r_x_2, r_b)Θ(b_b -R) e^- q⃗·r⃗_bwhich produces a long range correlation between the fragmentsx_1 and x_2 for weakly bound projectiles, due to the long tail of the wave function penetrating in the classically forbidden region. This give us a taste of thepossible röle of Efimov physics, for example when the projectile is atwo-neutron halo s-wave state such as ^11Li, in creating a long range correlation between the two neutrons. Theactual set of coordinates to compute (<ref>) can be visualizedin Fig. <ref>. Figure <ref> also supplies a glimpse of the effect of an extended weakly bound system on the correlation between the two fragments x_1 and x_2.The formula (<ref>) restricts b to beoutside the target absorptive region. However thetail of the projectile wave function can provide a long range correlation between the two fragments and the target. It is instructive to calculate Ŝ_b(r_x_1, r_x_2) in the case of two-fragment projectiles, such as the deuteron in the (d, p) reaction. Here b = p and x = n, and we obtain, within the same strong absorption (black disk)eikonal approximation,Ŝ_p(r_n) = ∫ dr_pΦ_d(r_n, r_p)Θ(b_p -R) e^- q⃗·r⃗_pwhich corresponds to an incomplete Fourier transform involving r_p and q.In the high energy regime, the cross section can be represented as an integral over impact parameter. Since Eq. (<ref>),E_x/k_xσ_R^x = ∫ dr_x|Ŝ_b(r_x)|^2[W(r_x)|χ^(+)_x(r_x)|^2] ,contains the integrand of a b-integral of the reaction cross section of x, [W(r_x)|χ^(+)_x(r_x)|^2],which can be replaced by [1 - |S_x(b_x)|^2], and thefactor |Ŝ_b(r_x)|^2 is basically the survival probability of the observed fragment, b, one can write, after rearranging terms,σ_R^x = 2π∫ db ⟨Φ_a||S_b(b)|^2[1 - |S_x(b)|^2]Φ_a⟩One can say that the above equation is the high energy eikonal limit of IAV or the HM formulae. At high energy, the source function Eq.(<ref>) for two-fragment projectiles reduces to ρ̂^IAV_x = ρ̂^HM_x = (χ^(-)_b|χ^(+)_bχ^(+)_xΦ_a⟩ = ⟨χ^(-)_b|χ^(+)_b⟩|χ^(+)_xΦ_a⟩. When used in the expression for the cross section Eq. (<ref>) or in its original form,d^2σ^INEB_b/dE_bdΩ_b =2/ħ v_aρ_b(E_b) ⟨ρ̂^HM_x|W_x|ρ̂^HM_x⟩ ,it is a simple exercise to reduce the expression to the Glauber cross section above. The above form of the cross section has been extensively used by <cit.> to calculate one nucleon stripping and pickup reactions. These authors, take R̂_1 = |Ŝ_b(r_x)|^2[W(r_x)|χ^(+)_x(r_x)|^2] as an operator and calculate the expectation value in the state from which the nucleon is removed or added, ⟨Φ_i|R̂_1|Φ_i⟩. This is then used to represent the cross section in a particularly transparent form. The partial cross section for removal of a nucleon, from a single-particle configuration j^π populating the residue final stateα with excitation energy E^⋆_α, is calculated as σ_α = (A/A - 1)^NC^2S(α, j^π) ⟨Φ_j^π|R̂_1|Φ_j^π⟩ where S_α^⋆= S_n,p + E_α^⋆ is the effective separation energy for the final state α and S_n,p is the ground-state to ground-state nucleon separation energy. Here the factor N, in theA-dependent center-of-mass correction factor that multi- plies the shell-model spectroscopic factors C^2S(α,j^π), is the number of oscillator quanta associated with the major shell of the removed particle <cit.>. The cross section for the removal of a nucleon from the j^π singleparticle shell is then the sum over all bound final states, α of the residual nucleus A -1Here, the radioactive projectile is now denoted by A. In this manner, useful nuclear structure information contained in the spectroscopic factor of the fragment in the projectile A, C^2S(α, j^π), can be extracted. Further, within the four-body theory, the practitioners of the eikonal theory extend their application to two nucleon removal or pickup. They use the following form of the reaction factor ⟨Φ_i|R̂_2|Φ_i⟩ = ⟨Φ_i||S_b|^2| (1 - |S_x_1|^2)(1 - |S_x_2|^2)|Φ_i⟩. This formula should come out from σ_R^3B of Eq. (<ref>) of our theory in the high energy limit. However, there are several shortcomings in the above model, as there is no reference to the individual absorptions of x_1 and x_2. Further, the model above misses the correlation between the two interacting fragments. Currently we are investigating this point.§ DOUBLE FRAGMENT INELASTIC PROCESSESThe inclusive cross-section expressed by σ_R^3B given by Eq. (<ref>) is new and corresponds togenuine three-body absorption processes to inelastic channels. This part of the cross-section is associated withathree-body optical potential U_3B dependingon the relative coordinates of the fragments x_1, x_2 and the target, and cannot be reduced to connected terms of the 3B transition matrix with two-body potentials in different subsystems and the target in the ground state.The structure of W_3B=[U_3B] results from conventional nuclear reaction theory. It includes, for example, processes like thevirtual excitation of the target by one of the fragments and its virtual de-excitation by the other, as well asother 3B processes irreducible to the re-scattering terms with the target in the ground state andwith the final result being the full capture, or complete fusion, of both fragments. This is illustrated schematically infigure <ref>. If a resonant process exists, such as the excitation of giant pairing vibrations from thetransfer of two neutrons from the projectile, it should furnish a large contribution to the opticalthree-body potential at the resonance energy.§.§ Hint on the 3B optical potential U_3B The schematic figure <ref> furnishes a path to formallybuild the the three-body optical potential by using projection operators outof the target ground state,U_3B = PV_x_1AQ (QG_xA(E_x)Q) QV_x_2AP +PV_x_2AQ (QG_xA(E_x)Q) QV_x_1AP ,where E_x=E_x_1+E_x_2 and the Q-projected 3B Green's function of the x_1 + x_2 + Asystem isQG_xAQ =[E_x - QH_0Q + QV_xA P G_0PV_xA Q + iε]^-1with V_xA≡ V_x_1A + V_x_2A.The imaginary part of the Q-projected Green's function of the x_1x_2A≡ xAsystem includes the virtual propagation n all states except for the elastic channel, Im[QG_xAQ] = -πΩ^(-)_QQδ(E_x - H_0)Q(Ω^(-)_Q)^†+ (QG_xAQ)^†QV_xA Pδ(E_x- H_0)PV_xAQG_xAQ Then, again using standardtechniques, the imaginary part of U_3B can be isolated in (<ref>) and can be written asW_3B=U_3B^†-U_3B = =π[PV_x_1AQ Ω^(-)_QQδ(E_x - H_0)Q(Ω^(-)_Q)^†QV_x_2AP+ PV_x_1AQ (QG_xAQ)^† Q V_xAP δ(E_x - H_0) P×P V_xA Q(QG_xAQ) QV_xAP]+(x_1↔ x_2)where one can identify the virtual propagation of the x_1x_2A system through the inelastic channels in the Q-space.§ BEYOND HM, UT, AND IAV FORMULAS The key points for going beyond the IAV formula for the inclusive inelastic cross-sectionare to include in the source term the full four-body dynamics and the three-body optical potential, by computing |Ψ^4B(+)_0⟩ without approximating the four-body problem. For the time being we will be content to generalize the three-body IAV formula to thefour-body case and present the CFH formula given in <cit.>. Thegeneralization given by CFH of the IAV formula to determine the inclusive cross-section observed by the detection of the spectator particle b, contains HM and UT like terms. The steps to derive a 4B version of the IAV formula follow<cit.> and we write|Ψ^4B(+)_0⟩=G^(+)_b, x_1, x_2, AV_x|Ψ^4B(+)_0⟩≈G^(+)_b, x_1, x_2, A V_x|χ^(+)_a , Φ_a⟩ where V_x=V_b x_1+V_b x_2 is the interaction between the detected fragment and the other two,|Φ_a⟩ the projectile internal wave function and χ^(+)_a the distorted wave of thecenter of mass. The four body Green's function isG^(+)_b, x_1, x_2, A = [E - T_b - T_x_1 - T_x_2 - V_x_1, x_2 - U_b - U_x_1 - U_x_2 + iε]^-1. where we have not explicitly included U_3B, which should also contribute to the Green's function. The generalization of the IAV three-body source function for the four-body case is:ρ̂^CFH_x_1,x_2 =(χ_b^(-)|Ψ_0^4B(+)⟩ =⟨χ_b^(-)|G^(+)_b, x_1, x_2, A V_x |χ^(+)_aΦ_a⟩By manipulating the above formula in analogy to the developmentprovided in Ref. <cit.>, one gets the 4B extension of the 3B Ichimura-Austern-Vincent asρ̂_x_1, x_2^CFH = ρ̂_x_1, x_2^UT +ρ̂_x_1, x_2^HMwhere the 4B Hussein-McVoy (HM) term isρ̂_x_1, x_2^HM = ⟨χ_b^(-)|χ_a^(+)Φ_a⟩and the 4B Udagawa-Tamura (UT) term is ρ̂_x_1, x_2^UT≡ G^(+)_x_1, x_2, A⟨χ_b^(-)|[U_b + U_x_1 + U_x_2 - U_a]|χ^(+)_aΦ_a⟩ .The Green's function G^(+)_x_1, x_2, A = [E - E_b - T_x_1 - T_x_2 - V_x_1, x_2 - U_x_1 - U_x_2 + iε]^-1should also contain in principle the three-body optical potential. Further studies of this particular pointwill be developed. §INCLUSIVE BREAKUP AND EFIMOV PHYSICSThe inclusive cross-section formulae for the elastic breakup (<ref>) andfor the inelastic process (<ref>)applied to a three-body halo nuclei, e.g., a weakly boundneutron-neutron-core nuclei close to the drip-line, contains a long-range correlation between the two neutrons and the core b, through the extended wave function of the halo and the scattering waveχ^(+)_x_1(r_x_1)χ^(+)_x_2(r_x_2)χ^(+)_x_b(r_x_b), which appears both in (<ref>) and in the source function in (<ref>), when such an approximation is made. However, we expect that interesting physics remains beyond such an approximation ifthe 4B scattering wave function,Ψ_0^4B(+)(r_b, r_x_1, r_x_2) is taken in full.It should contain dynamics of the continuum of the neutron-neutron-core system beyond the bound halo state. The reactionmechanism now includes Efimov physics <cit.> and with that the long range correlation between the halo fragments, which could be influenced by the presence of Efimov bound, virtual or resonant states <cit.> (see also <cit.> for a discussion of observing Efimov states in halo reactions). Indeed, the existence of Borromean Efimov states was observed about a decade ago <cit.> in the resonant three-body recombination of cold cesium atoms in magneto-optical traps.§ DIGRESSION ON TWO-FRAGMENT PROJECTILE INCLUSIVE BREAKUP CROSS SECTION It is important to remind the reader of the advances made in the application of the inclusive breakup theory of two-fragment projectiles, as was originally developed in <cit.>. Quite recently this theory was applied to the (d,p) reaction <cit.> as a mean to test the validity of the Surrogate Method <cit.>, employed to extract neutron capture and fission cross sections of actinide target nuclei, such as ^238U, ^232Th, of importance for the development of next generation fast breeder reactors. These neutron capture reactions on other targets are also important for the study of element formation following supernova explosion through the astrophysical s-process. The general conclusion of<cit.> was that the Surrogate Method, the extraction of the cross section for the formation and decay of the compound nucleus of the (n + A) subsystem, as well as the total capture cross section (the cross section for the formation of the compound nucleus),is justified as long as the direct part of the cross section is calculated and subtracted from the inclusive breakup total reaction cross section σ^(n + A)_R. Another topic of importance is the Trojan Horse Method <cit.> used to extract from the inclusive breakup data a direct reaction of interest to nuclear astrophysics following nova explosion, which otherwise would be difficult to measure in the laboratory. The THM uses the advantage that as the surrogate charged fragment x is brought by the primary projectile to the region of the target, the Coulomb barrier is already surmounted, and accordingly no hinderance due to barrier penetration and tunneling is present in the x induced reaction. Accoringly, the x-induced reaction proceeds above the Coulomb barrier. Being so, the electron screening problem is also avoided.The two methods, SM, and THM, are two pieces of the same quantity, namely σ^(x + A)_R in the breakup reaction, a + A → b + (x + A). Thus the THM can be easily justified within the theory alluded to above as being a process contributing to the direct part of σ^(x + A)_R. As a final remark, if the surrogate fragment x is a neutron then the SM and the THM are the same!It would certainly be important to extend these studies to the case of the three-fragment projectiles discussed in this contribution. In particular, it would be particularly interesting to extend the THM to cases involving three-fragment projectiles, where three types of reactions can be extracted, x_1 + A → y_1 + B_1, x_2 + A → y_2 + B_2, and x_1+x_2 + A → y_3 + B_3. Work along these lines is in progress.§ CONCLUSIONS AND PERSPECTIVES We have reported in this contribution that the general structure of the CFH cross section in the DWBA limit is similar in structure to the 3B one , with the full post form (or four-body IAV), which can be written as the sum of the prior four-bodyUT cross section plus the four-body HM one plus an interference term when the 4Bsource term detailed in Eq. (<ref>) is used to compute the inelastic part of the inclusive cross-section. The major difference between the 4B and 3Bcases resides in the structure of the reaction cross sections for the absorption of one of the interacting fragments, whichwe find to be damped by the absorption effect of the other fragment. It is expected that in a (t,p) reaction, an absorption cross section of the n + A subsystem would be smaller than the corresponding one in a (d,p) reaction. Another important new feature is the 3B absorption from a three-body optical potential, the formal structure of which we have sketched. The perspectives of our study are in the use of the Faddeev-Yakubovski equations <cit.> to develop an expansion of the four-body wave function, to attempt togo beyond the CFH/IAV formulas. This effort could also be accomplished by building the four-body CDCC wave functionto compute the source term. In addition, we can ask about the dynamics that are built in the three-body optical potential when giant pairing vibrations (GPV) are possible <cit.>.These collective states involve the coherent excitation ofparticle-particle pairs, in complete analogy to the coherent excitation of particle-hole pairs that constitutes the microscopic foundation of multipole giant resonances. For example, the GPV opens interesting prospects for nuclear structure studies in reactions of the type (t, p),2n Borromean cases such as (^6He, ^4He), (^11Li, ^9Li), (^14Be, ^12Be), (^22C, ^20C), and the 2p halo cases, (^17Ne, ^15O), and (^20Mg, ^18Ne). The theory we have developed is an appropriate frameworkto study these typesof collective nuclear excitations, associated with pairing correlations in the target.From the point of view of a two-neutron s-wave dominated halo target our theory naturally leads to an inquiry about the röle of Efimov physics in inclusive breakup cross-section, both in the elastic and inelastic contributions, which should have its place beyond the CFH formula. Further, the theory can be extended to allow for inclusive reactions where two or morefragments are detected. We expect that thedevelopments discussed in our recent workshould stimulateexperimental and theoretical works to seek more information aboutthe x_1 + x_2 system in the elastic and inelastic breakup cross sections, and in particularto theorists to extend the Surrogate Method and the Trojan Horse Method, both based on the inclusive non-elastic breakup partof the b spectrum in a two-fragment projectile induced reaction, to three-fragment projectiles. Acknowledgements.This work was partly supported by the Brazilian agencies, Fundação de Amparo à Pesquisa do Estado deSão Paulo (FAPESP), theConselho Nacional de Desenvolvimento Científico e Tecnológico(CNPq). MSH also acknowledges a Senior Visiting Professorship granted by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), through the CAPES/ITA-PVS program. 99CarPLB2017 B. V. Carlson, T. Frederico, M. S. Hussein, Phys. Lett. B 767, 53 (2017).IAV1985M. Ichimura, N. Austern, and C. M. Vincent, Phys. Rev. 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Dietrich, Phys. Rev. C 74, 054601 (2006) SM1 Jutta E. Escher, Jason T. Burke, Frank S. Dietrich, Nicholas D. Scielzo, Ian J. Thompson, and Walid Younes, Rev. Mod. Phys., 84, 353 (2012) SM2J. E. Escher, A. P. Tonchev, J. T. Burke, P. Bedrossian, R. J. Casperson, N. Cooper,, R. O. Hughes, P. Humby, R. S. Ilieva,, S. Ota1,, N. Pietralla,, N. D. Scielzo, and V.Werner, EPJ Web of Conferences 122, 12001 (2016) THM0 S. Typel and G. Baur, Ann. Phys. (NY)305,228 (2003) THM1C. Spitaleri et al., Phys. of Atomic Nuclei 74, 1725 (2011)THM2 A. Tumino et al., Few Body Systems, (2012) DOI 10.1007/s00601-012-0407-1 THM3 R. Tribble et al., Rep. Prog. Pays. 77, 106901(2014) | http://arxiv.org/abs/1704.08767v1 | {
"authors": [
"M. S. Hussein",
"L. A. Souza",
"E. Chimanski",
"B. V. Carlson",
"T. Frederico"
],
"categories": [
"nucl-th",
"nucl-ex"
],
"primary_category": "nucl-th",
"published": "20170427223959",
"title": "Inclusive Breakup Theory of Three-Body Halos"
} |
Graviton fluctuations erase the cosmological constant C. Wetterich[mailto:[email protected]@thphys.uni-heidelberg.de] Universität Heidelberg, Institut für Theoretische Physik, Philosophenweg 16, D-69120 Heidelberg ===================================================================================================Graviton fluctuations induce strong non-perturbative infrared renormalization effects for the cosmological constant. The functional renormalization flow drives a positive cosmological constant towards zero, solving the cosmological constant problem without the need to tune parameters. We propose a simple computation of the graviton contribution to the flow of the effective potential for scalar fields. Within variable gravity we find that the potential increases asymptotically at most quadratically with the scalar field. With effective Planck mass proportional to the scalar field, the solutions of the derived cosmological equations lead to an asymptotically vanishing cosmological “constant” in the infinite future, providing for dynamical dark energy in the present cosmological epoch. Beyond a solution of the cosmological constant problem, our simplified computation also entails a sizeable positive graviton-induced anomalous dimension for the quartic Higgs coupling in the ultraviolet regime, substantiating the successful prediction of the Higgs boson mass within the asymptotic safety scenario for quantum gravity.2The tiny value of the cosmological constant V, as compared to the Fermi scale of weak interactions, h_0 = 176, or the Planck mass M = 2.4e18, is an old puzzle <cit.>. In units of the Planck mass this value amounts to V/M^4 ≈e-120. Within quantum field theory the cosmological constant can be identified with the value that the effective potential of scalar fields takes at its minimum or for a given cosmological solution. Approximating the effective potential for the Higgs doublet h by U = λ_h (h^† h)^2/2 - λ_h h_0^2 h^† h + c_h, it is hard to understand why the difference V = c_h - λ_h h_0^4/2 should take a value ∼ (2e-3)^4, much smaller than h_0^4. In this note, we argue that quantum gravity effectsinduce a strong renormalization of the cosmological constant, making its value on cosmological scales almost independent of the value of c_h at the Fermi scale.In a general quantum field theory the cosmological constant or the effective scalar potential are functions of an infrared (IR) scale k. For a given k only fluctuations with momenta larger than k, or wavelength smaller than k^-1, are included in the computation of the renormalized quantities. For example, k may be set by the momentum of particles in a scattering process. For this case, the k-dependence or “running” of the strong coupling constant in quantum chromodynamics has been impressively confirmed by observation. We find that graviton fluctuations are responsible for a strong running of V(k). Even if V(k) is of the order h_0^4 or larger at the scale k = h_0, it will take a tiny value on a cosmological scale k ≈ H, as required by observation.A widespread view asserts that quantum gravity effects become important on short length scales of the order of the Planck length l_P = 1/M, while being negligible for much larger wavelengths of the fluctuations. One would then expect that quantum gravity effects can give only negligible contributions to the running of V between k = h_0 and k = H, suppressed by a factor M^-2. We will see that these simple considerations do not hold due to a potential instability of the graviton fluctuations. While the contribution of graviton fluctuations indeed involves a factor M^-2, there is also an enormous enhancement factor for the flow of V in the vicinity of the instability. As the instability is approached, the flow equations become singular, preventing in this way any unstable behavior. This “avoidance of instabilities” gives rise to strong IR effects in quantum gravity.Our method for the computation of the quantum gravity effects for the cosmological constant is based on the exact flow equation for the effective average action <cit.>. Applied to the ultraviolet (UV) behavior of quantum gravity <cit.>, it has already led to substantial evidence for asymptotic safety <cit.>, which would render quantum gravity non-perturbatively renormalizable. In this note, we concentrate on the IR effects of quantum gravity. While rather extensive information has been available on the UV flow for a long time <cit.>, a detailed exploration of the IR flow is only at its beginning <cit.>.Due to the importance of the gauge symmetry of diffeomorphism transformations, a full computation of the flow equation for V or U gets rather involved already for simple truncations of the effective average action. Our aim is here to demonstrate the key features of the gravitationally induced flow of the effective potential. For this purpose, we only include the contributions from the graviton fluctuations, leaving other degrees of freedom in the metric and ghosts aside. This leads to an intuitive and simple expression for the flow equation. The restriction to the graviton fluctuations also avoids several conceptual and technical complications. The graviton fluctuations are physical fluctuations that do not depend on the choice of a gauge. Analytic continuation between Euclidean and Minkowski signature is particularly simple for the graviton fluctuations. Furthermore, the graviton fluctuations cannot mix with scalar or vector fluctuations for any geometry with rotation invariance. This block diagonal form of the propagator matrix allows us to treat the graviton fluctuation effects independently of other fluctuations.Our proposal for the solution of the cosmological constant problem will be formulated within variable gravity, with effective Planck mass M(χ) or Newton's “constant” depending on a scalar field χ. The main ingredient states that the strong IR graviton fluctuations exclude for χ→∞ an increase of the effective potential U(χ) faster than M^2(χ). The observable cosmological constant is determined by the dimensionless ratio U(χ)/M^4(χ), which decays asymptotically ∼ M^-2(χ). The cosmological solutions of models with variable Planck mass lead to an asymptotic increase M(χ) →∞ for the infinite future, resulting in an asymptotically vanishing cosmological constant. The universal asymptotic properties of U(χ) are independent of microscopic parameters, as characteristic for an IR fixed point.The most prominent features of the graviton-induced renormalization effects can already be seen without the scalar field, and we therefore start with simple Einstein gravity in Euclidean flat space. The inclusion of scalar fields and extensions to Minkowski space and other geometries are discussed subsequently. We believe that our simple approach already contains all important elements for the strong IR-gravity effect. § SCALE-DEPENDENT COSMOLOGICAL CONSTANT Consider a flat Euclidean background geometry, g̅_μν = δ_μν. We restrict the metric fluctuations to the transversal traceless part,2 g_μ_ν = δ_μ_ν + t_μ_ν, t_μ_ν = t_ν_μ,t^μ_μ = δ^μν t_μν = 0, q^μ t_μν = 0.In momentum space t_μ_ν is a function of the Euclidean four-momentum q_ρ = (q_0,q⃗) with q^2 = q^ρ q_ρ. The Euclidean Einstein-Hilbert action with cosmological constant V, reduced Planck mass M and curvature scalar R̃ readsΓ = ∫_x√(g)(V - M^2/2R̃).The term quadratic in t_μ_ν,Γ_2 = 1/2∫_q,q^'t_μ_ν(-q^')Γ^(2)^μ^ν^ρ^τ(q^',q)t_ρ_τ(q),defines the second functional derivativeΓ^(2)^μ^ν^ρ^τ(q^',q) = (M^2q^2/4 - V/2) P^(t)^μ^ν^ρ^τ(q) δ(q^' - q).Here ∫_q = ∫^4 q/(2π)^4, δ(q^' - q) = (2π)^4 δ^4(q_μ^' - q_μ), and the projector on t_μ_ν, P^(t) = (P^(t))^2 is given by*P^(t)_μ_ν_ρ_τ = 1/2(P̃_μ_ρP̃_ν_τ + P̃_μ_τP̃_ν_ρ) - 1/3P̃_μ_νP̃_ρ_τ,P̃_μ^ν = δ_μ^ν - q_μ q^ν/q^2,P^(t) = *P^(t)_μ_ν^μ^ν = 5. The propagator or connected two-point function for t_μν is the inverse of Γ^(2) on the projected space <cit.>G_μ_ν_ρ_τ(q^',q) = 4/M^2(q^2 - 2V/M^2)^-1*P^(t)_μ_ν_ρ_τ(q)δ(q^' - q).We observe that the cosmological constant V acts like a mass term for the graviton,m^2 = -2V/M^2.For V>0 the propagator is tachyonic and one expects strong infrared instabilities. The negative sign in <ref> will be the crucial ingredient for the solution of the cosmological constant problem. We recall that for V ≠ 0 flat space is not a solution of the field equationM^2 (R̃_μ_ν - 12R̃ g_μ_ν) = -V g_μν.The metric propagator in a flat background is therefore an “off-shell propagator”, as needed for the functional renormalization flow.Functional renormalization investigates the dependence of the effective average action Γ_k on an infrared cutoff scale k. We work in the simple truncation eqn:einstein hilbert action, where V and M^2 are now k-dependent “running couplings”. We will see that the flow of V with t = ln(k/M) is such that the potential instability in the propagator eqn:propagator is avoided. Our starting point is the exact functional flow equation for the effective average action <cit.>∂_t Γ_k = 1/2{G_k∂_t R_k},with(Γ_k^(2) + R_k) G_k = 1.It involves the matrix Γ_k^(2) of second functional derivatives of Γ_k and the IR cutoff that we take asR_k^μ^ν^ρ^τ(q^',q) = M^2/4R_k(q)P^(t)^μ^ν^ρ^τ δ(q^' - q).The cutoff function will be chosen such that an addition of R_k(q) to to the kinetic term q^2 in Γ^(2) provides a type of positive (momentum-dependent) mass term ∼ k^2, but only for momenta q^2 ≲ k^2. Correspondingly, the propagator G_k in the presence of the IR cutoff replaces in <ref> q^2 → q^2 + R_k(q). We will start the flow for large enough k such that the IR cutoff prevents any singular behavior of G_k, e.g. k^2 ≫ |m^2|. As k is lowered, one gradually approaches the potential singular behavior.In a flat background t_μ_ν(q) is an irreducible tensor representation of the rotation group SO(4) (or Lorentz group for Minkowski signature). It therefore cannot mix on the quadratic level with scalar or vector representations. The transversal traceless fluctuations are physical metric fluctuations that do not mix with the gauge fluctuations <cit.>. These properties make the flow equation eqn:wetterich block-diagonal and allow us to deal with the t_μ_ν-contribution to the flow separately. Additional pieces in the flow equation from metric fluctuations beyond t_μ_ν, gauge fixing terms and ghosts <cit.> will be neglected for our discussion. They may partly cancel contributions from fluctuations in t_μ_ν, without changing the overall picture. With respect to the rotation group SO(3) the transversal traceless fluctuations contain the two degrees of freedom of the graviton. They also account for two vector and one scalar degree of freedom that do not correspond to propagating particles if the background obeys the field equations <cit.>. In a later part of this note we will focus on the propagating graviton fluctuations. This will not affect the main findings. In the first part on flat space we employ “graviton” in the wider sense of the full irreducible representation t_μ_ν.We are interested in the flow of the cosmological constant V. It is obtained directly by evaluating <ref> for vanishing metric fluctuations, g_μ_ν = δ_μ_ν, such that Γ = ∫_x V results in the simple one-loop expression∂_t V = k∂_k V = 5 I_k(-2 VM^2),withI_k(m^2) = 1/2∫_q (q^2 + R_k(q) + m^2)^-1∂_t R_k(q).Employing the Litim cutoff <cit.>R_k(q) = (k^2 - q^2)θ(k^2 - q^2)replaces q^2 + R_k → k^2 for q^2 < k^2, while not affecting the propagator (q^2 + m^2)^-1 for q^2 > k^2. It permits to solve the momentum integral analyticallyI_k(m^2) = 1/32 π^2k^6/k^2 + m^2. The flow equation eqn:cc flow for V has a very intuitive interpretation: one takes the one-loop formula for the contribution of the graviton fluctuations ∼∫_q ln(q^2 - 2 V/M^2), replaces q^2 → k^2 in the IR region, and takes a logarithmic k-derivative. We will see below that the same type of equation holds in Minkowski space, with q^2 = -ω^2 + q⃗^2, q^0 q_0 = -ω^2, in the momentum integration eqn:loop integral.The qualitative features of the solution of the flow equation are best discussed in terms of the dimensionless variablev = 2 V/M^2 k^2.Approximating first M by a constant, the flow equation reads∂_t v = β_v = - 2 v + 5 k^2/16 π^2 M^2 (1 - v)^-1.This is a central equation of this note. Its generalization to the flow of the effective potential for scalars is the basis for the proposed solution of the cosmological constant problem. We observe the pole at v = 1, which is multiplied by a small factor for k^2 ≪ M^2. The behavior of β_v is shown for two values of k^2/M^2 in <ref>. As k is lowered, the transition between the two regimes becomes very sharp. The singular second term in <ref> plays then a role only for v very close to one.< g r a p h i c s >figureBeta function for the dimensionless variable v = 2 V/M^2 k^2 for two values k/M = 1 and k/M = 0.4. The arrows denote the flow towards k → 0 approaching the zero close to v = 1 (cf. <ref>). We concentrate on a positive cosmological constant, v > 0, and the range v < 1 for which the integral eqn:loop integral is well defined. A small value of v flows towards larger values according to β_v ≈ - 2 v. On the other hand, values of v very close to the pole at v = 1 decrease and β_v is dominated by the positive second term. The solutions of the flow equation eqn:v flow are attracted towards a sliding approximate fixed point v_c(k) for which β_v vanishes,v_c (1 - v_c) = 5 k^2/32 π^2 M^2.This can be clearly seen by the numerical solutions of <ref> shown in <ref>. This figure demonstrates in a simple way how initial values are “forgotten”. The IR value v = 1 is reached independently of details of the initial conditions. In this way the cosmological constant problem will be solved without the need for fine-tuning of parameters.< g r a p h i c s >figureFlow of the dimensionless variable v = 2 V/M^2 k^2 with t = ln(k/M) for different initial conditions. The trajectories shown all approach the quasi-fixed point v_c ≈ 1 as k is lowered. For k^2 ≪ 32 π^2 M^2/5,the relevant v_c is close to 1, implying for the flow of the cosmological constantV_c(k) = M^2 k^2/2v_c(k) = M^2 k^2/2 - 5 k^4/64 π^2.As k goes to zero the cosmological constant approaches the pole at v = 1, but it never crosses it. The graviton-induced infrared flow of the cosmological constant drives it to zero! This holds actually even if the approximate fixed point v_c(k) is not reached, which can happen for initial values of v very close to zero or one. Since the flow is confined to the region 0 ≤ v ≤ 1, the flowing cosmological constant always obeys 0 ≤ V ≤ M^2 k^2/2.We conclude from this simple discussion that graviton fluctuations can have strong effects on the running of the cosmological constant, even in the region k^2 ≪ M^2. For k^2 ≫ 2 V/M^2 the flow ∂_t V ∼ k^4 would rapidly become insignificant as k goes to zero. This changes dramatically for k^2 ≈ 2 V/M^2. The pole in <ref> can enhance the β-function by a large factor and prevents V from remaining larger than M^2 k^2/2. The observation that a potential singularity in flow equations is avoided has been made earlier within functional renormalization. This feature explains the approach to convexity of the effective potential in scalar theories <cit.>. Rather than being just a formal mathematical construction, this approach to convexity due to fluctuations is a physical effect <cit.>, needed, for example, for the quantitative understanding of spontaneous nucleation in first-order phase transitions <cit.>.For quantum gravity singular features of the flow of the cosmological constant have been noted and discussed since early stages <cit.>, while consequences for the physical value of the cosmological constant and cosmological consequences remained unclear. In this note we argue that the “avoidance of instabilities” by the flow governs directly the value of the observable cosmological constant and determines the features of dark energy in cosmology.The length scale of the fluctuations driving the cosmological constant to zero can be far in the infrared. If we only include fluctuations with q^2 > k^2 the effective cosmological constant V has a positive value V ≈ M^2 k^2/2. With Minkowski signature, this would correspond to a de Sitter space with Hubble parameter H^2 = k^2/6. In other words, the fluctuations with wavelengths close to but somewhat smaller than the “would-be horizon” are needed in order to drive the cosmological constant to zero.At this point, we should mention two important implicit assumptions underlying our computation of the flow of V. The first is that the field equations for the graviton, the graviton propagator, and its interactions can be derived from a diffeomorphism invariant effective action involving only one metric field. The second states that this effective action can be approximated by a derivative expansion to second order. These assumptions imply <ref> at leading order. They are the basis of all computations in classical gravity and therefore well tested by experiment. Within functional renormalization it has been argued <cit.> that these properties indeed hold, provided one chooses a suitable physical gauge fixing or, equivalently, a constraint on conserved sources and physical fluctuations. These assumptions imply that the effective action for traceless transversal tensor fluctuations around flat space can be expanded asΓ = ∫_x √(g̅)(V - V/4t^μ^νt_μ_ν - M^2/8t^μ^ν∂^2 t_μ_ν + …)with √(g̅) = 1 for Euclidean signature and √(g̅) = i for Minkowski signature, and ∂^2 = g̅^μ^ν∂_μ∂_ν. The dots denote the graviton interactions. In the limit of vanishing momentum (∂_ργ^m^n = 0), the graviton propagator and all interactions are determined by a single constant V, which is also the constant part in Γ. This is the “diffeomorphism constraint” on the effective action for the graviton. We employ this constraint and evaluate the flow of V by the flow of the constant term.In pure gravity a k-independent value of M^2 is typically relevant for the IR running. Close to an ultraviolet (UV) fixed point one expects instead a scaling behavior M^2 = f k^2 <cit.>. Our discussion can easily be extended to M^2 depending explicitly on k. In this case, the factor M^2 in the cutoff eqn:ir cutoff leads to the replacement ∂_t R_k →∂_t R_k + (∂_t ln M^2) R_k in the integrand J. For M^2 = f k^2 one replaces <ref> by∂_t v = -4 v + 5/16 π^2 f (1 - v)^-1.(More precisely, the second term on the r.h.s. is multiplied by 4/3 if we replace in the IR cutoff eqn:ir cutoff M^2 → f k^2. Throughout this note, we will neglect terms from the k-dependence of M^2 in the cutoff eqn:ir cutoff.) The corresponding flow shows two fixed points, determined byv_∗ (1 - v_∗) = 5/64 π^2 f.The solution with smaller v_∗ corresponds to the UV fixed point, whereas the larger v_∗ is IR-attractive. In the UV scaling region, one has V ∼ k^4. For M^2(k) = f k^2 + M^2 the IR fixed point in <ref> is no longer realized if k^2 < M^2/f, and the flow switches to the behavior eqn:v flow. This reflects some of the qualitative features observed in more involved functional renormalization studies with a flat background geometry <cit.>. (Note that the concept and definition of the cosmological constant in <cit.> differs from the present one. The diffeomorphism constraint eqn:diffeomorphism constraint is not realized in this work.) § FLOWING SCALAR POTENTIAL Our simple discussion can be extended to the graviton contributions to the flow of the effective (average) potential U_k(χ) of some scalar field χ. (See <cit.> for more extended quantum gravity investigations with scalar fields.) One simply replaces in the flow equation eqn:cc flow V by U(χ). Here U(χ) is defined by evaluating Γ for vanishing metric fluctuations in flat space, and for constant χ, ∂_μχ = 0, i.e. Γ = ∫_x U(χ). The scalar potential describes all scalar interactions at zero momentum. The graviton contribution does not change if we replace V by U(χ). Flow equations for χ-derivatives, such as the scalar mass term ∂^2 U/∂^2 χ, can be obtained by taking χ-derivatives of the flow equation for U(χ). The implicit assumption underlying this framework states that the zero-momentum limit for the field equations, propagators and interactions of the physical graviton and scalar field fluctuations can be derived from a diffeomorphism-invariant effective action Γ = ∫_x √(g)U(χ).In addition to the graviton contribution, there are other contributions to the flow of U. For the example of a canonical kinetic term for χ, the combination of the fluctuations of the scalar χ and the graviton yields in a simple truncation∂_t U = k^6/32 π^2(5/k^2 - 2 U/M^2 + 1/k^2 + ∂^2 U/∂^2 χ).The second term <cit.> involves the χ-dependent scalar mass term, as given by the second derivative of U. We assume here a range for which the mixing of fluctuations in χ with scalar fluctuations in the metric can be neglected.We distinguish two regimes. As long as k ≫ |2 U/M^2|, the first term mainly contributes to the flow of a χ-independent additive constant. The flow of the χ-dependence of U is governed by the scalar fluctuations, with a negligible graviton contribution. This yields the standard flow of the effective average potential <cit.>. For models with a discrete symmetry χ→ -χ, the potential only depends on ρ = χ^2/2, with ∂^2 U/∂^2 χ = U^' + 2 ρ U^'' and primes denoting derivatives with respect to ρ. In a truncation where U^''' and higher ρ-derivatives are neglected, the flow of U^' and U^'' can be computed by taking ρ-derivatives of <ref>, i.e.∂_t U^' = -k^6/32 π^2{3 U^''/(k^2 + U^' + 2 ρ U^'')^2 - 10 U^'/M^2 (k^2 - 2 U/M^2)^2}and∂_t U^'' = k^6/16 π^2{9 U^''2/(k^2 + U^' + 2 ρ U^'')^3 + 5 U^''/M^2 (k^2 - 2 U/M^2)^2}.The graviton corrections are suppressed by a factor 1/M^2. (In <ref>, we keep onlythe leading power.) If we identify the quartic scalar coupling λ with U^''(ρ_0), with ρ_0 the location of the minimum of U(ρ), and take k^2 ≫ |U^' + 2 ρ U^''|, k^2 ≫ |2 U/M^2|, we recover the usual one-loop running up to a gravitational correction,∂_t λ = 9 λ^2/16 π^2 + 5 λ k^2/16 π^2 M^2. For particle physics experiments, the external momenta of the scattered particles typically set an effective IR cutoff. Identifying k roughly with this cutoff, it will be in therange. (A few orders of magnitude play no role here.) For a potentialU_SM = V + λ/2 (ρ - ρ_0)^2mimicking the one of the Higgs scalar, gravity effects are negligible in the range λρ^2 ≪ k^2 M^2. Similar features hold for extended models where Yukawa couplings to fermions or gauge interactions are taken into account.The second regime concerns the “singular region” k^2 ≈ 2 U/M^2. For a potential of the form eqn:higgs potential, this always becomes relevant for very large ρ, namely ρ≈ k M/√(λ). In this region, the graviton-induced renormalization effects become strong and the polynomial form of U is no longer maintained. Typically, the potential flattens for very large ρ, such that U(ρ) ≲ M^2 k^2/2 and the pole will not be crossed. A good approximation to the result of the flow is given byU(ρ) =U_SM(ρ) for ρ < ρ_c,M^2 k^2/2 for ρ > ρ_c,with ρ_c dependent on kaccording toU_SM(ρ_c) = M^2 k^2/2.The transition region for ρ near ρ_c will be smoothened, but the behavior for ρ≫ρ_c is given to a good accuracy by the flat part of the potential eqn:potential approx.As k is lowered, ρ_c(k) decreases and the field region where the “standard model” potential U_SM is valid shrinks. For V < 0 in <ref>, the standard model region remains finite with (ρ_c(k=0) - ρ_0)^2 = -2 V/λ. For V > 0, however, thepotential becomes entirely flat for small enough k, given by the k-dependent cosmological constant discussed previously. While the range of ρ where the flattening happens is not relevant for particle scattering, it may be very important for cosmology and the cosmological constant problem.We can also extend our simplified discussion to the scaling regime near a UV fixed point, where M^2 = f k^2. This results for <ref> in the gravity-induced anomalous dimension A_λ,∂_t λ = A_λλ + 9 λ^2/16 π^2,A_λ = 5/16 π^2 f = 5 g_∗/2 π.The anomalous dimension is positive, A_λ > 0, such that λ is driven towards zero as k is lowered. Typically, A_λ is of the order one, such that the approach to λ = 0 is fast. Values found for the scaling gravitational coupling in functional renormalization group investigations of gravity are in the range g_∗ = G_N k^2 = (8 π f)^-1 = 0.3 <cit.>, 1.4 <cit.>, 1.0 <cit.>, 2.0 <cit.>, 1.5 <cit.>, 0.8 - 1.8 <cit.>, 0.7 - 0.9 <cit.>, 0.8 <cit.>, 1.2 <cit.>. § COSMON POTENTIAL IN VARIABLE GRAVITY AND THE SOLUTION OF THE COSMOLOGICAL CONSTANT PROBLEM In general, M^2 is a function of the “cosmon” scalar field χ. For large χ, one expects M^2 ∼χ^2, and we will choose a normalization of the scalar field where M = χ. Adding a kinetic term for χ,Γ = ∫_x √(g){-M^2(χ)/2 R̃ + U(χ) + 1/2 K(χ) ∂^μχ∂_μχ}constitutes the effective action for “variable gravity” <cit.>.(Often M^2(χ) is denoted by F(χ) and U(χ) by V(χ).)For M^2 = χ^2 the graviton contribution to the flow of U takes the form∂_t U = 5 k^6/32 π^2 (k^2 - 2 U/χ^2)^-1 + …,where the dots denote contributions from the scalar fluctuations and other degrees of freedom. In the region of χ where the graviton contribution dominates, we can take over the discussion of the running cosmological constant, and <ref> turns for k^2 ≪χ^2 toU = k^2/2 χ^2.This region corresponds to the close vicinity of the pole in <ref> and applies to the asymptotic behavior for large χ. The cosmon potential increases for large χ quadratically, while a term U ∼χ^4 would lead to a crossing of the pole and is therefore not compatible with the flow of U. Strong gravity-induced renormalization effects prevent an increase of U (for increasing χ) faster than χ^2.The IR potential eqn:large chi u is universal. Due to the strong attraction to the fixed point it is independent of the initial conditions of the flow - no fine tuning of parameters is required. We will discuss below that the potential eqn:large chi u is also independent of the choice of the cutoff function R_k(q), up to a multiplicative constant k̅/k in the definition of k. For fixed k the IR potential governs the asymptotic behavior for large χ. For a more general dependence of M on χ the universal asymptotic form of the potential for χ→∞ becomesU = k̅^2/2 M^2(χ).This is the central ingredient for the proposed solution of the cosmological constant problem.For a scaling solution the dimensionless functions characterizing L̃ in Γ = k^4 ∫_x L̃ depend only on dimensionless combinations such as y = χ^2/k^2. In order to use the canonical dimension of mass for χ we have to select a particular k that sets the units. We denote this as k̅ = √(2)μ, such that for M = χ one has U = μ^2 χ^2. The value of μ is arbitrary and has no physical content - it only sets the units for χ. The ratio χ/μ can be used to denote the position on the flow trajectory corresponding to √(2 y) for the scaling solution. Cosmological observables will only depend on y. For a fixed μ the IR limit corresponds to χ→∞.Variable gravity with quadratic cosmon potential has been studied extensively for cosmology <cit.>. Depending on the precise form of the kinetic term, it can describe inflation for an early period where χ remains not too large, and dynamical dark energy for the region of large χ. Indeed, the cosmological field equations are solved by χ increasing from small values towards infinity in the infinite future. At late time cosmology therefore explores the IR limit. If χ increases to infinity for t →∞, the infinite future corresponds to the IR fixed point.This type of dynamics solves the cosmological constant problem and provides for a simple mechanism for dynamical dark energy. Indeed, the dimensionless effective cosmological constant vanishes asymptotically for large time <cit.>. It is given by the dimensionless ratio between the potential and the fourth power of the dynamical Planck mass,U/M^4(χ) = μ^2/M^2(χ) = μ^2/χ^2→ 0.(The last identity refers to M(χ) = χ.) The effective cosmological constant vanishes asymptotically for all cosmological solutions for which the dynamical Planck mass M(χ) diverges in the infinite future. In the Einstein frame, this corresponds (approximately) to an exponential decrease of the potential as a function of a scalar field φ with appropriately normalized kinetic term <cit.>.Since in the present epoch χ is still finite, the effective cosmological constant does not yet vanish. It decreases, however, with time, yielding the first prediction of dynamical dark energy or quintessence <cit.>. What is crucial for this dynamical solution of thecosmological constant problem is an increase of U for large χ slower than ∼ M^4(χ). In our present setting, this is enforced by the structure of the graviton contributions to the flow of U. The central result of this note simply states that for M ∼χ it is impossible that U increases ∼χ^4 for χ→∞. Such an increase would inevitably lead to a strong instability of the graviton fluctuation effects, which is avoided by the flow of the renormalized couplings.The contribution of scalar fluctuations omitted in <ref> is somewhat more complicated than in <ref>. This is due to mixing with the scalar degrees of freedom in the metric, as well as to a possible dependence of the kinetic coefficient K on χ. Near the pole, the graviton contributions will dominate, however, such that the scalar fluctuations will not modify the asymptotic behavior eqn:large chi u for large χ. This extends to other neglected fluctuation contributions as well.We notice that the IR fixed point leading to <ref> is not the only possible fixed point. An IR unstable fixed point can be obtained for U(χ→∞) ∼ k^4. This type of fixed point generalizes the solution of <ref> for the smaller v_∗, where the pole in the graviton contribution plays no role. Candidates for this type of scaling solution have been found in the discussion of dilaton quantum gravity in <cit.>. Our simplified discussion suggests that perturbations of this type of scaling solutions are unstable and trigger a flow to the IR fixed point with the behavior eqn:large chi u. In any case, a behavior U ∼μ^4 would also solve the cosmological constant problem dynamically.No matter what are the precise details in the variable gravity setting, the effective dynamical cosmological constant cannot be negative if |U(χ)| increases with some power of χ. A negative value of the relevant asymptotic potential U(χ→∞) would lead to an instability in the scalar sector that is not compatible with a consistent quantum field theory. While bosonic fluctuations drive U(χ) to smaller values as k is lowered, fermionic fluctuations tend to increase U. Near the pole in the graviton contribution, the bosonic graviton fluctuations always win and enforce the asymptotic behavior eqn:large chi u. If bosonic fluctuations also win in the region of small χ they may lead to negative values of U in this region. Nevertheless, the bosonic fluctuations cannot change the sign of the potential for large χ if the latter diverges for χ→∞. Only this asymptotic behavior counts for the observable effective cosmological “constant”. In summary, the strong IR fluctuation effects induced by the graviton predict a positive dynamical dark energy that vanishes in the asymptotic future. § GRAVITON CONTRIBUTIONS TO THE HIGGS POTENTIAL The cosmon field is not the only scalar field, and one may wonder what happens to the renormalization flow of the effective potential in the presence of several scalar fields. We investigate this issue in the context of variable gravity. We concentrate on the Higgs doublet h, with straightforward generalization to other scalar fields, including composite scalars such as the chiral condensate in QCD. (Similar results are found in gravity coupled only to the Higgs boson. One replaces below χ by a fixed M.)Let us assume that for k much smaller than χ, say k = 100, we can approximate for a suitable field range of h the potential byU = U_h(h,χ) + Δ U(χ),U_h = λ_h/2 (h^† h - ϵ_h χ^2)^2.This range includes the partial minimum of U with respect to h which occurs for h = (h_0,0), where h_0 = √(ϵ_χ)χ defines the Fermi scale. The Fermi scale is proportional to χ such that for constant Yukawa couplings and constant ϵ_h the ratio between the electron mass and Planck mass remains constant even for a cosmology with varying χ. The quartic coupling of the Higgs scalar λ_h(k,χ) is of the order 1 for k ∼100, χ∼e18, while λ_h ϵ_h is in this region of k and χ a tiny dimensionless coupling associated to the gauge hierarchy. The coefficient in front of the curvature scalar is given by the dynamical Planck mass that we take here asM^2 = χ^2 + ξ_h h^† h. We want to study what happens if k decreases further below 100. For h^† h sufficiently close to h_0^2, the graviton fluctuations give only a negligible contribution to the flow of U_h. Fluctuations of particles with mass smaller than k, such as electrons or quarks, lead to a flow of λ_h and ϵ_h according to the usual loop computation in particle physics. (The flow of λ_h ϵ_h is governed by an anomalous dimension <cit.>.) This flow stops effectively for k below the electron mass such that h_0(k)/χ reaches its final value. The graviton fluctuations contribute to the flow of Δ U(χ), however. For h = h_0 the flow of Δ U is the same as the one for U in <ref> if we take for simplicity ξ_h = 0. In summary, the flow of U_h is given by particle physics, while the flow of Δ U is strongly affected by the graviton fluctuations for the range of χ close to the pole in <ref>.This simple picture holds, however, only for a range of h^† h sufficiently close to h_0^2. Since U_h = λ_h Δ^2/2 increases quadratically with Δ = h^† h - h_0^2, graviton fluctuations become important at some critical Δ_c(k). The bound from the pole-like behavior reads nowU_h + Δ U ≤k^2/2 (χ^2 + ξ_h h^† h).It will be saturated for large enough Δ such that the asymptotic behavior of U_h is at most linear in Δ.We may estimate the critical value of Δ _c for thetransition from the approximation eqn:potential approximation to the behavior linear in Δ byλ_h Δ_c^2 = k^2 (χ^2 + ξ_h Δ_c), Δ_c ≈ k χ.This value depends on χ and varies with k. For k ≈100, χ≈e18 one has Δ_c ≈ (e10)^2, such that the particle physics potential eqn:potential approximation can be trusted for |h| ≲e10. For particle physics experiments, the graviton fluctuations play no role. On present cosmological length scales, k = e-33, the range of validity of the approximation eqn:potential approximation shrinks to Δ < (e-3)^2 or |h - h_0| < e-17. This range seems tiny at first sight. We should, however, compare the energy density for an excitation |h - h_0| = e-17, i.e. Δρ∼ h_0^3 |h - h_0| ∼ (e4)^4, with the present cosmological energy density ρ_c ∼ (2e-3)^4. It is 28 orders of magnitude larger, which makes it clear that all relevant excitations of h - h_0 on cosmological scales are far below the critical amplitude of e-17.It is often argued that the change in the effective potential due to spontaneous electroweak symmetry breaking is a puzzle for a vanishing cosmological constant, since it contributes to Δ U an amount ∼ h_0^4 ∼ (100)^4. It is interesting to see how our scenario of strong graviton-induced renormalization effects deals with this issue. At the scale k relevant for electroweak symmetry breaking, k ≈100, the potential Δ U(χ) still can reach values up to k^2 χ^2 ≈ (e10)^4 for χ∼e18. Thus h_0^4 induces only a tiny change. As k is lowered, Δ U follows its flow equation and reaches a value h_0^4 for k ≈e-5. The graviton fluctuations with momenta smaller than e-5 finally renormalize the complete Δ U to even smaller values, absorbing in this way the jump due to electroweak symmetry breaking.Beyond the IR regime we may approximateM^2 = χ^2 + f k^2.(As compared to <ref> we have set ξ_h = 0.) In this approximation we can summarize the qualitative features of graviton contributions to the flow of the effective potential by the simple formula∂_t U_g = 5 k^6/32 π^2(k^2 - 2 U/χ^2 + f k^2)^-1.<Ref> covers the whole range from the UV (k →∞) to the IR (k → 0). With ρ_h = h^† h and primes denoting now derivatives with respect to ρ_h, one obtains the graviton contribution to the flow of derivatives of U by taking corresponding derivatives of <ref>, e.g.∂_t U_g^' = A^(g) U^',A^(g) = 5 k^6/16 π^2 (χ^2 + f k^2)(k^2 - 2 U/χ^2 + f k^2)^-2.Similarly, one finds at U^' = 0∂_t U_g^''|_U^' = 0 = A^(g) U^''. We identify the second derivative with respect to ρ_h at the partial minimum of U with respect to ρ_h with a k- and χ-dependent quartic self-interaction of the Higgs boson, λ_h(k,χ). According to <ref> the graviton contribution to the flow of λ_h is given by the positive anomalous dimension A^(g). Depending on the value of k/χ, we observe three regimes. For the UV regime f k^2 ≫χ^2 one typically has U = u k^4 such thatA^(g) = 5/12 π^2 f (1 - v_∗)^2,v_∗ = 2 u/f.(We have included here the factor 4/3 from the full k-dependence of R_k.) With 8 π f = 1/g_∗ typically of the order one, the anomalous dimension is sizeable and not suppressed by any small parameter. Values u/f (often called λ) vary in the literature, u/f = 0.36 <cit.>, 0.26 <cit.>, 0.22 <cit.>, 0.22 <cit.>, 0.0 <cit.>, 0.1 <cit.>, 0.2 <cit.>, 0.25 <cit.>. These values, as well as the values for g_∗ = (8 π f)^-1 quoted above, are for pure gravity. They will change in the presence of matter particles. If we take for a rough estimate u/f = 0.25, g_∗ = 1, <ref> yields A^(g)≈ 4. Values of A^(g) exceeding 2 seem well within the possibilities.For the second regime one has f k^2 ≪χ^2, while the flow is not yet dominated by the pole-like behavior such that k^2 - 2 U/χ^2 is of the order k^2. As a consequence, A^(g) is suppressed by the small ratio k^2/χ^2. In this regime, the graviton contributions to the flow of λ_h or λ_h ϵ_h are tiny. The flow of these quantities is dominated by the particle physics contributions, following the perturbative running in the standard model or extensions thereof. For k^2 ≪ m_e^2, the flow eventually stops. Finally, the IR regime with strong gravitational infrared effects sets in when k^2 reaches the vicinity of 2 U/χ^2. The pole-like enhancement overwhelms the suppression k^2/χ^2 and the anomalous dimension eqn:graviton flow can get very large.The large and positive anomalous dimension A^(g) in the UV regime has important consequences for particle physics. First, A^(g) drives λ_h fast towards values close to zero. There are typically additional small particle physics contributions to the flow of λ_h that do not necessarily vanish for λ_h = 0. Nevertheless, the large gravitational anomalous dimension dominates, resulting in a very small value of λ_h at the scale k = χ/√(f) when the gravitational contributions die out. Subsequently, the contribution of the Yukawa coupling in the standard model lets λ_h grow. This simple structure has led to the successfulprediction <cit.> of the value of the Higgs boson mass of 126 with a fewof uncertainty. The main requirement for this prediction, namely the gravity-induced positive and sizable anomalous dimension for the quartic scalar coupling λ_h, is realized within our simple computation.For a second-order vacuum electroweak phase transition, the critical hypersurface cannot be crossed by the flow. The flow for a smalldeviation from the critical hypersurface is governed by an anomalous dimension <cit.>. With β_U^' = ∂_t U^' the anomalous dimension reads A = ∂β_U^'/∂ U^'|_c.s., where the subscript reminds that A has to be evaluated on the critical surface. If A exceeds 2, the small ratio between Fermi scale and Planck mass √(ϵ_h) can be naturally explained by a “resurgence mechanism” <cit.>. The dimensionless distance from the critical hypersurface γ first shrinks to a very small value due to the flow in the UV regime. It then increases again, essentially due to the canonical dimension, once k^2 < χ^2/f. Our computation yields a graviton contribution to A which is given by A^(g), cf. <ref>. Our simple estimate of its value eqn:anomalous dimension in the UV regime suggests that it may indeed exceed 2. Establishing A > 0 requires, of course, a more complete quantum gravity computation. For a Higgs-Yukawa model coupled to gravity A > 2 has been found in ref. <cit.>. § GENERAL INSTABILITY-INDUCED FLOW We have found strong quantum gravity effects in the infrared flow of the cosmological constant and the effective potential of the cosmon and the Higgs doublet. This perhaps surprising observation is related to an instability of the graviton propagator in flat space which occurs for (Euclidean) momenta q^2 < 2 V/M^2. Functional renormalization avoids this instability by a flow of V towards zero. For momenta q^2 ≈ k^2, the cosmological constant V(k) or the scalar potential U(χ,k) is small enough for 2 V/M^2 not to exceed k^2. (Here and in the following one can replace V by U and similarly v by 2 U/M^2 k^2.) This “avoidance of the instability” arises on the level of functional flow equations as a singular structure which prevents a flow into the unstable region. The effect is highly non-perturbative and will not be seen at any finite order of a perturbative expansion.The consequences of a simple estimate of this strong infrared effect in quantum gravity are rather impressive: * The cosmological constant is renormalized to zero.* Within variable gravity, dynamical dark energy is predicted.Extending the simple calculation of the graviton contribution to the UV regime and the functional flow of the Higgs potential leads to further results: * A positive, sizable anomalous dimension for the quartic coupling of the Higgs doublet leads to a successful prediction of the mass of the Higgs boson.* For a small enough flowing Planck mass in the UV scaling regime, the gauge hierarchy of the electroweak symmetry breaking can be explained.Needless to say that such dramatic consequences call for a critical assessment of the reliability of the simple estimate. We concentrate here on the IR regime and briefly address three questions: * Is the strong IR gravity effect in Euclidean flat space robust with respect to a change of cutoff and a more complete treatment of the metric fluctuations?* Does the result extend to Minkowski space?* Is the strong IR effect also relevant if a realistic cosmological solution replaces flat space? On the technical level, the singular structure in the flow equation eqn:v flow is the central ingredient. It turns β_v necessarily positive if v approaches one sufficiently closely. Since v cannot grow beyond the value where β_v turns positive, the singular behavior is avoided. For negative m^2 the divergence of I_k(m^2) in <ref> for a certain ratio k^2/|m^2| is indeed rather genuine for “ admissible”cutoff functions R_k(q). The cutoff should decrease with decreasing k such that ∂_t R_k > 0. Therefore I_k(m^2) is positive as long as P(q) = q^2 + R_k(q) is positive. If furthermore I_k(m^2) diverges as a critical value v_c for v = 2 V/M^2 k^2 is approached, one necessarily obtains β_v > 0 in the vicinity of the singularity for v_c, and v < v_c. Thus v either decreases (for β_v > 0) or it cannot increase beyond the point where β_v vanishes. In consequence, the singularity at v_c can never be crossed by the flow.For the strong IR gravity effect it is sufficient that I_k(v) has a singularity for finite v > 0. Let us denote x = q^2/k^2 and p(x) = (q^2 + R_k(q^2))/k^2, such thatI_k(v) = k^4/32 π^2∫_0^∞ x x (p(x) - v)^-1 (∂_t R_k/k^2).We consider cutoff functions for which p(x) has a minimum for some x̅≠ 0, p(x̅) = p̅, and is analytic in this region,p(x̅) = p̅ + a(x - x̅)^2 + …The case a = 0, p̅ = 1 corresponds to the Litim cutoff discussed above, and we extend the discussion now to a > 0. The singularity occurs for v_c = p̅. For ϵ = p̅ - v → 0, (∂_t R_k/k^2)(x̅) = 2 s̅, one can approximate the dominant integration region for x close to x̅ byI_k(v) = k^4 x̅s̅/16 π^2∫ x/ϵ + a (x - x̅)^2 = k^4 x̅s̅/16 π√(a ϵ).This integral indeed diverges with ϵ^-1/2 for ϵ→ 0, thus establishing that for this type of cutoff functions the singularity cannot be crossed <cit.>. As is well known from the investigation of the approach to convexity for scalar effective potentials, cutoff functions with minimum p(x) at x = 0 are less suited. The lack of a singularity in I_k(v) would entail a rather complex IR behavior that cannot be described anymore by a simple truncation.From <ref> the flow near the singularity is approximated by∂_t v = -2 v + 10/M^2 k^2I_k(v) ≈e̅/M^2 k^2 (p̅ - v)^-1/2 - 2 p̅,withe̅ = 5 x̅s̅/8 π√(a).The approximate solution readsv = p̅ - (e̅ k^2/2 p̅ M^2)^2,V = p̅ M^2 k^2/2 - e̅^2 k^6/8 p̅^2 M^2.The asymptotic value V_c = M^2 k̅^2/2, k̅^2 = p̅ k^2, is approached even closer than in <ref>. For admissible cutoffs the behavior eqn:approximate solution seems at first sight more generic than the limiting case of the Litim cutoff. We observe, however, that for small ϵ only a tiny range |x - x̅| ∼√(ϵ/a) contributes substantially to the integral eqn:dominant region, while for the Litim cutoff the weight is more equally distributed in the range x < 1. The precise approach to the universal IR value V = M^2 k̅^2/2 depends strongly on the cutoff. In contrast, the universal form V = M^2 k̅^2/2 is independent of the choice of the cutoff up to a multiplicative constant k̅/k related to the precise definition of k. Only the universal IR value of V is important for our purposes.What about additional contributions to ∂_t V from photons or other massless particles? Such fluctuations add to the flow of V a termΔ∂_t V = d k^4.This contribution is usually associated to a pledged “unnaturalness of a small cosmological constant due to quantum or vacuum fluctuations”. In the presence of the strong IR graviton fluctuations an additional term eqn:massless fluctuations in the flow of V has almost no influence in this region of the flow. It modifies V in <ref> only by a term ∼ k^8/M^4. For the Litim cutoff it adds to β_v in <ref> a term 2 d k^2/M^2 which is strongly suppressed for small k^2/M^2. As a result, the last term in <ref> is divided by a factor (1 - d k^2/M^2). The correction ∼ d k^6/M^2 is of the same order as other subleading terms in the expansion of V. We conclude that the effect of “vacuum fluctuations” of virtual particles is well present and of the order expected from simple estimates. It is, however, overwhelmed by the graviton fluctuations and cannot impede the approach of V towards zero as k is lowered. Fluctuations of particles with mass m_p are further suppressed by a factor k^2/m_p^2 in the range k ≪ m_p.So far, we have only included the contribution from the graviton fluctuations. We have paid little attention to issues such as gauge invariance, gauge fixing, other components of metric fluctuations, ghosts, or fluctuations of the cosmon χ or Higgs doublet h. The strong IR quantum gravity effects are related to a singular behavior of flow equations as an unstable region is approached. For the “avoidance of instabilities” we can neglect fluctuations which yield regular contributions in the field- and k-region relevant for the “singular flow”. This typically happens for matter fluctuations. (Additional singularities in the scalar sector may arise for non-convex parts of the potential. They are well understood <cit.> and will not be considered here.) The trace part of the physical metric fluctuations has a propagator G_k ∼(P(q) - V/(2 M^2))^-1. For P(q) ≥ 2 V/M^2, this potential singularity is not reached by the flow.The propagator of the gauge fluctuations in the metric is determined mainly by the gauge fixing term. If the gauge fixing parameter α goes to zero, the inverse propagator ∼ (q^2/α + finite pieces) does not lead to a singular behavior in the relevant range. Also mixing between physical fluctuations and gauge fluctuations becomes negligible, cf. ref. <cit.> for details. Finally, ghost contributions are not singular either. We conclude that the strong IR flow near the transition to instability is completely governed by the fluctuations in t_μ_ν. Of course, all other fluctuations contribute away from the singularity. They play a role for a precision estimate of the anomalous dimensions for the Higgs potential in the UV scaling regime. § POTENTIAL FLOW IN MINKOWSKI SPACE We next turn to the flow equation in Minkowski space with metric g̅_μ_ν = η_μ_ν. The factor √(g) in <ref> equals i, thus modifying <ref>,Γ^(2) = i (M^2q^2/4 - V/2) P^(t),with q^2 = q^μ q_μ = q⃗^2 - ω^2 e^2 i ϵ, q_μ = (-ω,q⃗), and ϵ→ 0_+ indicating the path taken in ω-integrations. Correspondingly, we also multiply the cutoff function R_k by a factor i, and G picks up a factor -i. The exact flow equation eqn:wetterich holds for arbitrary signature such that∂_t (iV) = 1/2∫_q ∂_t R_k G,with ∫_q = (2 π)^-4∫ω∫^3 q now performed with Minkowski signature andthe trace over internal indices. The contribution of the graviton fluctuations reads∂_t V= -5 i Ĩ_k, Ĩ_k= 1/2∫_q J(q) = 1/2∫_q⃗∫_ω J(ω,q⃗),J(q)= (q^2 + R_k(q) - 2 V/M^2)^-1∂_t R_k(q). The ω-integral can be considered as an integral along the real axis in the complex plane, ω = ω_R + i ω_I. With Euclidean momentum q_0 = -ω_I and assuming that J vanishes sufficiently fast for |ω| →∞, one hasĨ_k = i I_k - Δ_1 + Δ_3,with Euclidean integral I_k given by <ref>, and Δ_1, Δ_3 the clockwise contour integrals around the regions I and III. We define the regions as I: ω_R > 0, ω_I > 0, II: ω_R > 0, ω_I < 0, III: ω_R < 0, ω_I < 0, and IV: ω_R < 0, ω_I > 0. If J is analytic in the regions I and III, the flow equation for V is the same for Minkowski space and Euclidean flat space. If not, there are additional contributions ∼ 2 i (Δ_1 - Δ_3) that may contain an imaginary part.Let us consider a situation where J is analytic in regions I and III except for possible poles. With R_k depending only on q^2 the integrand J(ω,q⃗) only depends on ω^2. Possible poles come in pairs ±ω̅_j(q⃗). Near a pair of poles j one hasJ = r_j(ω^2)/ω^2 - ω̅_j^2 = r_j(ω̅_j^2)/2 ω̅_j(1/ω - ω̅_j - 1/ω + ω̅_j).The residua in regions I and III have therefore opposite signs, Δ_3 = -Δ_1. As a result one findsĨ_k = i(I_k + ∑_j K_j),K_j = ∫_q⃗r_j(ω̅_j^2)/2 ω̅_j,with j the sum over poles of J in region I, located at ω̅_j(q⃗). For v < 1 and suitable cutoff functions the integrals K_j do not show a singular behavior if r_j(ω̅_j^2)/ω̅_j remains finite in the whole integration region. The integrands should fall off fast enough for large |q⃗| due to the factor ∂_t R_k. The effect on the flow of any “finite contribution” in Ĩ_k is suppressed for small k by a factor k^2/M^2. In consequence, the singular structure in Ĩ_k for k̅^2 - 2 V/M^2 → 0 arises from the Euclidean integral I_k. The “avoidance of the singularity” is the same for Minkowski and Euclidean signature.We may map the regions in the complex variable x = q^2/k^2 = x_R + i x_I onto the complex ω-plane, recalling that a given x corresponds to two values of ω with opposite sign. With our i ϵ-definition of q^2 one finds that x_I < 0 maps into the ω-regions I, III, while x_I > 0 corresponds to regions II, IV. For x_I = 0 the real values of q^2 are mapped to regions II, IV if q⃗^2/k^2 > x_R, while they belong to regions I, III for q⃗^2/k^2 < x_R. If R_k(x) is a real function of x this also holds for J(x). Any pole of J(x) at z implies therefore the existence of another pole at z^∗. Poles in the regions I, III can therefore only by avoided for all q⃗^2 if they occur all for real negative x. An “ideal cutoff function” R_k would be such that J(x) remains analytic in the regions I, III.Infrared cutoff functions for Minkowski signature have been discussed by Floerchinger <cit.>, see also <cit.>. We may consider an algebraic cutoff of the formR_k(q) = b k^2 (k^2/k^2 + c q^2)^n,with positive real constants b and c. For |ω| →∞ one has J ∼ |ω|^-2(n+1) such that contours can indeed be closed at |ω| →∞. The poles of ∂_t R_k are in regions II and IV. If z = z_R + i z_I denotes one of the zeros of the polynomial S(x), x = q^2/k^2,S(x) = (x - v) (1 + c x)^n + b,e.g. S(z) = 0, poles are present in the regions I, III unless all zeros occur for real negative z. Typically, S(x) has zeros away from the real axis such that the choice eqn:cutoff does not correspond to an “ideal cutoff”. Since the contributions from Δ_1 - Δ_3 are subleading we concentrate on the Euclidean momentum integral for this cutoff. For n b c > 1, the minimum of p(x) = x + R_k/k^2 occurs at x̅ > 0. We may choose b such that p̅ = p(x̅) = 1, k̅ = k. The integral I_k has a singularity ∼ 1/√(ϵ) which will prevent the flow from entering the singular region v > 1, leading to the same strong graviton-induced renormalization effects for V as for Euclidean flat space.Our choice of b corresponds tox̅ = 1 - 1n+1(1 + 1c),b = 1n c (1 + c x̅)^n+1,where c > 1n, b > 1. An example is n = 2, c = 1, x̅ = 1/3, p(0) = b = 32/27, p(1) = 35/27. In the region 0 < x < 1 one finds only a small enhancement of p(x) as compared to the Litim cutoff p(x) = 1. Withp(x)= x + b (1 + c x)^-n,s(x)= ∂_k R_k/2 k^2 = b (1 + c x)^-n(n + 1 - n1 + c x),the Euclidean integrandJ(x) = 2 s(x)/p(x) - vhardly differs from the Litim cutoffJ_L(x) = 2/1 - vθ(1 - x).Our Euclidean estimate of the flow of V with a Litim cutoff therefore yields a valid approximation for the flow in Minkowski space with cutoff eqn:algebraic cutoff, except very near the singularity. The singular behavior is of the type eqn:analytic p,eqn:dominant region. § FLOW ON COSMOLOGICAL BACKGROUNDS The flow in a flat background metric clearly shows the strong IR gravity effect. One would like to extend the discussion to fluctuations in the vicinity of a realistic cosmological solution, with fixed homogeneous and isotropic background metric g̅_μ_ν = a^2(η) η_μ_ν and conformal time η. In variable gravity, this is accompanied by a fixed background solution for the scalar field χ̅(η). For the graviton fluctuations, the background metric enters the flow equation by replacing in Γ^(2) and R_k the dependence on q^2 by the covariant Laplacian, q^2 → -D^μ D_μ, as well as by additional geometric terms in Γ^(2) arising from a non-vanishing curvature scalar, for details see ref. <cit.>. In variable gravity, the background scalar field χ̅(η) appears in the factor M^2(χ̅) in the cutoff term. This differs from M^2(χ) in Γ^(2), which is evaluated for arbitrary χ. As for our simplified discussion in flat space, we omit the difference between χ̅ and χ and evaluate the effective action for χ̅ = χ.One is interested in the form of the effective average action Γ_k in the vicinity of a realistic cosmological solution a(η) and χ̅(η). For cosmology, this will be sufficient to derive quantum field equations and the power spectrum of primordial fluctuations <cit.>. Similarly to simple scalar models, where the best results of simple truncations are obtained by expanding around the ground state solution (e.g. the minimum of the effective potential), we expect for quantum gravity the most reliable results for expansions around the cosmological background. While Γ_k is a functional of arbitrary metrics g_μ_ν(x) and scalar fields χ(x), the background metric g̅_μ_ν and scalar field χ̅ may be considered fixed. (For alternatives cf. ref. <cit.>.) An optimal procedure adjusts g̅_μ_ν and χ̅ a posteriori to a solution of the field equations derived from the effective action for k = 0 (or k = k_0).For the discussion on a curved background we concentrate on the graviton fluctuations which correspond to the traceless transversal tensor with respect to the rotation group SO(3). In a homogeneous isotropic background the graviton can be identified <cit.> with the components of t_μ_ν in the three “space directions” m,n ∈{1,2,3} with constraintst_m_n = a^2 γ_m_n + …,q^n γ_m_n = 0, γ^m_m = 0.The projector P^(γ) onto the graviton mode reads*P^(γ)_m_n_p_q = 1/2(Q_m_pQ_n_q + Q_m_qQ_n_p - Q_m_nQ_p_q),Q_m^n = δ_m^n - q_m q^n/q^2,P^(γ) = *P^(γ)_m_n^m^n = 2,with P^(γ) = 0 if one index equals zero. In flat space a restriction to the graviton mode replaces effectively P^(t) by P^(γ). Thus the graviton part in the flow equation eqn:cc flow replaces the factor 5 by 2. Restricting the flat space discussion to the contribution of fluctuations of the “propagating graviton” γ_m_n multiplies the graviton contribution by a factor 2/5. This quantitative modification does not change any of our qualitative findings.For the effective action eqn:vg effective action the inverse graviton propagator becomes <cit.>Γ̃_γ^(2) = i/4[A^2 (D̂ + R_k(D̂)) + Δ_γ] P^(γ),withD̂ = ∂_η^2 + 2 ℋ̂∂_η + q⃗^2, q⃗^2 = δ^m^n q_m q_n,Δ_γ = -2 A^4 V̂ + 2 A^2 (ℋ̂^2 + 2 ∂_ηℋ̂) + A^2 K̂ (∂_ηχ̅)^2.Here we employ frame-invariant quantities, for M = M(χ), U = U(χ),A = M a, ℋ̂ = ∂_ηln A,V̂ = U/M^4, K̂ = K/M^2 + 3/2 M^4(∂ M^2/∂χ)^2.We work in three-dimensional Fourier space with spacelike comoving momenta q_m, and in position space for conformal time η, η^'. The graviton Green's function obeysΓ̃_γ^(2) G_γ(η,η^') = P^(γ)δ(η - η^').In <ref> we have omitted δ-functions for the comoving three-momenta. The trace in the flow equation becomes→∫ηη^' δ(η - η^') ∫_q⃗. The graviton contribution to the flow of the effective potential reads∂_t (a^4 U) = ∂_t (A^4 V̂)= -i ∫_q⃗∂_t R_k(q⃗,∂_η) G(q⃗,η,η^')|_η^'=η.In <ref> the local form in η reflects the general property in position space for Γ = ∫_x L(x), where∂_t Γ = ∫_x ∂_t L(x) = 1/2∫_x ∫_y ∂_t R_k(x,y) G(y,x)is solved by the local evolution equation∂_t L(x) = 1/2∫_y ∂_t R_k(x,y) G(y,x). We emphasize that both sides in <ref> involve only frame-invariant quantities <cit.>. In particular, the cutoff function involves the combinationA^2 k̂^2 = a^4 M^2 k^2, k̂ = a k,i.e. R_k(q) depends on k̂^2 and D̂. Concerning non-linear field transformations the computation of the flow equation is done in a specific frame or choice of fields, namely the one for which the infrared cutoff term is quadratic in the fluctuations. This is the analogue to the selection of a frame in a loop computation by the implicit assumption that no non-trivial Jacobian is present in the functional measure. For our purpose it is important that the cutoff is quadratic in the physical graviton fluctuations, e.g. those that couple to sources reflecting conserved energy-momentum tensors (“linear split”). Once the flow equation has been derived, arbitrary non-linear field transformations can be performed. With respect to conformal field transformations of the metric (Weyl scalings) this is reflected in the frame invariance of the flow equation, e.g. <ref>. The solution of the flow equation can be done in an arbitrary frame.For a flow in Minkowski space with g̅_μ_ν = a η_μ_ν and constant a, one replaces in the previous computation U → a^4 U, k^2 → a^2 k^2 such that J(q) in <ref> involves now k̂ instead of k. The flow equation forv = 2 U/M^2 k^2 = 2 A^2 V̂/k̂^2is independent of a. More generally, it is invariant under arbitrary k-independent Weyl scalings of the metric since the frame-invariant combinations remain unchanged. In other words, the flow equation eqn:flow due to gravitons is the same for all metrics g̅_μ_ν that can be transformed into each other by field-dependent but k-independent Weyl scalings or conformal transformations of g̅_μ_ν.For the flow in a curved background geometry one can approximate D̂ = ∂_η^2 + q⃗^2 and Δ_γ = -2 A^4 V̂ as long as k̂^2 ≫ℋ̂^2, ∂_ηℋ̂, ℋ̂∂_η, K̂(∂_ηχ̅)^2. This approximation corresponds to the flow in flat Minkowski space. Indeed, the propagator equation eqn:inverse for G,i A^2/4[D̂ + R_k(D̂) - 2 A^2 V̂] G_grav(η,η^') = δ(η - η^'),can be solved for η-independent A in Fourier space,G_grav(η,η^') = ∫_ω e^-i ω (η - η^') G_grav(ω),with (q^2 = q⃗^2 - ω^2 = a^2 q^μ q_μ)i A^2/4(q^2 + R_k(q^2) - 2 A^2 V̂) G_grav(ω) = 1.InsertingG_grav(η,η^') = -4 i/A^2∫_ω e^-i ω (η - η^')(q^2 + R_k(q^2) - 2 A^2 V̂)^-1into <ref>, one recovers the previously discussed flow in Minkowski space. As expected, the integration of modes with large momenta is not influenced by the background geometry if all characteristic length scales of the geometry are much larger than the inverse momentum of the fluctuations.As k̂^2 is lowered and reaches ℋ̂^2, the details of the background geometry influence the flow. A given background geometry could eventually stop the flow once k̂^2 ≲ℋ̂^2. This can happen when Δ_γ in <ref> vanishes or becomes positive, such that the potential instability is no longer present. Without an instability the graviton contributions are suppressed by k^2/M^2 and therefore negligible.As an illustration, we consider a model with gravity and a scalar field χ. We choose a background metric *g̅_μ_ν^c and background field χ̅_c(η) that solve the field equations which are derived by variation of the effective action at a given k̅. For definiteness we assume for Γ_k̅ the form eqn:vg effective action. The two independent field equations read <cit.>2 ℋ̂^2 + ∂_ηℋ̂ = A^2 V̂,andℋ̂^2 - ∂_ηℋ̂ = K̂/2 (∂_ηχ̅)^2.For the solution of the field equations eqn:ife1,eqn:ife2 the term Δ_γ in <ref> vanishes, corresponding to a massless on-shell graviton propagator. In such a geometry, the potential instability for V > 0 is cancelled by the geometric terms. The strong IR flow induced by the graviton is no longer present.The absence of a mass-like term in the “on-shell” propagator of the graviton is well known in general relativity and cosmology and has been discussed for the renormalization flow in refs. <cit.>. It is at the origin of scepticism about the relevance of the “avoidance of instabilities” for observable cosmology. We emphasize that the effective “on-shell stop” of the graviton-induced flow occurs only for a particular “on-shell” configuration of the scalar field, namely for χ(η) = χ̅_c(η) obeying the field equations. The flow of the effective potential and, more generally the effective action, is an off-shell issue. One evaluates Γ_k for arbitrary field configurations, and only a very small subset can obey the field equations derived from Γ_k. This has important consequences, as can be seen by simple examples: * We have already found that for k̂^2 ≫ℋ̂^2 the flow is well approximated by the flow in flat space and therefore exhibits the strong graviton-induced renormalization effects.* Consider a potential U(χ) increasing sufficiently fast with |χ|. For the range χ^2 ≫χ̅_c^2(η), the potential term will dominate the geometric terms in Δ_γ, inducing again a fast flow of U in this region. In variable gravity the asymptotic behavior U(χ) ∼χ^2 for χ→∞ cannot be changed by geometric effects.* For an η-independent field χ (instead of χ̅_c(η)), as appropriate for the flow of U, the term ∼K̂ in Δ_γ is missing. (This holds even if R_k employs χ̅_c(η).) Thus Δ_γ remains negative, creating again an instability barrier not to be crossed. Furthermore, we recall that interesting cosmological solutions are often not simple solutions of field equations for models of gravity coupled to a scalar field. For the matter-dominated universe in the Einstein frame (constant M), a ∼η^2 ∼ t^2/3, one has ℋ = 1/ 2 η, ℋ^2 + 2 ∂_ηℋ = 0 such that the term Δ_γ = -2 A^4 V̂ only involves the potential. For the radiation-dominated universe, a ∼η, ℋ∼ 1/η, ∂_ηℋ = -ℋ^2 one infers that Δ_γ = -2 A^4 V̂ - 2 A^2 ℋ^2 has a negative geometric contribution, adding to the instability.In summary, the choice of a realistic cosmological background will typically not modify qualitatively the strong IR effects related to the “avoidance of instability” for most parts of the flow. Geometric effects play a role, however, as the flow ends effectively due to the presence of a “physical IR cutoff” arising from the geometry of the cosmological solution. The exact flow equation does not depend on the choice of the cutoff. Details of the flow in flat space will, however, depend on the selection of the cutoff. This dependence should be compensated by the details of the stop of the flow due to a physical cutoff, which also depend on the choice of the cutoff. In many circumstances we expect that the role of the geometric effects for a curved background is related to the effective stop of the flow. § CONCLUSIONS We conclude that the strong infrared renormalization effects induced by the graviton fluctuations are generic for a positive effective potential U. They lead to a solution of the cosmological constant problem. The barrier preventing unstable behavior acts by erasing any microscopic value of the cosmological constant, replacing it by a universal scale-dependent infrared value.Let us concentrate on the action of this “graviton barrier” in variable gravity where M^2(χ) grows for large χ^2 ≫ k^2 proportional to χ^2. The flow equation will always induce a dependence of M^2 on χ. A scale-invariant coupling ∼ξχ^2 R̃ dominates the behavior of M^2 for large χ, as compared to any constant contribution. This is consistent with an IR fixed point reached for k^2/χ^2 → 0, i.e. for χ→∞ at fixed k. By multiplicative scaling of χ we can set ξ = 1, such that M^2(χ) = χ^2 becomes indeed a valid approximation. Since the χ-dependence of M is unavoidable, one should derive the flow equations within the framework of variable gravity. Only once the flow equations are derived within variable gravity, they can subsequently be translated to the Einstein frame. As we have argued, this translated flow will not be identical to the flow that one obtains by a direct computation with fixed M^2 as performed as a “warm-up example” at the beginning of this note. Only qualitative features may be expected to be similar.Assume that the cosmon potential U(χ) increases for increasing χ^2, with U(χ) > 0. The strong graviton-induced flow limits the asymptotic increase of U(χ→∞) to U ∼ k^2 χ^2 - in other words, the graviton barrier blocks any faster increase. As a consequence, the dimensionless ratio V̂(χ) = U(χ)/M^4(χ) = U(χ)/χ^4 decreases for increasing χ^2, V̂∼ k^2/χ^2. For cosmological solutions with χ^2 →∞ for the infinite future, the observable cosmological constant vanishes asymptotically <cit.>. If U(χ→∞) increases more slowly than χ^2, or goes to a constant, the decrease of V̂ with χ is even faster. An increase of |U(χ→∞)| with χ^2 towards negative values is not compatible with a potential bounded from below, as required for a consistent quantum field theory. We conclude that the graviton barrier implies an asymptotic decrease of the dimensionless cosmological “constant” V̂(χ→∞) to zero. An asymptotically vanishing cosmological constant obtains in more general settings as well. It is sufficient that V̂(χ) = U(χ)/M^4(χ) vanishes for χ→∞.Once the flow equation for A^4 V̂ is derived within variable gravity, we can transform this description into the Einstein frame with χ-independent M^2. This is achieved by a χ-dependent Weyl scaling of the metric. As long as the Weyl scaling is k-independent, the exact flow equation is the same for all frames. We have formulated the flow in terms of frame-invariant quantities such as V̂. In the Einstein frame, U = V̂ M^4 decreases ∼χ^-2 and goes precisely to zero (not a non-zero constant!) for χ→∞. For a rescaled scalar field φ with canonical kinetic term, the decrease of U is approximately exponential, U ∼ M^4 exp(-αφ/M). Such potentials give rise to dynamical dark energy or quintessence <cit.>, typically inducing “scaling” or “tracking” solutions for cosmology <cit.>.We infer that dynamical dark energy is a rather natural consequence of the graviton barrier. Since V̂ approaches zero for χ→∞ with V̂ > 0 for finite χ, the only possibility to avoid dynamical dark energy is a minimum of V̂ for a finite value χ_0. This requires a maximum for some finite χ_max > χ_0 as well. While this remains a possibility, the case of monotonic V̂ appears to be simpler. (We observe that in the case of a minimum, the value V = U(χ_0) would also be subject to strong IR renormalization effects if V > 0.) Our computation of the graviton-induced renormalization effects are in support of the ideas underlying the first proposal of quintessence <cit.>, of crossover variable gravity <cit.> and of the prediction of the Higgs mass within asymptotic safety <cit.>.We have only briefly discussed curved background geometries. Under certain circumstances, they may stop the IR flow for low k if metric and scalar fields are close to solutions of the field equations or if geometry provides an effective infrared cutoff in some other way. A quantitative understanding of this effect will require some technical effort in order to derive and solve flow equations in a time-dependent background. In view of our qualitative findings for the overall flow, we conclude that curved background geometries do not change our main conclusion: strong infrared quantum gravity effects solve the cosmological constant problem.Interesting open questions emerge: Which are the effects of the graviton barrier for other parts of the effective action for gravity, in particular in the low momentum domain? Are the infrared gravity effects consistent with the numerous precision tests of gravity which confirm the Einstein-Hilbert action? Could there be observable consequences, or implications for black holes? Functional flow equations evaluated on the corresponding geometries should be able to address these questions. This should also shed light on more phenomenological investigations <cit.> of possible consequences of the renormalization flow of the cosmological constant. If these issues show no conflict with observation, our findings solve the fundamental puzzle why our universe has grown large enough in a not too disruptive way, such that complex structures as galaxies, stars and life could emerge. The border of instability gives room for complexity. Acknowledgement The author would like to thank A. Eichhorn, S. Floerchinger, H. Gies, J. Pawlowski, R. Percacci, M. Reuter and F. Saueressig for comments and discussion. This work is supported by ERC-advanced grant http://cordis.europa.eu/project/rcn/101262_en.html290623 and the DFG Collaborative Research Centre “http://www.dfg.de/en/research_funding/programmes/list/projectdetails/index.jsp?id=273811115 sort=var_asc prg=SFBSFB 1225 (ISOQUANT)”. | http://arxiv.org/abs/1704.08040v2 | {
"authors": [
"C. Wetterich"
],
"categories": [
"gr-qc",
"astro-ph.CO",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170426100957",
"title": "Graviton fluctuations erase the cosmological constant"
} |
Point-shifts of Point Processes on Topological Groups James T. Murphy III James T. Murphy III, The University of Texas at Austin, [email protected] ================================================================================================================ This paper focuses on flow-adapted point-shifts of point processes on topological groups, which map points of a point process to other points of the point process in a translation invariant way. Foliations and connected components generated by point-shifts are studied, and the cardinality classification of connected components, previously known on Euclidean space, is generalized to unimodular groups. An explicit counterexample is also given on a non-unimodular group. Isomodularity of a point-shift is defined and identified as a key component in generalizations of Mecke's invariance theorem in the unimodular and non-unimodular cases. Isomodularity is also the deciding factor of when the reciprocal andreverse of a point-map corresponding to a bijective point-shift are equal in distribution. Next, sufficient conditions for separating points of a point process are given. Finally, connections between point-shifts of point processes and vertex-shifts of unimodular networks are given that allude to a deeper connection between the theories.AMS 2010 Mathematics Subject Classification: 37C85, 60G10, 60G55, 60G57, 05C80, 28C10. Keywords: Point process, Stationarity, Palm probability, Point-shift, Point-map, Mass transport principle, Foliation, Topological Group, Unimodular Network. § ACKNOWLEDGMENTSThis work was supported by a grant of the Simons Foundation (#197982 to The University of Texas at Austin).§ INTRODUCTION §.§ BackgroundStationary point processes are models of discrete subsets of ^d that exhibit “statistical homogeneity”. That is, their distributions are translation invariant. Such processes have a wide array of applications in population models, wireless networks, astrophysical models, or, more abstractly, as the set of vertices of a random graph. The use of stationarity is a way to encode the idea that no point in the point process is “special”. For models of physical systems this often suffices, but ^d is not always the most natural space on which to consider these models. A population model may be considered on S^2 since the Earth is approximately spherical. A computer network may be considered on a hyperbolic group since some studies suggest that distances on the internet are hyperbolic <cit.>. An astrophysical model concerned with structures on very large scales may be considered on ^3, on S^3, or on a 3-dimensional hyperbolic group, depending on the yet unknown spatial curvature of the universe. For these situations on a group , the concept of -stationarity, i.e. distributional invariance with respect to the action ofon a space, for a point process may be used. The general theory of point processes and random measures has been developed on any locally compact second-countable Hausdorff (LCSH) space , and -stationarity has been studied thoroughly whenis a LCSH topological group acting on a homogeneous space S, cf. <cit.>. The theory of -stationarity on homogeneous spaces is very general, but in many cases an -stationary point process on a homogeneous space S may be pushed forward in a natural way to an -stationary point process onitself, and this is the setting that is assumed here.This paper will make use of point processes on a general LCSH group , which is fixed for the remainder of the document. Denote the Borel sets ofby (). For sake of completeness, the following definition is included. A point process onis a random elementin the spaceof all locally finite counting measures on , whereis endowed with the cylindrical σ-algebra generated by the mappings μ∈↦μ(B) for each B∈(). All point processesin this paper are assumed to be simple, meaning every atom ofhas mass 1. In this case,can and will be identified with its support, which is a discrete subset of . Also, when both arguments need to be specified, the notation (ω,B) will be used instead of (ω)(B).The following framework for dealing with -stationary point processes was developed in <cit.>. A stationary framework (Ω, , θ,) onis a probability space(Ω,,) equipped with a measurable and -invariant left -action θ:×Ω→Ω, called a flow, which will be identified with the family of mappings {θ_x}_x ∈ defined by θ_xω := θ(x,ω) for x∈, ω∈Ω. A point processis flow-adapted if (θ_xω ,B) = (ω, x^-1B),∀ x∈,ω∈Ω,B∈().Another way of expressing (<ref>) is∘θ_x = T_x ,∀ x∈,where for μ∈ and x ∈ the translated measure T_xμ is defined by T_xμ(B) := μ(x^-1B) for all B ∈(). Under these assumptions, any flow-adaptedis -stationary in the usual sense thatand T_x have the same distribution for all x∈. For the remainder of the document fix a stationary framework (Ω,,θ,) on . All point processes introduced in this document are assumed to be flow-adapted. Many models are concerned with more than just the statistical properties of locations of points though. Often dynamics on the points are of primary interest. For instance, in a wireless network, one may be interested in a universally agreed upon protocol for determining an optimal route for packet transmission that depends only on local information. At a certain instant of time, such a protocol would determine a point-shift. A point-shift on a point processis a measurablemap :̋Ω×→ on the support of , i.e. for -a.e. ω∈Ω,(̋ω,X) ∈(ω),∀ X ∈(ω).From now on, all point-shifts considered in this document are assumed to be flow-adapted in the sense that(̋θ_yω,yx) =y(̋ω,x), ∀ x,y ∈,ω∈Ω.If unspecified, (̋ω,x):=x for x ∉(ω). Dependence on ω is usually dropped and (̋X) is written instead of (̋ω,X). Say that $̋ has a functional property, e.g. bijectivity, injectivity, surjectivity, if for-a.e.ω∈Ω,(̋ω,·)has the property on the support of(ω). Some references refer to point-shifts as point allocations as well.Point-shifts of point processes on^dand the random graphs they generate(by considering each point a vertex and a directed edge from each point to its image under the point-shift) have been well studied, cf. <cit.>. Point-shifts of a point process on a general LCSH group have not yet been studied, andthus the aforementioned models, e.g. wireless networks on hyperbolic groups, in which dynamics of points are of central importance, were not practical due to lack of theoretical machinery. The main focus of this document is to extend many known results of point-shifts of stationary point processes on^dto point-shifts of-stationary point processes on. In many ways this task is straightforward, but some proofs on^dmade specific use of the structural properties of, in particular that there is a translation-invariant order on. New proofs using mass transport techniques are given instead. The primary tools that will be used are Palm probabilities and the mass transport principle.The definition of Palm probability measures, as given in <cit.> and applied to the current setting for point processes, follows. Fix, for the remainder of the document, a left-invariant Haar measureon. Also, for the remainder of the section, supposeis a flow-adapted point processes with finite and nonzero intensity, that isΛ(·) := [(·)]is locally finite and not the zero measure. The Palm probability measure of, denoted^, is defined by,^(A) := 1/∫_ 1_θ_x^-1∈ A w(x) (dx), ∀ A ∈,where= [(B)]/(B)for anyB∈()with(B) ∈(0,∞), andw:→_+is any non-negative measurable function with∫_w d= 1. Note that∈(0,∞)is uniquely determined and^is independent of the choice ofw. Expectation with respect to^is denoted^. The Palm probability measure^makes rigorous what is meant by the view of the world from a typical point's perspective. Heuristically, it is the reference probability measure conditioned on the event thate ∈, whereeis the neutral element of.It is also possible to convert between-a.s. and^-a.s. events in the following manner. Intuitively, that which happens almost surely from the typical point's perspective happens almost surely from every point's perspective simultaneously, and vice-versa. Let A ∈. Then the following are equivalent:* ^(A) = 1,* ((x∈: θ_x^-1∉ A) =0)=1,* ^((x∈: θ_x^-1∉ A) =0)=1. A proof of <Ref> could not be found in the literature, so one is given in the appendix. <Ref> may be used to translate between definitions under^and definitions under. For example, a point-map onis a measurable map: Ω→such that(ω) ∈(ω)for^-a.e.ω∈Ω. There is a natural correspondence between point-shifts and point-maps. Namely, if$̋ is a point-shift, then (ω):=(̋ω,e) is a point-map, and ifis a point-map, then (̋ω,X):=X(θ_X^-1ω) is a point-shift, and these operations are inverses. <Ref> ensures that changing the definition of $̋ on a-null set will only change the correspondingon a^-null set, and vice-versa. The unfamiliar reader may see <Ref> in the appendix for the details of how toconvert definitions underand^more generally.Since point-shifts can be seen as special cases of mass transport kernels, the mass transport principle for-stationary point processes is also a crucial tool for this study. In what follows and for the rest of the document,Δ:→(0,∞)is the modular function of, i.e.(Bx) = Δ(x)(B)for allB ∈(). Applied to the current setting, the mass transport principle for-stationary point processes takes the following form.<cit.> For all diagonally invariantτ, i.e.measurable τ:Ω××→_+invariant in the sense thatτ(θ_z ω, zx,zy) = τ(ω,x,y) =: τ(x,y),ω∈Ω, x,y,z ∈,it holds that^∫_τ(e,y) (dy)= ^∫_τ(x,e)Δ(x^-1) (dx). Interpretτ(ω,x,y)as the amount of mass sent fromxtoyon the outcomeω. Under^,eis a point of. Thus the left side of (<ref>) is an average of mass sent out ofe ∈to all points. On the other hand, the right side of (<ref>) is a weighted average of mass received bye ∈from all points of. IfΔ(x) = 1for allx ∈, i.e. ifis unimodular, then the mass transport formula is the one expected from the case of translations on^d, which says that the average mass a typical point ofreceives equals the average mass a typical point ofsends. §.§ Results Since the modular functionΔofappears in the mass transport principle, unimodularity ofturns out to be an influential property for whether certain results about point-shifts of point processes on^dgeneralize. Moreover, the role thatΔplays in the story of point-shifts is crucial to understanding why exactly some results generalize and others do not. In <Ref>, a stark contrast between the behavior of point-shifts in unimodular and non-unimodular cases is shown. However,there is more subtlety in the behavior of point-shifts when one does not assume thatΔ(x) = 1for allx ∈, butrather only that a point-shift is, in a natural sense, compatible with the modular functionΔ. This motivates the primary new definition of this research.A point-shift $̋ on a point processis isomodular if-a.s. one hasΔ(X) = Δ((̋X))for allX ∈.Intuitively, althoughmay not be unimodular, an isomodular point-shiftacts on slices ofof the same modularity so that the mass transport principle works the same as it would on a unimodular space. <Ref> outline some of the subtleties of isomodularity.In <Ref>, the structure of the components of the random graph generated by a point-shift is studied whenis unimodular. For applications, it is interesting to study the behavior of points under repeated applications of a point-shift. How many points are in a given graph component? Given a point, how many other points will merge with it under repeated application of a point-shift? For=^d, the cardinality classification theorem proved in <cit.> asserts that answers to these questions fundamentally place every componentCinto one of three types:/, /,or/with(infinite) and(finite) representing the answers to the two questions respectively. The type of a componentCdetermines whether it is acyclic, the number of bi-infinite paths it contains, whether it can be linearly ordered in a flow-adapted way, and whether any points remain after an infinite number of applications of the point-shift. <Ref> shows that the classification when=^dextends verbatim towhenis unimodular.<Ref> shows that unimodularity ofis crucial to the cardinality classification theorem.In this section,is chosen to be an explicit non-unimodular group, theax+bgroup, and a point-shift$̋is given for which the previous cardinality classification fails in many respects.It is a classical result on^d, cf. <cit.>, that a point-shift preserves the Palm probability measure of a point process if and only if the point-shift is almost surely bijective on the support of the process. This result is commonly referred to as Mecke's invariance theorem or, in some cases, Mecke's point-stationarity theorem. In <Ref> it is shown that Mecke's invariance theorem holds ifis unimodular. Moreover, whenis not necessarily unimodular, the class of bijective point-shifts that preserve Palm probabilities is identified as the bijective isomodular point-shifts, the point-shifts that preserve the modular function of the group. Mecke's invariance theorem in the unimodular case is <Ref>, and the identification of isomodular point-shifts as being the ones that preserve Palm probabilities (which may be considered a generalization of Mecke's invariance theorem for the non-unimodular case)is <Ref>.<Ref> continues with the study of isomodularity and investigates for bijective point-shifts the distributional relationship between the reciprocal of the corresponding point-map and the reverse point-map, which corresponds to running the point-shift backwards in time. In particular, <Ref> shows that, amongst bijective point shifts, the isomodular point-shifts are exactly the ones for which the reciprocal of the point-map and the reverse point-map are equal in distribution. Note, however, that in most cases the reciprocal of a point-map onis itself not a point-map on .<Ref> studies different ways in which functions separate points of a point process. For example, given a function f:→ S for some set S and a point process , when are the values of f(X) distinct for all X ∈? <Ref> gives some sufficient conditions on f for separating points of the point process. This is useful to show that some point-shifts are well-defined when specifying where to send a point by comparing values of f(X) for different X ∈.Finally, many of the results of the previous sections have analogs to known results from another framework for dealing with random graphs and dynamics on their vertices. <Ref> studies this other framework, whose objects of study are called random (rooted) networks and where dynamics on the vertices of these networks are called vertex-shifts. Random networks are technical objects that will be introduced properly in <Ref>. Heuristically, they model random graphs where a vertex has been singled out and designated the root, and where vertices and edges may be endowed with extra associated information called marks. An analogous requirement to -stationarity for random networks is called unimodularity. In a unimodular random network, the root is picked “uniformly” from the vertex set in the sense that unimodular networks are defined as those that satisfy a certain mass transport principle for mass into and out of the root. This mass transport principle is similar to the mass transport theorem for point processes if the neutral element e ∈ under a Palm probability measure were considered the “root” of the point process. Because unimodular networks satisfy this mass transport principle, they exhibit analogs of many of the theorems of point processes such as Mecke's invariance theorem and the cardinality classification of components of point-shifts <cit.>. The parallels between point processes and random networks suggest the following motivating questions of the present research.*Given an -stationary point process, when can the Palm version of it be seen as an embedding of a unimodular network?*Given a unimodular network, when is it possible to find an -stationary point process such that the Palm version of the point process is an embedding of the given unimodular network?Some progress in the answering the first question is made in Section <ref>,where the problem is reduced to an invariant geometry problem on the underlying space, which is conjectured to always be solvable whenis unimodular. The results of previous sections also indicate that whenis not unimodular, one should not expect either question to be answered affirmatively.§ POINT-SHIFT BASICS AND NOTATION In this section, some notation and results that are used throughout the rest of the text are collected. Fix a point-shift $̋ with corresponding point-mapon a flow-adapted point processwith intensity∈ (0,∞)for the remainder of the section. In order to better suit the random graph setup desired in applications, define the functions (omittingωdependence): * Edge indicator: τ^(̋x,y) := 1_x,y∈,(̋x)=y for all x,y ∈.* Out-neighbors and in-neighbors of e under ^:h^+:= {Y ∈: τ^(̋e,Y)=1} = {},h^-:= {X ∈: τ^(̋X,e)=1}= {Y ∈: (θ_Y^-1ω) = Y^-1}.* Out-neighbors and in-neighbors underor ^:H^+(X):=X h^+(θ_X^-1) ={Y ∈: τ^(̋X,Y) = 1}= {(̋X)}, H^-(X):=X h^-(θ_X^-1) ={Y ∈: τ^(̋Y,X) = 1} = {Y ∈:(̋Y)=X}for all X ∈.* Preimage of e under ^: if ^-a.s. (h^-)=1, then ^- is defined to be the unique element in h^-. By <Ref>, this is equivalent to $̋ being bijective.* Reverse point-shift under: if-a.s.(H^-(X))=1for allX ∈(equivalently^-a.s.(h^-)=1), then^̋-(X)is defined to be the unique element inH^-(X). Note that^̋-is defined if and only if$̋ is bijective. In this case -a.s. (̋^̋-(X)) =^̋-((̋X)) = X for all X ∈. That is, $̋ and^̋-are inverses on the support of.With these definitions,τ^$̋ is diagonally invariant, and the definitions of h^+, h^-, ^- under ^ are equivalent to the definitionsof H^+, H^-, ^̋- underor ^ via <Ref>.With the mass transport theorem and <Ref>, the following may be obtained in a straightforward manner. The following hold:* -a.s. every X ∈ is the image under $̋ of at least (resp. at most)kdistinct points ofif and only if^-a.s.(h^-) ≥ k(resp.≤ k),*-a.s. everyX ∈is the image under$̋ of finitely (resp. infinitely) many distinct points ofif and only if ^-a.s. (h^-) < ∞ (resp. =∞),* -a.s. $̋ is bijective (resp. surjective, injective) if and only if^-a.s.(h^-)=1(resp.≥ 1, ≤ 1). In particular^-and^̋-are well-defined if and only if$̋ is bijective,* for all f:Ω→_+ measurable, ^[f(θ_^-1)Δ(^-1)] = ^[f(h^-)].In particular, the following mass flow relationship for point-shifts holds^[Δ(^-1)] = ^[(h^-)],* -a.s. every X ∈ is the image under $̋ of at least (resp. at most)kpoints ofif and only if for allf:Ω→_+measurable ^[f(θ_^-1)Δ(^-1)] ≥ k ^[f](resp. ≤ k^[f]),*(Test for Bijectivity)[G. Last also proves this and similar results, e.g. Corollary 10.1 in <cit.>.]$̋ is bijective if and only if for all f:Ω→_+ measurable^[f(θ_^-1)]Δ(^-1)] = ^[f],*If $̋ is bijective, also^[f(θ_^-1)] = ^[f/Δ(^-)],*If-a.s. everyX ∈is the image under$̋ of at least (resp. at most) k points of , then ^[Δ(^-1)] ≥ k (resp. ≤ k),*If ^[Δ(^-1)] < ∞, every X ∈ is the image of only finitely many Y ∈ under $̋,*If^[Δ(^-1)]=1, then$̋ is injective if and only if it is surjective. In particular, this is automatic ifis unimodular.(a),(b),(c): Direct application of <Ref>.(d): Apply the mass transport theorem with the diagonally invariant functionτ(ω,x,y) := f(θ_y^-1ω)1_x,y∈(ω), y=(̋x)Δ(y^-1x).(e): Apply (a) and (d).(f): Apply (e) with k:=1.(g): Replace f with f/Δ(^-) in (d) and use the fact that ^-a.s.^-(θ_^-1)=^-1(^-(θ_^-1)) =^-1^̋-() =^-1^̋-((̋e)) = ^-1.(h),(i): Take f:=1 in (d) and apply (a) or (b).(j): Take f:=1 in (d). Use (a), (c), and the fact that a random variable bounded above (or below) by 1 with expectation 1 must be constant 1 a.s.§ POINT-SHIFT FOLIATIONS§.§ The Cardinality Classification of ComponentsIn this section the cardinality classification components of point-shifts in <cit.> is extended to the general stationary framework for unimodular . The classification theorem is <Ref>, and the fundamental result used in its proof, which says it is impossible to pick out finite subsets of infinite sets in a flow-adapted manner, is <Ref>.Throughout this section,is assumed to be unimodular. Fix for the rest of the section a flow-adapted simple point processonwith intensity ∈ (0,∞), and a point-maponwithcorresponding point-shift $̋. The wording of proofs is substantially cut down by thinking of(̋X)as the father ofX. For example, the children ofXare theY ∈such that(̋Y) = X. Next appear the necessary ingredients needed for the classification theorem.The iterates ^̋n are defined by repeatedly applying the point-shift $̋. That is,^̋0(X) :=Xand^̋n+1(X) := (̋^̋n(X))for allX ∈. ElementsY∈that are in the image^̋n()for alln ∈are called primeval, and^̋∞()will denote the set of all primeval elements of. Here^̋n()is considered as a set, i.e. multiplicities are ignored, for alln ≤∞. Moreover,^̋n()is a flow-adapted simple point process for anyn ≤∞. Random graphs will be used throughout this section. Here a random (directed) graphGonis specified with a random variableNtaking values in∪{∞}and random elements{x_i}_i ∈inwithV(G) := { x_i : i ≤ N}, and measurable indicators{ξ_ij}_i,j ∈withE(G):= { (x_i,x_j) : i,j ≤ N, ξ_ij=1}. A random subsetCof vertices ofGis a map onΩtaking values in the subsets ofV(G)such that1_x_i ∈ Cis measurable for eachi. Similarly, a random (countable) collection = {C_i}_1 ≤ i ≤ N^of subsets of vertices ofGis identified with a random variableN^taking values in∪{∞}and random subsets{C_i}_i ∈with the elements ofbeing defined as{C_i : i ≤ N^}. In all cases of interest for the present study, the specific numbering of vertices in a random graph or elements of a random collection are of no interest and will not be given upon defining the graph or collection.The adjective flow-adapted has already been defined for point processes and point-shifts. The same adjective will also be used for random graphs and for random collections. A random graph G onis flow-adapted if for all ω∈Ω and all X,Y,z ∈ and one has X ∈ V(G(ω)) if and only if zX ∈ V(G(θ_zω)) and (X,Y) ∈ E(G(ω)) if and only if (zX,zY) ∈ E(G(θ_zω)). A random collection = {C_i}_1≤ i ≤ Nis flow-adapted if for all ω∈Ω,z ∈, one has N(θ_z ω) = N(ω) and there is a permutation π(ω) of 1,…, N(ω) such that C_i(θ_zω) = {z x: x ∈ C_π(ω)(ω)} for each i ≤ N(ω). That is, (θ_zω) contains the same elements as (ω), shifted by z, and possibly enumerated in a different order. Now the random graph generated by the point-shift$̋ is defined. The random graph G^$̋ is defined to have vertices at the points ofand directed edges from eachX ∈to(̋X). Two natural equivalence relations on the vertices ofG^$̋ are defined by connected components and foils. The set of undirected connected components of G^$̋ is denoted by^$̋ and the component of X ∈ is denoted C^(̋X). Then X, Y ∈ are in the same component if and only if there are n,m ∈ such that ^̋m(X) = ^̋n(Y). That is, C^(̋X) is the set of all relatives of X. The graph G^$̋ is flow-adapted, and hence so is^$̋.The foliationŁ^$̋ is defined to be the set of foilsL^(̋X)of$̋ for X ∈, which are equivalence classes under the equivalence relation where X,Y∈ are equivalent if and only if there is n ∈ such that ^̋n(X) = ^̋n(Y). That is, L^(̋X) is the relatives of X from the same generation as X. The foliation Ł^$̋ is flow-adapted, andŁ^$̋ is a subdivision of ^$̋. For a foilL, also denoteL_+ := L^(̋(̋X))for anyX ∈ L. Note that ifX,X' ∈ LthenL^(̋(̋X)) = L^(̋(̋X'))soL_+is well-defined. If there isY ∈such that(̋Y) ∈ L, then setL_- := L^(̋Y). ThenL_-is well-defined because ifY,Y'are both such that(̋Y),(̋Y') ∈ L, thenL(Y) = L(Y'). It holds that(L_+)_- = Land whenL_-exists(L_-)_+ = L.It will be important later to know that the graphG^$̋ is locally finite. The following result, generalizing one in <cit.>, guarantees this. It crucially relies on the unimodularity of . Let D_n(X) denote the n-th order descendants of X, i.e. D_n(X):= { Y ∈ : ^̋n(Y) = X }. Also let D(X) := ⋃_n=1^∞ D_n(X) be all descendants of X. Then with d_n(X):=(D_n(X)), d(X) :=(D(X)), one has for every n ≥ 0 that ^[d_n(e)] = 1. In particular, d_n(e) is ^-a.s. finite, or equivalently -a.s.every X ∈ has d_n(X) finite. If, in addition, G^$̋ is^-a.s. acyclic, then^[d(e)] = ∞. ^̋n is a point-shift in its own right, so the mass flow relationship (<ref>) implies ^[d_n(e)] = ^[ (D_n(e))] = 1 sinceis unimodular. Thus d_n(e)<∞, ^-a.s., and hence -a.s. d_n(X)<∞ for all X ∈ by <Ref>. Moreover, when G^$̋ is acyclic, theD_npartitionDand hence^[d(e)] = ∑_n=1^∞^[d_n(e)] = ∞.The primary tool needed to prove the classification theorem follows. It says that it is not possible to extract finite subsets of infinite subsets ofin a flow-adapted way. The proof is modified from the argument proving a similar result forunimodular networks given by Lemma 3.23 in <cit.>. Let 𝔑 = {𝔑_i }_1 ≤ i ≤ N be a flow-adapted collection of infinite measurable subsets ofand let k be the number of i such that e ∈𝔑_i. Suppose that ^[k] < ∞. Ifis a measurable flow-adapted subset offor which -a.s.(∩𝔑_i) < ∞ for each i, then -a.s. ∩𝔑_i = ∅ for all i. In particular, if ⊆⋃𝔑, then -a.s.= ∅.Defineτ(ω,x,y) := ∑_i=1^N(ω) 1_x,y ∈𝔑_i(ω),y∈(ω)1/((ω)∩𝔑_i(ω)).The assumptions about flow-adaptedness of 𝔑, , and , imply that τ is diagonally invariant. Then ∫_τ(e,y) (dy) = k by construction since e is in k of the 𝔑_i. Also ∫_τ(x,e) (dx)=∞ if e ∈∩𝔑_i for some i because the 𝔑_i are infinite. But the mass transport theorem implies^∫_τ(x,e) (dx) = ^∫_τ(e,y) (dy) =^[k]< ∞,and thus it must be that ^-a.s. e ∉∩𝔑_i for any i. Equivalently, -a.s. for all X ∈ it holds thatX ∉∩𝔑_i for any i. Since ∩𝔑_i ⊆ for each i, it follows that -a.s. ∩𝔑_i = ∅ for all i.Note that, by <Ref>, the condition^[k] < ∞appearing in <Ref> is automatically satisfied if the𝔑_iare pairwise disjoint, or more generally if there is a constantnsuch that almost surely noX ∈appears in more thannof the𝔑_i, as this would implyk ≤ n,^-a.s.More information follows about the structure of the locally finite graphG^$̋. In particular, cycles in components are unique, infinite components are acyclic, foils in infinite components can be ordered likeorin a flow-adapted way, and $̋ acts bijectively on the primeval elements.-a.s. a connected component C of G^$̋ is either an infinite tree or hasexactly one (directed) cycleK(C)for which for allY ∈ Cthere isn ∈such that^̋n(Y) ∈ K(C). Moreover,-a.s. there are no infinite components with a cycle.The fact that all elements in C are connected and have out-degree1 implies there can be at most one cycle. If there are no cycles then C must be infinite since applying $̋ to any element repeatedly must never repeat an element. Otherwise there is one cycleK(C)and connectedness implies for everyY ∈ Cthere isn ∈with^̋n(Y) ∈ K(C).Let𝔑be the set of infinite components ofG^$̋ with a cycle, and let ⊆⋃𝔑 be the union of all the cycles of these components. Since cycles are finite, it follows that ∩ C is finite for all components C ∈𝔑. By <Ref>= ∅ and hence there are no infinite components with a cycle -a.s. Within an infinite acyclic connected component C ∈^$̋, it is possible to define an order, called the foil order, on the foilsŁ^(̋C)that are subsets ofC. This is accomplished by declaringL^(̋X) < L_+^(̋X)for allX ∈ C. When thinking of(̋X)as being the father ofX, the orderis that of seniority.The foil order on an infinite acyclic component C is a total order on C similar to either the order ofor .Fix any X ∈ C. Let L_0 := L^(̋X) and recursively define L_n+1 := (L_n)_+ and if it exists L_-n-1 :=(L_-n)_- for n > 0. Let L be a foil in C, then it must be that L=L_i for some i. Indeed, let Y ∈ L and by definition of connectedness choose n,m such that ^̋n(Y) = ^̋m(X) ∈ L_m. It then follows by induction that Y ∈ L_m-n, and hence L=L^(̋Y) = L_m-n. Next it is shown that i ↦ L_i is injective. Suppose for contradiction that L_j= L_j+N. Then there are N pairs (X_i, Y_i+1) with X_i ∈ L_i,Y_i+1∈ L_i+1 such that (̋X_i) = Y_i+1 for j ≤ i ≤ j+N-1. Since L_j = L_j+N it follows that X_j,Y_j+N∈ L_j. Hence it is possible to choose n such that ^̋n(X_j) = ^̋n(Y_j+N) and ^̋n(X_i) = ^̋n(Y_i) for all j+1 ≤ i ≤ j+N-1. Assume by induction that for some k one has ^̋N(^̋n(X_j)) = ^̋N-k(^̋n(Y_j+k)). Then as long as k+1 ≤ N,^̋N(^̋n(X_j)) = ^̋N-k(^̋n(Y_j+k)) = ^̋N-k(^̋n(X_j+k))= ^̋N-k-1(^̋n((̋X_j+k)))= ^̋N-k-1(^̋n(Y_j+k+1)).Since ^̋N(^̋n(X_j)) = ^̋N-1(^̋n((̋X_j))) = ^̋N-1(^̋n(Y_j+1)) shows the base case k=1 holds, the induction is complete. Therefore, one finds ^̋N(^̋n(X_j)) = ^̋0(^̋n(Y_j+N)) = ^̋n(X_j), contradicting that C is acyclic. Thus i ↦ L_i is injective. If there is a smallest foil L_i_0 then i↦ L_i_0+i is an order isomorphism with , otherwise i ↦ L_i is an order isomorphism with .$̋ restricts to a bijective point-shift|̋_on the flow-adapted sub-process:=^̋∞()of primeval elements. $̋ naturally restricts to a point-shift|̋_onbecause ifX ∈^̋∞()then(̋X) ∈^̋∞(). By definition, primeval elements are in the image(̋), but moreover they are in the image(̋). Indeed, by <Ref>, points inhave only finitely many children. IfX ∈were such that none of its children were primeval, then there would ben∈large enough that none ofX's children are in the image^̋n(). But thenXwould not be in^̋n+1(), contradicting thatX ∈^̋∞(). Thus the restricted point-shift|̋_is surjective. Ifis not the empty process-a.s. then it has nonzero and finite intensity and^[Δ(^-1)] =1by unimodularity so that surjectivity and injectivity are equivalent by <Ref> (j), so|̋_is bijective.The main result of this section follows.-a.s. each connected component C of G^$̋ is in one of the three following classes:* Class /:C is finite, and hence so is each of its $̋-foils. In this case, when denoting by1 ≤ n = n(C) < ∞the number of its foils:* C has a unique cycle of length n;* ^̋∞()∩ C is the set of vertices of this cycle.* Class /:Cis infinite and each of its$̋-foils is finite. In this case:* C is acyclic;*Each foil has a junior foil;* ^̋∞()∩ C is a unique bi-infinite path, i.e. a sequence {X_n}_n ∈ of points of such that (̋X_n) = X_n+1 for all n.* Class /:C is infinite and all its $̋-foils are infinite. In this case:* C is acyclic;* ^̋∞() ∩ C = ∅.The properties of finite components C are immediate, so only infinite components are considered. Recall that by <Ref>-a.s. all infinite components are acyclic. Consider the collection 𝔑 of all infinite components that have both finite and infinite foils. Suppose C ∈𝔑. According to <Ref>, all X ∈ have only finitely many children, so that if L is an infinite foil, then L_+ is also infinite. It follows that there is a maximum finite foil L with respect to the foil order in C. Let ⊆⋃𝔑 be the union of these maximumfinite foils of each C ∈𝔑. By construction, ∩ C is finite for each C ∈𝔑, so <Ref> implies =∅ and hence 𝔑=∅, -a.s. Thus -a.s. each infinite component is either of class / or /.Next, redefine 𝔑 to be the set of infinite foils L of , and let :=^̋∞(). By construction ∩ L is finite for each L ∈𝔑 because a foil cannot have multiple primeval elements. If X≠ Y ∈ L were both primeval, then with n minimal such that ^̋n(X) = ^̋n(Y) one finds the primeval element ^̋n(X) is the image of two distinct primeval elements ^̋n-1(X),^̋n-1 (Y), contradicting injectivity of |̋_ guaranteed by <Ref>. Thus <Ref> implies -a.s. ∩ L = ∅ for all infinite foils L, and hence -a.s. ^̋∞() ∩ C ≠∅ implies C is of class /.Conversely, it will be shown that if C is class /, then ^̋∞() ∩ C≠∅. Indeed, redefine 𝔑 to be the collection of components C of class / that have a minimum foil in the foil order. Letting ⊆⋃𝔑 be the union of minimum foils in C, it holds that ∩ C is the (finite) minimum foil in C for each C ∈𝔑. Thus <Ref> implies =∅ and hence 𝔑=∅, -a.s. Now consider a C of class / and an arbitrary foil L of C. Since L is finite there is a minimum n such that ^̋n(L) is a single point. Let C_0 denote the subgraph of G^$̋ ofLtogether with all descendants of elements ofLand all forefathers of elements ofLup to^̋n(L). ThenC_0is an infinite connected graph with vertices of finite degree, and hence it contains an infinite simple path{X_i}_i ≤ 0with(̋X_i) = X_i+1for eachi<0by König's infinity lemma (c.f. Theorem 6 in <cit.>). Fori>0, defineX_i := ^̋i(X_0). Then{X_i}_i ∈is a bi-infinite path inCsatisfying(̋X_i) = X_i+1for alli ∈, and thus{X_i}_i ∈⊆^̋∞() ∩ C, in particular showing^̋∞() ∩ C ≠∅. It also holds that^̋∞() ∩ C ⊆{X_i}_i ∈since for anyX ∈^̋∞() ∩ Cit is possible to choosen,msuch that^̋n(X) = ^̋m(X_0) = X_m. Uniqueness of primeval children then impliesX = X_m-n. It follows that^̋∞() ∩ C = {X_i}_i ∈.Thus it is shown that-a.s. infinite componentsCare class/if and only if^̋∞() ∩ C ≠∅and in this case^̋∞() ∩ Cis a unique bi-infinite sequence{X_i}_i ∈satisfying(̋X_i) = X_i+1. Since/and/are the only possible choices, by process of elimination it follows that-a.s. infinite componentsCare of class/if and only if^̋∞() ∩ C = ∅. §.§ A Counterexample on a Non-unimodular Group This example serves to show that the cardinality classification (<Ref>) does not hold for non-unimodular spaces. It is an open question whether a more general classification for such spaces exists. Recall the standard first example of a non-unimodular group: theax+bgroup. In this section,= {a b 0 1: a>0, b ∈}with matrix multiplication and the topology inherited from^4.is identified with the right half-plane in^2by identifying(a,b)witha b 0 1. In this notation (a,b)(c,d) = (ac, ad+b),(a,b)^-1 = (1/a,-b/a).Then, cf. <cit.> Example 15.17 (g),has a left-invariant Haar measure(B) = ∬_B 1/a^2 da dband modular functionΔ(a,b) = 1/a.Letbe a homogeneous Poisson point process onwith intensity∈ (0,∞). Necessarilyis-stationary and simple.For all(a,b) ∈define the strip S(a,b) := [a,∞) × [b-δ a,b+δ a]for some fixedδ>0. Note that the definition is chosen so(a,b) S(1,0) = S(a,b), where here(1,0) = e ∈. Moreover, for any(a,b) ∈,(S(a,b))= ∫_b-δ a^b+δ a∫_a^∞1/x^2 dx dy = 1/a· ((b+δ a)-(b-δ a)) = 2 δso in particular(S(a,b)) < ∞a.s. By the Slivnyak-Mecke theorem (see <Ref> in the appendix),^! := -δ_eis Poisson under^with^[^!(B)] = (B). Hence^[^!(S(1,0))] = 2δand therefore(S(1,0)) < ∞,^-a.s. Equivalently,-a.s.(S(X)) < ∞for allX ∈by <Ref>. This leads to the strip point-shift$̋ where (̋X) is defined to be the right-most point ofin S(X) for each X ∈.One may theoretically resolve ties for right-most point using the lexicographic order on ^2, but the interested reader may note that results in <Ref> will show that there is no need because almost surely each point X∈ has a unique first coordinate.Now suppose that 2δ < 1. It will be shown that -a.s. ^̋n(X) eventually becomes constant as n →∞ for all X ∈. It suffices to show that under ^ it holds that ^̋n(e) eventually becomes constant. Recall that the n-th factorial moment measure of a counting measure μ with representation μ = ∑_iδ_x_i is defined as μ^(n) := ∑_i_1≠⋯≠ i_nδ_(x_i_1,…,x_i_n), where the notation i_1 ≠⋯≠ i_n means that i_1,…,i_n are all distinct. Then^∑_k=0^∞(S(^̋k(e))∖{^̋k(e)}) =∑_k=0^∞^(S(^̋k(e))∖{^̋k(e)})≤∑_k=0^∞^∫_^k+1 1_x_1 ∈ S(e)⋯ 1_x_k+1∈ S(x_k) (^!)^(k+1)(dx_1 ×⋯× dx_k+1)=∑_k=0^∞∫_^k+1 1_x_1 ∈ S(e)⋯ 1_x_k+1∈ S(x_k) ^(k+1)(dx_1 ×⋯× dx_k+1)=∑_k=0^∞∫_^k+1 1_x_1 ∈ S(e)⋯ 1_x_k+1∈ S(x_k)^k+1 (dx_k+1)⋯(dx_1)=∑_k=0^∞ (2δ)^k+1^k+1< ∞,where here the Slivnyak-Mecke theorem is used again, along with the fact that the factorial moment measures of a Poisson point process are just powers of the intensity measure, cf. Example 9.5 (d) in <cit.>. Thus it must be that (S(^̋k(e))∖{^̋k(e)}) = 0 for all k large, ^-a.s. That is, there are no points ofin S(^̋k(e)) besides ^̋k(e) itself. Consequently, ^̋k(e) is a fixed point of $̋ for largekand^̋k(e)is thus eventually constant ink. Equivalently,-a.s. for everyX ∈it holds that^̋k(X)is eventually constant ink.Next it will be shown that every fixed point of$̋ is the image of infinitely many X ∈. Again it is enough to show under ^ that if (̋e) = e then e is the image of infinitely many X ∈. This is accomplished by finding a region of points (x,y)∈ such that * (1,0) ∈ S(x,y), and* S(x,y) ∩ ([1,∞)×) ⊆ S(1,0),which implies $̋ would map a point ofat(x,y)to(1,0). The condition (i) says1 ≥ xandy-δ x ≤ 0 ≤ y+δ x, i.e.-δ x ≤ y ≤δ x. Condition (ii) is guaranteed if[y-δ x,y+δ x] ⊆ [-δ,δ], i.e.ify ≥δ(x-1)andy ≤δ(1-x). The constraints0 < x ≤ 1, -δ x ≤ y ≤δ x, y ≤δ(1-x), y ≥δ(x-1),bound a parallelogramDwith corners(0,0), (1/2,δ/2), (1,0), (1/2,-δ/2).Then^[^!(D)] = (D) ≥∫_0^1/2∫_-δ x^δ x1/x^2 dy dx = ∫_0^1/22δ/x dx = ∞so that the regionDcontains infinitely many points of,^-a.s. By construction, if(̋e)=ethen everyX ∈∩ Dhas(̋X) = e,proving the claim.Putting previous claims together, it holds that the foils and connected components are identical because every component contains a fixed point, and the foils and components are in bijection with the fixed points of$̋. The connected component of a fixed point Y of $̋ is allX ∈that are eventually sent toY. Thus all components and foils are infinite (class/). However, the components are not acyclic and^̋∞() = {X ∈ : (̋X) = X}≠∅, contrary to what the classification theorem would suggest for unimodular. It follows that the properties of the cardinality classification cannot be extended beyond the case of unimodular.§ PROPERTIES OF POINT-SHIFTS§.§ Mecke's Invariance Theorem In the case of=^d, Mecke's invariance theorem shows that Palm probabilitiesare preserved under bijective point-shifts. Even stronger, a point-shift is bijective if and only if it preserves Palm probabilities. It will be shown in <Ref> that ifis unimodular then this still holds. However, for non-unimodularthis is not so. Precisely, the notion of isomodularity defined in the introduction will be elaborated upon, and it will be shown that, amongst bijective point-shifts, isomodular ones are exactly those that preserve Palm probabilities (<Ref>).For the rest of the section, fix a flow-adapted simple point processof intensity∈ (0,∞), and a point-mapwith associated point-shift$̋. The notation for the corresponding functions τ^$̋,h^+,h^-,H^+,H^-,^-,^̋-mentioned in the preliminaries is retained.The simple case of Mecke's invariance theorem whenis unimodular follows.Suppose thatis unimodular. Then $̋ preserves^if and only if$̋ is bijective. That is, ^(θ_^-1∈ A) = ^(A) for all A ∈ if and only if $̋ is bijective.Apply <Ref> (f), the test for bijectivity, and use the fact that Δ(x) = 1 for all x ∈.With Mecke's invariance theorem for unimodularin place, one may ask about non-unimodular.For these, which bijective point-shifts preserve Palm probabilities? <Ref> shows that the obstruction is the factorΔ(^-1). This motivates the definition of isomodularity, which says that a point-shiftpreserves the value ofΔ(X)for eachX ∈. Isomodularity, defined already in the introduction, is a special case of invariance of a subgroup under$̋, which is defined presently. A measurable subgroup G ∈() ofis called $̋-invariant if-a.s.(̋X)is in the same coset asXfor allX ∈. Isomodularity of$̋ is the same as the assumption that the subgroup {Δ=1} is $̋-invariant. Also note that ifis unimodular, then$̋ is automatically isomodular.A brief detour is taken to go through the equivalent descriptions of $̋-invariance underand^. Let G∈() a measurable subgroup of , and for each x∈ let [x] := xG denote the coset of x. Then the following are equivalent* G is $̋-invariant, i.e.-a.s.[(̋X)] = [X]for allX ∈,*^-a.s.[] = [e],and if$̋ is bijective, the previous statements are also equivalent to* -a.s. [^̋-(X)] = [X] for all X ∈,* ^-a.s. [^-] = [e]. (a)(b): The equivalence follows from <Ref>, so that ^-a.s. [] = [e] is equivalent to-a.s. [(θ_X^-1)] = [e] for all X ∈, which is the same as [(̋X)] = [X] after multiplying by X.(a)(c): Using that $̋ and^̋-are inverses, replaceXwith^̋-(X)in (b) to get (c) or replaceXwith(̋X)in (c) to get (b).(c)(d): The proof is the same as (a)(b). Since isomodularity plays an important role in what follows, the previous result is restated forG:={Δ=1}in the bijective case. Let $̋ be bijective, then the following are equivalent* $̋ is isomodular, i.e.-a.s.Δ((̋X)) = Δ(X)for allX ∈,*^-a.s.Δ() = 1,*-a.s.Δ(^̋-(X)) = Δ(X)for allX ∈,*^-a.s.Δ(^-) = 1.Now the question of which bijective point-shifts preserve Palm probabilities is answerable. Suppose $̋ is bijective. Then$̋ preserves ^ if and only if $̋ is isomodular. That is,^(θ_^-1∈ A) = ^(A)for allA ∈if and only if$̋ is isomodular.Suppose $̋ is isomodular. ThenΔ(^-) = 1,^-a.s. by <Ref>. Hence (<ref>) immediately implies$̋ preserves ^. If $̋ is not isomodular, at least one of^(Δ(^-) > 1)and^(Δ(^-) < 1)is strictly positive. The cases are nearly identical, so assume^(Δ(^-)>1)>0and takeA:= {Δ(^-) > 1}. Then takef := 1_Ain (<ref>) to find ^(θ_^-1∈ A)= ^[1_Δ(^-) > 1/Δ(^-)] < ^[1_Δ(^-) > 1] = ^(A),showing that^is not preserved.§.§ Reciprocal and Reverse of a Point-mapA curious interplay between the reverse^-and the reciprocal^-1of a point-map is investigated, and a characterization of when the two have the same law under^is given.The notation of the previous section is retained. That is,is a flow-adapted simple point process of intensity∈ (0,∞), andis a point-map with associated point-shift$̋. The notation for the corresponding τ^$̋,h^+,h^-,H^+,H^-,^-,^̋-defined in the preliminaries is also retained. Next follows another result along the lines of <Ref> (f) and (g)which sparks interest in the distributional relationship between^-1and^-. Suppose $̋ is bijective. For allf:→_+measurable it holds that^[f(^-1)Δ(^-1)] = ^[f(^-)],^[f(^-1)] = ^[f(^-)/Δ(^-)].Use the fact that ^-a.s. ^-(θ_^-1) = ^-1(^-(θ_^-1)) = ^-1^̋-() =^-1^̋-((̋e))= ^-1and replace f by f(^-) in each of(<ref>) and (<ref>).One sees in (<ref>) that non-unimodularity ofis, as usual, an obstruction. Two more results relating the distributions ofΔ(^-)andΔ(^-1)are given. Then it is shown in <Ref> that amongstbijective point-shifts, the isomodular ones are precisely those for which^-1and^-have the same distribution under^. Recall that this is also the class of point-shifts that preserve Palm probabilities by <Ref>. Let $̋ be bijective, then for allr>0it holds thatr^(Δ(^-1)=r) = ^(Δ(^-)=r),and if this number is strictly positive then for allA ∈^(θ_^-1∈ A|Δ(^-1)=r) = ^(A |Δ(^-)=r).Fix r>0 and take f(x):= 1_Δ(x) = r in (<ref>). One finds ^(Δ(^-1)=r) = 1/r^(Δ(^-)=r) =: pshowing the first claim. Supposing that p>0, take f:=1_A 1_Δ(^-)=r in (<ref>) and use that ^-a.s. ^-(θ_^-1) = ^-1 to find^(θ_^-1∈ A, Δ(^-1)=r)= 1/r^(A, Δ(^-)=r).Division by p finishes the proof.Let $̋ be bijective, then for allα∈and0≤ r ≤ s ≤∞it holds that^[ Δ(^-1)^α1_r ≤Δ(^-1) ≤ s]= ^[ Δ(^-)^α-11_r ≤Δ(^-) ≤ s].Take f(x) := Δ(x)^α 1_r ≤Δ(x)≤ s in (<ref>). Let $̋ be bijective, then^-1and^-have the same law under^if and only if$̋ is isomodular.Suppose $̋ is isomodular. Then by <Ref>,^-a.s.Δ()=Δ(^-) =1and thus (<ref>) shows that^-1and^-have the same law under^.Next suppose that^-1and^-have the same law under^. Then ^[Δ(^-1)^α1_r ≤Δ(^-1) ≤ s]= ^[Δ(^-)^α1_r ≤Δ(^-) ≤ s]for allα∈and all0 ≤ r ≤ s ≤∞. But then for allα∈and all0 ≤ r ≤ s ≤∞^[Δ(^-1)^α+11_r ≤Δ(^-1) ≤ s] = ^[Δ(^-)^α1_r ≤Δ(^-) ≤ s] (by (<ref>))= ^[Δ(^-1)^α1_r ≤Δ(^-1) ≤ s](by (<ref>))= ^[Δ(^-)^α-11_r ≤Δ(^-1) ≤ s].(by (<ref>))Takingα:=1, r:=1, s:= ∞^[Δ(^-1)^21_1 ≤Δ(^-1)]= ^[Δ(^-1)1_1 ≤Δ(^-1)]which is absurd unlessΔ(^-1) ≤ 1,^-a.s. It also holds that withα:=1, r:= 0, s:= 1,^[Δ(^-1)^21_Δ(^-1) ≤ 1]= ^[Δ(^-1)1_Δ(^-1) ≤ 1],which is absurd unlessΔ(^-1) ≥ 1,^-a.s. It follows thatΔ(^-1) =1,^-a.s. By <Ref> the result follows.§.§ Separating Points of a Point Process In this section a notion of a function separating points of a point process is introduced. For the remainder of the section,is a simple and flow-adapted point process of intensity∈ (0,∞). Let S be a set, f:→ S, and suppose that -a.s. no distinct X,Y ∈ have f(X) = f(Y). Then say that fseparates points of . Similarly, say that a fixed partition {B_i}_i ∈ J of separates points ofif -a.s. no B_i contains more than 1 point of . When separation of points occurs is studied by proving a general result concerningwhen there cannot be ann-tuple of distinct points ofsatisfying a given constraint. Recall again thatμ^(n) = ∑_i_1 ≠⋯≠ i_nδ_(x_i_1,…,x_i_n)denotes then-th factorial moment measure of a measureμ = ∑_i δ_x_i, andμ^! = μ - δ_e. Let (S,Σ) be a measurable space and fix M ∈Σ. Let F:×^n → S be measurable, and suppose that for all y=(y_1,…,y_n) ∈ (∖{e})^n, or more generally that for ^[(^!)^(n)]-a.e. y ∈^n, (x ∈ : F(x,xy) ∈ M) = 0.Then -a.s. no n+1 distinct X,Y_1,…,Y_n∈ have F(X,Y_1,…,Y_n) ∈ M.By straight calculations,(∃ X ∈, Y ∈^(n): (X,Y) ∈^(n+1), F(X,Y) ∈ M)≤∫_ 1_∃ Y ∈^(n): ∀ i, Y_i ≠ x, F(x,Y) ∈ M (dx)≤∫_^(n)(θ_e,{y ∈^n: ∀ i, y_i≠ x, F(x,y) ∈ M}) (dx)=^∫_^(n)(θ_x,{y ∈^n,∀ i, y_i ≠ x, F(x,y) ∈ M}) (dx)=^∫_^(n)(θ_e,{x^-1y: y ∈^n, ∀ i, y_i ≠ x, F(x,y) ∈ M}) (dx)=^∫_^(n)(θ_e,{y∈^n, ∀ i, xy_i ≠ x, F(x,xy)∈ M}) (dx)=^∫_∫_ 1_F(x,xy) ∈ M (^!)^(n)(dy) (dx)=^∫_(x ∈ : F(x,xy)∈ M) (^!)^(n)(dy)= 0,where in the third equality the refined Campbell theorem, stated in the appendix as <Ref>, is used. This proves the claim.<Ref> immediately gives a condition for separating points of. Let (S,Σ) be a measurable space, f:→ S measurable, and suppose for all y ≠ e, or more generally for ^[^!]-a.e. y∈,(x ∈ : f(x) = f(xy)) = 0.Then f separates points of . Implicit in the previous line is the assumptionthat the sets {x ∈: f(x) = f(xy)} are measurable for all y ∈. This is automatic if (S,Σ) is a standard measurable space, or more generally if S× S has measurable diagonal.Take n:=1, F(x,y) := (f(x),f(y)) for all x,y ∈, and take M to be the diagonal of S × S, then apply <Ref>.<Ref> generalizes the well-known theorem in = ^dthat a stationary point process has not two points equidistant from0. That would be the case off(x) := |x|. Not allhave this property though. Indeed, ifis a countable group with the discrete distanced(x,y) := 1_x≠ y, then(x ∈: d(x,e) = d(xy,e))>0for ally ≠ eso the result does not apply ifhas more than one element.The next results can be used to show that there is no need to resolve ties when defining a point-shift in some situations. Intuitively, if a setBis small from the typical point's perspective, then no shift ofBwill contain more than one point of. Let B ∈() with e ∈ B. If ^[^!(B)] = 0, then -a.s. for all X ∈ it holds that (Xb: b ∈ B) = 1, i.e. X is the unique point ofinside {Xb : b ∈ B}.The hypotheses imply ^-a.s. (B∖{e}) = 0. By <Ref>, -a.s. all X ∈ are such that T_X^-1(B∖{e}) = 0, i.e. ({Xb:b ∈ B}∖{X}) = 0, and hence (Xb: b ∈ B) = 1.For example, recall the strip point-shift on theax+bgroup of <Ref>. It was defined by sending a pointXto the right-most point in a certain strip in the plane. In that case, takeB:= {1}×in the previous result to find that the points ofhave unique first coordinates. Hence there are no ties for right-most point.Finally, the previous result is restated in the case thatBis a subgroup and applied to see that the only way for$̋ to preserve a small subgroup from the typical point's perspective is to act as the identity. Let G∈() a subgroup of . If ^[^!(G)] = 0, then the cosets of G separate points of .Let G ∈() a subgroup of . If ^[^!(G)] = 0 but G is $̋-invariant for some point-shift$̋, then $̋ is the identity point-shift-a.s. G being $̋-invariant means(̋X)andXare in the same coset forX ∈, then by <Ref>$̋ is the identity point-shift.Let G ∈() a subgroup of . If (G) = 0 andis Poisson with intensity ∈ (0,∞), then the only $̋ for whichGis$̋-invariant is the identity.The Slivnyak-Mecke theorem, <Ref> in the appendix, implies that ^[^!(G)] = (G) = 0 and <Ref> applies. § CONNECTIONS WITH UNIMODULAR NETWORKS This section investigates the relationship between vertex-shifts on unimodular random networks and point-shifts of-stationary point processes whenis unimodular. See <cit.> for a general reference on unimodular networks, and <cit.> for a similar investigation to this one for point processes on^d.A graphwith vertex setand undirected edge setis written = (,). Write()and()for the vertices and edges of. A network is a graph=(,)together with a complete separable metric spaceΞcalled the mark space and mapsξ_:→Ξandξ_:{(,̌) ∈×:∼̌}→Ξcorresponding to vertex and edge marks, i.e. extra information attached to vertices and edges. Throughout this document, all networks are assumed to be connected and locally finite, i.e.is connected and all vertices ofhave finite degree. Graphs are special cases of networks that have every mark equal to some constant. Forr ≥ 0, the ball (with respect to graph distance) ofradius⌈ r ⌉with center∈̌()is denotedN_r(,)̌.An isomorphism of networks_1=(_1,_1)and_2=(_2,_2)with mark spaceΞ(one may always assume a common mark space^, or any other fixed Polish space)is a pair of bijectionsφ_:_1→_2, andφ_:_1→_2such that* {,̌'̌}∈_1 if and only if {φ_()̌,φ_('̌)}∈_2,* φ_({,̌'̌}) = {φ_()̌,φ_('̌)} for all {,̌'̌}∈_1,*the vertex mark maps ξ__1,ξ__2 satisfy ξ__1 = ξ__2∘φ_, and*the edge mark maps ξ__1,ξ__2 satisfy ξ__1= ξ__2∘φ_. A rooted network is a pair(,ø)in whichis a network andøis a distinguished vertex ofcalled the root. An isomorphism of rooted networks(,ø)and(',ø')is a network isomorphism such thatφ_(ø) = ø'. Letbe the set of isomorphism classes of networks,and letbe the set of isomorphism classes of rooted networks. Similarly definefor networks with a pair of distinguished vertices. The isomorphism class of a network(resp.(,ø)or(,ø,)̌) is denoted[](resp.[,ø]or[,ø,]̌).Equip(and similarly for) with a metric and its Borelσ-algebra. The distance between[,ø]and[',ø']is2^-α, whereαis the supremum of thoser>0such that there is a rooted isomorphism betweenN_r(,ø)andN_r(',ø')such that the distance of the marks of the corresponding elements is at most1/r. Thenis a complete separable metric space, and measurable functions onare those that can be identified by looking at finite neighborhoods of the root.A random network is a random element in. That is, it is a measurable map from(Ω,,)to. This map will be denoted[, ]. A random network[,]is unimodular if for all measurableg:→_+, the following mass transport principle is satisfied,∑_∈̌() g[,,]̌= ∑_∈̌()g[,,̌]. A vertex-shift, which is the analog of a point-shift, is a mapfthat associates to each networka functionf_:() →()such that*for all isomorphic networks ' with vertex isomorphism φ_:() →('), one hasf_'∘φ_ = φ_∘ f_, and* [,ø,]̌↦ 1_f_(ø)= is measurable on . A general situation is given under which a point-process, seen under its Palm probability measure and rooted at the identity, is a unimodular network. First, the appropriate notion of flow-adaptedness for networks must be given, then the result follows. Suppose β is a map on Ω such that for all ω∈Ω, β(ω) is a network whose vertex set V(β) ⊆. Then β is called flow-adapted if for all z ∈, β(θ_zω) is the shiftT_zβ(ω) of β(ω) by z, i.e.(β(θ_zω)) = {zX: X ∈(β(ω))}, and E(β(θ_zω)) = {{zX, zY}: {X,Y}∈ E(β(ω))}, and all marks are preserved.Supposeis unimodular. Letbe a flow-adapted simple point process with intensity ∈(0,∞). Let β be a map on Ω such that for all ω∈Ω, β(ω) is a network, and such that β satisfies* (β) =,* β is flow-adapted,* ω↦ [β(ω),e] is measurable on the event {e ∈(β)}.Then [β,e] is a unimodular network under ^ on the event { e ∈(β)} = {e ∈}.The assumption (iii) implies [β,e] is a random network under ^ on {e ∈(β)}, so one only needs to check unimodularity. Let g:→_+ be given. Then^∑_∈̌(β) g[β,e,]̌ =^∫_ g[β,e, x] (dx)=^∫_ g[β(θ_y^-1),e, y^-1] (dy) (mass transport)=^∫_ g[T_y^-1β,e, y^-1] (dy)=^∫_ g[β,y, e] (dy)=^∑_∈̌(β) g[β,,̌e],showing unimodularity. Recall that the symbolT_zforz ∈, used in the previous result as the shift operator on networks, is also used as the shift operator on counting measures and point processes, which is how it is used in the following. Let [,] be a unimodular network. An -embedding of [,] with respect to a probability measure 𝒫 on Ωis a map :→ with the following properties:*is measurable,* e ∈[,ø] for all rooted networks (,ø),*there is a measurable function :→ such that for every rooted network (,ø) and every vertex ∈̌(),[,]̌ = T_[,ø,]̌^-1[,ø], [,,̌ø]= [,ø,]̌^-1,* 𝒫-almost surely, the map defined on () by ↦̌[,,]̌ is a bijection between () and the support of[,].Say that the Palm version of a point processis an -embedding of [,] if there is an -embeddingwith respect to ^ such that ^-a.s. =[,]. Two of the motivating open questions of this research are:*For a fixed , is the Palm version ofan -embedding of some [,]?*For a fixed [,], is there ansuch that the Palm version ofis an -embedding of [,]? Whenis unimodular, the answer to the first question is “yes” forif it is possible to draw a connected graph onin a flow-adapted way.Call a flow-adapted simple point process connectible in a flow-adapted way if there exists a connectedflow-adapted locally finite graph =(ω) with ()== ∑_i δ_Y_i almost surely and such that 1_{Y_i,Y_j}∈() is measurable for all i,j.Supposeis unimodular and thatis a flow-adapted simple point process with intensity ∈ (0,∞) that is connectible in a flow-adapted way. Then the Palm version ofis an -embedding of some unimodular network.Choose a random graphwitnessing the fact thatis connectible. Consider some edge ∈() from X ∈ to Y ∈. Without loss of generality, assume the mark space Ξ =. Let the mark of (X,) be X^-1Y, and let β(ω) be the network with underlying graph (ω) and marks as just specified. Choose := e and let [,]:= [β,e]. Let ψ a rooted automorphism of (,)̌ be given, where $̌ is any vertex of. It will be shown thatψis the identity. Assume thatψfixes all vertices less than graph distancekfrom$̌. For each Y∈ of distance k+1 from $̌, there is anX ∈of distancekfrom$̌ and an edgefrom X to Y. The mark of (X,) is X^-1Y and this must equal the mark of ψ(X)=X between X and ψ(Y). But this mark is X^-1ψ(Y). Thus X^-1Y = X^-1ψ(Y) so that Y=ψ(Y). By induction and using thatis connected, one finds that ψ is the identity automorphism. By construction (β)=, β is flow-adapted, and ω↦ [β(ω),e] is measurable on the set {e ∈}. By <Ref>, [β,e] is a unimodular network under ^ on the set {e ∈}. It will be shown thatis an -embedding of [,].On the set {e ∈} one has thatcan be reconstructed from [,]. The reconstruction procedure will be used to define an -embedding . Indeed, let [,ø,]̌ := ∏_i=0^k-1ξ_(_̌i,_i) where _̌0_̌1⋯_̌k is a path between the two arbitrary vertices ø and $̌ of, and_iis the edge{_̌i,_̌i+1}, assuming the product is independent of the path chosen between. Let[,ø,]̌:= eotherwise. The fact that∏_i=0^k-1ξ_(_̌i,_i)is required to be independent of path implies that for any_̌1,_̌2,_̌3 ∈()one has[,_̌1,_̌3] = [,_̌1,_̌2] [,_̌2,_̌3]and in particular that[,,̌ø] = [,ø,]̌^-1for anyø,∈̌().For[,ø] ∈such that(,)̌has no rooted automorphisms for any∈̌(), define [,ø] := {[,ø,]̌ : ∈̌()},otherwise define[,ø] := { e }. Also for[,ø] ∈such that(,)̌has no rooted automorphisms for any∈̌(), one has for each∈̌()that[,]̌ ={[,,̌'̌] : '̌∈()}={[,ø,]̌^-1[,ø,]̌[,,̌'̌] : '̌∈()}={[,ø,]̌^-1[,ø,'̌] : '̌∈()}= T_[,ø,]̌^-1[,ø].If[,ø]is such that somev ∈()is such that(,v)has a rooted automorphism, then then same is true of[,]̌, so that[,]̌ = {e} = T_[,ø,]̌^-1[,ø]for the only vertex=̌ø∈().One may then recoverfrom[,]on the set{e ∈}in the following way. Consider a path_̌0_̌1⋯_̌kin[,]starting and ending at the root. Since there are no rooted automorphisms of(,)on the set{e ∈}one may uniquely chooseX_0:=e,X_1,…,X_k-1,X_k:=e ∈such thatξ_(_̌i,_i) =X_i^-1X_i+1for eachi. Then∏_i=0^k-1ξ_(_̌i,_i) = X_0^-1 X_k=eregardless of the choice of_̌0_̌1⋯_̌kso long as the path starts and ends at the root. It follows that for an arbitrary path_̌0_̌1⋯_̌kone has that∏_i=0^k-1ξ_(_̌i,_i)depends only on the endpoints_̌0and_̌k. Thus, for anyX ∈ = (), consider a path starting at the root∈and ending at:̌=X. Then[,,]̌ = e^-1 X = X. Thus = {[,,]̌ : ∈̌V()} = [,]almost surely on the set{e ∈}. Henceis a witness to the fact thatis an-embedding of[,]= [β, e]under^.The problem of finding which point processeshave Palm versions that are-embeddings of some unimodular network now reduces to finding whichadmit a connected flow-adapted locally finite graph on. It is conjectured that the requirement thatbe connectible in a flow-adapted way is automatic. Supposeis unimodular. Then all flow-adapted simple point processesonare connectible in a flow-adapted way. Finally, some special cases of the conjecture are known to hold. Letbe a flow-adapted simple point process of intensity ∈ (0,∞). Thenis connectible in a flow-adapted way in any of the following situations:*is compact,* =^d,*there exists a point-shift $̋ such thatG^$̋ is connected.Ifis compact then -a.s. ()<∞ and the complete graph onsuffices. If =^d, then Theorem 5.4 in <cit.> shows that the Delaunay graph ofsuffices. If there exists a point-shift $̋ such thatG^$̋ is connected, then the graph G^$̋ suffices.§ APPENDIX: PALM CALCULUS In this appendix, fix a flow-adapted point processof intensity∈ (0,∞). The necessity of this appendix is mostly to prove <Ref> and show how it may be used to translate definitions underand^, a technique that is used extensively is this research.The connection betweenand^is given by the refined Campbell theorem, abbreviated to C-L-M-M for Campbell, Little, Mecke, and Matthes.<cit.> For all f :Ω×→_+ measurable,∫_ f(θ_x^-1,x) (dx) = ^∫_ f(θ_e,x) (dx). It is possible to recover, up to the set on whichis the zero measure, via the following inversion formula. The zero measure onis denoted.<cit.> There exists a bounded measurable K:Ω×→_+ such that∫_ K(θ_e,x) (dx) = 1_≠,and for all K :Ω×→_+ (not necessarily bounded) -a.s. satisfying (<ref>), it holds that[1_≠ f] = ^∫_ f(θ_x)K(θ_x,x) (dx)for all measurable f:Ω→_+.If A ∈ is shift-invariant in the sense thatA = θ_x^-1 A for all x∈, then(A) = 1 ^(A) = 1 (A |≠) = 1.In particular, if { = }⊆ A then(A) = 1 ^(A) = 1.Suppose (A)=1. From the definition of Palm probabilities, for B ∈() such that (B) ∈ (0,∞),^(A) = 1/(B)∫_ 1_x ∈ B1_θ_x^-1∈ A (dx)= 1/(B)∫_ 1_x ∈ B1_A (dx)(shift-invariance of A)= 1/(B) [1_A (B)]= 1/(B)[ (B)] ((A)=1)= 1. Next suppose ^(A) = 1. Then from <Ref> there is measurable K:Ω×→ such that(A ∩{≠}) = [1_≠ 1_A] = ^∫_ 1_θ_x ∈ A K(θ_x,x) (dx) (inversion formula)= ^[1_A∫_ K(θ_x,x) (dx)] (shift-invariance of A)= ^[∫_ K(θ_x,x) (dx)](^(A)=1)= [1_≠· 1] (inversion formula)= (≠).Dividing by (≠) > 0 gives (A |≠) = 1, and if { = }⊆ A, then(A) = (A ∩{≠}) + (A ∩{ = }) = (≠) + ( = ) = 1. Let A ∈.Then^(A) = 1 ((x ∈: θ_x^-1∉ A) = 0) = 1.By replacing A with its complement it is equivalent to show ^(A) = 0 if and only if ((x ∈: θ_x^-1∈ A) > 0) = 0. Note that it is the joint measurability of the action (ω,x)↦θ_xω that lets one conclude for B ∈() that sets like{(x ∈: x∈ B, θ_x^-1∈ A) > 0}are measurable.If ^(A) = 0, then for B ∈() such that (B) ∈(0,∞),0=^(A) = 1/(B)∫_ 1_x ∈ B1_θ_x^-1∈ A (dx)= 1/(B)[(x ∈: x∈ B, θ_x^-1∈ A)].Thus[(x ∈: x∈ B, θ_x^-1∈ A)] = 0 and taking relatively compact B increasing toone finds [(x ∈: θ_x^-1∈ A)] = 0, so ((x ∈: θ_x^-1∈ A) > 0)=0.Conversely, suppose((x ∈: θ_x^-1∈ A) > 0)=0. Then for B ∈() with (B)∈ (0,∞),^(A)= 1/(B)∫_ 1_x ∈ B1_θ_x^-1∈ A (dx)= 1/(B)[(x ∈: x∈ B, θ_x^-1∈ A)]≤1/(B)[(x ∈: θ_x^-1∈ A)]= 0,completing the proof.It is now possible to prove <Ref>, which is restated here for clarity. Let A ∈. Then the following are equivalent:* ^(A) = 1,* ((x∈: θ_x^-1∉ A) =0)=1,* ^((x∈: θ_x^-1∉ A) =0)=1. theorem-1(a)(b):This is the content of <Ref>.(b)(c): This followsfrom <Ref> and the fact that the event {(x∈: θ_x^-1∉ A) =0 } contains { = } and is shift-invariant. To wit, for all y ∈,θ_y^-1ω∈{(x∈: θ_x^-1∉ A) =0 }(θ_y^-1ω,{x ∈:θ_x^-1θ_y^-1ω∉A})=0(ω, {yx: x ∈, θ_yx^-1ω∉ A})(ω, {x ∈, θ_x^-1ω∉ A})= 0ω∈{(x ∈, θ_x^-1ω∉ A)=0}. Fix some measurable space (S,Σ) and a measurable f:Ω→ S. Define F:Ω×→ S byF(ω,x) := f(θ_x^-1ω) for all ω∈Ω, X ∈(ω), and F(ω,x) may be defined arbitrarily otherwise. It will be shown that knowing F up to a - or ^-null set on the support ofis equivalent to knowing f up to a ^-null set. Indeed, suppose f=f', ^-a.s., then it will be shown that the corresponding F,F' agree ,^-a.s. on the support of . By <Ref>, - and ^-a.e. ω∈Ω has for all X ∈(ω) that f(θ_X^-1ω) = f'(θ_X^-1ω), i.e. F(ω,X)=F'(ω,X). Similarly, if either -a.e. or ^-a.e. ω∈Ω is such thatF(ω,X)=F'(ω,X) for all X ∈(ω), thenf(θ_X^-1ω) = F(ω,X) = F'(ω,X)= f'(θ_X^-1ω),for -a.e. or ^-a.e. ω∈Ω, X ∈(ω), so by <Ref> one finds that f = f', ^-a.s. Thus, f may be defined under ^ or F may be defined underor ^, whichever is more convenient. Finally, the following standard result is needed in <Ref>.<cit.> The distribution ofunder ^ is the same as the distribution of +δ_e underif and only ifis a homogeneous Poisson point process with intensityunder . § REFERENCESabbrv | http://arxiv.org/abs/1704.08333v2 | {
"authors": [
"James T. Murphy III"
],
"categories": [
"math.PR",
"37C85, 60G10, 60G55, 60G57, 05C80, 28C10"
],
"primary_category": "math.PR",
"published": "20170426200351",
"title": "Point-shifts of Point Processes on Topological Groups"
} |
SIT: A Lightweight Encryption Algorithm for Secure Internet of Things Muhammad Usman Faculty of Engineering Science and Technology,Iqra University, Defence View,Shaheed-e-Millat Road (Ext.), Karachi 75500, Pakistan.Email: [email protected] Irfan Ahmed and M. Imran Aslam Department of Electronic Engineering,NED University of Engineering and Technology,University Road, Karachi 75270, Pakistan.Email: [email protected], [email protected] Shujaat Khan Faculty of Engineering Science and Technology,Iqra University, Defence View,Shaheed-e-Millat Road (Ext.), Karachi 75500, Pakistan.Email: [email protected] S.M Usman Ali Department of Electronic Engineering,NED University of Engineering and Technology,University Road, Karachi 75270, Pakistan.Email: [email protected] December 30, 2023 =============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== A promising approach for obtaining improved approximation algorithms for Steiner tree is to use the bidirected cut relaxation (BCR). The integrality gap of this relaxation is at least 36/31, and it has long been conjectured that its true value is very close to this lower bound. However, the best upper bound for general graphs was an almost trivial 2. We improve this bound to 3/2 by a combinatorial algorithm based on the primal-dual schema. § INTRODUCTIONIn theproblem, we are given an undirected graph G(V,E), a cost function on the edges c:E →ℚ_≥ 0 and a subset of vertices R ⊆ V called terminals. The objective is to find a tree T of minimum cost c(T) := ∑_e ∈ E(T) c(e), which connects all the terminals. In general, the solution might contain vertices that are not in R. These vertices are often called Steiner vertices. The essence of the problem lies in the case where the vertices are assumed to be the points of a metric space. Thus, we work on the metric completion of the input graph. Theproblem takes a central place in the field of approximation algorithms. It is a natural generalization of the minimum spanning tree problem and appears as a special case of a large number of network design problems that are of great interest.Theproblem appears as an -hard problem in Karp's classic paper <cit.>. In fact, it is -hard to find an approximation ratio better than 96/95 <cit.>. The fact that a minimum spanning tree on the terminals is twice the cost of an optimum solution has been exploited by early authors <cit.> resulting in 2-approximation algorithms (see also <cit.>). The first algorithm breaking the barrier of factor 2 came from Zelikovsky <cit.> followed by a series of work <cit.> resulting in the best purely combinatorial algorithm of factor 1+ln3/2≈ 1.55 by Robins and Zelikovsky <cit.>. More recently, an LP-based algorithm was provided by Byrka et al. <cit.> achieving the approximation ratio ln4≈ 1.39, which stands as the current best result for approximating the problem.All the aforementioned improved algorithms are concerned with the concept of a k-restricted Steiner tree. A component is a tree whose leaves are all from the set of terminals R. A k-component is a component having at most k terminals as leaves. A k-restricted Steiner tree T is a collection of components whose union induces a Steiner tree. In this case, the cost c(T) of T is the total cost of its components, counting the duplicated edges with their multiplicity. A theorem of Borchers and Du <cit.> states that in order to get a good approximation ratio for , it is sufficient to consider k-restricted Steiner trees provided that k is large enough. In particular, let ρ_k be the supremum of the ratio between the cost of an optimal k-restricted Steiner tree and the cost of an optimal Steiner tree with no restrictions.<cit.> Given nonnegative integers r and s satisfying k=2^r+s and s<2^r, we have ρ_k = (r+1)2^r+s/r2^r+s≤ 1+1/⌊log_2 k ⌋.Given an optimal solution with a set of components, a specific component is optimal for the problem induced by its terminals. Besides, contracting the vertices in a specific component creates a tree which is also optimal for the residual problem. Using these and Theorem <ref>, a natural approach for approximatingis then following: Start from a minimum spanning tree on terminals (which is 2-approximate), identify by some greedy selection criteria, a k-component to contract, remove redundant edges and iterate until there is no improvement. The early combinatorial algorithms together with the LP rounding algorithm of ratio 1.39 using the directed-component cut relaxation (DCR) have used this general idea. One obvious disadvantage of this scheme is that it requires computing optimal k-components for large values of k. Although this can be done in polynomial-time, in order to attain the claimed approximation ratios, the running time has to be exorbitantly high. Another disadvantage is that the true nature of the algorithms are not fully apparent as there is, to the best of our knowledge, no way of constructing a tight example on which the algorithm is forced to give a solution with quality close to the proven bound.It is clear by the foregoing discussion that the case for theproblem with respect to approximability is rather unusual from what we have been used to see for some of the other fundamental combinatorial optimization problems. In the ideal setting, we have an LP relaxation for a problem together with an algorithm fully exploiting this relaxation in the sense that its approximation ratio is equal to the integrality gap of the relaxation. The approximation ratio of the algorithm is usually proven by comparing the cost of the solution to the optimum value of the LP relaxation or the value of a dual feasible solution. It is also not difficult to find a tight example for the algorithm, which would provide hints on improving the approximation ratio. A good example of this phenomenon is the more generalproblem with the undirected cut relaxation and the famous algorithms of <cit.>.The most promising approach along this line for theproblem is to use the bidirected cut relaxation (BCR), which we now describe. We first fix a root vertex r ∈ R. We then replace each edge e={u,v}∈ V by two directed edges (u,v) and (v,u) each with cost c(e). For a given cut S ∈ V, we define δ^+(S) = {(u,v) ∈ E: u ∈ S, v ∉ S}, i.e. the set of edges emanating from S. Then the following is a relaxation for theproblem: minimize ∑_e ∈E c(e) x_e (BCR) subject to ∑_e ∈δ^+(S) x_e ≥1,∀S ⊆V ∖{r}, S ∩R ≠∅, x_e ≥0,∀e ∈E. The BCR has been known for more than half a century <cit.>, yielding optimal results for finding minimum spanning trees and in general minimum cost branchings. There are results proving upper bounds on the integrality gap of the BCR for quasi-bipartite graphs. In particular, Rajagopalan and Vazirani <cit.> puts an upper bound of 3/2. Chakrabarty, Devanur and Vazirani <cit.> puts an upper bound of 4/3 (via an algorithm that does not run in strongly polynomial-time) for this class of graphs using a related relaxation they called simplex-embedding LP. Given the scarcity of the results thus far, the question of fully exploiting the BCR remains as a major open problem, especially given that the obtained approximation ratios are still far away from the best lower bound for the integrality gap. This is also related to one of the most important meta-problems in the field of approximation algorithms: the problem of whether one can always fully exploit a relaxation for a given problem, which was already mentioned by Vazirani in his book <cit.> in the Open Problems chapter, while discussing theproblem and whether one can obtain a significantly better approximation ratio using the BCR: “A more general issue along these lines is to clarify the mysterious connection between the integrality gap of an LP-relaxation and the approximation factor achievable using it”.This paper shows that the BCR can be exploited to a certain extent for general graphs. The integrality gap of the BCR is at most 3/2. Furthermore, there exists a polynomial-time primal-dual 3/2-approximation algorithm forbased on the BCR. This result is an improvement upon the previous combinatorial best approximation ratio 1+ln3/2≈ 1.55. It is based on a combinatorial algorithm utilizing the primal-dual schema, and does not require solving an LP. In particular, we extend the canonical primal-dual schema with synchronous dual growth by introducing the following idea: Instead of continuing the growth of all unsatisfied duals, we stop some of the high-degree duals if they already “connect” to another high-degree dual via a Steiner vertex. §.§ Related WorkA catalog of different LP relaxations foris given in <cit.>. Proving an integrality gap of smaller than 2 using one of these relaxations has attracted much attention. An upper bound of 1.55 was proven by <cit.> for the DCR. A simpler proof of this fact using an equivalent formulation appears in <cit.>. This upper bound was improved by <cit.> to 1.39, which also proves an upper bound of 73/60 for quasi-bipartite graphs (graphs in which there are no edges between the Steiner vertices), and provides a deterministic algorithm with the same performance ratio as that of <cit.> by solving the LP only once. This upper bound is also valid for the BCR as will be noted below. However, it is not obtained via the usual primal-dual schema.The relationship between the BCR and the DCR has also been studied. One of the motivations for this is that using the BCR results in much more efficient algorithms due to its size. The first such result was given by <cit.>, which shows that the two relaxations are equivalent on quasi-bipartite graphs. An efficient procedure converting a solution from the BCR to the DCR was then given by <cit.>. This equivalence is strengthened to the case of graphs that do not have a claw on Steiner nodes by <cit.>. These are instances which do not contain a Steiner vertex with three Steiner neighbors.§ THE NAIVE PRIMAL-DUAL ALGORITHM: 2-APPROXIMATIONThe following is the dual of the BCR: maximize ∑_S ⊆V ∖{r}, S ∩R ≠∅ y_S (BCR-D) subject to ∑_S: e ∈δ^+(S) y_S ≤c(e),∀e ∈E, y_S ≥0,∀S ⊆V ∖{r}, S ∩R ≠∅. We first present the straightforward primal-dual algorithm employing the well-known idea of growing dual variables (or simply duals) uniformly and synchronously at unit rate. In the course of the algorithm, given a running solution T, a cut S (synonymously a dual) satisfying the following properties is called a minimal violated set over T: * S ⊆ V ∖{r}, S ∩ R ≠∅; * The degree of S on T, δ_T(S) := |δ^+(S) ∩ T| = 0; * S is minimal with respect to inclusion. The running solution T is initialized to ∅. The minimal violated sets are initially the singleton vertices in R∖{r}. The algorithm uniformly and synchronously grows the duals corresponding to these sets until a constraint in (BCR-D) corresponding to an edge (or simply an edge) becomes tight. The edge is then included into T, and the minimal violated sets over T are recomputed. We conceive the growth of duals as a continuous process over time, with one unit cost covered in one unit of time. Accordingly, the grown duals are said to cover an edge, or some part of an edge it has grown over. The iterations continue until there is a directed path in T from each terminal in R∖{r} to r, i.e., T is feasible. This is followed by the pruning phase, which is an execution of the so-called reverse-delete step utilized by many primal-dual algorithms. It considers deletion of the edges in T ordered with non-increasing inclusion time. It discards an edge e if T ∖{e} remains feasible.We provide two example executions of the algorithm to point out its finer details. The first one underlines its difference from the usual application of the primal-dual schema using the undirected cut relaxation. The second one is an example showing that the algorithm cannot provide an approximation factor better than 2. Figure <ref> shows an input graph together with the initial minimal violated sets considered by the algorithm. We have the configuration in Figure <ref> at time t=1 in which only the selected edges are shown. At this point, the minimal violated sets are {r_1,s}, and {r_2,s}. At time t=3/2, the algorithm arrives at a feasible solution as shown in Figure <ref>. On the edge taken in the interim, there are two distinct duals growing on the edge (s,r), corresponding to the aforementioned minimal violated sets, so that it takes half unit of time to cover it. Note also that the edge (r_2,r) of cost 3/2+ϵ is not selected, which is in contrast to an execution of the primal-dual schema using the undirected cut relaxation. The total value of all the duals (the ones in Figure <ref> and Figure <ref>) is 3, which is equal to the cost of the directed edges from the terminals to r. Thus, the solution returned by the algorithm is optimal. Figure <ref> and Figure <ref> depict another execution. Figure <ref> shows the input graph together with the initial minimal violated sets. The terminals r' and r” are connected to all the Steiner vertices s_1,,s_k with a cost of 1, where k is a large number. The terminal r_i is connected to s_i, for i ∈{1,,k} with a cost of 1. The cost of the edges (s_i,r) is 3-ϵ, for i ∈{1,,k}. The augmentation phase selects all the edges directed to the root. On the edge (s_i, r), there are three duals growing in the time period [1,2-ϵ/3]. These correspond to the cuts {r_i,s_i}, {r',s_1,,s_k}, and {r”,s_1,,s_k}. In the pruning phase, none of the edges of cost 3-ϵ is deleted, and T is shown in Figure <ref>. Its cost is (4-ϵ)k+2, whereas the optimal solution has cost 2k+5-ϵ, as shown in Figure <ref>. § THE ALGORITHM WITH THE ENHANCED PRIMAL-DUAL SCHEMA: 3/2-APPROXIMATIONThe problem with the naive primal-dual schema is that there might be more than one high-degree dual (with respect to T) growing on an edge, as exemplified in the previous section. There is a simple modification, which forbids this, and in turn implies an improved approximation ratio. Let 𝒮_ℓ be the set of duals to be grown at the beginning of an iteration ℓ. Given a dual S ∈𝒮_ℓ, define d(S,ℓ):=|{v ∈ S ∩(⋃_S' ∈𝒮_ℓ∖ S S')}| For a dual S with d(S,ℓ) ≥ 2, the dual degree of S at iteration ℓ, is defined to be Δ(S,ℓ):= |{S' ∈𝒮_ℓ| d(S',ℓ) ≥ 2, S ∩ S' ≠∅}| To give an example, the duals {r',s_1,,s_k} and {r”,s_1,,s_k} in Figure <ref> in the time period [1,2-ϵ/3] have both dual degree 1. Form the graph of duals G_D = (V_D, E_D) at iteration ℓ as follows. V_D consists of duals S with Δ(S,ℓ) ≥ 1, and there is an edge in E_D between S and S' if they share at least one vertex, i.e., S ∩ S' ≠∅. In this case, we say that S and S' are adjacent to each other. The algorithm computes a maximal independent set on the graph of duals at each iteration, and stops the growth of all the duals that are not in this independent set. All the other duals continue to grow, and the pruning phase is applied as usual. The running solution T after the augmentation phase of Algorithm 2 is feasible. Let S be a dual with Δ(S,ℓ) ≥ 1 for some iteration ℓ, and y_S = 0 after the augmentation phase. By the maximality of the independent set computed by the algorithm at each iteration, S is adjacent to some S' which grows in iteration ℓ. Thus, the terminals in S are connected to the root via the edges covered by S'.Note that Algorithm 2 breaks the tight example given in Figure <ref> and Figure <ref>. After time t=1, only one of the duals {r',s_1,,s_k} and {r”,s_1,,s_k} is grown. After this point, all the duals {r_i,s_i} grow concurrently only with this dual, thus having the freedom to grow until t= 5/2-ϵ/2. This results in the inclusion of all the edges (r',s_i) before the costly edges (s_i,r), thereby finding the optimal solution. §.§ Implementation DetailsWe give a straightforward polynomial-time implementation of the algorithm. More compact and faster implementations are possible, which is not the focus of this paper. During the course of the algorithm, we explicitly store all the vertices in a given minimal violated set. Initially, there are |R|-1 such lists, each containing a single vertex in R∖{r}. We also keep the edges in a list of size O(|E|) with respect to non-decreasing inclusion time. In what follows, we do not describe the cost of computing a maximal independent set upon inclusion of a new edge, as it can be performed in linear time, and its complexity is subsumed by that of other operations.We first describe how to select the next edge and how to update the minimal violated sets in the loop of the augmentation phase. In order to find the next tight edge, we keep a priority queue for edges. The key values of the edges are the times at which they will go tight. Initially, all the edges that are not incident to the terminals might be set to ∞, and the key values of the immediately accessible edges are set to their correct values by examining their costs. For each edge, we also keep a list of duals growing on that edge. This is convenient in updating the key values. The initialization of the priority queue takes O(|E|) time. At each iteration, we extract the minimum from the priority queue and update all the other edges in the queue with the information obtained from the new set of minimal violated sets. This takes at most O(|E|log |V|) time, since we consider at most |E| edges to update. Upon inclusion of the edges in an iteration, we update the list of vertices in the sets by performing a standard graph traversal procedure such as BFS, which takes time O((|E|+|V|)|R|)=O(|E||R|). Notice that not all of these sets might be minimally violated, i.e. there might be a set which is a proper subset of another. Initially declare all the sets active, i.e. consider them as minimal violated sets.In order to determine which one of these are actual minimal violated sets, we perform the following operation starting from the smallest cardinality set (assume that the lists keep their sizes). Compare the elements in the set with all the other sets, and if another set turns out to be a strict superset of this set, declare the larger set inactive, i.e. not a minimal violated set. Comparing sets can be performed in expected time O(|V|) by hashing the values of one set and looping over the second set to see if they contain the same elements. Hence, for a single set, we spend O(|V||R|) time in expectation. The total time requirement for this operation is then O(|V||R|^2). If the two sets compared are identical, we merge them into a new minimal violated set and declare it active (See Figure <ref> for an example of this procedure and merging). The number of iterations is at most O(|V|). So the execution of the whole loop takes time O(|E||V|log|V|+|E||V||R|+|V|^2|R|^2) in expectation.For each edge considered in the loop of the pruning phase, the algorithm checks if there is a path between r and all the other terminals even if the edge is discarded. This takes time O((|E|+|V|)|R|=O(|E||R|) with a standard graph traversal algorithm. Since there are at most O(|E|) edges to consider, the total running time is O(|E|^2|R|). Together with the augmentation phase then, the algorithm can overall be implemented in time O(|E||V|log|V|+|E|^2|R|+|V|^2|R|^2).§ PROOF OF THEOREM <REF>We start with the following lemma whose proof crucially uses the fact that the cost of an edge in both directions is the same. Let v and v' be vertices in R ∖{r}. If there is a directed path from v to v' covered by Algorithm 2 at time t, then the directed path from v' to v is covered by Algorithm 2 at the same time t. Let d(u,u') denote the distance between u and u' in V. Define d:=min_u,u' ∈ V d(u,u'). Select some ϵ such that 0 < ϵ < d. Scale all the edge costs in E by |R|/ϵ. In what follows, we argue on these new edge costs. Given w ∈ R ∖{r}, let t(w) denote the time at which all the edges (u,w) ∈ T are covered, i.e., the incoming edges to w. Define t(G):=max_wt(w). We argue by induction on ℓ(G):=⌊ t(G) ⌋. The case ℓ=0 is obvious. Assume the claim holds for some ℓ≥ 1. Let G_w be the graph obtained by reducing the costs of all the edges incident to w by |R|. Since in general there are at most |R|-1 duals grown by the algorithm concurrently, there exists at least one w ∈ R ∖{r} such that ℓ(G_w) ≤ℓ-1. Otherwise, we get a contradiction by reducing all the costs to 0 without changing t(G). Let w be such a vertex with t(G)=t(w). Let u ∈ V such that (u,w) ∈ T, and p be a point on (u,w) such that d(p,w)=|R|. Suppose there is a single dual growing towards w after time t_p at which p is covered. By the choice of w, ⌊ t_p ⌋≤ℓ-1. Assuming the induction hypothesis, there remains the cost of d(p,w)=|R|, which is to be covered by two single duals in both directions, one of them being the dual that starts to grow from w to p. Clearly, it takes the same amount of time to cover them. Suppose now that there is a set S of k ≥ 2 duals growing towards w after time t_p. Let w' ∈ R ∖{r} be another terminal, (u',w') ∈ T, and p' be a point on (u',w') such that d(p',w')=|R|. There is a single dual that starts to grow from w to p, and a single dual that starts to grow from w' to p'. Since there are k duals that grow from p to w concurrently, to complete the induction, it remains to see that there are k duals that grow from p' to w' concurrently. By the algorithm and the edge costs, all the k-1 duals in S excluding the one containing w' are among these duals together with the one originating from w, making a total of k duals. Conversely, any other dual among these duals must be in S. Thus, there are exactly k duals that grow from p' to w' concurrently, which completes the induction and the proof. Let S_1,,S_k be cuts, e = (v,w) be an edge such that e ∈δ^+(S_i) ∩ T, and δ_T(S_i) ≥ 2, for all i ∈{1,,k}. Then the following hold: * v ∉ R. * There exists at most one S_i growing on e at a given time t. Consider the union U of δ^+(S_i) ∩ T, for all i ∈{1,,k}, so that e = (v,w) ∈ U. Assume first for a contradiction that v ∈ R. It is clear by definition of the cuts that there exists v' ∈ R ∩ S_i for some i such that there is a directed path from v' to v established by the algorithm at some time t. By Lemma <ref>, there is a directed path from v to v' established by the algorithm at time t. Note next that e is included into the running solution T after time t, since otherwise we have a contradiction to the existence of S_i. Given this, e becomes redundant, since its deletion is considered before the edges in S_i, and there exists a feasible solution including these edges and excluding e (See Figure <ref>). Given that v ∉ R, the rest of the claim immediately follows by the algorithm, as there are no adjacent duals of dual degree at least 1 growing at the same time (See Figure <ref>). To complete the proof, we show that the ratio of c(T) with the total value of the duals constructed by Algorithm 2 is bounded by 3/2. Lemma <ref> implies that a cut S with δ_T(S) ≥ 2 grows concurrently on an edge with only duals of degree 1 on T. Let S_1,,S_k be such degree-1 duals. Suppose S and all the S_i grow concurrently on the same set of edges e_i=(v_i,w_i), respectively, for a time period [t_1,t_2], where t=t_2-t_1 is maximal in the sense that for t' > t, there is no set of edges on which S and S_i grow concurrently on. Recall by Lemma <ref> that none of the v_i is in R. This implies the following: * There exist degree-1 duals S_i' ⊆ S_i that have grown for a time period of length at least t alone, i.e., without any other dual growing concurrently with it (See Figure <ref> for an illustration). * The mapping (S,S_i) ↦ (S_i',[t_1,t_2]) is an injection, so that each (S_i',[t_1,t_2]) is considered for a separate set of S and S_i.Otherwise, by the argument in the proof of Lemma <ref>, we derive a contradiction to the fact that S and S_i grow concurrently for a time period of length t, since in that case S would transform into a larger cut before completing the time period. An illustration for this is given in Figure <ref>. Given this, the cost covered by y_S, y_S_i and y_S_i' is 3kt, whereas the sum of these duals is (2k+1)t. This analysis excludes all the remaining degree-1 duals, which obviously pay for the cost they cover. Since the constructed dual solution is feasible, this completes the proof of Theorem <ref>. § TIGHT EXAMPLEA tight example for the algorithm is given in Figure <ref> and Figure <ref>, where k is a large number. There is a single high-degree dual in this example, and the cost of the solution found by the algorithm is (3-ϵ)k+1. The optimal solution has cost 2k+2-ϵ.plain | http://arxiv.org/abs/1704.08680v9 | {
"authors": [
"Ali Çivril",
"Muhammed Mirza Biçer",
"Berkay Tahsin Tunca",
"Muhammet Yasin Kangal"
],
"categories": [
"cs.DS"
],
"primary_category": "cs.DS",
"published": "20170427175020",
"title": "An Improved Integrality Gap for Steiner Tree"
} |
Given an involutive automorphism θ of a Coxeter system (W,S), let ℑ(θ) ⊆ W denote the set of twisted involutions. We provide a minimal set of moves that can be added to the braid moves, in order to connect all reduced S-expressions (also known as admissible sequences, reduced I_θ-expressions, or involution words) for any given w ∈ℑ(θ). This can be viewed as an analogue of the well-known word property for Coxeter groups. It improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types due to Hu, Zhang, Wu, and Marberg. Point-shifts of Point Processes on Topological Groups James T. Murphy III James T. Murphy III, The University of Texas at Austin, [email protected] ================================================================================================================§ INTRODUCTIONA fundamental result in the theory of Coxeter groups is the word property, due to Matsumoto <cit.> and Tits <cit.>. Given a Coxeter system (W,S) and two generators s,s' ∈ S such that ss' has finite order m(s,s'), let _s,s' denote the word ss's of length m(s,s'). Operating on words in the free monoid S^*, call the replacement of _s,s' by _s',s a braid move. A word in S^* representing w ∈ W is called a reduced word for w if w cannot be represented by a product of fewer elements of S.[Word property ]Let (W,S) be a Coxeter system and let w ∈ W. Then any two reduced words for w can be connected by a sequence of braid moves. Given an involutive automorphismof (W,S), let()={w ∈ W |(w)=w^-1}be the set of twisted involutions. Note that (𝕀) is the set of ordinary involutions in W. Like W, () can be described in terms of “words” and “reduced expressions”. Namely, with the alphabet ={| s ∈ S}, every word in ^* represents a twisted involution, and conversely, every twisted involution can be represented in this way; see Section <ref> for the details. The shortest possible representatives of w ∈() are called reduced -expressions for w; let _(w) be the set of all such representatives.In combinatorial approaches to Coxeter group theory, the study of (reduced) words is central. Analogously, (reduced) -expressions form the combinatorial foundation for (); systematic study of () was initiated by Richardson and Springer <cit.> because of its connections with Borel orbit decompositions of symmetric varieties. In the former case, <ref> is a cornerstone. The goal of this paper is to identify the correct analogue in the latter setting.In the context of -expressions, a braid move is the replacement ofby . It is known that braid moves preserve _(w) (it follows from <cit.> when W is a Weyl group, and from <cit.> for general W), but they do not in general suffice to connect all of _(w). For example, when W is the symmetric group S_4, s_i=(i,i+1), and =𝕀, the reduced -expressions _2_3_1_2 and _3_2_1_2 both represent the same (ordinary) involution, namely the longest group element, but they cannot be connected using only braid moves.For the purposes of the introduction, a half-braid move amounts to replacingwith ' (orwith ') if these are the first letters in a reduced -expression, and m(s,s')=3 (m(s,s')=4). A more general definition is given in Section <ref>.When W is of type A and =𝕀, Hu and Zhang <cit.> proved that braid moves and half-braid moves suffice to connect _(w). They extended this result to types B and D (also with =𝕀) in <cit.> ( 3.10 and 4.8); in either of these types, braid moves, half-braid moves, and one special, additional move (involving more than two different letters) are enough. With Wu <cit.>, they presented a similar assertion for type F_4.Recently, Marberg <cit.> generalised these results to all finite and affine Coxeter groups, for arbitrary . Aided by a computer, he identified moves that are necessary and sufficient in such groups. He also presented several conjectures about the general situation.Before <cit.>, Hamaker, Marberg, and Pawlowski <cit.> provided an answer for general W and . Under certain circumstances, they allow replacingwith ' even if the number of letters is less than m(s,s'). The set of all such involution braid relations is always sufficient for connecting _(w).As noted in <cit.>, in type A, <cit.> requires the addition of many more moves to the braid moves than just the half-braid moves, so it reduces to a weaker statement than <cit.>. The same is true in types B and D: the set of all involution braid relations is much larger than necessary for connecting _(w). Our main result provides, for any Coxeter system (W,S) with any involutive automorphism , a minimal set of moves that must be added to the braid moves in order to connect _(w). In fact, we show that Marberg's moves from <cit.> suffice in arbitrary Coxeter systems with arbitrary . In particular, we confirm his aforementioned conjectures. Our approach is different from Marberg's, and thus provides an independent road to his results as a special case.An initial move is the replacement of one element in _(v) (for some v ∈()) by another, in the beginning of a reduced -expression for some w ∈(). Both half-braid moves and the two special Hu-Zhang moves in types B and D are initial moves. In the former case, v is the longest element of a dihedral parabolic subgroup (I_2(3) in types A and D, and I_2(3) or I_2(4) in type B), and in the latter case, v is the longest element of a parabolic subgroup of type B_3 or D_4, respectively.The following is the main result of the present paper. It describes precisely the initial moves that are necessary and sufficient to add to the braid moves, in order to connect all of _(w). As such, it can be viewed as an analogue of <ref> for ().If JS generates a finite parabolic subgroup W_J of W, let w_0(J) denote its longest element. When (J)=J, _J is the restriction ofto W_J.Let (W,S) be a Coxeter system with an involutive automorphism , and let w ∈(). Then any two reduced -expressions for w can be connected by a sequence of braid moves and initial moves that replace x with y if x,y ∈_(w_0(J)) for some -stable JS. The following W_J and _J are necessary and sufficient: * W_J of type A_3 with _J ≠𝕀;* W_J of type B_3;* W_J of type D_4 with _J=𝕀;* W_J of type H_3;* W_J of type I_2(m), 3 ≤ m<, with _J=𝕀;* W_J of type I_2(m), 2 ≤ m<, with _J ≠𝕀. It should be observed that the listed W_J (apart from I_2(2)) are precisely the finite types for which the complement of the Coxeter graph is disconnected. This fact is explained in the proof of Lemma <ref>.Note that there is no need to specify _J in types B_3 and H_3 since they admit no non-trivial Coxeter system automorphism. Also, observe that the reducible dihedral group I_2(2) does not appear in the list when _J=𝕀. The corresponding move is allowed, but it is a braid move in this case. We conclude this section with an outline of the paper. In Section <ref>, the necessary definitions and previous results are recalled. Then, in Section <ref>, we prove <ref>. It follows from the proof that, in fact, only one move of each type is needed, if x and y are appropriately chosen. An explanation is given in Section <ref>, where a list of such x and y is also provided. This clarifies how the results from <cit.> are recovered from ours. Finally, in Section <ref>, consequences of <ref> in some special cases are discussed.§ NOTATION AND PRELIMINARIESAs general references on Coxeter group theory, the reader could consult <cit.> or <cit.>. We assume familiarity with the basics but reiterate some of it here in order to agree on notation. Some useful tools for dealing with twisted involutions are also reviewed. For finite Coxeter groups, they appear in Richardson and Springer <cit.>. In the general case, everything can be found in <cit.> from which our notation is taken.Let (W,S) be a Coxeter system. In the sequel, we shall often find it important to distinguish notationally between words in the monoid S^* and the elements of W they represent. From now on, a sequence of generators inside square brackets indicates an element of S^*, whereas a sequence without brackets is an element of W. Thus, if s_i ∈ S, then [s_1s_k]∈ S^* and s_1s_k∈ W.For w ∈ W, its length ℓ(w) is the smallest integer k such that w=s_1s_k for some s_i ∈ S; [s_1s_k] is then called a reduced word (or a reduced expression) for w.Letbe an involutive automorphism of (W,S). Recall the set of twisted involutions () defined in the introduction. Like W, () can be described using words, this time in the monoid ^*.The free monoid ^* acts from the right on the set W byw=wsif (s)ws=w, (s)wsotherwise,and w_1 _k=(((w_1)_2))_k. Observe that w=w for all w ∈ W and all s ∈ S. We write _1 _k for e_1 _k, where e ∈ W is the identity element. The orbit of e under this action is precisely ():[]We have()={_1 _k |_1,,_k ∈, k ∈}.Again, we shall henceforth use square brackets to indicate that a sequence of letters should be interpreted as an element of the monoid. If w=_1 _k for some _i ∈, we call [_1 _k] ∈^* an -expression for w. It is reduced if no -expression for w consists of fewer than k elements of . In this case, ρ(w)=k is the rank of w.Given an -expression ϵ∈^*, (ϵ) denotes the element of S^* obtained by expanding ϵ according to <ref>. Let W=S_4 with s_i=(i,i+1). If =𝕀, ([_1_2_3_2])=[s_3s_2s_1s_2s_3s_2], which is a reduced expression for the longest element w_0 ∈ W. On the other hand, if ≠𝕀 (i.e., (s_i)=s_5-i), we have ([_1_2_3_2])=[s_2s_2s_3s_1s_2s_3s_2], which is not reduced. Some remarks about the terminology are in order. We follow the notation used in <cit.>. What we call an -expression is the right handed version of an “I_*-expression” <cit.>. Reduced -expressions are the same as (again, right handed versions of) “admissible sequences” <cit.> and “involution words” <cit.>. One should furthermore note that Richardson and Springer mostly use a monoid action, denoted by * in <cit.>, which is different from that defined in <ref> above although they coincide on reduced -expressions (i.e., admissible sequences).Let [_1 _k] be a reduced -expression for w ∈(). Then the twisted absolute length of w, denoted ℓ^(w), is the number of indices i ∈ [k] such that _1 _i-1_i=_1 _i-1s_i. It follows from <cit.> that this definition of ℓ^(w) is independent of the choice of reduced -expression for w. When =𝕀, ℓ^(w) coincides with the absolute length of w, i.e., the smallest number of reflections whose product is w, see <cit.>.Given w ∈ W, let D_R(w)={s ∈ S |ℓ(ws)<ℓ(w)} and D_L(w)={s ∈ S |ℓ(sw)<ℓ(w)} be the sets of right descents and left descents, respectively. Observe that D_R(w)=D_L((w)) if w ∈().[]If w ∈() and s ∈ S, then ρ(w)=ρ(w) ± 1. Moreover, the following statements are equivalent: * s ∈ D_R(w);* ρ(w)=ρ(w)-1;* some reduced -expression for w ends with .The exchange property is a fundamental property of Coxeter groups; in fact, it characterises them among groups generated by involutions, see <cit.>.[Exchange property] If [s_1s_k] is a reduced expression and s ∈ D_R(s_1s_k), then s_1s_ks=s_1s_i-1s_i+1 s_k for some i ∈ [k]. An analogous result holds for twisted involutions:[]If [_1 _k] is a reduced -expression and s ∈ D_R(_1 _k), then _1 _k=_1 _i-1_i+1_k for some i ∈ [k]. The subword property, due to Chevalley <cit.>, characterises the Bruhat order on W.[Subword property]Let u,w ∈ W and suppose [s_1s_k] is a reduced expression for w. Then u ≤ w in the Bruhat order if and only if [s_i_1 s_i_m] is a reduced expression for u, for some 1 ≤ i_1<<i_m ≤ k. Again there is an analogous result for twisted involutions:[]Let u,w ∈() and suppose [_1 _k] is a reduced -expression for w. Then u ≤ w in the Bruhat order if and only if [_i_1_i_m] is a reduced -expression for u, for some 1 ≤ i_1<<i_m ≤ k. § A WORD PROPERTY FOR REDUCED -EXPRESSIONS §.§ LemmasIn order to establish <ref>, it is crucial to understand when w ∈() admits reduced -expressions that end with the longest possible alternating sequence of two given right descents.Given w ∈() and s,s' ∈ D_R(w), s ≠ s', say that w is (s,s')-maximal if it has a reduced -expression of the form[_1 _k'_m(s,s')] ∈_(w).Since braid moves in a reduced -expression preserve the twisted involution, any (s,s')-maximal element is also (s',s)-maximal. A more general assertion is provided by Lemma <ref> below.The main goal of this subsection is to provide a characterisation of the (s,s')-maximal elements w ∈(); this is Lemma <ref> below. A closely related description, in terms of the minimal coset representative of w in W/W_{s,s'}, can be gleaned from the proof of <cit.>.[Namely, with the notation of <cit.>, said proof shows that the number m(s,t,_a) is the length of the longest sequence s,t,s, which can be attached to a, if s,t ∉ D_R(δ̂_*(a)), without destroying reducedness.] In particular, the following lemma is a consequence of that description. We provide a short independent argument.Suppose w ∈() and s,s' ∈ D_R(w). Ifϵ=[_1 _k'_ letters] ∈_(w)withmaximal, thenϵ'=[_1 _k_ letters] ∈_(w). By Lemma <ref>, a reduced -expression ϵ” for w=w is obtained by deleting a letter from ϵ and appendingat the end. Maximality ofand rank considerations show that the deleted letter necessarily is the one immediately to the right of _k. Hence, ϵ”=ϵ'. Let ϵ be a reduced -expression. Then (ϵ) is also reduced. If, moreover,ϵ=[_1 _k'_ letters],then(ϵ)=[(s')(s)(s') _ letters(_1 _k) s'ss'_ letters]for some -2 ≤≤.By <cit.>, ℓ=2ρ-ℓ^. The first claim thus follows from <ref>. For the second, observe that for any u ∈() and s ∈ S, ℓ^(u)>ℓ^(u) implies D_R(u) ⊃ D_R(u). Since ϵ is reduced, this means that at most two of its rightmostletters (namely, the first and the last) can contribute to ℓ^(_1 _k '). This yields the second claim. If ws=(s')w for some w ∈() and s,s' ∈ S, then ws'=(s)w.We have ws'=(s'(w))^-1=(((s')w))^-1=((ws))^-1=(s)w. Let w ∈() and s,s' ∈ D_R(w), and suppose w is not (s,s')-maximal. Then ws=(s)w if and only if ws'=(s')w.By Lemmas <ref> and <ref>, w has reduced -expressionsϵ=[_1 _k'_] andϵ'=[_1 _k_]with corresponding reduced words(ϵ)=[(s')(s)(s') _(_1 _k) s'ss'_]and(ϵ')=[(s)(s')(s) _(_1 _k) ss's_]for some ≤. Now, ws=(s)ww=wsw=(w)s, which is the case if and only if (ϵ') does not begin with (s). Similarly, ws'=(s')w precisely when (ϵ) does not begin with (s'). Since <m(s,s'), and (ϵ) and (ϵ') represent the same element w, they begin with different letters. Hence, it cannot be that exactly one of ws=(s)w and ws'=(s')w holds. The promised characterisation of (s,s')-maximal elements can now be delivered. It is the main technical ingredient in the proofs of Lemmas <ref> and <ref>, which are needed in order to establish <ref>.Given w ∈(), let s,s' ∈ D_R(w) with s ≠ s'. Then w is (s,s')-maximal if and only if either * m(s,s')=2 and ws ≠(s')wor * m(s,s') ≥ 3 and {ws,ws'}≠{(s)w,(s')w}. Choose a reduced -expression for w,ϵ=[_1 _k'_] ∈_(w),withmaximal.The proof is divided into three parts. In the first part, we show that if {ws,ws'}≠{(s)w,(s')w}, then =m(s,s'). Then, in the second part, we show that if ws=(s')w and ws'=(s)w, then <m(s,s'). Finally, we show that if ws=(s)w and ws'=(s')w, then =m(s,s') if m(s,s')=2, and <m(s,s') if m(s,s') ≥ 3.First, assume {ws,ws'}≠{(s)w,(s')w}. In order to obtain a contradiction, suppose <m(s,s'). Then, by Lemmas <ref> and <ref>, (s)w ∉{ws,ws'} and (s')w ∉{ws,ws'}. From Lemma <ref>, we know that (ϵ) is reduced, and since ws' ≠(s')w, we have(ϵ)=[(s')(s)(s') _(_1 _k) s'ss'_]for some 1 ≤≤.Consider the element ws. By the exchange property, a reduced word for ws is obtained by deleting a letter in (ϵ). Since <m(s,s'), s is not a right descent of _1 _ks'ss'. Hence the deleted letter is one of the leftmostgenerators. It is not the very first one, because ws ≠(s')w. It cannot be any of the other ones not adjacent to (_1 _k) since ℓ(ws)=ℓ(w)-1. Hence it is the generator immediately to the left of (_1 _k). Thus,w=(ws)s=(s')(s)(s') _-1(_1 _k) s'ss's_+1.Now considering ws', and continuing in this way, we finally obtainw=(s')(s)(s') _+-m(s,s')(_1 _k) s'ss's _m(s,s').Observe that (s) ∈ D_L(w) and +-m(s,s')<m(s,s'). By the exchange property and similar reasoning as above, this however means that(s)w=(s')(s)(s') _+-m(s,s')(_1 _k) s'ss's _m(s,s')-1,which is equal to either ws or ws', a contradiction. Hence, =m(s,s').Next, assume ws=(s')w and ws'=(s)w. We have(ϵ)=[(s')(s)(s') _(_1 _k) s'ss'_]with 1 ≤≤. Thus,w=(s')ws=(s)(s') _-1_1 _k s'ss's_+1.Since (ϵ) is reduced, <m(s,s').Finally, suppose ws=(s)w and ws'=(s')w. This implies that(ϵ)=[(s)(s')(s) _(_1 _k) s'ss'_]for some 0 ≤<. In case ≥ 1, we obtainw=(s)ws=(s')(s) _-1_1 _k s'ss's_+1,which exactly as above leads to <m(s,s'), and hence, m(s,s') ≥ 3. If =0, then w=_1 _ks'ss'=_1 _kss's, whence m(s,s')=. By Lemma <ref>, ≤ 2, so m(s,s')==2. It is convenient to encode the information conveyed by Lemma <ref> in a graph. To this end, let G(w,) be the graph on vertex set D_R(w) in which {s,s'} is an edge if and only if w is (s,s')-maximal.Figure <ref> displays G(w,) when w is the longest element in S_4. In this case, D_R(w)={s_1,s_2,s_3} with s_i=(i,i+1). We have ws_1=s_3w, ws_2=s_2w, and ws_3=s_1w. Let us use Lemma <ref> to determine the edge set. Consider first =𝕀. Then ws_1=(s_3)w, so w is not (s_1,s_3)-maximal. On the other hand, ws_1 ∉{(s_1)w,(s_2)w} and ws_3 ∉{(s_2)w,(s_3)w}, whence w is (s_1,s_2)- and (s_2,s_3)-maximal. Now assume ≠𝕀. Then ws_1 ≠(s_3)w, so w is (s_1,s_3)-maximal. However, ws_1=(s_1)w and ws_2=(s_2)w, whence w is not (s_1,s_2)-maximal. Similarly, w is not (s_2,s_3)-maximal either.§.§ Proof of the main resultBefore commencing to prove <ref>, let us describe the basic strategy. Starting with a list comprised of all braid moves, we work by induction on the rank of a given twisted involution w. If the moves that are so far collected do not suffice to connect all the reduced -expressions for w, sufficient new ones are added to the list. As we shall see, new moves must be added precisely when G(w,) is disconnected. Moreover, this can only happen when w is the longest element of a finite, -stable parabolic subgroup W_J (see Lemma <ref>). Using the classification of finite Coxeter groups, we obtain the list of such W_J (see Lemma <ref>).We now turn to the actual proof. First, the connectedness of G(w,) is investigated.Let w ∈(). If G(w,) is disconnected, then w is the longest element w_0(J) of a finite, -stable parabolic subgroup W_J, JS.Say that s ∈ S passes through w if ws=s̃w for some s̃∈ S. It is an immediate consequence of Lemma <ref> that s ∈ D_R(w) is adjacent to all other vertices of G(w,) if s does not pass through w. Hence, if G(w,) is disconnected, then every right descent of w passes through w. We claim that in this case, w is the longest element of the (hence finite) parabolic subgroup generated by S(w), where S(w) is the set of generators that appear in some reduced word for w; by the word property, it is independent of the choice of reduced expression. Indeed, if [s_1s_k] is a reduced word for w, we havew=s_1s_k=s̃_ks_1s_k-1=s̃_k-1s̃_ks_1s_k-2==s̃_2s̃_3 s̃_ks_1,implying that S(w)=D_R(w), a property which is equivalent to w=w_0(S(w)), see <cit.>.Clearly, S(w) is -stable for any w ∈(). Therefore, G(w,) is always connected, unless, possibly, w=w_0(J) for some -stable JS. If W is finite with longest element w_0, then G(w_0,) is disconnected in exactly the following cases: * W of type A_3 with ≠𝕀;* W of type B_3;* W of type D_4 with =𝕀;* W of type H_3;* W of type I_2(m), 3 ≤ m<, with =𝕀.* W of type I_2(m), 2 ≤ m<, with ≠𝕀.Moreover, when it is disconnected, it has exactly two connected components.Letdenote the involutive automorphism x ↦ w_0xw_0. It is convenient to reformulate the conditions stated in Lemma <ref> in terms of . Namely, notice that w_0s ≠(s')w_0 is equivalent to (s) ≠(s') and that {w_0s,w_0s'}≠{(s)w_0,(s')w_0} if and only if {(s),(s')}≠{(s),(s')}.Now,preserves irreducible group components. Hence, by Lemma <ref>, s is non-adjacent to at most one vertex in a different component, s'=((s)) being the only candidate. If s' is indeed in a different component, theninterchanges the component containing s with that which contains s'. In this case, s is adjacent to every vertex in its irreducible group component. It follows that the only reducible Coxeter system with disconnected G(w_0,) is I_2(2) withinterchanging the two generators.From now on, let us restrict attention to finite, irreducible Coxeter systems. It is known, see, e.g., <cit.>, thatis not the identity involution if and only if the system has an even exponent, i.e., if and only if it is of type A_n (n ≥ 2), D_2m+1, E_6, or I_2(2m+1), for integral m.First, suppose =. Then Lemma <ref> shows that G(w_0,) is the complement of the Coxeter graph. That is, s and s' are connected if and only if they commute. The finite, irreducible Coxeter systems with disconnected complement of the Coxeter graph are those of rank 2 and 3 and that of type D_4. With the requirement = they comprise the following list: A_3 (≠𝕀), B_3, D_4 (=𝕀), H_3, I_2(2m) (=𝕀), and I_2(2m+1) (≠𝕀). In all cases, the complement of the Coxeter graph has two components.Second, assume ≠. Only type D_4 admits distinct non-trivial, involutive automorphisms; in this type, however,is trivial. Thus, exactly one ofandmust be the identity involution. Let ψ∈{,} denote the non-trivial involution. By Lemma <ref>, s and s' are adjacent in G(w_0,) unless either ψ(s)=s', or s and s' are both fixed by ψ and m(s,s') ≥ 3. It follows that G(w_0,) is disconnected if and only if |S|=2. This accounts for the remaining dihedral cases I_2(2m) (≠𝕀) and I_2(2m+1) (=𝕀), and concludes the proof. Recall that an initial move is the replacement of one element in _(v) (for some v ∈()) by another, in the beginning of a reduced -expression for some w ∈(). Let a list initial move be an initial move in which v is the longest element of a -stable parabolic subgroup of one of the types listed in Lemma <ref>.Having established all the necessary preliminaries, we are now in position to prove the main result. Fix w ∈(). The result is trivially true if w is the identity element. In order to induct on the rank, assume the result holds for all twisted involutions of rank less than k=ρ(w). Consider first two reduced -expressions for w, ϵ and ϵ', that end with the same letter. By the induction hypothesis, they are related by a sequence of braid moves and list initial moves of rank at most k-1 that never interfere with the last letter; let ϵind.ϵ' indicate this property.If s and s' are connected by an edge in G(w,), then there are two reduced -expressions ϵ and ϵ' for w which are related by a braid move and end withand ', respectively; let ϵbr.ϵ' indicate this relationship.Now, choose two arbitrary reduced -expressions for w, ϵ=[_1 _k] and ϵ'=[_1' _k']. If there is a path s_k=z_0 → z_1 →→ z_t=s_k' in G(w,), we have reduced -expressions for w related in the following way:ϵ ind. [u_0_1_0_m(z_0,z_1)] br. [u_0_0_1_m(z_0,z_1)] ind. [u_1_2_1_m(z_1,z_2)] br. [u_1_1_2_m(z_1,z_2)] ind.ind. [u_t-1_t_t-1_m(z_t-1,z_t)] br. [u_t-1_t-1_t_m(z_t-1,z_t)] ind.ϵ',where the u_i are reduced -expressions. Hence, ϵ and ϵ' are related by a sequence of braid moves, and list initial moves of rank at most k-1. On the other hand, if there is no such path connecting s_k and s_k', then it follows from Lemmas <ref> and <ref> that ϵ and ϵ' are related by a list initial move of length k. § NECESSARY LIST INITIAL MOVESIn a reduced -expression, any operation that trades a prefix representing w_0(J) for another is among those listed in Theorem <ref>, if W_J is one of the specified parabolic subgroups. However, it is far from necessary to allow all of these list initial moves if the only objective is to connect all reduced -expressions that represent the same twisted involution. In fact, it follows from the proof of Theorem <ref> that it is necessary and sufficient to allow the replacement of one fixed prefix whose last letter is in one connected component of G(w_0(J),), whenever this graph is disconnected, by one whose last letter is in the (only) other connected component. We next present one possible list of such replacements.Let (W,S) be a Coxeter system with an involutive automorphism . Suppose JS is -stable. Consider the following moves, with generator indexing as in Figure <ref>: * When W_J is of type A_3 and _J ≠𝕀:[_2_3_1_2]⟷[_2_3_2_1] * When W_J is of type B_3:[_1_2_3_1_2_1]⟷[_1_2_3_2_1_2] * When W_J is of type D_4 and _J=𝕀:[_4_2_1_3_2_1_3_4]⟷[_4_2_1_3_2_1_4_3] * When W_J is of type H_3:[_1_3_2_1_3_2_1_3_2]⟷[_1_3_2_1_3_2_1_2_3] * When W_J is of type I_2(m), 3 ≤ m<, and _J=𝕀:[_1_2_1 ]_⌈(m(s_1,s_2)+1)/2⌉ letters⟷[_2_1_2 ]_⌈(m(s_1,s_2)+1)/2⌉ letters * When W_J is of type I_2(m), 2 ≤ m<, and _J ≠𝕀:[_1_2_1 ]_⌈m(s_1,s_2)/2⌉ letters⟷[_2_1_2 ]_⌈m(s_1,s_2)/2⌉ letters If w ∈(), then any two reduced -expressions for w can be connected by a sequence of braid moves and initial moves of the listed kinds. Of course, many other choices than those stated in <ref> are possible. A different selection can be found in Marberg <cit.>. Those we have chosen are involution braid relations in the sense of <cit.>; recall the discussion about those from the introduction. Thus, the theorem conveys a minimal set of involution braid relations to add to the ordinary braid relations in order to connect all reduced -expressions of any twisted involution. As an example, Figure <ref> illustrates how <ref> connects the various reduced -expressions for the longest element in type A_3.§ SPECIAL CASESThere are many situations where there are few list initial moves possible. In this final section, we present some consequences of the main result that arise in such settings. A half-braid move is a list initial move of type I_2(m), see the list given in <ref>.Let (W,S) be a Coxeter system with an involutive automorphism , and let w ∈(). Suppose (W,S) does not have a -stable parabolic subgroup W_J of type B_3, D_4, or H_3 with _J=𝕀, nor one of type A_3 with _J ≠𝕀. Then any two reduced -expressions for w can be connected by a sequence of braid moves and half-braid moves. Marberg <cit.> conjectured that the conclusion of <ref> holds wheneverfixes no element of S. Since the hypotheses are satisfied in that situation, we have confirmed the conjecture. Marberg's 1.8 and 1.9 also follow directly from <ref>.[<cit.> predicts that every _(w) can be connected using moves that satisfy certain assumptions. These assumptions imply that all braid moves and all list initial moves can be performed.]Another interesting consequence of <ref> concerns right-angled Coxeter systems, i.e., those that satisfy m(s,s') ∈{2,} for all generators s ≠ s'. Observe that if =𝕀, no list initial moves are available in a right-angled group. Let I(W) denote the set of involutions in W, and recall that (𝕀)=I(W).If W is right-angled and =𝕀, then the map [s_1s_k] ↦ [_1 _k] sends reduced words to reduced -expressions, and it induces a bijection W → I(W).In order to obtain a contradiction, assume that [s_1s_k] is reduced and [_1 _k] is not. If k is minimal among all expressions with this property, s_k ∈ D_R(_1 _k-1). Hence, by <ref>, [_1 _k-1] is related to a reduced -expression ending with _k by a sequence of braid moves. But then the same sequence of braid moves transforms [s_1s_k-1] into a word ending with s_k. This contradicts the reducedness of [s_1s_k], proving the first claim. It is then clear that [s_1s_k] ↦ [_1 _k] provides a bijection between reduced words and reduced -expressions. Since it respects braid moves, the second claim follows. Given a subset XW, let (X) denote the poset on X with the order induced by the Bruhat order on W.If (W,S) is right-angled, then (W) and (I(W)) are isomorphic as posets.This follows from <ref> together with Lemmas <ref> and <ref>. The only finite, right-angled (W,S) are of type A_1 ×× A_1. In these groups, I(W)=W, so <ref> is not particularly amusing. If W is infinite, however, the inclusion I(W) ⊂ W is proper. Hence a copy of (W) sits inside (W) as a proper subposet. Thus (W) contains an infinite sequence of induced subposets P_i, all of them isomorphic to (W), such that(W)=P_0 ⊃ P_1 ⊃ P_2 ⊃. amsplain | http://arxiv.org/abs/1704.08329v1 | {
"authors": [
"Mikael Hansson",
"Axel Hultman"
],
"categories": [
"math.CO"
],
"primary_category": "math.CO",
"published": "20170426195414",
"title": "A word property for twisted involutions in Coxeter groups"
} |
Representativity and waist of cable knots Román Aranda, Seungwon Kim and Maggy Tomova=============================================== We study the incompressible surfaces in the exterior of a cable knot and use this to compute the representativity and waist of most cable knots. § INTRODUCTION AND DEFINITIONS Let K⊂ S^3 be a knot and let S be a closed orientable surface containing K. Following <cit.>, we define the representativity of the pair (S,K) as the minimal intersection number |K∩∂ D| over all the compressing discs for S in S^3. Denoted as r(S,K), the representativity of (S,K) measures how many times the knot is “wrapping" the surface S. The representativity of K is the maximal number of r(F,K) among all the closed orientable surfaces F⊂ S^3 that contain the knot. In other words, r(K)=max_K⊂ Fmin_D|∂ D∩ K| This knot invariant had been studied by M. Ozawa and it is known for several classes of knots: torus knots, 2-bridge knots, and composite knots. There are also bounds for algebraic knots and knots with Conway spheres, see <cit.>. Recently, Kindred determined that all alternating knots have representativity 2, <cit.>. Similarly, if F⊂ S^3 is a closed incompressible surface in the exterior of a knot K, the waist of K, waist(F,K) of (F,K) is the minimum number of intersection |D∩ K|, between K and the compressing discs for F in S^3. The waist of K is the maximum number waist(F,K) among all the closed incompresible surfaces in the exterior of K. In other words, waist(K)=max_K⊂ Fmin_D|∂ D∩ K|It is known that there are many classes of knots with waist one: 2-bridge knots <cit.>, torus knots <cit.>, twisted torus knots with twists on 2-strands <cit.>, small knots, alternating knots <cit.>, almost alternating knots <cit.>, toroidally alternating knots <cit.>, 3-braid knots <cit.>, Montesinos knots <cit.>, and algebraically alternating knots <cit.>. Let V a solid torus in S^3. A cable knot K is an embedded circle in ∂ V with slope p/q with respect to the Seifert framing for V, such that gcd(p,q)=1 and q>1. The number q is called the index of K and V is called the companion of K. In this paper we study the behavior of both invariants, representativity and waist, under the cabling operation. Let K be a (p,q)-cable with companion solid torus V. The boundary torus of V allows us to obtain the bounds r(K)≥ p and waist(K)≥ p· waist(J), where J is a core of V.We will show that, most of the time (see below), these estimates are exact. A cable knot is called inconsistent if the knot on the companion torus is not a boundary slope of the companion knot; i.e., a cable knot is inconsistent if there is no essential surface in the complement of the torus whose boundary is the knot.Let K be an inconsistent cable knot with index p. Then r(K) = p and the companion torus is the unique surface that realizes the representativity. In <cit.>, Pardon showed that the distortion of a knot, δ(K), is at least 1/160r(K). Thus, the following corollary follows immediately: Let K be an inconsistent cable knot with index p. Then δ(K) ≥p/160. Let K be an inconsistent cable knot with index p. Then waist(K)=p· waist(J) where J is a companion knot for K.Recall that by <cit.>, the set of boundary slopes for any knot is finite therefore the above theorem applies to almost every cable.In particular for (p,q)-torus knots the set of all boundary slopes is {0,pq} and for 2-bridge knots the set of boundary slopes is a subset of the even integers, <cit.>. Let J be either a 2-bridge or torus knot. Then for every cable K of index p along J, r(K)=p and waist(K)=p· waist(K). Every cable knot K with a pattern that is a 2-bridge knot or a torus knot has a finite, non-integer slope therefore K is inconsistent. § MAIN RESULTS A properly embedded surface F in a 3-manifold M is incompressible if it does not have any compressing discs. The surface is peripheral if it is boundary parallel. A sphere in a 3-manifold is essential if it does not bound a ball. The following two lemmas are well known:<cit.> Every connected orientable incompressible surface in a solid torus is either a peripheral disc, a peripheral annulus, or a meridian disc.<cit.> Suppose F is a connected, orientable, incompressible surface properly embedded in a thickened torus T × I. Then F is one of the following:* A peripheral disc,* A peripheral annulus,* γ× I where γ is an essential simple loop of T,* T ×{i}, where i ∈ I.We will often use the following set up. Suppose K is a cable knot with companion torus V and let T=∂ V. Let η(K) be an open regular neighborhood of K in S^3 and E(K) = S^3 - η(K). Consider a regular neighborhood of T in S^3. The boundary of this neighborhood consists of two tori, T̃ will be the component contained outside of V. Let Ṽ be the 3-manifold bounded by T̃ that contains V with η (K) removed. Observe that Ṽ is a solid torus intersected with the exterior of K. Let F̃=F-Ṽ. Let A_K = T - η(K). The boundary of A_K partitions ∂ E(K) into two annuli ∂^+ E(K) and ∂^- E(K). Let T^± = A_K ∪∂^± E(K). Without loss of generality, we assume that T^- is the torus which bounds a solid torus V^- ⊂ V which does not contains K. Also let W be a thickened torus which is bounded by T^+ and T̃. Let F^W = F ∩ W and F^- = F ∩ V^-.Let K be a cable knot with companion torus V and let F be an incompressible and boundary incompressible surface in S^3-η(K) possibly with boundary. There exists an isotopy of F which minimizes (|F∩T̃| ,|F ∩ A_K|) so that F satisfies the following: * Every component of F̃ is incompressible and boundary incompressible in S^3- Ṽ. * Every component of F^W is incompressible in W. * Every component of F^- is incompressible in V^-. * Every component of F ∩Ṽ is incompressible and boundary incompressible in Ṽ. First, we show that we can isotope F so that F̃ is incompressible in S^3 - Ṽ.Isotope F so that |F ∩T̃| is minimal. Suppose that F̃ is compressible in S^3- Ṽ with a compressing disc D. Since F is incompressible, ∂ D bounds a disc D' in F such that D' ∩T̃≠∅. Since E(K) is irreducible, D ∪ D' bounds a ball. So we can isotope D' to remove the intersection D' ∩ T, reducing the intersection T̃∩ F, which contradicts |F∩T̃| is minimal. Every component of F̃ is not boundary compressible since the only properly embedded incompressible, boundary compressible surface in solid torus complement is peripheral annulus, which can be isotoped to reduce the number of intersections.Note that by the same argument, F ∩Ṽ is incompressible in Ṽ.Now, we show that F ∩Ṽ is boundary incompressible. Suppose that F ∩Ṽ is boundary compressible. Since F is already boundary incompressible in E(K), there is a boundary compressing disk D with ∂ D = α∪β, α⊂T̃ and β⊂ F∩Ṽ an essential arc. Since |F∩T̃| is minimal, α must connect the same component σ of F∩T̃, and because T̃ is a torus and F is orientable, α∪σ must bound a bigon on T̃. By slightly pushing into Ṽ the union of this bigon with D, we obtain a disc for F∩Ṽ. But F ∩Ṽ is incompressible so exist a disc in F∩Ṽ with the same boundary that the latter compression, inducing a parallelism between β into the boundary of F ∩Ṽ, a contradiction. Second, we show that we can isotope F ∩Ṽ so that F^W is incompressible in W. In order to do this, isotope F∩Ṽ, fixing F ∩T̃ so that |F∩ A_K| is minimal. Then, using the same argument of the beginning of the proof, F^- is incompressible in V^-, using the fact that |F ∩ T^-| is minimal. It follows that F^W is also incompressible in W. Suppose K is an inconsistent cable knot with a companion solid torus V. We will use the notation we established in the paragraph before Lemma <ref>. Let F be a surface such that K ⊂ F and assume F has been isotoped to satisfy the conclusions of Lemma <ref>.As K⊂ F and K ⊂Ṽ, there are two possibilities; either F is contained in Ṽ or it intersects T̃.F ⊂Ṽ, i.e., F ∩T̃ = ∅. In this case we will prove that r(K, F) ≤ p. Let f : Ṽ→ S^1 be a Morse function and assume that K and F are in Morse position and K has no critical points with respect to f. The preimage of every regular value of f is a meridian disc for Ṽ. Let D_t be one such disc and consider its intersection with F. As F ∩T̃= ∅ this intersection consists entirely of simple closed curves. As D_t contains exactly p points of K and K ⊂ F, there are exactly p points of F contained in the curves F ∩ D_t. In particular, an innermost such curve contains at most p points of K. Note that such an innermost curve bounds a compressing disc for F and therefore r(K, F) ≤ p. F ∩T̃≠∅. In this case we will prove that r(K, F) ≤ 2.Consider F ∩T̃ which is a collection of simple closed curves. If any of these curves are inessential on T̃, an innermost such would bound a compressing disc for F disjoint from Kand so we can assume that F ∩T̃ is a set of parallel loops essential in T̃. By Lemma <ref> it follows F̃, F^W and F^- are all incompressible and F̃ is boundary incompressible.Recall that (|F∩T̃| , |F ∩ A_K|) has been minimized. Claim:∂ A_K ∩ F ≠∅. Proof of Claim: Suppose ∂ A_K ∩ F =∅. Note that F intersects A_K only in loops by the hypothesis of the Claim and these loops are essential by Lemma <ref>. The surface F_W has boundary on both T̃ and T_+. By the connectivity of F and Lemma <ref>, there exists a non-peripheral annulus. Since one of the boundaries of this annulus is isotopic to K, every boundary of F_Won T_+ is isotopic to K. Therefore F∩T̃ is isotopic to K. We showed that F ∩T̃ bounds a incompressible surface in S^3 - Ṽ, so each component of F ∩T̃ is a boundary slope of a companion knot. However, this is not possible because K is inconsistent. By the Claim, we may assume that ∂ A_K ∩ F ≠∅.Note that A_K ∩ F is a set of simple loops and properly embedded simple arcs on A_K. By the minimality assumption in Lemma <ref>, allloops of intersection must be essential. Suppose that there exists a simple arc in A_K ∩ F which bounds a disc in A_K. Then an outermost such arc bounds a boundary compressing disc for F, which implies that r(K,F) ≤ 1. Hence, we can assume that every simple arc of intersection is essential, hence a spanning arc in A_K. This also implies that there are no essential simple loops of intersection. Let Γ^± =F^±∩ T^±. Then Γ^± are sets of essential simple loops on the tori T^±. By Lemma <ref> F^- is incompressible in V^- and therefore Γ^- is a set of essential simple loops in T^- and so F^- is a set of either peripheral annuli, or meridian discs. Suppose that F^- is a set of peripheral annuli. Take an outermost such annulus 𝒜 so that it cuts V^- into two solid tori, one of them not containing F^-. Notice that ℬ=F∩η(K)is also a peripheral annulus which coincides with 𝒜 in pairs of consecutive arcs in ∂ ^- E(K). When pushing both 𝒜 and ℬ towards ∂ ^- E(K), either their projections on∂ ^- E(K) coincide or not. If the projections do not coincide, taking a curve like in Figure <ref> we obtain a compression for F intersecting K once. If some of their projections coincide, since 𝒜 and ℬ agree on their boundaries, both parallelisms induce a compressing disc for F which intersects K once. Hence r(K,F)=1.Suppose now that F^- is a set of meridian discs. Consider two adjacent meridian discs D_1 and D_2. The curves ∂ D_1 and ∂ D_2 cobound an annulus 𝒜⊂ T^- with interior disjoint from F. As above, take the peripheral annulus ℬ=F∩η (K) and project it towards ∂ E(K). Let C be a square component of 𝒜∩ A_K; each of the arc components of ∂ C∩∂ E(K) either coincide with the projection of ℬ or not. If one arc coincides and the other does not, we can find a disc such that its boundary intersects K geometrically and algebraically twice (see Figure <ref>). Hence, this disc is a compressing disc of F, and r(K, F) ≤ 2. If none of the arcs coincide, we can find two discs which intersect K geometrically twice but algebraically 0 times (see figure <ref>). However, the boundaries of the two discs intersect once, so both are compressing discs of F. Hence, r(K,F) ≤ 2. Notice that if both arcs of ∂ C∩∂ E(K) coincide, we can replace D_2 with the other meridian disc in F^- adjacent to D_1 and reach the same conclusion. Therefore r(F,K)≤ 2.Let F⊂ S^3 be a closed surface disjoint from K such that F is incompressible in S^3-η(K). We will continue to use the notation established in the paragraph before Lemma <ref>. If F∩Ṽ=∅ then F is contained in the complement of Ṽ. If D is a compressing disc for F, it will intersect Ṽ in meridians and so |D∩ K|=p· |D∩ J| where J is the core of V. Thus waist(K,F)=waist(J,F)· p.Suppose now that F∩Ṽ≠∅, and recall that F^-, F^W and F̃ are incompressible in V^-, W and S^3-Ṽ, respectively (Lemma <ref>). Moreover, since F is disjoint from K, ∂(A_k)∩ F= ∅. By Lemma <ref>, F^W is either isotopic to T×{i} (in such case waist(K,F)=p), or F^W contains a non-peripheral annulus. Suppose the latter, then the components of F∩ T are isotopic to K and, since Kis inconsistent, F̃ must be the union of boundary parallel annuli in S^3-V which can be pushed inside Ṽ. Hence, F^W ∪F̃≃ F^W is the union of annuli parallel to annuli in T^+. Finally, Lemma <ref> implies that F^- is union of peripheral annuli and so F=F^W∪ F^- is parallel to ∂η (K), and thus waist(F,K)=1. 10almost alternating knotsC. Adams, J. Brock, J. Bugbee, T. Comar, K. Faigin, A. Huston, A. Joseph and D. Pesikoff, Almost alternating links, Topology Appl. 46 (1992) 151–165.toroidally alternating knotsC. Adams, Toroidally alternating knots and links, Topology 33 (1994) 353– 369. S(K) is finiteA. Hatcher, On the boundary curves of incompressible surfaces, Pac. J. Math. 99 (1982), 373–377.inc surfs 2bridge knotsA. Hatcher and W. Thurston, Incompressible surfaces in 2-bridge knot complements, Inv Math 79 (1985), 225–246.torus knotsW. Jaco, Lectures on Three Manifold Topology, AMS Conference board of Math. No. 43, 1980.KindT. Kindred, Alternating links have representativity 2, arXiv:1703.03393.braid knotsM. T. Lozano and J. H. Przytycki, Incompressible surfaces in the exterior of a closed 3-braid I, surfaces with horizontal boundary components, Math. Proc. Camb. Phil. Soc. 98 (1985) 275–299. alternating knotsW. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984) 37–44. twisted torus knotsK. Morimoto, Essential surfaces in the exteriors of torus knots with twists on 2-strands, preprintMontesinos knotsU. Oertel, Closed incompressible surfaces in complements of star links, Pacific J. Math. 111 (1984) 209– 230.Algebraically alternating knotsM. Ozawa, Rational structure on algebraic tangles and closed incompressible surfaces in the complements of algebraically alternating knots and links, Topology Appl. 157 (2010), 12, 1937–1948. OzawaM. Ozawa, Bridge position and representativity of spatial graphs, Topology Appl. 159 (2012), 4, 936–947. PardonJohn Pardon, On the distortion of knots on embedded surfaces, Ann. of Math. (2) 174 (2011), 1, 637–646.W1F. Waldhausen,Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II, Invent. Math. 3 (1967), 308–333.ibid.4 (1967), 87–117.W2F. Waldhausen, On irreducible 3-manifolds which are sufficiently large, Ann. of Math. (2), (1968), 56–88. | http://arxiv.org/abs/1704.08414v1 | {
"authors": [
"Román Aranda",
"Seungwon Kim",
"Maggy Tomova"
],
"categories": [
"math.GT",
"57M25, 57M27"
],
"primary_category": "math.GT",
"published": "20170427024100",
"title": "Representativity and waist of cable knots"
} |
Existence and Stability of Four-Vortex Collinear RelativeEquilibria with Three Equal Vorticities Brian [email protected] Gareth E. [email protected]. of Mathematics and Computer Science College of the Holy Cross December 30, 2023 ================================================================================================================================================================== We study collinear relative equilibria of the planar four-vortex problem where three of the four vortex strengths are identical. The S_3 invariance obtained from the equality of vorticities is used to reduce the defining equations and organize the solutions into two distinct groups based on the ordering of the vortices along the line.The number and type of solutions are given, along with a discussion of the bifurcations that occur.The linear stability of all solutions is investigated rigorously and stable solutions are found to exist for cases where the vorticities have mixed signs. We employ a combination of analysis and computational algebraic geometry to prove our results.Key Words:Relative equilibria, n-vortex problem, linear stability, symmetry § INTRODUCTION The planar n-vortex problem is a Hamiltonian system describing the motion of n point vortices in the plane acting under a logarithmic potential function.It is a well-known model for approximating vorticity evolution in fluid dynamics <cit.>.One of the most fruitful approaches to the problem is to study stationary configurations, solutions where the initial configuration of vortices is maintained throughout the motion.As explained by O'Neil <cit.>, there are four possibilities: equilibria, relative equilibria (uniform rotations), rigidly translating configurations, and collapse configurations. Much attention has been given to relative equilibria since numerical simulations of certain physical processes (e.g., the eyewall of hurricanes <cit.>) often produce rigidly rotating configurations of vortices. Analyzing the stability of relative equilibria improves our understanding of the local behavior of the flow;it also has some practical significance given the persistence of these solutions in numerical models of hurricane eyewalls. Other physical examples are provided in <cit.>. There are many examples of stable relative equilibria in the planar n-vortex problem.Perhaps the most well known is the equilateraltriangle solution, where three vortices of arbitrary circulations are placed at the vertices of an equilateral triangle.If the sum of the circulations does not vanish, then the triangle rotates rigidly about the center of vorticity.This periodic solution is linearly (and nonlinearly) stable provided that the total vortex angular momentum L =∑_i < jΓ_i Γ_j is positive <cit.>, where Γ_i ∈ℝ - {0} represents the circulation or vorticity of the ith vortex. Other stable examples include the regular n-gon for 4 ≤ n ≤ 7 (equal-strength circulations required) <cit.>; the 1 + n-gon for n ≥ 3 (a regular n-gon with an additional vortex at the center) <cit.>; the isosceles trapezoid <cit.>; a family of rhombus configurations <cit.>; and configurations with one “dominant” vortex and n small vortices encircling the larger one <cit.>. In <cit.>, Aref provides a comprehensive study of three-vortex collinear relative equilibria, finding linearly stable solutions for certain cases when the vortex strengths have mixed signs. The rhombus configuration studied in <cit.> and some particular solutions of the (1+3)-vortex problem discussed in <cit.>provide some additional examples of stable solutions with circulations of opposite signs. Relative equilibria can be interpreted as critical points of the Hamiltonian H restricted to a level surface of the angular impulse I.This gives a promising topological viewpoint to approach the problem <cit.>. If all vortices have the same sign, then a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum of H restricted to I = <cit.>.Moreover, because I is a conserved quantity, a technique of Dirichlet's applies to show that any linearly stable relative equilibrium with same-signed circulations is also nonlinearly stable. In this paper we apply methods from computational algebraic geometry to investigatethe existence and stability of collinear relative equilibria in the four-vortex problem.To make the problem more tractable, we restrict to the case where three of the vortices are assumed to have the same circulation.Specifically, if Γ_i is the circulation of the ith vortex, then we assume that Γ_1 = Γ_2 = Γ_3 = 1 and Γ_4 = m, where m ∈ℝ - {0} is treated as a parameter. Solutions to this problem come in groups of six due to the invariance that arises from permuting the three equal-strength vortices. We use this invariance to simplify the problem considerably, obtaining a complete classification of the number and type of solutions in terms of m. We also provide a straight-forward algorithm to rigorously find all solutions for a fixed m-value. When counting the number of solutions, we follow the usual convention (inherited from the companion problem in celestial mechanics)of identifying solutions that are equivalent under rotation, scaling, or translation.In other words, we count equivalence classes of relative equilibria.In general, there are n!/2 ways to arrange n vortices on a common line, where the factor of 1/2 occurs because configurations equivalent under a 180^∘ rotation are identified. The 12 possible orderings in our setting are organized into two groups.Group I contains the 6 arrangements where the unequal vortex (vortex 4) is positioned exterior to the other three;Group II consists of the 6 orderings where vortex 4 is located between two equal-strength vortices.We show that for any m > -1/2, there are exactly 12 collinear relative equilibria,one for each possible ordering of the vortices.As m decreases through -1/2, the solutions in Group II disappear;there areprecisely 6 solutions for each m ∈ (-1, -1/2], one for each ordering in Group I.There are no collinear relative equilibria for m ≤ -1. We also consider the linear stability of the collinear relative equilibria in the planar setting.Due to the integrals and symmetry that naturally arise for any relative equilibrium, there are always four trivial eigenvalues 0, 0, ± i (after a suitable scaling). For the case n = 4, there are four nontrivial eigenvalues remaining that determine stability.We explain how the nontrivial eigenvalues can be computed from the trace T and determinant D of a particular 2 × 2 matrix and provide useful formulasfor T and D as well as conditions that guarantee linear stability. By applying these formulas and conditions to our specific problem, we are able to rigorously analyze the linear stability of all solutions in Groups I and II. We show that the Group II solutions are always unstable, with two real pairs of nontrivial eigenvalues ±λ_1, ±λ_2. The Group I solutions go through two bifurcations, at m = m_c ≈ -0.0175 and m = m^∗≈ -0.8564.These important parameter values are roots of a particular sixth-degree polynomial in m with integer coefficients.For m > m_c, the Group I solutions are unstable, with two real pairs of eigenvalues.At m = m_c, these pairs merge and then bifurcate into a complex quartuplet ±α± i β for m ∈ (m^∗, m_c).The Group I solutions are linearly stable for m ∈ (-1, m^∗) and spectrally stable at m = m^∗. The linear stability of the Group I solutions is somewhat surprising since four of the six solutions limit on a configuration with a pair of binary collisions as m → -1^+. This problem has recently been explored in <cit.>, where the intent was to classify all relative equilibria, not just the collinear configurations. Unfortunately, there are some errors in this paper.For example, Theorem 4 claims the existence of two families of rhombus configurations. However, this violates the main theorem in <cit.>, which states that a convex relative equilibrium is symmetric with respect to one diagonal if and only if the circulations of the vortices on the other diagonal are equal. To obtain a rhombus, there must be two pairs of equal-strength vortices, one pair for each diagonal(see Section 7.4 in <cit.> for the complete solution).If three vortices have equal circulations, then the only possible rhombus relative equilibriumis a square. In this article we treat the collinear case in much greater depth than in <cit.> and focus on the linear stability of solutions (the stability question is not considered in <cit.>).Much of our work relies on the theory and computation of Gröbner bases andwould not be feasible without the assistance of symbolic computing software. Computations were performed using Maple <cit.> andmany results were checked numerically with Matlab <cit.>. The award-winning text by Cox, Little, and O'Shea <cit.> is an excellent referencefor the theory and techniques used in this paper involving modern and computational algebraic geometry.The paper is organized as follows.In the next section we introduce relative equilibria and provide the set up for our particular family of collinear configurations. We then explain how the solutions come in groups of six and use the invariance inherent in the problem to locate, count, and classify solutions in terms of the parameter m.In Section 3 we provide the relevant theory and techniques for studying the linear stability of relative equilibria in the planar n-vortex problem. Applying these ideas in our specific setting, we obtain reductions that reduce the stability problem to the calculation of two quantities, T and D.This leads to the discovery of linearly stable solutions and the bifurcation values that signify a change in eigenvalue structure.§ COLLINEAR RELATIVE EQUILIBRIA WITH THREE EQUAL VORTICITIES We begin with some essential background.The planar n-vortex problem was first described as a Hamiltonian system by Kirchhoff <cit.>. Let z_i ∈ℝ^2 denote the position of the ith vortex and let r_ij = z_i - z_j represent the distance between the ith and jth vortices. The mutual distances r_ij are useful variables. The motion of the ith vortex is determined byΓ_iż_i=J ∂ H/∂ z_i= J∑_j ≠ i^n Γ_i Γ_j/r_ij^2(z_j - z_i),1 ≤i ≤ n ,whereJ= [01; -10 ] is the standard 2 × 2 symplectic matrix andH = -∑_i<jΓ_i Γ_j ln (r_ij)is the Hamiltonian function for the system. The total circulation of the system is Γ = ∑_iΓ_i, and as long as Γ≠ 0, the center of vorticityc =1/Γ∑_i Γ_i z_i is well-defined.§.§ General facts about relative equilibria A relative equilibrium is a periodic solution of (<ref>) where each vortex rotates about c with the same angular velocity ω≠ 0. Specifically, we havez_i(t) = c + e^- ω J t (z_i(0) - c) ,i ∈{1, … , n}.It is straight-forward to check that the mutual distances r_ij in a relative equilibrium are unchanged throughout the motion, so that the initial configuration of vortices is preserved. Upon substitution into system (<ref>), we see that the initial positions of a relative equilibrium must satisfy the following system of algebraic equations- ω Γ_i (z_i- c) =∂ H/∂ z_i=∑_j ≠ i^n Γ_i Γ_j/r_ij^2 (z_j - z_i), i ∈{1, …, n}.Suppose that z_0 = (z_1(0), …, z_n(0)) represents the initial positions of a relative equilibrium. Although it is a periodic solution, it is customary to treat a relative equilibrium as a point z_0 ∈ℝ^2n (e.g., a fixed point in rotating coordinates). We will adopt this approach here. From equation (<ref>), we see that any translation, scaling, or rotation of z_0 leads to another relative equilibrium (perhaps with a different value of c or ω). Thus, relative equilibria are never isolated and it makes sense to consider them as members of an equivalence class [[ z_0 ]], where w_0 ∼ z_0 provided that w_0 is obtained from z_0 by translation, scaling, or rotation. The stability type of z_0 is the same for all members of [[ z_0 ]].Reflections of z_0 are also relative equilibria (e.g., multiplying the first coordinate of c and each z_i by -1), but these will not be regarded as identical when counting solutions.The quantity I=∑_i=1^nΓ_i z_i - c^2=1/Γ∑_i < jΓ_i Γ_j r_ij^2can be regarded as a measure of the relative size of the system. It is known as the angular impulse with respect to the center of vorticity, the analog of the moment of inertia in the n-body problem.The angular impulse is an integral of motion for the planar n-vortex problem <cit.>. One important property of I is that relative equilibria (regarded as points in ℝ^2n) are critical points of the Hamiltonian restricted to a level surface of I.This can be seen by rewritingsystem (<ref>) as∇ H(z) + ω/2∇ I(z) = 0 ,where ∇ is the usual gradient operator. Here we treat the constant ω/2 as a Lagrange multiplier. This gives a very useful topological approach to the study of relative equilibria.The main result of <cit.> is that, for positive vorticities, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum of H restricted to I =. Using equation (<ref>), it is straight-forward to derive the formula ω = L/I. §.§ Defining equationsWe now focus on four-vortex relative equilibria whose configurations are collinear, that is, all vortices lie on a common line.To make the problem tractable, we assume that three of the four vortex strengths are identical. Set Γ_1 = Γ_2 = Γ_3=1, and Γ_4=m, where m ∈ℝ is a parameter. Without loss of generality, we take the positions of the relative equilibrium to be on the x-axis, z_i = (x_i, 0), and translate and scale the configuration so that x_1 = -1 and x_2 = 1.This produces a simpler system to solve than other approaches (e.g., setting c = 0 and ω = 1).It also helps elucidate the inherent symmetries in the problem.In our set up, the center of vorticity c and angular vorticity ω will vary,but the coordinates of the first two vortices will remain fixed (see Figure <ref>).According to equation (<ref>), a relative equilibrium in this form must satisfy the following system of equations:ω(-1 - c) + 1/2 + 1/x_3 + 1 +m/x_4 + 1= 0, ω(1 - c) - 1/2 + 1/x_3 - 1 +m/x_4 - 1= 0, ω(x_3 - c) - 1/x_3 + 1 - 1/x_3 - 1 +m/x_4 - x_3= 0, ω(x_4 - c) - 1/x_4 + 1 - 1/x_4 - 1 -1/x_4 - x_3= 0.Let the numerators of the left-hand side of each equation above be denoted by f_1, f_2, f_3, and f_4 respectively, and append the two polynomialsf_5 = u(x_4 - 1) - 1f_6 = v(x_4 + 1) - 1,in order to eliminate solutions with collisions (i.e., x_4 = ± 1).Let F be the polynomial ideal generated by f_1, …, f_6 in ℚ[ω, c, u, v, x_4, x_3, m].Computing a Gröbner basis of F, denoted by GB, with respect to the lexicographic order ω > c > u > v > x_4 > x_3 > m, yields a basis with 26 elements. The first of these is a 12th-degree polynomial in x_3 with coefficients in m, given by[ P(x_3, m) = (m+1)(2m+1)(m+2)^2 x_3^12 - (32m^5 + 224m^4 + 635m^3 + 873m^2 + 576m + 144)x_3^10;+ (640m^5+4066m^4+10126m^3+12546m^2+7776m+1944)x_3^8;- (3776m^5 + 23984m^4 + 60278m^3+ 75042m^2 + 46656m +11664)x_3^6; + (5760m^5+40806m^4+115191m^3+158841m^2+104976m+26244)x_3^4; -27m^2(96m^3+464m^2+717m+351)x_3^2 + 54m^4. ]Although P appears intimidating to analyze, we will use the equality of the vorticities and invariant group theory to factor it into four cubic polynomials in x_3. Before performing this reduction, we repeatedly apply the Extension Theorem to insure that a zero of P can be extended to a full solution of system (<ref>).The third term in GB is Q(x_4, x_3, m) =192m^3(4m + 5)(x_3^2 + 3)x_4 +q_1(x_3, m),where q_1 is a polynomial in the variables x_3 and m.If m ≠ 0 and m ≠ -5/4, then the Extension Theorem applies to extend a zero of P, call it (x_3^∗, m^∗), to a solution (x_4^∗, x_3^∗, m^∗).Moreover, since Q is linear in x_4, there is a unique such extension. The 20th term in GB is (m+1)(c(m+3) - x_3 - mx_4),which implies thatc =x_3 + mx_4/m+3as long as m ≠ -1, -3.As expected, this agrees with the formula for the center of vorticity in our set up.Similar arguments work to extend any zero of P uniquely to a solution of the full system (<ref>).Note that we have not ruled out the case that x_3^∗ = x_4^∗, a collision between the third and fourth vortices.We have proven the following lemma. Fix an m ∈ℝ with m ≠ -3, -5/4, -1, 0.Then any solution x_3^∗ to P = 0 can be extended uniquely to a full solution of system (<ref>).§.§ Classifying solutionsSince vortices 1, 2, and 3 have the same vorticity, we can interchange their positions (a relabeling of the vortices) to create a new relative equilibrium. However, because x_1 = -1 and x_2 = 1 are always assumed, it is necessary to apply a scaling and translation in order to convert a relabeled solution into our specific coordinate system.This creates a map between solutions of system (<ref>).To make these ideas precise, we will keep track of how the vortices are arranged under different permutations. If vortices i, j, k, and l are positioned so that x_i < x_j < x_k < x_l, then the corresponding ordering is denoted ijkl. To illustrate the inherent invariance in system (<ref>), suppose that we have a relative equilibriumwith coordinates x =(x_1, x_2, x_3, x_4)=(-1, 1, a, b),1 < a < b.This corresponds to the ordering 1234.Relabeling the vortices in the order 3124 gives another relative equilibrium, but with coordinatesx' =(x_1', x_2', x_3', x_4')=(1, a, -1, b),which does not match our setup.The linear map ϕ(x_i) = 2 x_i - a - 1/a - 1 satisfies ϕ(1) = -1 and ϕ(a) = 1.Consequently, applying ϕto each entry in (<ref>) will convert x' into the correct form.Because ϕ is a scaling and translation, the resultingcoordinate vector,ϕ(x') =( -1, 1, 3+a/1 - a, -2b+1+a/1 - a), is also a relative equilibrium, and its first two coordinates match our setup. This gives a new solution to system (<ref>), one with ordering 3124. The above argument demonstrates an important invariance for the ideal F. It shows that for a fixed value of m, if (x_3, x_4) = (a,b) is a partial solution in the variety of F, then so is (x_3, x_4) = (3+a/1 - a, -2b+1+a/1 - a). Put another way, if F = F ∩ℚ[x_3, x_4, m] is an elimination ideal, then F is invariant under the mapS(x_3, x_4) =( 3 + x_3/1 - x_3, -2x_4 + 1 + x_3/1 - x_3),after clearing denominators.(Here we can assume that x_3 ≠ 1 because x_3 = 1 is a collision between vortices 2 and 3.)Another symmetry, which is easy to discern, arises by reflecting all four positions about the origin:R(x_3, x_4) =(-x_3, -x_4).However, this operation reverses the ordering of the vortices (e.g., ordering 1234 maps to 4321), and thus requires that vortices 1 and 2 be interchanged in order to insure that x_1 = -1 and x_2 = 1 is maintained (e.g., ordering 4321 becomes 4312). We can also consider the composition of R and S to generate additional invariants for F.As expected, this yields a total of six invariants (including the identity) for F, as R and S generate a group of order six that is isomorphic to S_3, the symmetric group on three symbols.Let G be the group generated by the maps R and S under composition, where R and S are given by (<ref>) and (<ref>), respectively. Then G is isomorphic to S_3 and the elimination ideal F = F ∩ℚ[x_3, x_4, m] arising from system (<ref>) is invariant under G. Consequently, solutions to system (<ref>) come in groups of six.As explained above, F is invariant under both R and S.This was confirmed using Maple by checking that,for each polynomial p in a Gröbner basis of F, p(R(x_3, x_4)) and the numerator of p(S(x_3, x_4)) are also in F. It follows that F is also invariant under any composition of these maps.Let e represent the identity function e(x_3, x_4) = (x_3, x_4). We compute that R^2 = R ∘ R = e, S^3 = S ∘ S ∘ S = e, and (R ∘ S)^2 = e.This is sufficient to show that G is isomorphic to S_3.Since F is invariant under G and the order of G is six, it follows that one solution in the variety of F leads to five others. The only possible exception occurs when G has fixed points, that is, points in ℝ^2 that are mapped to the same place under different group transformations. A straight-forward calculation reveals that the only possible fixed points are (0,0), (3,1), (-3,-1), (-1,b), and (1,b), where b ∈ℝ is arbitrary. However, each of these corresponds to a collision between two vortices, and is thus excluded.Therefore, the six solutions generated by G are distinct.Suppose that we have a relative equilibrium solution with (x_3, x_4) = (a, b), where 1 < a < b.This solution has ordering 1234.Applying the transformations from G generates five additional solutions, each with a different ordering of the vortices. These solutions and their corresponding orderings are shown in the first two columns of Table <ref> and will be denoted as Group I. Likewise, for a solution with 1 < b < a, which corresponds to the ordering 1243, there are five other solutions generated by G whose orderings are displayed in the third column of Table <ref>. These solutions will be referred to as Group II. The 12 orderings from the union of the two groups are the only allowable orderings because we have assumed that x_1 < x_2, thereby eliminating half of the 24 permutations in S_4. Note that each of the orderings in Group I have vortex 4 positioned exterior to the three equal-strength vortices, while for Group II, the fourth vortex always lies between two of the equal-strength vortices. Recall that S_3 is isomorphic to D_3, the dihedral group of degree three.Since G ≃ S_3 ≃ D_3, the transformations S and S^2 from Table <ref> correspond to rotations, while the remaining non-identity elements represent reflections. §.§ Using invariant group theory to find solutions Based on the discussion in the previous section, we can apply invariant group theory to rigorously study the solutions to system (<ref>) in terms of the parameter m.Let r_1 = x_3, r_2 = (x_3-3)/(x_3+1), and r_3 = (3+x_3)/(1-x_3) denote the three values of x_3 corresponding to the group elements e, S^2, and S, respectively (the rotations).The cubic polynomial with these three roots should be a factor of P(x_3, m), the first polynomial in the lex Gröbner basis GB arising from system (<ref>). We introduce the coordinates σ, τ, and ρ, defined by the elementary symmetric functions on the roots r_1, r_2, and r_3:σ=r_1 + r_2 + r_3, τ=r_1 r_2+ r_1 r_3 + r_2 r_3,ρ=r_1 r_2 r_3 .Consider the ideal in ℚ[σ, τ, ρ, x_3, m] generated by equations (<ref>), (<ref>), and (<ref>) (after clearing denominators)and the twelfth-degree polynomial P(x_3, m).Computing a lex Gröbner basis (denoted GB^∗) with respect to the ordering τ < σ < x_3 < ρ < m yields a basis with four polynomials, the first two of which areP_1 =(m+1)(2m+1)(m+2)^2 ρ^4- m^2(32m^3 +152m^2 + 239m + 117)ρ^2 +54m^4, P_2 =x_3^3 + ρ x_3^2- 9x_3 - ρ.The polynomial P_2 is just the expanded version of equation (<ref>).The fact that P_1 is even in ρ is expected from the reflection symmetry R. Note that (ρ, x_3) ↦ (-ρ, -x_3) is a symmetry for P_2 = 0. This Gröbner basis calculation effectively factors P into the product of four cubics in the form of P_2.Indeed, using Maple, we confirm thatP =(m+1)(2m+1)(m+2)^2∏_i = 1^4(x_3^3 + ρ_i x_3^2- 9x_3 - ρ_i),where ρ_i, i ∈{1, 2, 3, 4}, are the four roots of P_1 (each a function of m). We now analyze the roots of the quartic P_1 as a function of m.The roots ρ_i(m) of P_1 satisfy the following properties: * For m > -1/2 and m ≠ 0, P_1 has four real roots.* For -1 < m ≤ -1/2, P_1 has exactly two real roots.* At m = 0, P_1 has precisely one real root at zero of multiplicity four.* At m = -1/2, P_1 reduces to a quadratic function with two real roots at ±√(3/7).* For m ≤ -1, P_1 has no real roots.Introduce the variable ξ = ρ^2.The results follow by treating P_1 as a quadratic function of ξ with coefficients in ℚ[m]. The discriminant of P_1(ξ) is given byΔ_1 =m^4 (4m+5)^2 (64m^4 + 448m^3 + 1153m^2 + 1278m + 513),which is clearly positive for m > 0.Using Sturm's Theorem <cit.>, it is straight-forward to check that Δ_1 > 0 for -1 < m < 0 as well.Therefore, the roots of P_1(ξ) are real for m > -1.If m > -1/2 and m ≠ 0, the leading coefficient and constant term of P_1(ξ) are positive, while the middle term has a negative coefficient. Since the roots are real, Descartes' Rule of Signs shows that P_1(ξ) has two positive roots, which implies that P_1(ρ) has four real roots of the form ±ρ_1(m), ±ρ_2(m).The leading coefficient of P_1 becomes negative for -1 < m < -1/2, while the middle coefficient flips sign at m ≈ -0.942. Thus, the sign pattern for the coefficients of P_1(ξ) is either - - + or - + +.In either case there is just one sign change, so Descartes' Rule implies that P_1(ξ) has only one positive root.Thus, for -1 < m < -1/2,P_1(ρ) has precisely two real roots of the form ±ρ_1(m).For m < -1, all three coefficients are positive so we have P_1(ξ) > 0 for ξ≥ 0.Consequently, P_1(ρ) has no real roots. The remaining facts listed for the specific m-values 0, -1/2, and -1 are easily confirmed.The roots of P_2(x_3) = x_3^3 + ρ x_3^2- 9x_3 - ρ are real and distinct for any ρ∈ℝ.If r_1 = a is a root, then the other two roots are given byr_2 =a - 3/a + 1 r_3 =3 + a/1 - a.Let r_1 = a denote the largest root. If ρ > 0, then the roots satisfy 1 < r_1 < 3, -1 < r_2 < 0, and r_3 < -3.If ρ < 0, then the roots satisfy r_1 > 3, 0 < r_2 < 1, and -3 < r_3 < -1. The discriminant of P_2 with respect to x_3 is 4(27 + ρ^2)^2, which is always positive.Consequently, the roots of P_2 are always real. If a is a root of P_2, then we have ρ = (a^3 - 9a)/(1 - a^2) = r_1 r_2 r_3, as expected.Then, it is straight-forward to check that P_2 factors as (x - r_1)(x - r_2)(x - r_3).Next we note that P_2(1) = -8 and P_2(3) = 8 ρ.By the Intermediate Value Theorem, we have a root a satisfying 1 < a < 3 if ρ > 0, or a > 3 if ρ < 0.In the first case, we see that -1 < r_2 < 0 and r_3 < -3 by straight-forward algebra.This also serves to show that r_1 = a is the largest root.For the case ρ < 0 and a > 3, we have 0 < r_2 < 1 and -3 < r_3 < -1, as desired. Lemma <ref> is important because it provides specific information on the location of the third vortex without having to work with the complicated expressions that arise from Cardano's cubic formula.Note that if ρ is a complex number, then the roots of P_2 must also be complex. Algorithm for computing solutions:Applying Lemma <ref> and the reductions outlined above, we have the following algorithm for computing the positions of all relative equilibria for a fixed value of m.In theory, the calculations are exact because they only require solving, in order, a quadratic, cubic, and linear equation.1.Compute the real roots ρ_i of the even quartic P_1. 2.For each real value of ρ_i, substitute into the cubic P_2 and find the largest root to obtain x_3. 3.Substitute x_3 and m into Q and solve Q = 0 for x_4.(Recall that Q is linear in x_4.) 4.Two additional solutions for (x_3, x_4) are obtained by using the formulas in the bottom two rows of Table <ref>.* Each choice of ρ leads to three distinct solutions with different orderings. By symmetry, using both ρ and -ρ yields six solutions that correspond to six orderings in a particular group (either Group I or Group II). Thus, two positive roots of the quadratic P_1(ξ) will generate 12 solutions, while one positive root leads to six solutions.If P_1(ξ) has no positive roots, then there are no solutions. * The remaining two polynomials in the Gröbner basis GB^∗ reveal some peculiar properties of solutions.The third polynomial in GB^∗ is simply σ + ρ, which implies that the sum of the roots of P_2 is the negative of the product of the roots.The remaining entry in GB^∗ is just τ + 9, which reveals that the symmetric product of the three roots is always equal to -9.These facts can also be verified by examining the coefficients of P_2 and are apparently an artifact of our special choice of coordinates x_1 = -1, x_2 = 1. Next we demonstrate our algorithm for finding all relative equilibria solutions in two important cases. The case m=1.If all four vortices have the same strength Γ_i = 1, then the four roots of P_1 are ρ_i = ± (√(3)±√(2)). Taking ρ = √(3) - √(2),the three roots of P_2 are-√(3) - √(2) ≈-3.146,1 + √(2) - √(6) ≈-0.035,-1 + √(2) + √(6) ≈2.864 .Notice that the sum and product of these roots equals -ρ and ρ, respectively, in accordance with part 2 of Remark <ref>. Since r_12 = 2, we expect the solution with ordering 1234 to have r_34 = 2 by symmetry. Choosing x_3 = -1 + √(2) + √(6) and x_4 = 1 + √(2) + √(6)≈ 4.864 gives the desired solution. After computing the corresponding values of c and ω,this solution was confirmed by substituting it into the Gröbner basis GB as well as into system (<ref>). The center of vorticity is c = (√(2) + √(6))/2 ≈ 1.932 and the angular velocity is ω = 3/(6 + 2√(3)) ≈ 0.317. The other two roots of P_2 shown in equation (<ref>) yield solutions with orderings3124 and 4132, with x_4 coordinates √(3) + √(2) and -1 + √(2) - √(6), respectively. These solutions concur with those obtained by using the symmetry transformations indicated on the bottom two rows of Table <ref>. If we choose ρ = -√(3) - √(2) instead, then we obtain x_3 =1 + √(2) + √(6)≈ 4.864 as the largest root of P_2. Then x_4 = -1 + √(2) + √(6)≈ 2.864 gives the coordinate of the fourth vortex.This solution corresponds to ordering 1243. The other two roots of P_2 lead to solutions with orderings 3142 and 1432.The remaining two values of ρ lead to six other solutions corresponding to the orderings in rows 2, 3, and 4 of Table <ref>. All 12 solutions are symmetric, with the distance between the first pair of vortices equal to the distance between the second pair (i.e., for the ordering ijkl, we have r_ij = r_kl).The ratio of this distance over the distance between the inner pair of vortices is always (√(3)+√(2)-1)/2 ≈ 1.073.All 12 solutions are geometrically equivalent.These results agree with those given in Section 5.1 of <cit.>.The case m=0.The solutions for the case Γ_4 = 0 are relative equilibria of the restricted four-vortex problem.The three equal-strength vortices are akin to the large masses (called primaries) in the celestial mechanics setting. It is known that the primaries must form a relative equilibrium on their own <cit.>. Thus, based on symmetry, we expect the first three vortices to be equally spaced. Setting m=0 in P_1 gives ρ_i = 0 ∀ i.The cubic P_2 reduces to x_3^3 - 9x_3 and thus P(x_3, m) factors as P(x_3, 0) = 4 x_3^4 (x_3 - 3)^4 (x_3 + 3)^4,with roots x_3 = -3, 0, 3 each repeated four times. As expected, each of the three possible values for x_3 yield an equally-spaced configuration for the equal-strength vortices. The repeated roots and the fact that Lemma <ref> does not apply when m=0 suggest a bifurcation. Surprisingly, this does not happen:there are still 12 different solutions, one for each possible ordering in Table <ref>. While the first nine elements of the Gröbner basis GB, including Q, vanish entirely at m=0 and x_3 = 3, the tenth element yields a quartic polynomial in x_4 with four distinct real roots.The same feature occurs if x_3 = -3 or x_3 = 0. We obtain 12 solutions given by(x_3, x_4) =(a_i,a_i/3±(a_i^2 + 9) √(54 ± 6 √(57))/54),where a_i = -3, 0, or 3, and all four sign combinations occur.These solutions were checked to insure that each satisfied system (<ref>). We now have enough information to count and classify all solutions in terms of the parameter m. The number and type of four-vortex collinear relative equilibria with circulations Γ_1 = Γ_2 = Γ_3 = 1 and Γ_4 = m are given as follows:(i)If m > -1/2, then there are 12 relative equilibria, one for each possible ordering of the vortices (both Groups I and II are realized);(ii)If -1 < m ≤ -1/2, there are 6 relative equilibria, one for each ordering in Group I;(iii)If m ≤ -1, there are no solutions.First, we compute the discriminant of P(x_3, m) as a polynomial in x_3, and find that for m > -1, the discriminant vanishes only if m = -1/2 or m = 0.Consequently, the roots of P are distinct for m > -1 except in these two special cases.(i)Suppose that m is fixed with m > -1/2 and m ≠ 0.Lemmas <ref> and <ref> combine to show that P has 12 distinct roots,and Lemma <ref> implies that each of these x_3-values can be extended to a full solution of system (<ref>).Thus there are 12 relative equilibria. We now show that each possible ordering in Table <ref> is realized.Recall from Example <ref> that for the case m=1, choosing ρ = √(3) - √(2) (the smaller positive root of P_1) leads to a solution with 1 < x_3 < x_4 (ordering 1234). This solution can be continued analytically as m varies away from 1 by following the solution corresponding to the smaller positive root of P_1 and the largest root of P_2. By continuity, the only way for the ordering 1234 to disappear is for there to be a collision of vortices, either with x_3 = x_2 = 1 or x_3 = x_4, at some particular m-value.The first of these possibilities is ruled out by Lemma <ref>. The second possibility is eliminated by taking the third and tenth polynomials in GB and making the substitution x_4 = x_3. Computing a Gröbner basis for these two polynomials, along with P_1 and P_2, produces the polynomial 1. Consequently, there are no solutions in the variety of F with x_3 = x_4.We note that neither x_3 nor x_4 can become infinite for a particular m-value when m > -1/2.This follows fromLemmas <ref> and <ref> and from the fact that Q is linear in x_4. Some care must be taken to continue the solution with ordering 1234 to m < 0 because ρ_i = 0∀ i at m = 0 and P has repeated roots. However, as explained in Example <ref>, there are 12 different solutions for the case m = 0, one for each possible ordering. A similar calculation to the one outlined in Example <ref> shows that for the case m = -1/4, there are also 12 solutions, one for each ordering. In this case, we take ρ to be the smaller (in absolute value) negative root of P_1 in order to obtain the solution with 1 < x_3 < x_4. As explained above, this solution can be continued throughout the interval -1/2 < m < 0 because x_3 = x_2 = 1 and x_3 = x_4 are impossible. Thus, the solution with ordering 1234 varies continuously as m decreases through 0, as the corresponding choice for ρ transitions from the smallest positive root of P_1 to the smallest negative root of P_1, passing through ρ = 0 at m = 0. Applying Theorem <ref>, we have shown that the six orderings from Group I are realized for any m > -1/2. A plot of the solution curve in the x_3x_4-plane that corresponds to the ordering 1234 for -1/2 ≤ m ≤ 10 is shown to the left in Figure <ref>.The argument for the ordering 1243 and its five cousins is similar. This time we follow the larger (in absolute value) negative root of P_1 for m > 0 because ρ = -√(3) - √(2) corresponds to the solution with 1 < x_4 < x_3 when m = 1.For -1/2 < m < 0, we follow the larger positive root of P_1,making a continuous transition through ρ = 0 at m = 0. We know the solution satisfying 1 < x_4 < x_3 persists for all m > -1/2 because the only possible collisions are at x_4 = x_3and x_4 = 1.The first of these was eliminated by the Gröbner basis calculation mentioned above,while the second is impossible because f_5 = u(x_4 - 1) - 1 was included in the original calculation of GB. By Theorem <ref>, it follows that all six orderings from Group II are realized. A plot of the solution curve in the x_3x_4-plane that corresponds to the ordering 1243 for -1/2 < m ≤ 10 is shown on the right in Figure <ref>. Although the curve appears to be linear, it is not.This completes the proof of item (i). (ii)A bifurcation occurs at m = -1/2 as the quartic P_1 becomes a quadratic with roots at ±√(3/7).Here, P reduces to a tenth degree polynomial with repeated roots at -1 and 1, each with multiplicity two.These correspond to collisions between x_3 and x_1, or x_3 and x_2, respectively. The remaining six roots of P give six relative equilibria with the orderings from Group I.This can be shown rigorously by calculating the largest root of P_2 when ρ = -√(3/7) and then finding x_4 from Q.We find that 1 < x_3 < x_4, so this solution has ordering 1234. Theorem <ref> then yields the remaining five solutions from Group I. For -1 < m < -1/2, Lemmas <ref> and <ref> combine to show that P has six distinct real roots and six complex roots. By Lemma <ref>, the six real roots can be extended to a full solution of system (<ref>).As with case (i), rigorously justifying that the six solutions belong to the orderings from Group I involves picking a sample test case (we choose m = -3/4) and showing that the solution with ordering 1234 persists for all m in the open interval (-1, -1/2).Here we follow the negative root of P_1. The argument is similar to that used in case (i).(iii)If m ≤ -1, then all of the roots of P_1 are complex.Applying Lemma <ref> shows that P_2 has no real roots, so there are no solutions.For m > 0, the existence of a unique collinear relative equilibrium for each possible ordering is a consequence of a well-known result from the Newtonian n-body problem due to Moulton <cit.>.For any choice of positive masses, there are exactly n!/2 collinear relative equilibria, one for each possible ordering (see Section 2.1.5 of <cit.> or Section 2.9 of <cit.>).The result generalizes to the vortex setting as long as the circulations are positive <cit.>.For the case -1 < m < 0, a result due to O'Neil implies that there are at least six solutions (Theorem 6.2.1 in <cit.>). §.§ Bifurcations We now discuss the bifurcations at m = -1/2 and m = -1 in greater detail, focusing on the behavior of those solutions which disappear after the bifurcation.As m decreases toward -1/2, four of the solutions with orderings from Group II head toward triple collision, while the remaining two orderings have the third vortex escaping to ±∞.To see this, note that two of the roots of P_1 are heading off to ±∞ as m → -1/2^+.Focusing on the ordering 1243, we track the solution corresponding to ρ→ + ∞.Since ρ is positive, we have 1 <x_4 < x_3 < 3 for this particular solution.Moreover, sincex_3^3 + ρ x_3^2- 9x_3 - ρ = 0⟹x_3^2 - 1 =x_3(9 - x_3^2)/ρ,we see that x_3 →1^+ as ρ→∞.By the Squeeze Theorem, we also have that x_4 → 1^+, and thus the limiting configuration has a triple collision between vortices 2, 3, and 4.To track the other five solutions from Group II, we use the formulas in Table <ref> and take limits as a → 1^+ and b → 1^+.The solution with ordering 1342 also limits on triple collision between vortices 2, 3, and 4. Orderings 3412 and 1432 limit on triple collision between vortices 1, 3, and 4.The solutions with orderings 1423 and 3142 have x_3 →∞ and x_3 → -∞, respectively, but the x_4-coordinate takes the form 0/0. To determine the fate of the fourth vortex for the orderings 1423 and 3142, we first compute an asymptotic expansion for x_3 and x_4 corresponding to the ordering 1243. Introduce the small parameter ϵ by setting m = -1/2 + ϵ^2, and let κ = 1/ρ be a new variable. As ϵ→ 0, we have κ→ 0, while x_3 and x_4 each approach 1.Rewriting P_1 = 0 and P_2 = 0 with κ, we can expand κ and x_3 in powers of ϵ. This, in turn, leads to an expansion for x_4.We find thatκ=√(14)/7 ϵ+ 289/1029√(14) ϵ^3 + 24373/43218√(14) ϵ^5 + 𝒪(ϵ^7),x_3 =1 + 4/7√(14) ϵ+ 8/7 ϵ^2 - 20/1029√(14) ϵ^3 - 416/1029 ϵ^4 + 𝒪(ϵ^5),x_4 =1 + 2/7√(14) ϵ+ 4/7 ϵ^2 - 10/1029√(14) ϵ^3 - 320/1029 ϵ^4 + 𝒪(ϵ^5)are expansions for the solution with ordering 1243 near m = -1/2. Using these expansions, we have-2x_4 + x_3 + 1/x_3 - 1=4/147√(14) ϵ^3+ 𝒪(ϵ^4),which implies that x_4 → 0 as m → -1/2^+ for both solutions with orderings 1423 and 3142. Notice that x_4 - 1 ≈ (1/2)(x_3 - 1), an observation which supports the nearly linear relationship shown in the right-hand graph of Figure <ref>. Finally, the expansions given above are perfectly valid for ϵ < 0 as well.In this case, they correspond to the solution with ordering 1342 near m = -1/2. As m decreases through -1, the solutions with orderings from Group I vanish; however, in this case, four of the limiting configurations end with a pair of binary collisions.As m → -1^+, the remaining real roots of P_1 are heading off to ±∞. Focusing on the ordering 1234, we track the solution corresponding to ρ→ - ∞.By Lemma <ref> we have 3 < x_3 < x_4 for this solution.Since P_2(-ρ) = 8 ρ < 0 and the leading coefficient of P_2 is positive, we see that P_2 has a root larger than -ρ.In fact, for m close to -1, x_3 = -ρ is an excellent approximation to this root.Thus, for the ordering 1234,x_3 →∞, which implies x_4 →∞ as well. A similar fate occurs for the solution with ordering 4312, except that here, x_3 and x_4 approach - ∞ as m → -1^+. For the remaining four orderings in Group I, the formulas in Table <ref> show that the x_3-coordinate is approaching 1 or -1, while the x_4-coordinate is an indeterminate form.Substituting m = -1 and x_3 = 1 into Q quickly yields x_4 = -1, while inserting m = -1 and x_3 = -1 into Q gives x_4 = 1.Thus, as m → -1^+, the solutions with orderings 4123 and 4132 have vortex four approaching vortex one, and vortex three approaching vortex two. For the orderings 1324 and 3124, the opposite collisions occur,with vortex three approaching vortex one, and vortex four approaching vortex two.A plot of the solution curve in the x_3x_4-plane that corresponds to the ordering 1324 for -1 < m ≤ -1/2 is shown in Figure <ref>. We will give an asymptotic expansion about m = -1 for this solution in Section <ref>.§ LINEAR STABILITY OF SOLUTIONSWe now turn to the linear stability of the collinear relative equilibria found in Section 2, investigating the eigenvalues as the parameter m varies. Through our analysis, we discover an important polynomial,Ψ=64m^6 + 320m^5 +96m^4 - 220m^3 +505m^2 + 522m + 9,whose roots include two new bifurcation values. Using Sturm's Theorem, Ψ has four real roots, all of which are negative, and precisely two real roots between -1 and 0. The root closest to -1 ism^∗≈ -0.8564136.Note that -6/7 is a fairly good approximation to this root;it is the second convergent in the continued fraction expansion for m^∗. The root m_c ≈ -0.0175413 will also be significant. We will prove that the collinear relative equilibria in Group I are linearly stable for -1 < m < m^∗.For all other m-values, the relative equilibria in both groups are unstable. §.§ Background and a useful lemmaWe first review some key definitions and properties concerning the linear stability of a relative equilibrium z_0 in the planar n-vortex problem.We follow the approach and setup described in <cit.>. The natural setting for determining the stability of z_0 is to change to rotating coordinates and treatz_0 as a rest point of the corresponding flow. We will assume that z_0 has been translated so that its center of vorticity c is located at the origin. Denote M =diag{Γ_1, Γ_1, …, Γ_n, Γ_n } as the 2n × 2n matrix of circulations, and let K be the 2n × 2n block diagonal matrix containingJ= [01; -10 ] on the diagonal. The matrix that determines the linear stability of a relative equilibrium z_0 with angular velocity ω is given byB =K( M^-1 D^2 H(z_0) + ω I ) ,where D^2H(z_0) is the Hessian of the Hamiltonian evaluated at z_0 and I is the 2n × 2n identity matrix. Since we are working with a Hamiltonian system, the eigenvalues of B come in pairs ±λ.For a solution to be linearly stable, the eigenvalues must lie on the imaginary axis. One important property of the Hessian is that it anti-commutes with K, that is,D^2H(z) K =-K D^2H(z) .From this, it is straight-forward to see that the characteristic polynomial of M^-1 D^2 H(z_0) is even.In addition, if v is an eigenvectorof M^-1 D^2 H(z_0) with eigenvalue μ, then Kv is also an eigenvector with eigenvalue - μ (see Lemma 2.4 in <cit.>).This fact cuts the dimension of the problem in half. In order to compare the collinear relative equilibria within a particular group, it is easier to work with the scaled stability matrixω^-1 B =K( ω^-1 M^-1 D^2H(z_0) + I) .This scaling has no effect on the stability of z_0 because the characteristic polynomial of B is even. We will refer to the eigenvalues of ω^-1 B as normalized eigenvalues. The following lemma explains how to compute the normalized eigenvalues from the eigenvalues of ω^-1 M^-1 D^2H(z_0). Let p(λ) denote the characteristic polynomial of the scaled stability matrix ω^-1 B. (i) Suppose that v is a real eigenvector ofω^-1 M^-1 D^2H(z_0) with eigenvalue μ.Then { v, Kv } is a real invariant subspace of ω^-1 B and the restriction of ω^-1 B to { v, Kv } is[ 0 μ - 1;μ +1 0; ].Consequently, p(λ) has a quadratic factor of the form λ^2 + 1- μ^2.(ii) Suppose that v = v_1 + i v_2 is a complex eigenvector ofω^-1 M^-1 D^2H(z_0)with complex eigenvalue μ = α + i β.Then { v_1,v_2, Kv_1, Kv_2 } is a real invariant subspace of ω^-1 Band the restriction of ω^-1 B to this space is[ 0 0 α - 1 β; 0 0-β α - 1; α + 1 β 0 0;-β α + 1 0 0 ].Consequently, p(λ) has a quartic factor of the form(λ^2 + 1 - μ^2)(λ^2 + 1 - μ^2), where μ = α - i β.(i) Since ω^-1 M^-1 D^2H(z_0) v = μ v, we have ω^-1 M^-1 D^2H(z_0) K v = - μ K v by equation (<ref>). This implies that ω^-1 B v = (μ + 1) K v and ω^-1 B (Kv) =(μ -1) v, verifying matrix (<ref>).The characteristic polynomial of matrix (<ref>) is λ^2 + 1 - μ^2 and therefore, this quadratic is a factor of p(λ). (ii)If v_1 + i v_2 is a complex eigenvector with eigenvalue α + i β, then we have ω^-1 M^-1 D^2H(z_0) v_1 = α v_1 - β v_2 and ω^-1 M^-1 D^2H(z_0) v_2 = α v_2 + β v_1. Using equation (<ref>), this implies that ω^-1 B v_1 = (α+ 1) K v_1 - β K v_2, ω^-1 B v_2 =β K v_1 + (α + 1) K v_2, ω^-1 B (Kv_1) =(α - 1) v_1 - β v_2, and ω^-1 B (Kv_2) =β v_1 + (α - 1) v_2, which confirms matrix (<ref>).The characteristic polynomial of matrix (<ref>) is (λ^2 + 1 - μ^2)(λ^2 + 1 - μ^2) and hence, this quartic is a factor of p(λ).Due to the conserved quantities of the n-vortex problem, any relative equilibrium will have the four normalized eigenvalues0, 0, ± i.We call these eigenvalues trivial.The eigenvalues ± i arise from the center of vorticity integral and can be derived by noting that the vector [1, 0, 1, 0, …, 1, 0] is in the kernel ofω^-1 M^-1 D^2H(z_0).The two zero eigenvalues appear because relative equilibria are not isolated rest points.For any relative equilibrium z_0, the vector K z_0 is in the kernel of ω^-1 B. This vector is tangent to the periodic orbitdetermined by z_0 at t=0.This follows from the identityω^-1 M^-1 D^2H(z_0) z_0 =z_0,and part (i) of Lemma <ref>. Thus, in the full phase space, a relative equilibrium is always degenerate. One method for dealing with the issues that arise from the symmetries of the problem is to work in a reduced phase space (e.g., quotienting out the rotational symmetry). However, it is typically easier to make the computations in ℝ^2n and then define linear stability by restricting to the appropriate subspace. This is the approach we follow here. Let V = {z_0, Kz_0} and denote V^⊥ as the M-orthogonal complement of V, that is,V^⊥={ w ∈ℝ^2n : w^T M v = 0 ∀ v ∈ V } .The invariant subspace V accounts for the two zero eigenvalues; the vector space V^⊥ has dimension 2n - 2 and is invariant under ω^-1 B. We also have that V ∩ V^⊥ = {0} provided L ≠ 0.This motivates the following definition for linear stability.A relative equilibrium z_0 always has the four trivial normalized eigenvalues 0, 0, ± i.We call z_0 nondegenerate if the remaining 2n - 4 eigenvalues are nonzero.A nondegenerate relative equilibriumis spectrally stable if the nontrivialeigenvalues lie on the imaginary axis, and linearly stable if, in addition, the restriction of the scaled stability matrix ω^-1 B to V^⊥has a block-diagonal Jordan form with blocks [0β_i; -β_i0;].As noted in <cit.>, if Γ_i > 0∀ i, then ω^-1 M^-1 D^2H(z_0) is symmetric with respect to an M-orthonormal basis and has a full set of linearly independent real eigenvectors.Consequently, part (i) of Lemma <ref> applies repeatedly and the characteristic polynomial factors asp(λ) =λ^2 (λ^2 + 1) ∏_j = 1^n-2 (λ^2 + 1 - μ_j^2),where the μ_j are the nontrivial eigenvalues of ω^-1 M^-1 D^2H(z_0). It follows that the relative equilibrium is linearly stable if and only if |μ_j| < 1∀ j. If the circulations Γ_i are of mixed sign, then ω^-1 M^-1 D^2H(z_0)may have complex eigenvalues, leading to quartic factors of the characteristic polynomial, as explained in part (ii) of Lemma <ref>.This is the case for certain values of m < 0 in our problem.Note that when μ_j ∈ℂ - ℝ, the corresponding eigenvalues of the relative equilibrium form a complex quartuplet ±α' ± i β'.This implies instability unless α' = 0, which only occurs when Re(μ_j) = 0.§.§ Finding the nontrivial eigenvalues of a collinear relative equilbriumWe now focus on the stability of a collinear relative equilibrium z_0 = (z_1, z_2, …, z_n) ∈ℝ^2n, where z_i = (x_i, 0)∀ i.Rearranging the coordinates from (x_1, y_1, x_2, y_2, …, x_n, y_n) to(x_1, x_2, …, x_n, y_1, y_2, …, y_n), we find that ω^-1 M^-1 D^2H(z_0) takes the special formω^-1 M^-1 D^2H(z_0) =[A0;0 -A ] ,where A is an n × n matrix with entries A_ij = - ω^-1Γ_jr_ij^-2 if i ≠ j and A_ii = - ∑_j ≠ iA_ij.This reduces our calculations from a 2n-dimensional vector space to an n-dimensional one. For the case n=4, we haveA = ω^-1[ Γ_2/r_12^2 + Γ_3/r_13^2 + Γ_4/r_14^2-Γ_2/r_12^2-Γ_3/r_13^2-Γ_4/r_14^2;-Γ_1/r_12^2 Γ_1/r_12^2 + Γ_3/r_23^2 + Γ_4/r_24^2-Γ_3/r_23^2-Γ_4/r_24^2;-Γ_1/r_13^2-Γ_2/r_23^2 Γ_1/r_13^2 + Γ_2/r_23^2 + Γ_4/r_34^2-Γ_4/r_34^2;-Γ_1/r_14^2-Γ_2/r_24^2-Γ_3/r_34^2 Γ_1/r_14^2 + Γ_2/r_24^2 + Γ_3/r_34^2 ] .Note that the vectors s = [1, 1, 1, 1]^ and x = [x_1, x_2, x_3, x_4]^ are eigenvectors of A with eigenvalues 0 and 1, respectively. These are the two trivial eigenvalues of A arising from the center of vorticity integral and the rotational symmetry.The remaining two eigenvalues of A determine the linear stability of z_0.Specifically, applying both parts of Lemma <ref>, the normalized nontrivial eigenvalues of z_0 are given byλ_1 =±√(μ_1^2 - 1)λ_2 =±√(μ_2^2 - 1) ,where μ_1 and μ_2 are the nontrivial eigenvalues of A. Let W = {s, x} and let W^⊥ denote the M-orthogonal complement of W where M =diag{Γ_1, Γ_2, Γ_3, Γ_4 }.The subspace W^⊥ is invariant under A. To find μ_1 and μ_2, we compute the restriction of A to W^⊥. It is straight-forward to check that the two vectorsw_1 =[Γ_2 Γ_3 (x_3 - x_2), Γ_1 Γ_3 (x_1 - x_3), Γ_1 Γ_2 (x_2 - x_1),0]^,w_2 =[Γ_2 Γ_4 (x_4 - x_2), Γ_1 Γ_4 (x_1 - x_4), 0,Γ_1 Γ_2 (x_2 - x_1)]^form a basis for W^⊥, as w_i^ M s = 0 and w_i^ M x = 0 for each i.However, it is not an M-orthogonal basis since w_1^ M w_2 ≠ 0. Let C denote the restriction of A to W^⊥ and writeC =[ C_11 C_12; C_21 C_22 ] .To find the entries of C, note that C_11 w_1 + C_21 w_2 = [∗, ∗, C_11Γ_1 Γ_2 (x_2 - x_1), C_21Γ_1 Γ_2 (x_2 - x_1)]^. Thus, we see that C_11Γ_1 Γ_2 (x_2 - x_1) and C_21Γ_1 Γ_2 (x_2 - x_1) are equal to the third and fourth coordinates, respectively,of the vector Aw_1.A similar fact applies for C_12 and C_22. After some computation, we find thatC_11=ω^-1[ Γ_1 + Γ_3/r_13^2 + Γ_2 + Γ_3/r_23^2 + Γ_4/r_34^2 + Γ_3/(x_3 - x_1)(x_3 - x_2)], C_22=ω^-1[ Γ_1 + Γ_4/r_14^2 + Γ_2 + Γ_4/r_24^2 + Γ_3/r_34^2 + Γ_4/(x_4 - x_1)(x_4 - x_2)], C_21=- ω^-1Γ_3/x_2 - x_1[x_3 - x_2/r_14^2 + x_1 - x_3/r_24^2 + x_2 - x_1/r_34^2], C_12=- ω^-1Γ_4/x_2 - x_1[x_4 - x_2/r_13^2 + x_1 - x_4/r_23^2 + x_2 - x_1/r_34^2].The nontrivial eigenvalues of A, μ_1 and μ_2, are equivalent to the eigenvalues of C. They are easily expressed in terms of the trace and determinant of C. The quantity δ, defined by δ= (Γ_1 + Γ_2)(Γ_3 + Γ_4) /r_12^2 r_34^2 + (Γ_1 + Γ_3)(Γ_2 + Γ_4) /r_13^2 r_24^2+(Γ_1 + Γ_4)(Γ_2 + Γ_3) /r_14^2 r_23^2+ ∑_i=1^4 ∑_ j < k j, k ≠ i^4 Γ_i (Γ_i + Γ_j + Γ_k) /r_ij^2 r_ik^2,is important in the computation of the determinant of C.Let T and D denote the trace and determinant, respectively, of C.We haveT=ω^-1∑_i < j^4Γ_i + Γ_j/r_ij^2-1 ,D =-T + ω^-2δ. By definition, if (x_1, x_2, x_3, x_4) are the coordinates of a collinear relative equilibrium with center of vorticity at the origin,then equation (<ref>) implies thatΓ_2/x_2 - x_1 + Γ_3/x_3 - x_1 + Γ_4/x_4 - x_1 + ω x_1 = 0, Γ_1/x_1 - x_2 + Γ_3/x_3 - x_2 + Γ_4/x_4 - x_2 + ω x_2 = 0.Subtracting equation (<ref>) from equation (<ref>) and multiplying through by 1/(x_2 - x_1) givesω=Γ_1 + Γ_2/r_12^2- Γ_3/(x_3 - x_1)(x_3 - x_2) - Γ_4/(x_4 - x_1)(x_4 - x_2) .It then follows thatT =C_11 + C_22=ω^-1∑_i < j^4Γ_i + Γ_j/r_ij^2-1.Alternatively, recall that the trace of a matrix is equal to the sum of its eigenvalues.Applying this fact to both matrices A and C gives1 + T =0 + 1 + μ_1 + μ_2 =(A) =ω^-1∑_i < j^4Γ_i + Γ_j/r_ij^2 ,which gives an alternative proof of formula (<ref>). Computing the determinant D from the entries of C gives a messy expression.A more useful formula can be obtained by utilizing the fact that the sum of the product of all pairs of eigenvalues of A is equal to half the quantity ((A))^2 - (A^2).This yields0 · 1 + 0 ·μ_1 + 0 ·μ_2 + 1 ·μ_1 + 1 ·μ_2 + μ_1 ·μ_2 =1/2[ω^-2(∑_i < j^4Γ_i + Γ_j/r_ij^2)^2 - (A^2) ],which, after some calculation, givesT + D =ω^-2δ. Recall that the angular velocity of a relative equilibrium is given by ω = L/I, where L = ∑_i < jΓ_i Γ_j andI = (1/Γ) ∑_i < jΓ_i Γ_j r_ij^2. It follows from formulas (<ref>) and (<ref>), that T and D depend only on the circulations Γ_i and the mutual distances r_ij. Thus, as we would expect, the stability of a relative equilibrium is unaffected by translation, and we may retain our original coordinates (e.g., x_1 = -1, x_2 = 1) when calculating T and D, rather than shifting the configuration so that the center of vorticity is at the origin. The nontrivial eigenvalues of A are the roots of λ^2 - T λ + D, where T and D are given by (<ref>) and (<ref>), respectively. They are identical for any solution within a particular group of orderings.Sufficient conditions for linear stability of the relative equilibrium are(i) -2 < T < 2, (ii) D < T^2/4,(iii)D > T - 1, and (iv) D > -T - 1. As explained above, the nontrivial eigenvalues of A are equivalent to the eigenvalues of C, and the characteristic polynomial of C is λ^2 - T λ + D.It is straight-forward to check that T and D are invariant under the maps S and R defined in equations (<ref>) and (<ref>), respectively.This was also confirmed using Maple. It follows that T and D are invariant under the group G and thus, the nontrivial normalized eigenvalues are identical for all six solutions in a given group of orderings.If μ_i ∈ℝ, then formula (<ref>) shows that |μ_1| < 1 and|μ_2| < 1 are both required for stability. This is guaranteed if conditions (i) though (iv) are satisfied (see Figure <ref>).Note that linear stability follows as well becausecondition (ii) insures that μ_1 ≠μ_2, so there are no repeated nontrivial eigenvalues.In addition to the blue region in Figure <ref>, solutions are also linearly stable along the positive D-axis (T = 0, D > 0). In this case the eigenvalues of C are pure imaginary and the nontrivial eigenvaluesof the relative equilibrium are ± i √(1 + D), ± i √(1 + D). Using matrix (<ref>), it is straight-forward to show thatthe Jordan form of ω^-1 B has no off-diagonal blocks, and thus the solution is linearly stable.The case m=1.Recall from Example <ref> that in the case of equal-strength vortices, the solution with ordering 1234 has positions (x_1, x_2, x_3, x_4) =(-1, 1, -1 + √(2) + √(6), 1 + √(2) + √(6)).This gives T = 5 and D = 6, so the nontrivial eigenvalues of A are μ_1 = 2 and μ_2 = 3 and the relative equilibrium is unstable.The same result holds for the ordering 1243. By formula (<ref>), the nontrivial normalized eigenvalues for either group are ±√(3) and ± 2√(2). The case m=0.If Γ_4 = 0, we found in Example <ref> that the solution with ordering 1234 has positions (x_1, x_2, x_3, x_4) =(-1, 1, 3, 1 + (1/3)√( 54 + 6 √(57))).From these values, we find that μ_1 = 2 and μ_2 = (15 - √(57) )/4, so the relative equilibrium is unstable.The nontrivial normalized eigenvalues are ±√(3)≈± 1.732 and ± (1/4) √( 266 - 30√(57))≈± 1.571.A similar result holds for the solution with ordering 1243, except that the eigenvalues are much further apart.The nontrivial normalized eigenvalues for this solution are ±√(3) and± (1/4) √( 266 + 30√(57))≈± 5.548 .§.§ Stability, bifurcations, and eigenvalue structureWe now apply Theorem <ref> and formulas (<ref>) and (<ref>) to investigate the linear stability of our two families of relative equilibriaas m varies.Recall from Theorem <ref> that there are two groups of solutions varying continuously in m: Group I exists for all m > -1 and Group II exists for all m > -1/2.Moreover, for a fixed m, the valuesof x_3 and x_4, and thereby the trace T and determinant D, can be found analytically by working with the roots of a cubic equation. We first compute an asymptotic expansion for the solution with ordering 1324 about m = -1.Recall that as m → -1^+, the limiting configuration for this specific ordering contains a pair of binary collisions, that is, x_3 → -1 and x_4 → 1 (see Figure <ref>). Following the same approach used in Section <ref>, we let m = -1 + ϵ^2, where ϵ is a small positive parameter, and repeatedly solve the equations P_1 = 0, P_2 = 0, and Q = 0 to obtain the expansionsx_3 =-1 + 2√(2) ϵ-2 ϵ^2 - 3 √(2) ϵ^3 + 7 ϵ^4 + ⋯ - 71179/16√(2) ϵ^11 + 226695/16ϵ^12 + 𝒪(ϵ^13), x_4 =1 + 2 ϵ^2- √(2) ϵ^3 - 5 ϵ^4 + 9/2√(2) ϵ^5 + ⋯ - 24863/16√(2) ϵ^11 - 192805/16ϵ^12 + 𝒪(ϵ^13).Substituting these expressions into the formula ω = L/I shows that 1/ω = 4 ϵ^2 + 𝒪(ϵ^3), which implies that the angular velocity becomes infinite as the vortices approach collision. Using formulas (<ref>) and (<ref>), we find the following expansions for T and D:T = 1 + 6 ϵ^2 - 12 ϵ^4 + 60 ϵ^6 - 426 ϵ^8 +𝒪(ϵ^10),D =6 ϵ^2 - 12 ϵ^4 + 78 ϵ^6 - 588 ϵ^8 + 9453/2ϵ^10 + 𝒪(ϵ^12).The fact that each series contains only even powers of ϵ is a consequence of the invariance described in Theorem <ref>. Choosing ϵ < 0 in formulas (<ref>) and (<ref>) is perfectly valid; in fact, it provides an expansion for the solution withordering 3124, which is also a member of the Group I orderings.Since T and D are invariant within a specific group, they must be even functions of the parameter ϵ. Recall that m^∗≈ -0.8564136 and m_c ≈ -0.0175413 are important roots of the polynomial Ψ(m) =64m^6 + 320m^5 +96m^4 - 220m^3 +505m^2 + 522m + 9.The linear stability and nontrivial eigenvalue structure for the four-vortex collinear relative equilibria with circulations Γ_1 = Γ_2 = Γ_3 = 1 and Γ_4 = m are as follows:(i) The solutions from Group I are linearly stable for -1 < m < m^∗, spectrally stable at m = m^∗, and unstable for m > m^∗.(ii) For m > m_c, the nontrivial eigenvalues for the Group I solutions consist of two real pairs, and at m = m_c, these pairs merge to form a real pair with multiplicity two. As m →∞, the normalized Group I eigenvalues approach ± 2 √(6) and ± 2 √(2). For m^∗ < m < m_c, the nontrivial eigenvalues form a complex quartuplet ±α± i β.As m → -1^+, the nontrivial normalized eigenvalues approach 0, 0, ± i.(iii)The solutions from Group II (m > -1/2) are always unstable with two real pairs of nontrivial normalized eigenvalues ±λ_1, ±λ_2. As m → -1/2^+, λ_1 →∞ and λ_2 → 2√(14)/5. As m →∞,λ_1 → 2 √(2) and λ_2 → 0. We begin by focusing on the solutions with the orderings in Group I.By Theorem <ref> we can restrict our attention to one particular solution from this group. Since the solution varies continuously in m (m > -1), so do the values of T = T(m) and D = D(m) that govern stability. Figure <ref> shows a plot of D versus T as m varies, including two bifurcations at m = m_c and m = m^∗. Our intent is to justify this picture rigorously.Using the asymptotic expansion for the solution with ordering 1324, we find from equations (<ref>) and (<ref>) that T and D are approaching 1 and 0, respectively, as m → -1^+. By formula (<ref>), the nontrivial normalized eigenvalues are limiting on 0, 0, ± i. We also find that D - (T - 1) =18 ϵ^6- 162 ϵ^8 +𝒪(ϵ^10) ,so that D > T - 1 for m sufficiently close to -1. Since the other three conditions in Theorem <ref> are also satisfied,the solution is linearly stable. This proves that the trace-determinant curve for the Group I solutions lies in the stability region for m sufficiently close to -1.At m = 1, this curve reaches the point (T=5, D=6) in the unstable region II (two real pairs of eigenvalues).To ascertain how stability is lost, we search for bifurcations, that is, we look for intersections between the trace-determinant curve and the boundaries of the stability region. Adding the polynomial obtained from the numerator of T^2/4 - D to our defining system of equations {f_1, f_2, …, f_6} yields a system of polynomials whose solutions contain those relative equilibria with repeated eigenvalues.Fortunately, it is possible to compute a lex Gröbner basis for this augmented system and eliminate all variables except for the parameter m. The first polynomial in this basis is (m+1)^2 (m+3)^2 Ψ(m).Therefore, repeated eigenvalues may only occur if m = m_c or m = m_∗.Applying back-substitution into the Gröbner basis, we find that there are six solutions corresponding to the orderings in Group I at both m = m_c and m = m^∗.At m = m_c, the solutions have a T-value larger than 2 (T ≈ 3.8344) and are therefore unstable. On the other hand, the value of T for the solutions at m = m^∗ is T^∗≈ 1.7054. Applying formula (<ref>), the Group I relative equilibria are spectrally stable with repeated nontrivial normalized eigenvalues ± i √(1 - (T^∗/2)^2). The relative equilibria at m = m^∗ are not linearly stable because the matrix C is not a scalar multiple of the identity matrix. This was confirmed by appending the numerator of C_21 to the augmented system described above and computing a Gröbner basis. Since the polynomial 1 was obtained, the value of C_21 at either bifurcation is nonzero. Computing the trace and determinant for the Group I solution at m = -1/2 (ρ = -√(3/7)), we find that D > T^2/4 and thus the nontrivial eigenvalues μ_1 and μ_2 are complex. It follows that the trace-determinant curve for the Group I solution lies in region IV for m^∗ < m < m_c. By part (ii) of Lemma <ref>, the normalized eigenvalues form a complex quartuplet ±α± i β for these m-values.Next, we add the numerator of D - (T - 1) to the system {f_1, f_2, …, f_6} and compute a lex Gröbner basis for this augmented system. The first term in the Gröbner basis is simply (m+1)^2 (m+3)^2.The m+1 term is expected because T=1 and D=0 are the limiting values as m → -1^+.The fact that there are no other roots for m > -1 shows that D > T - 1 for all solutions (using continuity) becausethe values at m = 1 satisfy this inequality (see Example <ref>). This is true for solutions from either Group I or II. A similar Gröbner basis calculation shows that T > 1 for all solutions.It follows that the trace-determinant curve lies in the first quadrant and never in region III.Moreover, the only possible bifurcations occur at m = m_c and m = m^∗, where the curve crosses the repeated root parabola D = T^2/4.Thus, we have shown that the trace-determinant curve for the Group I solutions lies in the stability region for -1 < m < m^∗, in region IV for m^∗ < m < m_c, and in region II for m > m_c. To determine the fate of the Group I normalized eigenvalues as m →∞, we compute an asymptotic expansion for the solution with ordering 1234.Setting m = 1/ϵ^2 and treating ϵ as a small parameter, we findx_3 =3- √(3)/3ϵ + 1/12ϵ^2 + 1025/1296√(3) ϵ^3 - 2059/5184ϵ^4 + 𝒪(ϵ^5), x_4 =4/3√(3) ϵ^-1 + 2/3+ 35/27√(3) ϵ + 307/648ϵ^2 + 𝒪(ϵ^3), T =8 - 45/4ϵ^2 + 𝒪(ϵ^4),D =15 - 171/4ϵ^2 + 𝒪(ϵ^4) .Thus, μ_1 = (T + √(T^2 - 4D))/2 approaches 5 and μ_2 = (T - √(T^2 - 4D))/2 approaches 3 as m →∞. The limiting behavior of the nontrivial normalized eigenvalues now follows from formula (<ref>). We note that the solution with this particular ordering limits on a configuration with the three equal-strength vortices equally spaced (r_12 = r_23 = 2) and the fourth vortex infinitely far away.This completes the proof of items (i) and (ii) of the theorem.The Gröbner basis calculations above show that the Group II solutions never bifurcate.Since we also have T = 5 and D = 6 at m=1 for this group of orderings, it follows that the trace-determinant curve for the Group II solutions is always contained in region II.Consequently, the nontrivial eigenvalues form two real pairs ±λ_1, ±λ_2 for all m > -1/2.Using the asymptotic expansions (<ref>) and (<ref>) for the Group II solution 1243, we find that T =21/10ϵ^-2 + 578/175 +𝒪(ϵ^2) D =189/50ϵ^-2 + 3459/875 +𝒪(ϵ^2)are expansions for the trace and determinant of the Group II solutions for m close to -1/2. It follows that μ_1 ≈ (21/10) ϵ^-2 approaches ∞, while μ_2 approaches 9/5 as m → -1/2^+.The limiting behavior of the nontrivial normalized eigenvalues now follows from formula (<ref>). To determine the fate of the Group II normalized eigenvalues as m →∞, we compute an asymptotic expansion for the solution with ordering 1432.Setting m = 1/ϵ^2 and treating ϵ as a small parameter, we findx_3 =1-ϵ + 1/4ϵ^2 + 1/16ϵ^3 - 3/64ϵ^4 + 𝒪(ϵ^5), x_4 =-1/4ϵ - 3/8ϵ^2 + 1/8ϵ^3 + 𝒪(ϵ^4), T =4 + 3/4ϵ^2 + 𝒪(ϵ^4),D =3 + 21/4ϵ^2 + 𝒪(ϵ^4) .It follows that μ_1 → 3 and μ_2 → 1^+ as m →∞. The limiting behavior of the nontrivial normalized eigenvalues now follows from formula (<ref>). We note that the solution with this particular ordering limits on a configuration containing a collision between vortices 2 and 3, with the fourth vortex located in the middle of vortices 1 and 2 (r_14 = r_24 = 1). This completes the proof of item (iii). * Using Gröbner bases, it is possible to express the x_3-coordinate at m = m^∗ as the rootof an even 36th-degree polynomial in one variable with integer coefficients. The same is true for the x_4-coordinate (same degree, different polynomial). * The fact that both the Group I and II collinear relative equilibria are unstable for m > 0 agrees with Corollary 3.5 in <cit.>. In general, any collinear relative equilibrium of n vortices, where all circulations have the same sign, is always unstable and has n-2 nontrivial real eigenvalue pairs ±λ_j.This follows by generalizing a clever argument of Conley's from the collinear n-body setting (see Pacella <cit.> or Moeckel <cit.> for details in the n-body case). * Recall that relative equilibria are critical points of the Hamiltonian H restricted to a level surface of the angular impulse I. Numerical calculations in Matlab indicate that all of our solutions (both stable and unstable) are saddles (the Morse index is always 2, except for m = 0). Thus, in contrast to the case of same-signed circulations, with mixed signs it is possible for a saddle to be linearly stable. A similar observation, using a modified potential function, was also made in <cit.>. § CONCLUSIONWe have used ideas from modern and computational algebraic geometry to rigorously study the collinear relative equilibria in the four-vortex problem where three circulation strengths are assumed identical. Exploiting the S_3 invariance in the problem, we simplified the defining equations and obtained a specific count on the number and type of solutions in terms of the fourth vorticity Γ_4 = m. The linear stability of solutions in the full plane was investigated and stable solutions were discoveredfor m negative. Reductions were made to simplify the stability calculations and useful formulas were derived that apply to any four-vortex collinear relative equilibrium.Asymptotic expansions were computed to rigorously justify the behavior of solutions near collision.Gröbner bases were used to locate key bifurcation values. It is hoped that the reductions employed here involving symmetry and invariant group theory will prove useful in similar problems.Acknowledgments:The authors would like to thank the National Science Foundation (grant DMS-1211675) and the Holy Cross Summer Research Program for their support. amsplain 99albouyAlbouy, A., Fu, Y., Sun, S., Symmetry of planar four-body convex central configurations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.464 (2008), no. 2093, 1355–1365.aref-equil Aref, H., On the equilibrium and stability of a row of point vortices, J. Fluid Mech. 290 (1995), 167–181.aref-int Aref, H., Integrable, chaotic, and turbulent vortex motion in two-dimensional flows, Ann. Rev. Fluid. Mech. 15 (1983), 345–389. aref-stab3 Aref, H., Stability of relative equilibria of three vortices, Phys. Fludis 21 (2009), 094101. aref-newton Aref, H., Newton, P. K., Stremler, M. A., Tokieda, T., Vainchtein, D. L., Vortex crystals, Adv. Appl. Mech. 39 (2003), 1–79. bhw Barry, A. M., Hall, G. R., Wayne, C. E.,Relative equilibria of the (1+n)-vortex problem, J. Nonlinear Sci. 22 (2012), 63–83. bhl Barry, A. M., Hoyer-Leitzel, A., Existence, stability, and symmetry of relative equilibria with a dominant vortex, SIAM J. Appl. Dyn. Syst. 15, no. 4 (2016), 1783–1805.cs Cabral, H. E., Schmidt, D. S., Stability of relative equilibria in the problem of N+1 vortices, SIAM J. Math Anal. 31, no. 2 (1999), 231–250. CLOCox, D. A., Little, J. B., O'Shea, D., Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd ed., Springer, Berlin (2007).davis Davis, C., Wang, W., Chen, S. S.,Chen, Y., Corbosiero, K., DeMaria, M.,Dudhia, J., Holland, G., Klemp, J., Michalakes, J., Reeves, H., Rotunno, R., Snyder, C., Xiao, Q., Prediction of Landfalling Hurricanes with the Advanced Hurricane WRF Model, Monthly Weather Review 136 (2007), 1990–2005.HRS Hampton, M., Roberts, G. E., Santoprete, M., Relative equilibria in the four-vortex problem with two pairs of equal vorticities, J. Nonlinear Sci. 24 (2014), 39-92. have Havelock, T. H., The stability of motion of rectilinear vortices in ring formation, Philosophical Magazine 11, no. 7 (1931), 617–633. kirchhoffKirchhoff G., Vorlesungen über Mathematische Physik, I,Teubner, Leipzig, 1876. KossSchub Kossin, J. P., Schubert, W. H., Mesovortices, polygonal flow patterns,and rapid pressure falls in hurricane-like vortices, J. Atmos. Sci. 58 (2001), 2196–2209.maple Maple, version 15.00, (2011), Maplesoft, Waterloo Maple Inc.matlab MATLAB, version 7.10.0.499 (R2010a), (2010), The MathWorks, Inc. meyer Meyer, K. R., Hall, G. R., Offin, D., Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, 2nd ed., Applied Mathematical Sciences, 90, Springer, New York (2009). rick-bookMoeckel, R., Central configurations, in Central Configurations, Periodic Orbits, and Hamiltonian Systems, Llibre, J., Moeckel, R., Simó, C, Birkhäuser (2015), 105–167. moultonMoulton, F. R.,The straight line solutions of the problem of n bodies,Ann. of Math. (2) 12, no. 1 (1910), 1–17.newton Newton, P. K., The N-Vortex Problem: Analytic Techniques,Springer, New York (2001).oneilO'Neil, K. A.,Stationary configurations of point vortices,Trans. Amer. Math. Soc.302, no. 2 (1987),383–425.pacellaPacella, F., Central configurations of the N-bodyproblem via equivariant Morse theory,Arch. Ration. Mech. Anal. 97 (1987), 59–74. palmorePalmore, J.,Relative equilibria of vortices in two dimensions, Proc. Natl. Acad. Sci. USA 79 (Jan. 1982), 716–718.perezPérez-Chavela, E., Santoprete, M., Tamayo, C.,Symmetric relative equilibria in the four-vortex problem with three equal vorticities, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal.22, no. 3 (2015), 189–209.g:stabilityRoberts, G. E., Stability of relative equilibria in the planar n-vortex problem, SIAM J. Appl. Dyn. Syst. 12, no. 2 (2013), 1114–1134. schm Schmidt, D., The stability of the Thomson heptagon, Regul. Chaotic Dyn. 9, no. 4 (2004), 519–528. spring Spring, D., On the second derivative test for constrained local extrema, Amer. Math. Monthly 92 (1985), no. 9, 631–643. sturmSturmfels, B., Solving Systems of Polynomial Equations, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, no. 97, Amer. Math. Soc.(2002). synge Synge, J. L., On the motion of three vortices, Can. J. Math 1 (1949), 257–270. thomsonThomson, J. J., A Treatise on the Motion of Vortex Rings: An essay to which the Adamsprize was adjudged in 1882, University of Cambridge, Macmillan, London (1883). xia Xia, Z., Central configurations with many small masses, J. Differential Equations 91 (1991), 168–179. | http://arxiv.org/abs/1704.08647v1 | {
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mymainaddress]Marcos Paulo Belançonmycorrespondingauthor [mycorrespondingauthor]Corresponding author [email protected][mymainaddress]Universidade Tecnológica Federal do Paraná Câmpus Pato Branco -Grupo de Física de Materiais CEP 85503-390, Via do conhecimento Km 01, Pato Branco, Paraná, Brazil In this “Letter”, I do introduce my point of view about the role of science in the development and organization of our societies, emphasizing that science is too far from our societies. In Brazil we follow a way similarly to that of USA and France concerning this phenomenon, marked by the lack of participation of our academies and scientific community in politics and economy, i.e. we can see that strategic decisions are been hold by only electoral bias and, when there is the presence of academic science, usually it is restricted to the fields of economical, political and social sciences. I conclude that the strategic planning of a country, or of the planet as a whole, needs that academy stay close of society, been this one necessary both not enough condition to the proper functioning of any democracy. In the Brazil case more specifically, I propose that this political change in our academies begin by one profound thought about the Brazilian Science with a consequent refoundation of the CNPQ (Nacional Council for Science and Technology development). - - - -Nesta “carta”, apresento o meu ponto de vista sobre o papel da ciência no desenvolvimento e organização de nossas sociedades, enfatizando que ciência está muito distante da sociedade. No Brasil trilhamos um caminho semelhante ao dos EUA e França no que diz respeito a este fenômeno, marcado pela falta de participação das academias e da comunidade científica na política e economia, i.e. verificamos que decisões estratégicas são tomadas com viés eleitoral e, quando há presença da ciência acadêmica, habitualmente ela se restringe a esfera da ciência econômica, política e social. Concluo que o planejamento estratégico de um país, ou do planeta todo, precisa que a academia se aproxime da sociedade, sendo esta uma condição necessária e não determinante para o bom funcionamento de qualquer democracia. No caso do Brasil mais especificamente, proponho que essa mudança política das academias comesse por uma profunda reflexão sobre a ciência brasileira com a consequente refundação do CNPQ.FísicaPolítica Sociedade CiênciaAcademia A física dirigiu as revoluções científicas nos últimos séculos. Do método científico de Galileu até a difração de elétrons e a equação de Schörodinger; passando pela gravitação de Newton, os raios-x de Röntgene o méson π de César Lattes, entre tantas outras descobertas. A revolução industrial começa com o domínio da máquina a vapor, passa pela exploração do eletromagnetismo; e então a física do estado sólido trás os semicondutores, chegam o laser e as comunicações ópticas. Na sequência de cada um destes desenvolvimentos, vieram muitos avanços na biologia, medicina, e até mesmo na história e arqueologia. A física abriu o caminho para essas e outras áreas da ciência se desenvolverem.O século XXI começa muito diferente de qualquer século anterior. Os sistemas educacionais do planeta cresceram ainda mais do que a população, de maneira que o acesso as universidades não é mais uma exclusividade de poucos homens e quase nenhuma mulher. A universidade de Zurich tinha pouco mais de 2000 alunos quando Einstein se candidatou a uma vaga, e Marie Curie não tinha uma infinidade de opções já que a universidade de Paris era uma das poucas que “aceitava” mulheres no final do século XIX. Hoje a Universidade de São Paulo sozinha possui mais professores do que Zurich tinha como alunos, e no Brasil as mulheres ocupam cerca de metade das vagas universitárias. Entretanto muitos dos problemas deste século deveriam passar por um intenso debate científico; na prática o que continuamos a ver são discussões e tomada de decisões em termos exclusivamente econômicos e políticos.Num mundo tão diferente do passado, a física talvez seja vítima de seu sucesso. Somos tentados a justificar toda e qualquer pesquisa com a ideia de que é pelo bem comum de todos, como se o financiamento de nossas pesquisas fossem um elemento sagrado do orçamento. Assim dizemos que os governos deveriam investir em pesquisa, mesmo em tempos de crise. Certamente eu acredito que a ciência, sobretudo a brasileira precisa desenvolver-se; entretanto, que tipo de ciência queremos para o nosso país? Nossa sociedade está de fato se servindo de nosso conhecimento? Estamos ensinando o que a população? Vale lembrar que no mundo todo o financiamento da ciência é predominantemente público, e em geral o acesso ao próprio conteúdo produzido em nossas academias não é aberto, i.e. o estado paga para ter acesso aos artigos que financiou.Vejamos alguns exemplos dos países ditos desenvolvidos que justificam heuristicamente o meu ponto. Todos reconhecemos que os Estados Unidos são o líder mundial em praticamente qualquer parâmetro que almeje medir o desempenho científico; seja pelo número de publicações, de patentes ou de prêmios Nobel. Entretanto, no ano de 2017 ainda é possível que esse país eleja um presidente que ignora toda a sua comunidade científica e faça uma opção na direção dos combustíveis fósseis em detrimento de fontes alternativas ou do investimento no desenvolvimentos de novas fontes para sua matríz. É fato que o sistema educacional americano não se destaca pelo mundo; programas como o “No Child Left Behind” foram implementados e pouca ou nenhuma evolução foi observada. Por fim podemos concluir que o mais premiado sistema universitário do planeta não reflete seu desempenho na população do país, que por sua vez não vê um grande problema em declarações absurdas de um presidenciável[Ele há pouco tempo acusou a administração Obama, que tinha o Nobel Steven Chu como ministro de energia de enfraquecer os EUA ao acreditar na invenção chinesa do aquecimento global.]. Dezenas de milhões de americanos ignoram o que sua academia tem a dizer sobre o assunto.É fato que o carvão tem sido o combustível responsável pelo crescimento econômico mundial neste século, sobretudo a partir da China. Para cada MWh de energia renovável, 35MWh de carvão foram adicionados a matríz mundial nos anos 2000. Enquanto isso, na Europa muitos países começam a se mover na direção das energias renováveis, sem sequer considerar as limitações físicas da questão. Por um lado temos a promessa, principalmente dos “partidos verdes” de que o futuro será movido a energia eólica e fotovoltaíca, entretanto as tecnologias que dispomos hoje parecem limitar-se inclusive na disponibilidade de matéria prima em nosso planeta<cit.>; e pra mudar o fato de que Prata, Telúrio e Neodímio são raros, precisariamos mudar o “Big Bang”. E claro, a contribuição das fotovoltaicas tem sido mais a de dar esperança a algunspaíses, como a Alemanha, do que de fato resolver o problema; em boa parte da Europa a energia solar produz no inverno 1/5 de sua capacidade, e isso justamente na época do ano em que o consumo de energia aumenta. O que dizem as academias européias sobre essa questão? Praticamente nada, pois algum barulho vêm das academias apenas para reclamar das medidas de austeridade.O caso da França e seu programa nuclear é ainda mais interessante. Na década de 1970 se implementou uma opção pela energia nuclear, entre outras coisas convencendo boa parte de sua população de que era a única maneira de construir uma independência energética para o país; na época o mundo era sacudido pela crise do preço do petróleo. Hoje 80% de sua eletricidade é de origem nuclear, e somando-se a isso o monópolio de tecnologia construído sob a propaganda do tratado de não proliferação, a França consegue participar de projetos de reatores e usinas de reprocessamento por todo mundo. Por este lado o programa parece ter sido um sucesso, entretanto, nem mesmo em território Francês o problema dos resíduos radioativos foi resolvido. A França tem em “La Hague”<cit.> um dos maiores estoques de lixo a procura de um destino, e muitos defendem que a usina de reprocessamento foi um erro e deveria ser fechada; de lá saem isótopos radioativos gasosos e líquidos que espalham-se pelo norte do país, sem que se saiba quais serão as consequências. O programa nuclear francês, sem dar explicações claras a população levou 11 anos e 60 bilhões de francos para construir seu reator “fast breeder”, chamado de “Superphènix”, que eliminaria o problema do Plutônio; o reator funcionou por 10 anos e foi desligado permanentemente depois de produzir 2 bilhões de francos de eletricidade. Se não bastasse este episódio, apenas 5 anos depois a França anuncia a criação de uma mega empresa do setor nuclear, a AREVA, que iria unir o “know-how” francês para liderar projetos nucleares pacíficos pelo mundo, vendendo produtos e serviços que garantiriam o desenvolvimento econômico. Em 2005 anunciam um contrato para a construção de um reator de terceira geração na Finlândia, que seria o primeiro grande empreendimento da AREVA, no valor de 3 bilhões de Euros. O reator que deveria estar operacional em 2010 continua em construção, e já consumiu 9 bilhões de euros que sairam dos cofres franceses. No começo dos anos 2010, num escândalo de corrupção, a AREVA começou a ser investigada por ter comprado a canadense “UraMin”; no negócio, dizem os investigadores, o preço da UraMin foi inflado e parte do dinheiro foi utilizado como propina para que magnatas no continente africano garantissem a AREVA contratos para a construção de novos reatores em seus países. O domínio da física nuclear tem ainda exemplos mais graves. O projeto Manhattan e a subsequente corrida armamentista deixaram em “Hanford site”, as margens do rio Columbia,um dos maiores depósitos de lixo radioativo do planeta em condições precárias. O Reino Unido e outros países despejaram centenas de milhares de toneladas de lixo radioativo no Atlântico. A Rússia aposentou submarinos nucleares com seus reatores no fundo do mar do ártico. O Japão construiu uma usina idêntica a de “La Hague”, e agora é proprietário de dezenas de toneladas de plutônio a procura de um destino, já que inclusive a maioria dos reatores japoneses que poderiam utilizá-lo está desligada. Outros exemplos semelhantes não faltam.Qual o papel da ciência nesse tipo de questão? Não somos culpados, mas somos responsáveis por muitos destes problemas; inclusive por nos omitirmos. No caso do Brasil, onde está acontecendo a discussão sobre o destino do lixo de Angra? Onde está acontecendo a discussão sobre novos reatores nucleares? Infelizmente não é nas universidades, tão pouco é no CNPQ; A questão do lixo está nos tribunais, que agora obrigam o estado brasileiro a encontrar um destino definitivo para ele, ao mesmo tempo que os novos reatores devem ser de mais interesse de empreiteiras, investidores de multinacionais do setor nuclear e, por consequência, de partidos políticos. Instituições como o CNEM, a Eletronuclear ou o Ibama não tem como função incluir a sociedade neste debate. As Universidades Públicas do Brasil deveriam cumprir essa função.Vale lembrar que o nosso CNPQ foi fundado depois da segunda guerra, quando o nosso representante na comissão de energia atômica da recém criada ONU, o Almirante Álvaro Alberto da Motta e Silva, recomendou ao governo a criação de um conselho nacional de pesquisa. E o que tem feito esse conselho nos dias de hoje? Ainda que o CNPQ tenha participado de louváveis feitos, sendo uma das principais agências de fomento e contribuindo portanto para o expressivo aumento do número de doutores no Brasil, é preciso que se discuta o futuro dessa importante instituição.Ás perguntas que paíram sobre a minha cabeça são, entre outras: O “conselho nacional de pesquisa” deve continuar no papel de distribuir uma centena de milhões de reais quantizados em pacotes de 30 ou 50 mil reais? Deve continuar fomentando a numerologia da nossa produção científica? Deve distribuir os “títulos de nobreza” na forma de bolsas de produtividade? Isto é o que tinham em mente aquele grupo de militares e cientístas que criaram o projeto científico nacional quando o Brasil tinha 80% de analfabetismo na década de 1950? Convenhamos, essa discussão precisava ter acontecido há décadas; ou melhor, previsava acontecer de maneira contínua. Mas na academia, enquanto o minguado orçamento de ciência e tecnologia não é cortado, sobretudo uma grande parcela da nobreza da produtividade não faz nenhuma questão de discutir a ciência brasileira. Afinal, estão fazendo a parte deles: autores publicam artigos mensalmente, mesmo quando se tem certeza de que às promessas da introdução e da conclusão dos trabalhos é uma fantasia. É necessário que exista pesquisa em ciência básica, sem pretenções imediatas de atingir o público; entretanto, em “hard science” é claro que a imensa maioria das pesquisas está desligada da realidade, ou pelo menos da realidade brasileira. A busca por indicadores de produção já atingiu os doutorandos, que não tem futuro se não publicarem uma dezena de artigos; atingiu os mestrandos, que num comprimido espaço de 2 anos devem fazer seu mestrado e publicar um artigo. Em muitos casos nem mesmo a publicação de uma dissertação em português é mais necessária, bastando a publicação do artigo em inglês, aumentando ainda mais a distância entre a ciência e a sociedade que a está financiando; por isso tudo não é surpresa encontrar alunos recém graduados desesperados atrás de indicadores de produção que lhe garantam uma vaga de mestrado.É preciso não diminuir as realizações da ciência brasileira, simbolizadas pela história do CNPQ. Começamos a fazer ciência num contexto de instabilidade política que perdurou da segunda guerra até a década de 1990; aumentamos nossos indicadores, o número de teses defendidas por ano foi multiplicado por 8. Entretanto, o próprio CNPQ reconhece que o investimento privado em pesquisa no Brasil é pífio; o que, convenhamos mais uma vez, não deveria ser uma surpresa: nosso conselho nacional de pesquisa fez o que pôde para não ser engolido em meio a tantos desafios, mas no caminho desfigurou-se em agência de fomento a medida que deixou de dar conselhos ao estado brasileiro e a toda a população sobre que direção devemos tomar.Não deveria o CNPQ e nossas academias discutirem agora se queremos extrair gás de xisto em nosso território?Ou se vamos construir uma grande hidrelétrica ou mais alguns reatores nucleares? Não deveríamos estar a frente dessa discussão? Nós não estamos a frente de nenhuma dessas e de tantas outras, porque estamos preocupados com o que o CNPQ e a CAPES querem que seja nossa preocupação. Não deveríamos esperar as usinas do Xingu e de Angra para brigar pela paralisação de um canteiro de obras; deveríamos propor hoje quais serão as usinas que daqui a 20 anos vão garantir a necessidade energética brasileira de maneira que não seja necessário começar um canteiro de uma obra a qual nos opomos. Uma audiência no Senado com meia dúzia de especialistas não é o mesmo que ouvir e debater com a sociedade.Por todos os motivos justificados nessa carta, manifesto o meu ponto de vista da necessidade de uma refundação da política nacional que vá muito além da “reforma eleitoral” em discussão em Brasília. A comunidade academica nacional precisa fazer política, o CNPQ precisa voltar a ser um conselho que oriente políticas estatáis e privadas de desenvolvimento, e nós pesquisadores precisamos voltar a se preocupar com o avanço da ciência, com os problemas e anseios de nossa sociedade. Só precisamos explicar para a sociedade que ela está sendo ameaçada pelos cortes no ministério de ciência e tecnologia porque a sociedade não se sente ameaçada; de fato, o que está ameaçado além do já minguado orçamento do CNPQ são as bolsas de produtividade e os indicadores de produção que a sociedade sequer conhece. | http://arxiv.org/abs/1704.08610v1 | {
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A Generalization of Convolutional Neural Networks to Graph-Structured Data Yotam Hechtlinger, Purvasha Chakravarti & Jining Qin Department of Statistics Carnegie Mellon University{yhechtli,pchakrav,jiningq}@stat.cmu.eduDecember 30, 2023 ================================================================================================================================================================================In the field of statistical prediction, the tasks of model selection and model evaluation have received extensive treatment in the literature.Among the possible approaches for model selection and evaluation are those based on covariance penalties, which date back to at least 1960s, and are still widely used today. Most of the literature on this topic is based on what we call the “Fixed-X” assumption, where covariate values are assumed to be nonrandom. By contrast, in most modern predictive modeling applications, it is more reasonable to take a “Random-X” view, where the covariate values (both those used in training and for future predictions) are random. In the current work, we study the applicability of covariance penalties in the Random-X setting. We propose a decomposition of Random-X prediction error in which the randomness in the covariates has contributions to both the bias and variance components of the error decomposition.This decomposition is general, and for concreteness, we examine it in detail in the fundamental case of least squares regression. We prove that, for the least squares estimator, the move from Fixed-X to Random-X prediction always results in an increase in both the bias and variance components of the prediction error. When the covariates are normally distributed and the linear model is unbiased, all terms in this decomposition are explicitly computable, which leads us to propose an extension of Mallows' Cp <cit.> that we call .Whileprovides an unbiased estimate of Random-X prediction error for normal covariates, we also show using standard random matrix theory that it is asymptotically unbiased for certain classes of nonnormal covariates. When the noise variance is unknown, plugging in the usual unbiased estimate leads to an approach that we call , which turns out to be closely related to the existing methods Sp <cit.>, and GCV (generalized cross-validation, ). As for the excess bias, we propose an estimate based on the well-known “shortcut-formula” for ordinary leave-one-out cross-validation (OCV), resulting in a hybrid approach we call . We give both theoretical arguments and numerical simulations to demonstrate that this approach is typically superior to OCV, though the difference is usually small. Lastly, we examine the excess bias and excess variance of other estimators, namely, ridge regression and some common estimators for nonparametric regression. The surprising result we get for ridge is that, in the heavily-regularized regime, the Random-X prediction variance is guaranteed to be smaller than the Fixed-X variance, which can even lead to smaller overall Random-X prediction error. In statistical prediction, classical approaches for model selection and model evaluation based on covariance penalties are still widely used. Most of the literature on this topic is based on what we call the “Fixed-X” assumption, where covariate values are assumed to be nonrandom. By contrast, it is often more reasonable to take a “Random-X” view, where the covariate values are independently drawn for both training and prediction. To study the applicability of covariance penalties in this setting, we propose a decomposition of Random-X prediction error in which the randomness in the covariates contributes to both the bias and variance components.This decomposition is general, but we concentrate on the fundamental case of least squares regression. We prove that in this setting the move from Fixed-X to Random-X prediction results in an increase in both bias and variance. When the covariates are normally distributed and the linear model is unbiased, all terms in this decomposition are explicitly computable, which yields an extension of Mallows' Cp that we call .also holds asymptotically for certain classes of nonnormal covariates.When the noise variance is unknown, plugging in the usual unbiased estimate leads to an approach that we call , which is closely related to Sp (Tukey 1967), and GCV (Craven and Wahba 1978). For excess bias, we propose an estimate based on the “shortcut-formula” for ordinary cross-validation (OCV), resulting in an approach we call . Theoretical arguments and numerical simulations suggest thatis typically superior to OCV, though the difference is small.We further examine the Random-X error of other popular estimators. The surprising result we get for ridge regression is that, in the heavily-regularized regime, Random-X variance is smaller than Fixed-X variance, which can lead to smaller overall Random-X error.§ INTRODUCTION A statistical regression model seeks to describe the relationship between a response y ∈ and a covariate vector x ∈^p, based on training data comprised of paired observations (x_1,y_1),…,(x_n,y_n). Many modern regression models are ultimately aimed at prediction: given a new covariate value x_0, we apply the model to predict the corresponding response value y_0. Inference on the prediction error of regression models is a central part of model evaluation and model selection in statistical learning (e.g., ). A common assumption that is used in the estimation of prediction error is what we call a “Fixed-X” assumption, where the training covariate values x_1,…,x_n are treated as fixed, i.e., nonrandom, as are the covariate values at which predictions are to be made, x_01,…,x_0n, which are also assumed to equal the training values. In the Fixed-X setting, the celebrated notions of optimism and degrees of freedom lead to covariance penalty approaches to estimate the prediction performance of a model <cit.>, extending and generalizing classical approaches like Mallows' Cp <cit.> and AIC <cit.>.The Fixed-X setting is one of the most common views on regression (arguably the predominant view), and it can be found at all points on the spectrum from cutting-edge research to introductory teaching in statistics. This setting combines the following two assumptions about the problem. (i) The covariate values x_1,…,x_n used in training are not random (e.g., designed), and the only randomness in training is due to the responses y_1,…,y_n.(ii) The covariates x_01,…,x_0n used for prediction exactly match x_1,…,x_n, respectively, and the corresponding responses y_01,…,y_0n are independent copies of y_1,…,y_n, respectively.Relaxing assumption (i), i.e., acknowledging randomness in the training covariates x_1,…,x_n, and taking this randomness into account when performing inference on estimated parameters and fitted models, has received a good deal of attention in the literature. But, as we see it, assumption (ii) is the critical onethat needs to be relaxed in most realistic prediction setups. To emphasize this, we define two settings beyond the Fixed-X one, that we call the “Same-X” and “Random-X” settings.The Same-X setting drops assumption (i), but does not account for new covariate values at prediction time.The Random-X setting drops both assumptions, and deals with predictions at new covariates values.These will be defined more precisely in the next subsection. §.§ Notation and assumptions We assume that the training data (x_1,y_1),…,(x_n,y_n) are i.i.d. according to some joint distribution P. This is an innocuous assumption, and it means that we can posit a relationship for the training data,y_i = f(x_i) + ϵ_i,i=1,…,nwhere f(x)=(y|x), and the expectation here is taken with respect to a draw (x,y) ∼ P.We also assume that for (x,y) ∼ P,ϵ=y-f(x) is independent of x,which is less innocuous, and precludes, e.g., heteroskedasticity in the data.We let σ^2 = (y|x) denote the constant conditional variance.It is worth pointing out that some results in this paper can be adjusted or modified to hold when (<ref>) is not assumed; but since other results hinge critically on (<ref>), we find it is more convenient to assume (<ref>) up front.For brevity, we write Y=(y_1,…,y_n) ∈^n for the vector of training responses, and X ∈^n× p for the matrix of training covariates with ith row x_i, i=1,…,n. We also write Q for the marginal distribution of x when (x,y) ∼ P, and Q^n=Q ×⋯× Q (n times) for the distribution of X when its n rows are drawn i.i.d. from Q. We denote by ỹ_i an independent copy of y_i, i.e., an independent draw from the conditional law of y_i|x_i, for i=1,…,n, and we abbreviate Ỹ=(ỹ_1,…,ỹ_n) ∈^n. These are the responses considered in the Same-X setting, defined below. We denote by (x_0,y_0) an independent draw from P.This the covariate-response pair evaluated in the Random-X setting, also defined below.Now consider a model building procedure that uses the training data (X,Y) to build a prediction function _n : ^p →.We can associate to this procedure two notions of prediction error:= _X,Y,Ỹ[1/n∑_i=1^n ( ỹ_i - _n(x_i))^2] and = _X,Y,x_0,y_0( y_0 - _n(x_0) )^2,where the subscripts on the expectations highlight the random variables over which expectations are taken.(We omit subscripts when the scope of the expectation is clearly understood by the context.) The Same-X and Random-X settings differ only in the quantity we use to measure prediction error: in Same-X, we use , and in Random-X, we use .We callthe Same-X prediction error andthe Random-X prediction error, though we note these are also commonly called in-sample and out-of-sample prediction error, respectively.We also note that by exchangeability,= _X,Y,ỹ_1( ỹ_1 - _n(x_1))^2.Lastly, the Fixed-X setting is defined by the same model assumptions as above, but with x_1,…,x_n viewed as nonrandom, i.e., we assume the responses are drawn from (<ref>), with the errors being i.i.d. We can equivalently view this as the Same-X setting, but where we condition on x_1,…,x_n. In the Fixed-X setting, prediction error is defined by= _Y,Ỹ[ 1/n∑_i=1^n ( ỹ_i - _n(x_i))^2].(Without x_1,…,x_n being random, the terms in the sum above are no longer exchangeable, and sodoes not simplify asdid.) §.§ Related work From our perpsective, much of the work encountered in statistical modeling takes a Fixed-X view, or when treating the covariates as random, a Same-X view.Indeed, when concerned with parameter estimates and parameter inferences in regression models, the randomness of new prediction points plays no role, and so the Same-X view seems entirely appropriate.But, when focused on prediction, the Random-X view seems more realistic as a study ground for what happens in most applications.On the other hand, while the Fixed-X view is common, the Same-X and Random-X views have not exactly been ignored, either, and several groups of researchers in statistics, but also in machine learning and econometrics, fully adopt and argue for such random covariate views. A scholarly and highly informative treatment of how randomness in the covariates affects parameter estimates and inferences in regression models is given in <cit.>.We also refer the reader to these papers for a nice review of the history of work in statistics and econometrics on random covariate models. It is also worth mentioning that in nonparametric regression theory, it is common to treat the covariates as random, e.g., the book by <cit.>, and the random covariate view is the standard in what machine learning researchers call statistical learning theory, e.g., the book by <cit.>. Further, a stream of recent papers in high-dimensional regression adopt a random covariate perspective, to give just a few examples: <cit.>.In discussing statistical models with random covariates, one should differentiate between what may be called the “i.i.d. pairs” model and “signal-plus-noise” model.The former assumes i.i.d. draws (x_i,y_i), i=1,…,n from a common distribution P, or equivalently i.i.d. draws from the model (<ref>); the latter assumes i.i.d. draws from (<ref>), and additionally assumes (<ref>).The additional assumption (<ref>) is not a light one, and it does not allow for, e.g., heteroskedasticity. The books by <cit.> assume the i.i.d. pairs model, and do not require (<ref>) (though their results often require a bound on the maximum of (y|x) over all x.)More specifically related to the focus of our paper is the seminal work of <cit.>, who considered Random-X prediction error mostly from an intuitive and empirical point of view. A major line of work on practical covariance penalties for Random-X prediction error in least squares regression begins with <cit.> and <cit.>, and continues onwards throughout the late 1970s and early 1980s with <cit.>. Some more recent contributions are found in <cit.>. A common theme to these works is the assumption that (x,y) is jointly normal.This is a strong assumption, and is one that we avoid in our paper (though for some results we assume x is marginally normal); we will discuss comparisons to these works later. Through personal communication, we are aware of work in progress by Larry Brown, Andreas Buja, and coauthors on a variant of Mallows' Cp for a setting in which covariates are random.It is out understanding that they take somewhat of a broader view than we do in our proposals ,,, each designed for a more specific scenario, but resort to asymptotics in order to do so.Finally, we must mention that an important alternative to covariance penalties for Random-X model evaluation and selection are resampling-based techniques, like cross-validation and bootstrap methods (e.g., ). In particular, ordinary leave-one-out cross-validation or OCV evaluates a model by actually building n separate prediction models, each one using n-1 observations for training, and one held-out observation for model evaluation. OCV naturally provides an almost-unbiased estimate of Random-X prediction error of a modeling approach (“almost”, since training set sizes are n-1 instead of n), albeit, at a somewhat high price in terms of variance and inaccuracy (e.g., see ). Altogether, OCV is an important benchmark for comparing the results of any proposed Random-X model evaluation approach.§ DECOMPOSING AND ESTIMATING PREDICTION ERROR §.§ Bias-variance decompositions Consider first the Fixed-X setting, where x_1,…,x_n are nonrandom. Recall the well-known decomposition of Fixed-X prediction error (e.g., ):= σ^2 + 1/n∑_i=1^n (_n(x_i) - f(x_i))^2 + 1/n∑_i=1^n (_n(x_i))where the latter two terms on the right-hand side above are called the (squared) bias and variance of the estimator _n, respectively. In the Same-X setting, the same decomposition holds conditional on x_1,…,x_n.Integrating out over x_1,…,x_n, and using exchangeability, we conclude= σ^2 + _X ((_n(x_1) | X) - f(x_1))^2_B + _X (_n(x_1) | X) _V.The last two terms on the right-hand side above are integrated bias and variance terms associated with _n, which we denote by B and V, respectively.Importantly, whenever the Fixed-X variance of the estimator _n in question is unaffected by the form of f(x)=(y|x) (e.g., as is the case in least squares regression), then so is the integrated variance V.For Random-X, we can condition on x_1,…,x_n and x_0, and then use similar arguments to yield the decomposition= σ^2 + _X,x_0( (_n(x_0) | X,x_0) - f(x_0))^2 + _X,x_0(_n(x_0) | X,x_0).For reasons that will become clear in what follows, it suits our purpose to rearrange this as= σ^2 + B + V+ _X,x_0((_n(x_0) | X,x_0) - f(x_0))^2 - _X ((_n(x_1) | X) - f(x_1))^2_ + _X,x_0(_n(x_0) | X,x_0) - _X (_n(x_1) | X)_.We call the quantities in (<ref>), (<ref>) the excess bias and excess variance of _n (“excess” here referring to the extra amount of bias and variance that can be attributed to the randomness of x_0), denoted byand , respectively. We note that, by construction,-=+ ,thus, e.g., +≥ 0 implies the Random-X (out-of-sample) prediction error of _n is no smaller than its Same-X (in-sample) prediction error. Moreover, asis easily estimated following standard practice for estimating , discussed next, we see that estimates or bounds , lead to estimates or bounds on . §.§ Optimism for Fixed-X and Same-X Starting with the Fixed-X setting again, we recall the definition of optimism, e.g., as in <cit.>,= _Y,Ỹ[1/n∑_i=1^n (ỹ_i-_n(x_i))^2 - 1/n∑_i=1^n (y_i-_n(x_i))^2],which is the difference in prediction error and training error. Optimism can also be expressed as the following elegant sum of self-influence terms,= 2/n∑_i=1^n (y_i, _n(x_i)),and furthermore, under a normal regression model (i.e., the data model (<ref>) with ϵ∼ N(0,σ^2)) and some regularity conditions on _n (i.e., continuity and almost differentiability as a function of y),= 2σ^2/n∑_i=1^n [∂_n(x_i)/∂ y_i],which is often called Stein's formula <cit.>.Optimism is an interesting and important concept because an unbiased estimateof(say, from Stein's formula or direct calculation) leads to an unbiased estimate of prediction error:1/n∑_i=1^n (y_i-_n(x_i))^2 + .When _n is given by the least squares regression of Y on X (and X has full column rank), so that _n(x_i)=x_i^T (X^T X)^-1 X^T Y, i=1,…,n, it is not hard to check that =2σ^2p/n. This is exact and hence “even better” than an unbiased estimate; plugging in this result above forgives us Mallows' Cp <cit.>.In the Same-X setting, optimism can be defined similarly, except additionally integrated over the distribution of x_1,…,x_n,= _X,Y,Ỹ[1/n∑_i=1^n (ỹ_i-_n(x_i))^2 - 1/n∑_i=1^n (y_i-_n(x_i))^2] = 1/n∑_i=1^n _X (y_i, _n(x_i)|X).Some simple results immediately follow. (i) If T(X,Y) is an unbiased estimator ofin the Fixed-X setting, for any X in the support of Q^n, then it is also unbiased forin the Same-X setting.(ii) Ifin the Fixed-X setting does not depend on X (e.g., as is true in least squares regression), then it is equal toin the Same-X setting.Some consequences of this proposition are as follows. * For the least squares regression estimator of Y on X (and X having full column rank almost surely under Q^n), we have ==2σ^2 p/ n.* For a linear smoother, where _n(x_i)=s(x_i)^T Y, i=1,…,n and we denote by S(X) ∈^n× n the matrix with rows s(x_1),…,s(x_n), we have (by direct calculation) = 2σ^2(S(X)) / n and = 2σ^2_X[(S(X))]/n.* For the lasso regression estimator of Y on X (and X being in general position almost surely under Q^n), and a normal data model (i.e., the model in (<ref>), (<ref>) with ϵ∼ N(0,σ^2)), <cit.> prove that for any value of the lasso tuning parameter λ > 0 and any X, the Fixed-X optimism is just = 2σ^2 _Y|A_λ(X,Y)| /n, where A_λ(X,Y) is the active set at the lasso solution at λ and |A_λ(X,Y)| is its size; therefore we also have = 2σ^2 _X,Y|A_λ(X,Y)|/n. Overall, we conclude that for the estimation of prediction error, the Same-X setting is basically identical to Fixed-X. We will see next that the situation is different for Random-X. §.§ Optimism for Random-X For the definition of Random-X optimism, we have to now integrate over all sources of uncertainty,= _X,Y,x_0,y_0[(y_0-_n(x_0))^2 - (y_1-_n(x_1))^2].The definitions of , are both given by a type of prediction error (Same-X or Random-X) minus training error, and there is just one common way to define training error. Hence, by subtracting training error from both sides in the decomposition (<ref>), (<ref>), (<ref>), we obtain the relationship:=++ ,where , are the excess bias and variance as defined in (<ref>), (<ref>), respectively.As a consequence of our definitions, Random-X optimism is tied to Same-X optimism by excess bias and variance terms, as in (<ref>). The practical utility of this relationship: an unbiased estimate of Same-X optimism (which, as pointed out in the last subsection, follows straightforwardly from an unbiased estimate of Fixed-X optimism), combined with estimates of excess bias and variance, leads to an estimate for Random-X prediction error.§ EXCESS BIAS AND VARIANCE FOR LEAST SQUARES REGRESSIONIn this section, we examine the case when _n is defined by least squares regression of Y on X, where we assume X has full column rank (or, when viewed as random, has full column rank almost surely under its marginal distribution Q^n). §.§ Nonnegativity of , Our first result concerns the signs ofand .For _n the least squares regression estimator, we have both ≥ 0 and ≥ 0.We prove the result separately forand .Nonnegativity of . For a function g : ^p →, we will write g(X)=(g(x_1),…,g(x_n))∈^n, the vector whose components are given by applying g to the rows of X. Letting X_0 ∈^n× p be a matrix of test covariate values, whose rows are i.i.d. draws from Q, we note that excess variance in (<ref>) can be equivalently expressed as= _X,X_01/n[ ( _n(X_0)|X,X_0 )] - _X 1/n[ ( _n(X)|X )].Note that the second term here is just _X [(σ^2/n)(X(X^T X)^-1X^T)] = σ^2 p/n. The first term isσ^2/n(_X,X_0[ (X^T X)^-1 X_0^TX_0 ])= σ^2/n([ (X^T X)^-1][ X_0^TX_0]) = σ^2/n([ (X^T X)^-1][ X^TX]),where in the first equality we used the independence of X and X_0, and in the second equality we used the identical distribution of X and X_0. Now, by a result of <cit.>, we know that [(X^tX)^-1]- [ (X^tX)]^-1 is positive semidefinite. Thus we haveσ^2/n([ (X^T X)^-1][X^TX]) ≥σ^2/n([ (X^T X) ]^-1 [X^TX]) = σ^2 p/n.This proves ≥ 0.Nonnegativity of . This result is actually a special case of Theorem <ref>, and its proof follows from the proof of the latter.An immediate consequence of this, from the relationship between Random-X and Same-X prediction error in (<ref>), (<ref>), (<ref>), is the following. For _n the least squares regression estimator, we have ≥. This simple result, that the Random-X (out-of-sample) prediction error is always larger than the Same-X (in-sample) prediction error for least squares regression, is perhaps not suprising; however, we have not been able to find it proven elsewhere in the literature at the same level of generality.We emphasize that our result only assumes (<ref>), (<ref>) and places no other assumptions on the distribution of errors, distribution of covariates, or the form of f(x)=(y|x).We also note that, while this relationship may seem obvious, it is in fact not universal. Later in Section <ref>, we show that the excess variancein heavily-regularized ridge regression is guaranteed to be negative, and this can even lead to <. §.§ Exact calculation offor normal covariates Beyond the nonnegativity of ,, it is actually easy to quantifyexactly in the case that the covariates follow a normal distribution.Assume that Q = N(0,Σ), where Σ∈^p× p is invertible, and p<n-1.Then for the least squares regression estimator,= σ^2 p/np+1/n-p-1. As the rows of X are i.i.d. from N(0,Σ), we have X^T X ∼ W(Σ,n), which denotes a Wishart distribution with n degrees of freedom, and so (X^T X)=nΣ. Similarly, (X^T X)^-1∼ W^-1 (Σ^-1,n), denoting an inverse Wishart with n degrees of freedom, and hence [(X^T X)^-1]=Σ^-1/(n-p-1). From the arguments in the proof of Theorem <ref>,= σ^2/n([ (X^T X)^-1][X^TX]) - σ^2 p/n = σ^2/n( I_p× pn/n-p-1) - σ^2 p/n = σ^2p/np+1/n-p-1,completing the proof. Interestingly, as we see, the excess variancedoes not depend on the covariance matrix Σ in the case of normal covariates. Moreover, we stress that (as a consequence of our decomposition and definition of ,), the above calculation does not rely on linearity of f(x)=(y|x).When f(x) is linear, i.e., the linear model is unbiased, it is not hard to see that =0, and the next result follows from (<ref>).Assume the conditions of Theorem <ref>, and further, assume that f(x)=x^T β, a linear function of x. Then for the least squares regression estimator,=+ σ^2 p/np+1/n-p-1 = σ^2 p/n(2+p+1/n-p-1)For the unbiased case considered in Corollary <ref>, the same result can be found in previous works, in particular in <cit.>, where it is proven in the appendix. It is also similar to older results from <cit.>, which assume the pair (x,y) is jointly normal (and thus also assume the linear model to be unbiased). We return to these older classical results in the next section.When bias is present, our decomposition is required, so that the appropriate result would still apply to . §.§ Asymptotic calculation offor nonnormal covariates Using standard results from random matrix theory, the result of Theorem <ref> can be generalized to an asymptotic result over a wide class of distributions.[We thank Edgar Dobriban for help in formulating and proving this result.]Assume that x ∼ Q is generated as follows: we draw z ∈^p, having i.i.d. components z_i ∼ F, i=1,…,p, where F is anydistribution with zero mean and unit variance, and then set x=Σ^1/2 z, where Σ∈^p × p is positive definite and Σ^1/2 is its symmetric square root. Consider an asymptotic setup where p/n →γ∈ (0,1) as n →∞. Then→σ^2 γ^2/1-γas n →∞. Denote by X_n = Z_n Σ^1/2 the training covariate matrix, where Z_n has rows z_1,…,z_n, and we use subscripts of X_n,Z_n to denote the dependence on n in our asymptotic calculations below. Then as in the proof of Theorem <ref>,= σ^2/n([ (X_n^T X_n)^-1][X_n^TX_n]) = σ^2/n([ (Z_n^T Z_n)^-1][Z_n^TZ_n]) = σ^2/n( n [ (Z_n^T Z_n)^-1] ).The second equality used the relationship X_n=Z_nΣ^1/2, and the third equality used the fact that the entries of Z_n are i.i.d.with mean 0 and variance 1. This confirms thatdoes not depend on the covariance matrix Σ.Further, by the Marchenko-Pastur theorem, the distribution of eigenvalues λ_1,…,λ_p of Z_n^T Z_n/n converges to a fixed law, independent of F; more precisely, the random measure μ_n, defined byμ_n(A) = 1/p∑_i=1^p 1{λ_i ∈ A},converges weakly to the Marchenko-Pastor law μ.We note that μ has density bounded away from zero when γ<1.As the eigenvalues of n(Z_n^T Z_n)^-1 are simply 1/λ_1,…,1/λ_p, we also have that the random measure μ̃_n, defined byμ̃_n(A) = 1/p∑_i=1^p 1{1/λ_i ∈ A},converges to a fixed law, call it μ̃. Denoting the mean of μ̃ by m, we now have= σ^2/n( n [ (Z_n^T Z_n)^-1] ) = σ^2 p/n[1/p∑_i=1^p 1/λ_i] →σ^2 γ m as n →∞.As this same asymptotic limit, independent of F, must agree with specific the case in which F=N(0,1), we can conclude from Theorem <ref> that m=γ/(1-γ), which proves the result. The next result is stated for completeness.Assume the conditions of Theorem <ref>, and moreover, assume that the linear model is unbiased for n large enough.Then→σ^2 γ2-γ/1-γas n →∞.It should be noted that the requirement of Theorem <ref> that the covariate vector x be expressible as Σ^1/2 z with the entries of z i.i.d. is not a minor one, and limits the set of covariate distributions for which this result applies, as has been discussed in the literature on random matrix theory (e.g., ). In particular, left multiplication by the square root matrix Σ^1/2 performs a kind of averaging operation. Consequently, the covariates x can either have long-tailed distributions, or have complex dependence structures, but not both, since then the averaging will mitigate any long tail of the distribution F. In our simulations in Section <ref>, we examine some settings that combine both elements, and indeed the value ofin such settings can deviate substantially from what this theory suggests.§ COVARIANCE PENALTIES FOR RANDOM-X LEAST SQUARESWe maintain the setting of the last section, taking _n to be the least squares regression estimator of Y on X, where X has full column rank (almost surely under its marginal distribution Q). §.§ A Random-X version of Mallows' Cp Let us denote RSS=Y-_n(X)_2^2, and recall Mallows' Cp <cit.>, which is defined as Cp=RSS/n+2σ^2p/n. The results in Theorems <ref> and <ref> lead us to define the following generalized covariance penalty criterion we term := Cp += RSS/n + σ^2 p/n(2+p+1/n-p-1).An asymptotic approximation is given by ≈RSS/n + σ^2 γ (2+γ/(1-γ)), in a problem scaling where p/n →γ∈ (0,1). is an unbiased estimate of Random-X prediction error when the linear model is unbiased and the covariates are normally distributed, and an asymptotically unbiased estimate of Random-X prediction error when the conditions of Theorem <ref> hold. As we demonstrate below, it is also quite an effective measure, in the sense that it has much lower variance (in the appropriate settings for the covariate distributions) compared to other almost-unbiased measures of Random-X prediction error, such as OCV (ordinary leave-one-out cross-validation) and GCV (generalized cross-validation).However, in addition to the dependence on the covariate distribution as in Theorems <ref> and <ref>, two other major drawbacks to the use ofin practice should be acknowledged.(i) The assumption that σ^2 is known. This obviously affects the use of Cp in Fixed-X situations as well, as has been noted in the literature.(ii) The assumption of no bias. It is critical to note here the difference from Fixed-X or Same-X situations, where(i.e., Cp) is independent of the bias in the model and must only correct for the “overfitting” incurred by model fitting. In contrast, in Random-X, the existence of , which is a component ofnot captured by the training error, requires taking it into account in the penalty, if we hope to obtain low-bias estimates of prediction error. Moreover, it is often desirable to assume nothing about the form of the true model f(x)=(y|x),hence it seems unlikely that theoretical considerations like those presented in Theorems <ref> and <ref> can lead to estimates of . We now propose enhancements that deal with each of these problems separately. §.§ Accounting for unknown σ^2 in unbiased least squares Here, we assume that the linear model is unbiased, f(x)=x^T β, but the variance σ^2 of the noise in (<ref>) is unknown. In the Fixed-X setting, it is customary to replace σ^2 in covariance penalty approach like Cp with the unbiased estimate ^2 = RSS/(n-p). An obvious choice is to also use ^2 in place of σ^2 in , leading to a generalized covariance penalty criterion we call := RSS/n + ^2 p/n(2+p+1/n-p-1) = RSS(n-1)/(n-p)(n-p-1).An asymptotic approximation, under the scaling p/n →γ∈ (0,1), is ≈RSS/(n (1-γ)^2).This penalty, as it turns out, is exactly equivalent to the Sp criterion of <cit.>; see also <cit.>.These authors all studied the case in which (x,y) is jointly normal, and therefore the linear model is assumed correct for the full model and any submodel. The asymptotic approximation, on other hand, is equivalent to the GCV (generalized cross-validation) criterion of <cit.>, though the motivation behind the derivation of GCV is somewhat different.Comparingtoas a model evaluation criterion, we can see the price of estimating σ^2 as opposed to knowing it, in their asymptotic approximations. Their expectations are similar when the linear model is true, but the variance of (the asymptotic form) ofis roughly 1/(1-γ)^4 times larger than that of (the asymptotic form) of .So when, e.g., γ = 0.5, the price of not knowing σ^2 translates roughly into a 16-fold increase in the variance of the model evaluation metric. This is clearly demonstrated in our simulation results in the next section. §.§ Accounting for bias and estimatingNext, we move to assuming nothing about the underlying regression function f(x)=(y|x), and we examine methods that account for the resulting bias . First we consider the behavior of(or equivalently Sp) in the case that bias is present.Though this criterion was not designed to account for bias at all, we will see it still performs an inherent bias correction. A straightforward calculation shows that in this case_X,YRSS = (n-p) σ^2 + nB,where recall B=_X (_n(X) | X)-f(X)^2/n, generally nonzero in the current setting, and thus_X,Y = σ^2 n-1/n-p-1 + B n(n-1)/(n-p)(n-p-1)≈σ^2/1-γ + B/(1-γ)^2,the last step using an asymptotic approximation, under the scaling p/n →γ∈ (0,1).Note that the second term on the right-hand side above is the (rough) implicit estimate of integrated Random-X bias used by , which is larger than the integrated Same-X bias B by a factor of 1/(1-γ)^2. Put differently,implicitly assumes thatis (roughly) 1/(1-γ)^2-1 times as big as the Same-X bias.We see no reason to believe that this relationship (between Random-X and Same-X biases) is generally correct, but it is not totally naive either, as we will see empirically thatstill provides reasonably good estimates of Random-X prediction error in biased situations in Section <ref>. A partial explanation is available through a connection to OCV, as discussed, e.g., in the derivation of GCV in <cit.>. We return to this issue in Section <ref>.We describe a more principled approach to estimating the integrated Random-X bias, B+, assuming knowledge of σ^2, and leveraging a bias estimate implicit to OCV. Recall that OCV builds n models, each time leaving one observation out, applying the fitted model to that observation, and using these n holdout predictions to estimate prediction error. Thus it gives us an almost-unbiased estimate of Random-X prediction error(“almost”, because its training sets are all of size n-1 rather than n). For least squares regression (and other estimators), the well-known “shortcut-trick” for OCV (e.g., ) allows us to represent the OCV residuals in terms ofweighted training residuals. Write _n^(-i) for the least squares estimator trained on all but (x_i,y_i), and h_ii the ith diagonal element of X(X^T X)^-1X^T, for i=1,…,n. Then this trick tells us thaty_i - _n^(-i)(x_i) = y_i-_n(x_i)/1-h_ii,which can be checked by applying the Sherman-Morrison update formula for relating the inverse of a matrix to the inverse of its rank-one pertubation.Hence the OCV error can be expressed asOCV = 1/n∑_i=1^n (y_i-_n^(-i)(x_i))^2 = 1/n∑_i=1^n (y_i-_n(x_i)/1-h_ii)^2.Taking an expectation conditional on X, we find that(OCV | X)= 1/n∑_i=1^n (y_i-_n(x_i)|X)/(1-h_ii)^2 + 1/n∑_i=1^n [f(x_i)-(_n(x_i)|X)]^2/(1-h_ii)^2=σ^2/n∑_i=1^n 1/1-h_ii + 1/n∑_i=1^n [f(x_i)-(_n(x_i)|X)]^2/(1-h_ii)^2,where the second line uses (y_i-_n(x_i)|X)=(1-h_ii)σ^2, i=1,…,n. The above display showsOCV-σ^2/n∑_i=1^n 1/1-h_ii = 1/n∑_i=1^n ( (y_i-_n(x_i))^2 - (1-h_ii) σ^2) 1/(1-h_ii)^2is an almost-unbiased estimate of the integrated Random-X prediction bias, B+ (it is almost-unbiased, due to the almost-unbiased status of OCV as an estimate of Random-X prediction error). Meanwhile, an unbiased estimate of the integrated Same-X prediction bias B isRSS/n - σ^2 (n-p)/n = 1/n∑_i=1^n ((y_i-_n(x_i))^2 - (1-h_ii) σ^2 ).Subtracting the last display from the second to last delivers= 1/n∑_i=1^n ((y_i-_n(x_i))^2 - (1-h_ii) σ^2 ) (1/(1-h_ii)^2 - 1),an almost-unbiased estimate of the excess bias . We now define a generalized covariance penalty criterion that we callby adding this to := RCp + = OCV - σ^2/n∑_i=1^n h_ii/1-h_ii+ σ^2 p/n(1+p+1/n-p-1).It is worth pointing out that, likeand , assumes that we are in a setting covered by Theorem <ref> or asymptotically by Theorem <ref>, as it takes advantage of the value ofprescribed by these theorems.A key question, of course, is: what have we achieved by moving from OCV to , i.e., can we explicitly show thatis preferable to OCV for estimating Random-X prediction error when its assumptions hold? We give a partial positive answer next. §.§ Comparing and OCVAs already discussed, OCV is by an almost-unbiased estimate of Random-X prediction error (or an unbiased estimate of Random-X prediction error for the procedure in question, here least squares, applied to a training set of size n-1).The decomposition in (<ref>) demonstrates its variance and bias components, respectively, conditional on X.It should be emphasized that OCV has the significant advantage overof not requiring knowledge of σ^2 or assumptions on Q. Assuming that σ^2 is known and Q is well-behaved, we can compare the two criteria for estimating Random-X prediction error in least squares.OCV is generally slightly conservative as an estimate of Random-X prediction error, as models trained on more observations are generally expected to be better. does not suffer from such slight conservativeness in the variance component, relying on the integrated variance from theory, and in that regard it may already be seen as an improvement.However we will choose to ignore this issue of conservativeness, as the difference in training on n-1 versus n observations is clearly small when n is large. Thus, we can approximate the mean squared error or MSE of each method, as an estimate of Random-X prediction error, as(OCV-)^2≈_X ((OCV|X)) + _X ((OCV|X)), ( - )^2≈_X ((|X)) + _X ((|X)),where these two approximations would be equalities if OCV andwere exactly unbiased estimates of .Note that conditioned on X, the difference between OCV and , is nonrandom (conditioned on X, all diagonal entries h_ii, i=1,…,p are nonrandom).Hence _X(OCV|X)=_X(|X), and we are left to compare _X(OCV|X) and _X(|X), according to the (approximate) expansions above, to compare the MSEs of OCV and .Denote the two terms in (<ref>) by v(X) and b(X), respectively, so that (OCV|X)=v(X)+b(X) can be viewed as a decomposition into variance and bias components, and note that by construction_X ((OCV|X)) = _X(v(X)+b(X)) and_X ((|X)) = _X(b(X)).It seems reasonable to believe that _X(OCV|X) ≥_X(|X) would hold in most cases, thuswould be no worse than OCV. One situation in which this occurs is the case when the linear model is unbiased, hence b(X)=0 and consequently _X ((|X)) = _X(b(x)) = 0. In general, _X(OCV|X) ≥_X(|X) is guaranteed when _X(v(X),b(X)) ≥ 0. This means that choices of X that give large variance tend to also give large bias, which seems reasonable to assume and indeed appears to be true in our experiments.But, this covariance depends on the underlying mean function f(x)=(y|x) in complicated ways, and at the moment it eludes rigorous analysis.§ SIMULATIONS FOR LEAST SQUARES REGRESSIONWe empirically study the decomposition of Random-X prediction error into its various components for least squares regression in different problem settings, and examine the performance of the various model evaluation criteria in these settings. The only criterion which is assumption-free and should invariably give unbiased estimates of Random-X prediction error is OCV (modulo the slight bias in using n-1 rather than n training observations). Thus we may consider OCV as the “gold standard” approach, and we will hold the other methods up to its standard under different conditions, either when the assumptions they use hold or are violated.Before diving into the details, here is a high-level summary of the results:performs very well in unbiased settings (when the mean is linear), but very poorly in biased ones (when the mean is nonlinear);andpeform well overall, withhaving an advantage and even holding a small advantage over OCV, in essentially all settings, unbiased and biased. This is perhaps a bit surprising sinceis not designed to account for bias, but then again, not as surprising once we recall thatis closely related to GCV.We perform experiments in a total of six data generating mechanisms, based on three different distributions Q for the covariate vector x, and two models for f(x)=(y|x), one unbiased (linear) and the other biased (nonlinear). The three generating models for x are as follows.* Normal.We choose Q=N(0,Σ), where Σ is block-diagonal, containing five blocks such that all variables in a block have pairwise correlation 0.9.* Uniform. We define Q by taking N(0,Σ) as above, then apply the inverse normal distribution function componentwise. In other words, this can be seen as a Gaussian copula with uniform marginals.* t(4). We define Q by taking N(0,Σ) as above, then adjust the marginal distributions appropriately, again a Gaussian copula with t(4) marginals.Note that Theorem <ref> covers the normal setting (and in fact, the covariance matrix Σ plays no role in theestimate), while the uniform and t(4) settings do not comply with either Theorems <ref> or <ref>.Also, the latter two settings differ considerably in the nature of the distribution Q: finite support versus long tails, respectively. The two generating models for y|x both use ϵ∼ N(0,20^2), but differ in the specification for the mean function f(x)=(y|x), as follows.* Unbiased. We set f(x)=∑_j=1^px_j.* Biased. We set f(x) = C ∑_j=1^p |x_j|. The simulations discussed in the coming subsections all use n=100 training observations. In the “high-dimensional” case, we use p=50 variables and C=0.75, while in the “low-dimensional, extreme bias” case, we use p=10 and C=100. In both cases, we use a test set of 10^4 observations to evaluate Random-X quantities like ,,. Lastly, all figures show results averaged over 5000 repetitions. §.§ The components of Random-X prediction error We empirically evaluate B,V,, for least squares regression fitted in the six settings (three for the distribution of x times two for (y|x)) in the high-dimensional case, with n=100 and p=50.The results are shown in Figure <ref>.We can see the value ofimplied by Theorem <ref> is extremely accurate for the normal setting, and also very accurate for the short-tailed uniform setting. However for the t(4) setting, the value ofis quite a bit higher than what the theory implies. In terms of bias, we observe that for the biased settings the value ofis bigger than the Same-X bias B, and so it must be taken into account if we hope to obtain reasonable estimates of Random-X prediction error . §.§ Comparison of performances in estimating prediction error Next we compare the performance of the proposed criteria for estimating the Random-X prediction error of least squares over the six simulation settings. The results in Figures <ref> and <ref> correspond to the “high-dimensional” case with n=100 and p=50 and the “low-dimensional, extreme bias” case with n=100 and p=10, respectively.Displayed are the MSEs in estimating the Random-X prediction error, relative to OCV; also, the MSE for each method are broken down into squared bias and variance components.In the high-dimensional case in Figure <ref>, we see that for the true linear models (three leftmost scenarios),has byfar the lowest MSE in estimating Random-X prediction error, much better than OCV. For the normal and uniform covariate distributions, it also has no bias in estimating this error, as warranted by Theorem <ref> for the normal setting.For the t(4) distribution, there is already significant bias in the prediction error estimates generated by , as is expected from the results in Figure <ref>; however, if the linear model is correct then we seestill has three- to five-fold lower MSE compared to all other methods. The situation changes dramatically when bias is added (three rightmost scenarios). Now,is by far the worse method, failing completely to account for large , and its relative MSE compared to OCV reaches as high as 10.As forandin the high-dimensional case, we see thatindeed has lower error than OCV under the normal models as argued in Section <ref>, and also in the uniform models. This is true regardless of the presence of bias. The difference, however is small: between 0.1% and 0.7%. In these settings, we can seehas even lower MSE than , with no evident bias in dealing with the biased models. For the long-tailed t(4) distribution, bothandsuffer some bias in estimating prediction error, as expected. Interestingly, in the nonlinear model with t(4) covariates (rightmost scenario),does suffer significant bias in estimating prediction error, as opposed to . However, this bias does not offset the increased variance due to /OCV. In the low-dimensional case in Figure <ref>, many of the same conclusions apply:does well if the linear model is correct, even with the long-tailed covariate distribution, but fails completely in the presence of nonlinearity. Also,performs almost identically to OCV throughout. The most important distinction is the failure ofin the normal covariate, biased setting, where it suffers significant bias in estimating the prediction error (see circled region in the plot). This demonstrates that the heuristic correction for employed bycan fail when the linear model does not hold, as opposed toand OCV. We discuss this further in Section <ref>. § THE EFFECTS OF RIDGE REGULARIZATIONIn this section, we examine ridge regression, which behaves similarly in some ways to least squares regression, and differently in others. In particular, like least squares, it has nonnegative excess bias, but unlike least squares, it can have negative excess variance, increasingly so for larger amounts of regularization.These results are established in the subsections below, where we study excess bias and variance separately.Throughout, we will write _n for the estimator from the ridge regression of Y on X, i.e., _n(x)=x^T (X^T X + λ I)^-1 X^T Y, where the tuning parameter λ > 0 is considered arbitrary (and for simplicity, we make the dependence of _n on λ implicit). When λ=0, we must assume that X has full column rank (almost surely under its marginal distribution Q^n), but when λ>0, no assumption is needed on X. §.§ Nonnegativity ofWe prove an extension to the excess bias result in Theorem <ref> for least squares regression that the excess bias in ridge regression is nonnegative.For _n the ridge regression estimator, we have ≥ 0.This result is actually itself a special case of Theorem <ref>; the latter is phrased in somewhat of a different (functional) notation, so for concreteness, we give a direct proof of the result for ridge regression here. Let X_0 ∈^n× p be a matrix of test covariate values, with rows i.i.d. from Q, and let Y_0 ∈^n be a vector of associated test response value. Then excess bias in (<ref>) can be written as= _X,X_01/n(_n(X_0)|X,X_0 )-f(X_0)_2^2 - _X 1/n(_n(X)|X )-f(X)_2^2.Note _n(X)=X(X^T X+λ I)^-1 X^T Y, and by linearity, (_n(X)|X)=X(X^T X+λ I)^-1 X^T f(X). Recalling the optimization problem underlying ridge regression, we thus have(_n(X)|X)= _Xβ∈^nf(X) - Xβ_2^2 + λβ_2^2.An analogous statement holds for _0n, which we write to denote the result from the ridge regression Y_0 on X_0; we have(_0n(X_0)|X_0)= _X_0β∈^nf(X_0) - X_0β_2^2 + λβ_2^2.Now write β_n=(X^T X + λ I)^-1 X^T f(X) and β_0n=(X_0^T X_0 + λ I)^-1 X_0^T f(X_0) for convenience. By optimality of X_0β_0n for the minimization problem in the last display,X_0 β_n - f(X_0)_2^2 + λβ_n_2^2 ≥ X_0 β_0n - f(X_0)_2^2 + λβ_0n_2^2,and taking an expectation over X,X_0 gives_X,X_0[ X_0 β_n - f(X_0)_2^2 + λβ_n_2^2]≥_X_0[ X_0 β_0n - f(X_0)_2^2 + λβ_0n_2^2] = _X [ X β_n - f(X)_2^2 + λβ_n_2^2],where in the last line we used the fact that (X,Y) and (X_0,Y_0) are identical in distribution.Cancelling out the common term of λ _X β_n_2^2 in the first and third lines above establishes the result, since (_n(X_0)|X,X_0) = X_0β_n and (_n(X)|X)=Xβ_n.§.§ Negativity offor large λHere we present two complementary results on the variance side.For _n the ridge regression estimator, the integrated Random-X prediction variance,V+ = _X,x_0(_n(x_0)|X,x_0),is a nonincreasing function of λ.As in the proofs of Theorems <ref> and <ref>, let X_0 ∈^n× p be a test covariate matrix, and notice that we can write the integrated Random-X variance asV+ = _X,X_01/n[ ( _n(X_0)|X,X_0 )].For a given value of X,X_0, we have1/n[ ( _n(X_0)|X,X_0 )]= σ^2/n( X_0(X^T X + λ I)^-1 X^T X (X^T X + λ I)^-1 X_0^T ) = σ^2/n( X_0^T X_0 ∑_i=1^p u_iu_i^T d_i^2/(d_i^2 + λ)^2),where the second line uses an eigendecomposition X^T X = U D U^T, with U ∈^p× p having orthonormal columns u_1,…,u_p and D=diag(d_1^2,…,d_p^2). Taking a derivative with respect to λ, we seed/dλ( 1/n[ ( _n(X_0)|X,X_0 )]) = -2σ^2/n( X_0^T X_0 ∑_i=1^p u_iu_i^T λ d_i^2/(d_i^2 + λ)^3) ≤ 0,the inequality due to the fact that (AB) ≥ 0 if A,B are positive semidefinite matrices. Taking an expectation and switching the order of integration and differentiation (which is possible because the integrand is a continuously differentiable function of λ>0) givesd/dλ( _X,X_01/n[ ( _n(X_0)|X,X_0 )]) = _X,X_0d/dλ( 1/n[ ( _n(X_0)|X,X_0 )]) ≤ 0,the desired result. The proposition shows that adding regularization guarantees a decrease in variance for Random-X prediction.The same is true of the variance in Same-X prediction.However, as we show next, as the amount of regularization increases these two variances decrease at different rates, a phenomenon that manifests itself in the fact that and the Random-X prediction variance is guaranteed to be smaller than the Same-X prediction variance for large enough λ.For _n the ridge regression estimator, the integrated Same-X prediction variance and integrated Random-X prediction variance both approach zero as λ→∞.Moreover, the limit of their ratio satisfieslim_λ→∞_X,x_0(_n(x_0)|X,x_0)/_X(_n(x_1)|X) = [(X^TX) (X^TX)]/[(X^TX X^TX)]≤ 1,the last inequality reducing to an equality if and only if x ∼ Q is deterministic and has no variance.Again, as in the proof of the last proposition as well as Theorems <ref> and <ref>, let X_0 ∈^n× p be a test covariate matrix, and write the integrated Same-X and Random-X prediction variances as_X 1/n[ (_n(X)|X)] and_X,X_01/n[ (_n(X_0)|X,X_0 )],respectively.From the arguments in the proof of Proposition <ref>, letting X^T X = U D U^T be an eigendecomposition with U ∈^p× p having orthonormal columns u_1,…,u_p and D=diag(d_1^2,…,d_p^2), we havelim_λ→∞_X,X_01/n[ (_n(X_0)|X,X_0 )]= lim_λ→∞_X,X_0σ^2/n( X_0^T X_0 ∑_i=1^p u_iu_i^T d_i^2/(d_i^2 + λ)^2) = _X,X_0lim_λ→∞σ^2/n( X_0^T X_0 ∑_i=1^p u_iu_i^T d_i^2/(d_i^2 + λ)^2) = 0,where in the second line we used the dominated convergence theorem to exchange the limit and the expectation (since _X,X_0 [X_0^T X_0 ∑_i=1^p u_iu_i^T (d_i^2/(d_i^2+λ)^2)] ≤_X,X_0(X_0^T X_0 X^T X) < ∞). Similar arguments show that the integrated Same-X prediction variance also tends to zero.Now we consider the limiting ratio of the integrated variances,lim_λ→∞_X,X_0[ (_n(X_0)|X,X_0)]/_X [ (_n(X)|X)] = lim_λ→∞_X,X_0[ X_0(X^T X + λ I)^-1 X^T X (X^T X + λ I)^-1 X_0^T]/_X [X(X^T X + λ I)^-1 X^T X (X^T X + λ I)^-1 X^T]= lim_λ→∞_X,X_0[ λ^2 X_0(X^T X + λ I)^-1 X^T X (X^T X + λ I)^-1 X_0^T]/_X [λ^2 X(X^T X + λ I)^-1 X^T X (X^T X + λ I)^-1 X^T]= lim_λ→∞_X,X_0[ λ^2 X_0(X^T X + λ I)^-1 X^T X (X^T X + λ I)^-1 X_0^T]/lim_λ→∞_X [λ^2 X(X^T X + λ I)^-1 X^T X (X^T X + λ I)^-1 X^T],where the last line holds provided that the numerator and denominator both converge to finite nonzero limits, as will be confirmed by our arguments below. We study the numerator first.Noting that λ^2 (X^T X + λ I)^-1 X^T X (X^T X + λ I)^-1-X^T X has eigenvaluesd_i^2 (λ^2/(d_i^2 + λ)^2-1), i=1,…,p,we have that λ^2 (X^T X + λ I)^-1 X^T X (X^T X + λ I)^-1→ X^T X as λ→∞, in (say) the operator norm, implying [λ^2 X_0 (X^T X + λ I)^-1 X^T X (X^T X + λ I)^-1 X_0^T] →(X_0 X^T X X_0^T) as λ→∞.Hencelim_λ→∞_X,X_0[ λ^2 X_0(X^T X + λ I)^-1 X^T X (X^T X + λ I)^-1 X_0^T] = _X,X_0lim_λ→∞[ λ^2 X_0(X^T X + λ I)^-1 X^T X (X^T X + λ I)^-1 X_0^T] = _X,X_0(X_0 X^T X X_0^T) = [ _X(X^T X)_X_0(X_0^T X_0) ] = [ _X(X^T X) _X(X^T X)].Here, in the first line, we applied the dominated convergence theorem as previously, in the third we used the independence of X,X_0, and in the last we used the identical distribution of X,X_0. Similar arguments lead to the conclusion for the denominatorlim_λ→∞_X [ λ^2 X(X^T X + λ I)^-1 X^T X (X^T X + λ I)^-1 X^T] = [ _X(X^T XX^T X)],and thus we have shown thatlim_λ→∞_X,X_0[ (_n(X_0)|X,X_0)]/_X [ (_n(X)|X)] =[ _X(X^T X) _X(X^T X)]/ [ _X(X^T XX^T X)],as desired. To see that the ratio on the right-hand side is at most 1, considerA = (X^TX X^TX) - (X^TX) (X^TX),which is a symmetric matrix whose trace is(A) = ∑_i,j=1^p ((X^T X)_i,j) ≥ 0.Furthermore, the trace is zero if and only if all summands are zero, which occurs if and only if all components of x ∼ Q have no variance. In words, the theorem shows that the excess varianceof ridge regression approaches zero as λ→∞, but it does so from the left (negative side) of zero.As we can have cases in which the excess bias is very small or even zero (for example, a null model like in our simulations below), we see that -= can be negative for ridge regression with a large level of regularization; this is a striking contrast to the behavior of this gap for least squares, where it is always nonnegative.We finish by demonstrating this result empirically, using a simple simulation setup with p=100 covariates drawn from Q=N(0,I), and training and test sets each of size n=300.The underlying regression function was f(x)=(y|x)=0, i.e., there was no signal, and the errors were also standard normal.We drew training and test data from this simulation setup, fit ridge regression estimators to the training at various levels of λ, and calculated the ratio of the sample versions of the Random-X and Same-X integrated variances.We repeated this 100 times, and averaged the results. As shown in Figure <ref>, for values of λ larger than about 250, the Random-X integrated variance is smaller than the Same-X integrated variance, and consequently the same is true of the prediction errors (as there is no signal, the Same-X and Random-X integrated biases are both zero). Also shown in the figure is the theoretical limiting ratio of the integrated variances according to Theorem <ref>, which in this case can be calculated from the properties of Wishart distributions to be n^2p/(n^2p+np^2+np) ≈ 0.7481, and is in very good agreement with the empirical limiting ratio.§ NONPARAMETRIC REGRESSION ESTIMATORSWe present a brief study of the excess bias and variance of some common nonparametric regression estimators.In Section <ref>, we give a high-level discussion of the view on the gap between Random-X and Same-X prediction errors from the perspective of empirical process theory, which is a topic that is well-studied by researchers in nonparametric regression.§.§ Reproducing kernel Hilbert spaces Consider an estimator _n defined by the general-form functional optimization problem_n = _g ∈𝒢 ∑_i=1^n (y_i-g(x_i))^2 + J(g),where 𝒢 is a function class and J is a roughness penalty on functions.Examples estimators of this form include the (cubic) smoothing spline estimator in p=1 dimensions, in which 𝒢 is the space of all functions that are twice differentiable and whose second derivative is in square integrable, and J(g)=∫ g”(t)^2dt; and more broadly, reproducing kernel Hilbert space or RKHS estimators (in an arbitrary dimension p), in which 𝒢 is an RKHS and J(g)=f_𝒢 is the corresponding RKHS norm.Provided that _n defined by (<ref>) is a linear smoother, which means _n(x)=s(x)^T Y for a weight function s:^p → (that can and will generally also depend on X), we now show that the excess bias of _n is always nonnegative.We note that this result applies to smoothing splines and RKHS estimators, since these are linear smoothers; it also covers ridge regression, and thus generalizes the result in Theorem <ref>.For _n a linear smoother defined by a problem of the form (<ref>), we have ≥ 0.Let us introduce a test covariate matrix X_0 ∈^n× p and associated response vector Y_0∈^n, and write the excess bias in (<ref>) as= _X,X_01/n(_n(X_0)|X,X_0 )-f(X_0)_2^2 - _X 1/n(_n(X)|X )-f(X)_2^2.Writing _n(x)=s(x)^T Y for a weight function s:^p→, let S(X) ∈^n× n be a smoother matrix that has rows s(x_1),…,s(x_n). Thus _n(X)=S(X) Y, and by linearity, (_n(X)|X )=S(X)f(X). This in fact means that we can express g_n=(_n|X ), a function defined by g(x)=s(x)^T f(X), as the solution of an optimization problem of the form (<ref>),g_n = _g ∈𝒢 f(X)-g(X)_2^2+ J(g),where we have rewritten the loss term in a more convenient notation. Analogously, if we denote by _0n the estimator of the form (<ref>), but fit to the test data X_0,Y_0 instead of the training data X,Y, and g_0n=(_0n|X_0), theng_0n = _g ∈𝒢 f(X_0)-g(X_0)_2^2 + J(g).By virtue of optimality of g_0n for the problem in the last display, we haveg_n(X_0) - f(X_0)_2^2 + J(g_n) ≥g_0n(X_0) - f(X_0)_2^2 + J(g_0n)and taking an expectation over X,X_0 gives_X,X_0[g_n(X_0) - f(X_0)_2^2 + J(g_n)]≥_X_0[g_0n(X_0) - f(X_0)_2^2 + J(g_0n)] = _X [g_n(X) - f(X)_2^2 + J(g_n)],where in the equality step we used the fact that X,X_0 are identical in distribution. Cancelling out the common term of _X J(g_n) from the the first and third expressions proves the result, because (_n(X_0)|X,X_0) = g_n(X_0) and (_n(X)|X)=g_n(X).§.§ k-nearest-neighbors regression Consider _n the k-nearest-neighbors or kNN regression estimator, defined by_n(x) = 1/k∑_i ∈ N_k(x) y_i,where N_k(x) returns the indices of the k nearest points among x_1,…,x_n to x.It is immediate that the excess variance of kNN regression is zero.For _n the kNN regression estimator, we have =0.Simply compute( _n(x_1)| X ) = σ^2/k,by independence of y_1,…,y_n, and hence the points in the nearest neighbor set N_k(x_1), conditional on X.Similarly,( _n(x_0)| X,x_0 ) = σ^2/k.On the other hand, the excess bias is not easily computable, and is not covered by Theorem <ref>, since kNN cannot be written as an estimator of the form (<ref>) (though it is a linear smoother). The next result sheds some light on the nature of the excess bias.For _n the kNN regression estimator, we have= B_n,k - (1-1/k^2) B_n-1,k-1,where B_n,k denotes the integrated Random-X prediction bias of the kNN estimator fit to a training set of size n, and with tuning parameter (number of neighbors) k.Observe((_n(x_0) | X,x_0)-f(x_0))^2= ( 1/k∑_i ∈ N_k(x_0)(f(x_i)-f(x_0)))^2,and by definition, _X,x_0[(_n(x_0) | X,x_0)-f(x_0)]^2=B_n,k. Meanwhile((_n(x_1) | X)-f(x_1))^2= ( 1/k∑_i ∈ N_k(x_1)(f(x_i)-f(x_1)))^2 = ( 1/k∑_i ∈ N_k^-1(x_1)(f(x_i)-f(x_1)))^2,where N_k-1^-1(x_1) gives the indices of the k-1 nearest points among x_2,…,x_n to x_1 (which equals N_k(x_1) as x_1 is trivially one of its own k nearest neighbors).Now notice that x_1 plays the role of the test point x_0 in the last display, and therefore, _X[(_n(x_1) | X)-f(x_1)]^2=((k-1)/k)^2B_n-1,k-1. This proves the result. The above proposition suggests that, for moderate values of k, the excess bias in kNN regression is likely positive. We are comparing the integrated Random-X bias of a kNN model with n training points and k neighbors to that of a model n-1 points and k-1 neighbors; for large n and moderate k, it seems that the former should be larger than the latter, and in addition, the factor of (1-1/k^2) multiplying the latter term makes it even more likely that the difference B_n,k - (1-1/k^2)B_n-1,k-1 is positive.Rephrased, using the zero excess variance result of Proposition <ref>: the gap in Random-X and Same-X prediction errors, -=B_n,k - (1-1/k^2)B_n-1,k-1, is likely positive for large n and moderate k.Of course, this is not a formal proof; aside from the choice of k, the shape of the underlying mean function f(x)=(y|x) obviously plays an important role here too.As a concrete problem setting, we might try analyzing the Random-X bias B_n,k for f Lipschitz and a scaling for k such that k →∞ but k/n → 0 as n →∞, e.g., k ≍√(n), which ensures consistency of kNN.Typical analyses provide upper bounds on the kNN bias in this problem setting (e.g., see ), but a more refined analysis would be needed to compare B_n,k to B_n-1,k-1.§ DISCUSSIONWe have proposed and studied a division of Random-X prediction error into components: the irreducible error σ^2, the traditional (Fixed-X or Same-X) integrated bias B and integrated variance V components, and our newly defined excess biasand excess variancecomponents, such that B+ gives the Random-X integrated bias and V+ the Random-X integrated variance.For least squares regression, we were able to quantifyexactly when the covariates are normal and asymptotically when they are drawn from a linear transformation of a product distribution, leading to our definition of .To account for unknown error variance σ^2, we definedbased on the usual plug-in estimate, which turns out to be asymptotically identical to GCV, giving this classic method a novel interpretation. To account for(when σ^2 is known and the distribution Q of the covariates is well-behaved), we defined , by leveraging a Random-X bias estimate implicit to OCV. We also briefly considered methods beyond least squares, proving thatis nonnegative in all settings considered, whilecan become negative in the presence of heavy regularization.We reflect on some issues surrounding our findings and possible directions for future work. Ability ofto account for bias. An intriguing phenomenon that we observe is the ability of /Sp and its close (asymptotic) relative GCV to deal to some extent within estimating Random-X prediction error, through the inflation it performs on the squared training residuals. For GCV in particular, where recall GCV = RSS/(n(1-γ)^2), we see that this inflation a simple form: if the linear model is biased, then the squared bias component in each residual is inflated by 1/(1-γ)^2. Comparing this to the inflation that OCV performs, which is 1/(1-h_ii)^2, on the ith residual, for i=1,…,n, we can interpret GCV as inflating the bias for each residual by some “averaged” version of the elementwise factors used by OCV.As OCV provides an almost-unbiased estimate offor Random-X prediction, GCV can get close when the diagonal elements h_ii, i=1,…,n do not vary too wildly. When they do vary greatly, GCV can fail to account for ,as in the circled region in Figure <ref>. Alternative bias-variance decompositions. The integrated terms we defined are expectations of conditional bias and variance terms, where we conditioned on both training and testing covariates X,x_0. One could also consider other conditioning schemes, leading to different decompositions. An interesting option would be to condition on the prediction point x_0 only and calculate the bias and variance unconditional on the training points X before integrating, as in _x_0((_n(x_0) | x_0)-f(x_0))^2 and _x_0((_n(x_0) | x_0)) for these alternative notions of Random-X bias and variance, respectively.It is easy to see that this would cause the bias (and thus excess bias) to decrease and variance (and thus excess variance) to increase. However, it is not clear to us that computing or bounding such new definitions of (excess) bias and (excess) variance would be possible even for least squares regression. Investigating the tractability of this approach and any insights it might offer is an interesting topic for future study. Alternative definitions of prediction error. The overall goal in our work was to estimate the prediction error, defined as =_X,Y,x_0,y_0(y_0-_n(x_0))^2, the squared error integrated over all of the random variables available in training and testing. Alternative definitions have been suggested by some authors. <cit.> generalized the Fixed-X setting in a manner that led them to define _Y,x_0,y_0[(y_0-_n(x_0))^2|X] as the prediction error quantity of interest, which can be interpreted as the Random-X prediction error of a Fixed-X model. <cit.> emphasized the importance of the quantity_x_0,y_0[(y_0-_n(x_0))^2|X,Y], which is the out-of-sample error of the specific model we have trained on the given training data X,Y.Of these two alternate definitions, the second one is more interesting in our opinion, but investigating it rigorously requires a different approach than what we have developed here. Alternative types of cross-validation. Our exposition has concentrated on comparing OCV to generalized covariance penalty methods. We have not discussed other cross-validation approaches, in particular, K-fold cross-validation (KCV) method with K ≪ n (e.g., K=5 or 10). A supposedly well-known problem with OCV is that its estimates of prediction error have very high variance; we indeed observe this phenomenon in our simulations (and for least squares estimation, the analytical form of OCV clarifies the source of this high variance). There are some claims in the literature that KCV can have lower variance than OCV (, and others), and should be considered as the preferred CV variant for estimation of Random-X prediction error. Systematic investigations of this issue for least squares regression such as <cit.> actually reach the opposite conclusion—that high variance is further compounded by reducing K. Our own simulations also support this view (results not shown), therefore we do not consider KCV to be an important benchmark to consider beyond OCV. Model selection for prediction. Our analysis and simulations have focused on the accuracy of prediction error estimates provided by various approaches. We have not considered their utility for model selection, i.e., for identifying the best predictive model, which differs from model evaluation in an important way. A method can do well in the model selection task even when it is inaccurate or biased for model evaluation, as long as such inaccuracies are consistent across different models and do not affect its ability to select the better predictive model. Hence the correlation of model evaluations using the same training data across different models plays a central role in model selection performance. An investigation of the correlation between model evaluations that each of the approaches we considered here creates is of major interest, and is left to future work. Semi-supervised settings. Given the important role that the marginal distribution Q of x plays in evaluating Random-X prediction error (as expressed, e.g., in Theorems <ref> and <ref>), it is of interest to consider situations where, in addition to the training data, we have large quantities of additional observations with x only and no response y. In the machine learning literature this situation is often considered under then names semi-supervised learning or transductive learning. Such data could be used, e.g., to directly estimate the excess variance from expressions like (<ref>). General view from empirical process theory. This paper was focused in large part on estimating or bounding the excess bias and variance in specific problem settings, which led to estimates or bounds on the gap in Random-X and Same-X prediction error, as -=+.This gap is indeed a familiar concept to those well-versed in the theory of nonparametric regression, and roughly speaking, standard results from empirical process theory suggest that we should in general expect - to be small, i.e., much smaller than either oforto begin with.The connection is as follows. Note that-= _X,Y,x_0( f(x_0) - _n(x_0))^2 -_X,Y[1/n∑_i=1^n ( f(x_i) -_n(x_i))^2] = _X,Y[f-_n_L_2(Q)^2 -f-_n_L_2(Q_n)^2 ],where we are using standard notation from nonparametric regression for “population” and “empirical”norms, ·_L_2(Q) and ·_L_2(Q_n), respectively. For an appropriate function class 𝒢, empirical process theory can be used to control the deviations between g_L_2(Q) and g_L_2(Q_n), uniformly over all functions g ∈𝒢. Such uniformity is important, because it gives us control on the difference in population and empirical norms for the (random) function g=f-_n (provided of course this function lies in 𝒢). This theory applies to finite-dimensional 𝒢 (e.g., linear functions, which would be relevant to thecase when f is assumed to be linear and _n is chosen to be linear), and even to infinite-dimensional classes 𝒢, provided we have some understanding of the entropy or Rademacher complexity of 𝒢 (e.g., this is true of Lipschitz functions, which would be relevant to the analysis of k-nearest-neighbors regression or kernel estimators).Under appropriate conditions, we typically find _X,Y|f-_n_L_2(Q)^2-f-_n_L_2(Q_n)^2| =O(C_n), where C_n is the L_2(Q) convergence rate of _n to f.This is even true in an asymptotic setting in which p grows with n (so C_n here gets replaced by C_n,p), but such high-dimensional results usually require more restrictions on the distribution Q of covariates. The takeaway message: in most cases where _n is consistent with rate C_n, we should expect to see the gap being- = O(C_n), whereas ,≥σ^2, so the difference in Same-X and Random-X prediction error is quite small (as small as the Same-X and Random-X risk) compared to these prediction errors themselves; said differently, we should expect to see , being of the same order (or smaller than) B,V.It is worth pointing out that several interesting aspects of our study really lie outside what can be inferred from empirical process theory. One aspect to mention is the precision of the results: in some settings we can characterize , individually, but (as described above), empirical process theory would only provide a handle on their sum. Moreover, for least squares regression estimators, with p/n converging to a nonzero constant, we are able to characterize the exact asymptotic excess variance under some conditions on Q (essentially, requiring Q to be something like a rotation of a product distribution), in Theorem <ref>; note that this is a problem setting in which least squares is not consistent, and could not be treated by standard results empirical process theory.Lastly, empirical process theory tells us nothing about the sign off-_n_L_2(Q)^2 -f-_n_L_2(Q_n)^2,or its expectation under P (which equals -=+, as described above). This 1-bit quantity is of interest to us, since it tells us if the Same-X (in-sample) prediction error is optimistic compared to the Random-X (out-of-sample) prediction error. Theorems <ref>, <ref>, <ref>, <ref> and Propositions <ref>, <ref> all pertain to this quantity. plainnat36 urlstyle[Akaike(1973)]akaike1973information Hirotogu Akaike. 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0pt by -by -6.5in 9in =13pt =3pt plus 1pt minus .5pt0pt 0.5in#10=#1 .05em0-0 -.025em.0433em0 myheadings Wen-Xu Wang, Ying-Cheng Lai, and Celso Grebogicntdlistroman(cntd) cntdlistarabiccntd. cntdlistbull∙ Data Based Identification and Prediction of Nonlinear andComplex Dynamical SystemsWen-Xu Wang^a,b, Ying-Cheng Lai^c,d,e, and Celso Grebogi^ea School of Systems Science, Beijing Normal University, Beijing, 100875, Chinab Business School, University of Shanghai for Science andTechnology, Shanghai 200093, China c School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA d Department of Physics, Arizona State University, Tempe, Arizona 85287, USA e Institute for Complex Systems and Mathematical Biology, King's College, University of Aberdeen, Aberdeen AB24 3UE, UK AbstractThe problem of reconstructing nonlinear and complex dynamical systemsfrom measured data or time series is central to many scientific disciplinesincluding physical, biological, computer, and social sciences, as well asengineering and economics. The classic approach to phase-space reconstructionthrough the methodology of delay-coordinate embedding has been practiced formore than three decades, but the paradigm is effective mostly forlow-dimensional dynamical systems. Often, the methodology yieldsonly a topological correspondence of the original system. There aresituations in various fields of science and engineering where the systemsof interest are complex and high dimensional with many interacting components.A complex system typically exhibits a rich variety of collective dynamics,and it is of great interest to be able to detect, classify, understand,predict, and control the dynamics using data that are becoming increasinglyaccessible due to the advances of modern information technology. To accomplishthese tasks, especially prediction and control, an accurate reconstructionof the original system is required.Nonlinear and complex systems identification aims at inferring, from data,the mathematical equations that govern the dynamical evolution and thecomplex interaction patterns, or topology, among the various components ofthe system. With successful reconstruction of the system equations and theconnecting topology, it may be possible to address challengingand significant problems such as identification of causal relations amongthe interacting components and detection of hidden nodes. The “inverse”problem thus presents a grand challenge, requiring new paradigms beyond thetraditional delay-coordinate embedding methodology.The past fifteen years have witnessed rapid development of contemporarycomplex graph theory with broad applications in interdisciplinary scienceand engineering. The combination of graph, information, and nonlineardynamical systems theories with tools from statistical physics,optimization, engineering control, applied mathematics, and scientificcomputing enables the development of a number of paradigms to addressthe problem of nonlinear and complex systems reconstruction. In thisReview, we review the recent advances in this forefront and rapidlyevolving field, with a focus on compressive sensing based methods.In particular, compressive sensing is a paradigm developed in recent yearsin applied mathematics, electrical engineering, and nonlinear physics toreconstruct sparse signals using only limited data. It has broad applicationsranging from image compression/reconstruction to the analysis of large-scalesensor networks, and it has become a powerful technique to obtain high-fidelitysignals for applications where sufficient observations are not available. We will describe in detail how compressive sensing can be exploited toaddress a diverse array of problems in data based reconstruction of nonlinearand complex networked systems. The problems include identification ofchaotic systems and prediction of catastrophic bifurcations, forecasting future attractors of time-varying nonlinear systems, reconstruction ofcomplex networks with oscillatory and evolutionary game dynamics, detection of hidden nodes, identification of chaotic elements in neuronal networks, and reconstruction of complex geospatial networks and nodalpositioning. A number of alternative methods, such as those based onsystem response to external driving, synchronization, noise-induced dynamical correlation, will also be discussed. Due to the high relevance of network reconstruction to biological sciences, a special Section is devoted to abrief survey of the current methods to infer biological networks. Finally, anumber of open problems including control and controllability of complex nonlinear dynamical networks are discussed. The methods reviewed in this Review are principled on various conceptsin complexity science and engineering such as phase transitions,bifurcations, stabilities, and robustness. The methodologies have the potential to significantly improve our ability to understand a varietyof complex dynamical systems ranging from gene regulatory systems tosocial networks towards the ultimate goal of controlling such systems. § INTRODUCTION An outstanding problem in interdisciplinaryscience is nonlinear and complex systems identification, prediction, and control. Given a complex dynamical system, the various types ofdynamical processes are of great interest. The ultimate goal in the study of complex systems is to devise practically implementablestrategies to control the collective dynamics. A great challenge isthat the network structure and the nodal dynamics are often unknownbut only limited measured time series are available. To control thesystem dynamics, it is imperative to map out the systemdetails from data. Reconstructing complex network structure and dynamicsfrom data, the inverse problem, has thus become a central issue incontemporary network science and engineering <cit.>.There are broad applications of the solutions of the networkreconstruction problem, due to the ubiquity of complex interacting patternsarising from many systems in a variety of disciplines <cit.>. §.§ Existing works on data based reconstruction ofnonlinear dynamical systemsThe traditional paradigm of nonlinear time series analysis is thedelay-coordinate embedding method, the mathematical foundation ofwhich was laid by Takens more than three decades ago <cit.>.He proved that, under fairly general conditions, the underlying dynamicalsystem can be faithfully reconstructed from time series in the sense thata one-to-one correspondence can be established between the reconstructedand the true but unknown dynamical systems. Based on the reconstruction,quantities of importance for understanding the system can be estimated,such as the relative weights of deterministicity and stochasticity ofthe underlying system, its dimensionality, the Lyapunov exponents, andunstable periodic orbits that constitute the skeleton of the invariantset responsible for the observed dynamics.There exists a large body of literature on the application of the delay-coordinate embedding technique to nonlinear/chaotic dynamicalsystems <cit.>. A pioneering work in this field is Ref. <cit.>. The problem of determining the proper time delay was investigated <cit.>, with a firm theoretical foundationestablished by exploiting the statistics for testing continuity anddifferentiability from chaotic time series <cit.>. The mathematical foundation for the required embedding dimension for chaotic attractors was laid inRef. <cit.>. There were works on the analysis of transientchaotic time series <cit.>, on the reconstruction of dynamical systems with time delay <cit.>, on detecting unstable periodic orbits from time series <cit.>, on computing the fractal dimensions from chaotic data <cit.>, and on estimating the Lyapunov exponents <cit.>.There were also works on forecasting nonlinear dynamicalsystems <cit.>. A conventional approach is to approximate a nonlinear system with a large collection of linear equations in different regionsof the phase space to reconstruct the Jacobian matrices on a proper grid <cit.> or fit ordinary differential equations to chaotic data <cit.>. Approaches based on chaotic synchronization <cit.> or geneticalgorithms <cit.> to parameter estimationwere also investigated. In most existing works, short-term predictionsof a dynamical system can be achieved by employing the classicaldelay-coordinate embedding paradigm <cit.>. For nonstationary systems, the method of over-embedding wasintroduced <cit.> in which the time-varying parameters weretreated as independent dynamical variables so that the essential aspectsof determinism of the underlying system can be restored.A recently developed framework based on compressive sensing wasable to predict the exact forms of both system equations andparameter functions based on available time series forstationary <cit.> and time-varying dynamical systems <cit.>. §.§ Existing works on data based reconstruction of complexnetworks and dynamical processesData based reconstruction of complex networks in general is deemed to be an important but difficult problem and has attracted continuous interest,where the goal is to uncover the full topology of the network based onsimultaneously measured time series <cit.>. There were previous efforts in nonlinear systems identification and parameter estimation for coupled oscillators and spatiotemporal systems, such as the auto-synchronization method <cit.>.There were also works on revealing the connection patterns of networks.For example, methods were proposed to estimate the network topologycontrolled by feedback or delayed feedback <cit.>. Network connectivity can be reconstructed from the collective dynamicaltrajectories using response dynamics <cit.>. The approach ofrandom phase resetting was introduced to reconstruct the details ofthe network structure <cit.>. For neuronal systems, there wasa statistical method to track the structural changes <cit.>.Some earlier methods required more information about the network than just data.For example, the following two approaches require completeinformation about the dynamical processes, e.g., equations governingthe evolutions of all nodes on the network. (1) In Ref. <cit.>,the detailed dynamics at each node is assumed to be known. A replica ofthe network, or a computational model of this “target” network, canthen be constructed, with the exception that the interaction strengths among the nodes are chosen randomly. It has been demonstrated that insituations where a Lyapunov function for the network dynamics exists,the connectivity of the model network converges to that of the targetnetwork <cit.>. (2) In Ref. <cit.>, a Kuramoto-typeof phase dynamics <cit.> on the network isassumed, where a steady-state solution exists. By linearizing the networkdynamics about the steady-state solution, the associated Jacobian matrixcan be obtained, which reflects the network topology and connectivity.Besides requiring complete information about the nodal dynamics,the amount of computations required tends to increase dramatically withthe size of the network <cit.>. For example, suppose thenodal dynamics is described by a set of differential equations. For anetwork of size N, in order for its structure to be predicted, thenumber of differential equations to be solved typically increases withN as N^2. For nonlinear dynamical networks, there was a method <cit.>based on chaotic time-series analysis through estimating the elements of the Jacobian matrix, which are the mutual partial derivatives of thedynamical variables on different nodes in the network. A statistically significant entry in the matrix implies a connection between the two nodes specified by the row and the column indices of that entry. Becauseof the mathematical nature of the Jacobian matrix, i.e., it is meaningfulonly for infinitesimal tangent vectors, linearization of the dynamicsin the neighborhoods of the reconstructed phase-space points is needed,for which constrained optimization techniques <cit.> were found to be effective <cit.>. Estimating the Jacobian matrices, however, hasbeen a challenging problem in nonlinear dynamics <cit.>and its reliability can be ensured but only for low-dimensional,deterministic dynamical systems. The method <cit.> appeared thusto be limited to small networks with sparse connections.While many of the earlier works required complete or partial informationabout the intrinsic dynamics of the nodes and their coupling functions,completely data-driven and model-free methods exist. For example, The global climate network was reconstructed using the mutual information method, enabling energy and information flow in the network to be studied <cit.>. The sampling bias of DNA sequences in viruses from different regions can be used to reveal the geospatial topologies of the influenza networks <cit.>. Network structure can also beobtained by calculating the causal influences among the time series basedon the Granger causality <cit.> method <cit.>, the overarchingframework <cit.>, the transfer entropymethod <cit.>, or the method of inner composition alignment <cit.>. However, such causality based methods are unable to reveal information about the nodal dynamical equations. In addition, there were regression-based methods <cit.> for systems identification based on the least squares approximation through theKronecker-product representation <cit.>, which would requirelarge amounts of data. In systems biology, reverse engineering of gene regulatory networks from expression data is a fundamentally important problem, andit attracts a tremendous amount of interest <cit.>. The wide spectrum of methods for modelinggenetic regulatory networks can be categorized based on the level ofdetails with which the genetic interactions and dynamics aremodeled <cit.>. One of the classical mathematical formalisms used to model the dynamics of biological processes is differential equations, which can capture the dynamics of each component in a system at a detailed level <cit.>. A major limitation of this approach is its overwhelmingcomplexity and the resulting computational requirement, which limitsits applicability to small-scale systems. In contrast, Booleannetwork models assume that the states of components in the system arebinary and the state transitions are governed by logicoperations <cit.>. Sincethe gene expressions are usually described by their expression levels and the interaction patterns between genes may not be logic operations, in some cases the Boolean network models may not be biologicallyappropriate. Another class of methods explicitly model biological systems as a graph in which the vertices represent basic units in the system and the edges characterize the relationships between the units. The graph itself can be constructed either by directly comparing the measurements for the vertices based on certain metric, such as the Euclidean distance, mutual information, or correlation coefficient <cit.>, or by some probabilistic approaches for Bayesian networklearning <cit.>. However, inferring regulatory interactions based on Bayesian networks is an intractable problem <cit.>. The linear regression models for learning regulatory networks assume that expression level of a gene can be approximated by a linear combination of the expressions of other genes <cit.>, and such models form a middle ground betweenthe models based on differential equations and Boolean logic.§.§ Compressive sensing based reconstruction of nonlinear and complex dynamical systemsA recent line of research <cit.>exploited compressive sensing <cit.>. The basic principle is that thedynamics of many natural and man-made systems are determined by smoothenough functions that can be approximated by finite series expansions.The task then becomes that of estimating the coefficients in the seriesrepresentation of the vector field governing the system dynamics. Ingeneral, the series can contain high order terms, and the total number of coefficients to be estimated can be quite large. While this is achallenging problem, if most coefficients are zero (or negligible), thevector constituting all the coefficients will be sparse. In addition, a generic feature of complex networks in the real world is that theyare sparse <cit.>. Thus for realistic nonlinear dynamical networks, the vectors to be reconstructed are typically sparse, andthe problem of sparse vector estimation can then be solved by theparadigm of compressive sensing <cit.> that reconstructs a sparse signal from limitedobservations. Since the observation requirements can be relaxedconsiderably as compared to those associated with conventional signal reconstruction schemes, compressive sensing has evolved into a powerfultechnique to reconstruct sparse signal from small amounts of observationsthat are much less than those required in conventional approaches. Compressive sensing has been introduced to the field of networkreconstruction for discrete time and continuous time nodaldynamics <cit.>, for evolutionary gamedynamics <cit.>, for detecting hiddennodes <cit.>, for predicting and controlling synchronization dynamics <cit.>, and for reconstructing spreading dynamics based on binary data <cit.>. Compressive sensing also finds applications in quantum measurementscience, e.g., to exponentially reduce the experimental configurationsrequired for quantum tomography <cit.>. §.§ Plan of this reviewThis Review presents the recent advances in the forefront and rapidly evolving field of nonlinear and complex dynamical systems identificationand prediction. Our focus will be on the compressive sensing based approaches. Alternative approaches will also be discussed, whichinclude noised-induced dynamical mapping, perturbations, reverse engineering,synchronization, inner composition alignment, global silencing, Grangercausality, and alternative optimization algorithms. In Sec. <ref>, we first introduce the principle ofcompressive sensing and discuss nonlinear dynamical systems identificationand prediction. We next discuss a compressive sensing based approach topredicting catastrophes in nonlinear dynamical systems under theassumption that the system equations are completely unknown and onlytime series reflecting the evolution of the dynamical variables of thesystem are available. We then turn to time-varying nonlinear dynamicalsystems, motivated by the fact that systems with one or a few parametersvarying slowly with time are of considerable interest in many areas of science and engineering. In such a system, the attractors in the future can be characteristically different from those at the present. To predict the possible future attractors based on available information at the present is thus a well-defined and meaningful problem, which is challenging especially when the system equations are not known but only time-series measurements are available. We review a compressive-sensing based methodfor time-varying systems. This framework allows us to reconstruct thesystem equations and the time dependence of parameters based on limitedmeasurements so that the future attractors of the system can be predictedthrough computation. Section <ref> focuses on compressivesensing based reconstruction of complex networked systems. The followingproblems will be discussed in detail.Reconstruction of coupled oscillator networks. The basic idea is that the mathematical functions determining thedynamical couplings in a physical network can be expressed bypower-series expansions. The task is then to estimate all thenonzero coefficients, which can be accomplished by exploitingcompressive sensing <cit.>.Reconstruction of social networks based on evolutionary-game data via compressive sensing. Evolutionary games are a common type interactions in a variety of complex networked, natural and social systems. Given such asystem, uncovering the interacting structure of the underlyingnetwork is key to understanding its collective dynamics. We discuss a compressive sensing based method to uncover thenetwork topology using evolutionary-game data. In particular, in a typical game, agents use different strategies in order to gain the maximum payoff. The strategies can be dividedinto two types: cooperation and defection. It was shown <cit.>that, even when the available information about each agent's strategy and payoff is limited, the compressive-sensing based approach can yield precise knowledge of the node-to-node interaction patterns in a highly efficient manner. In addition to numericalvalidation of the method with model complex networks, we discuss an actual social experiment in which participants forming afriendship network played a typical game to generate short sequencesof strategy and payoff data. The high prediction accuracy achievedand the unique requirement of extremely small data set suggest that the method can be appealing to potential applications to reveal “hidden”networks embedded in various social and economic systems.Detecting hidden nodes in complex networks from timeseries. The power of science lies in its ability to infer and predictthe existence of objects from which no direct information can beobtained experimentally or observationally. A well known example isto ascertain the existence of black holes of various masses indifferent parts of the universe from indirect evidence, such as X-ray emissions. In complex networks, the problem of detecting hidden nodes can be stated, as follows. Consider a networkwhose topology is completely unknown but whose nodes consist of twotypes: one accessible and another inaccessible from the outside world.The accessible nodes can be observed or monitored, and we assume thattime series are available from each node in this group. The inaccessiblenodes are shielded from the outside and they are essentially “hidden.”The question is, can we infer, based solely on the available time seriesfrom the accessible nodes, the existence and locations of the hiddennodes? Since no data from the hidden nodes are available, nor can theybe observed directly, they act as some sort of “black box” from theoutside world. Solution of the network hidden node detection problemhas potential applications in different fields of significant currentinterest. For example, to uncover the topology of a terroristorganization and especially, various ring leaders of the network isa critical task in defense. The leaders may be hidden in the sense thatno direct information about them can be obtained, yet they may rely ona number of couriers to operate, which are often subject to surveillance. Similar situations arise in epidemiology, where the original carrierof a virus may be hidden, or in a biology network where one wishes todetect the most influential node from which no direct observation canbe made. We discuss in detail a compressive sensing basedmethod <cit.> to ascertain hidden nodes in complexnetworks and to distinguish them from various noise sources. Identifying chaotic elements in complex neuronal networks. We discuss a completely data-driven approach <cit.> to reconstructingcoupled neuronal networks that contain a small subset of chaotic neurons. Suchchaotic elements can be the result of parameter shift in their individualdynamical systems, and may lead to abnormal functions of the network.To accurately identify the chaotic neurons may thus be necessary andimportant, for example, for applying appropriate controls to bring thenetwork to a normal state. However, due to couplings among the nodesthe measured time series even from non-chaotic neurons would appearrandom, rendering inapplicable traditional nonlinear time-seriesanalysis, such as the delay-coordinate embedding method, which yieldsinformation about the global dynamics of the entire network. Themethod to be discussed is based on compressive sensing. In particular,identifying chaotic elements can be formulated as a general problem ofreconstructing the nodal dynamical systems, network connections, and allcoupling functions as well as their weights. Data based reconstruction of complex geospatial networks, nodal positioning, and detection of hidden node. Complex geospatial networks with components distributed in the realgeophysical space are an important part of the modern infrastructure.Given a complex geospatial network with nodes distributed in a two-dimensional region of the physical space, can the locations of the nodes be determined and their connection patterns be uncovered based solely on data? In realistic applications, time series/signalscan be collected from a single location. A key challenge is that thesignals collected are necessarily time delayed, due to the varyingphysical distances from the nodes to the data collection center.We discuss a compressive sensing based approach <cit.> thatenables reconstruction of the full topology of the underlying geospatialnetwork and more importantly, accurate estimate of the time delays. Astandard triangularization algorithm can then be employed to find the physical locations of the nodes in the network. A hidden source orthreat, from which no signal can be obtained, can also be detected through accurate detection of all its neighboring nodes. As a geospatialnetwork has the feature that a node tends to connect with geophysicallynearby nodes, the localized region that contains the hidden node canbe identified.Reconstructing complex spreading networks with naturaldiversity and identifying hidden source. Among the various types of collective dynamics on complex networks, propagation or spreading dynamics is of paramount importance as it is directly relevant to issues of tremendous interests such as epidemic and disease outbreak in the human society and virus spreading on computer networks. We discuss a theoretical and computational framework based on compressive sensing to reconstruct networks of arbitrary topologies, in which spreading dynamics with heterogeneous diffusion probabilities take place <cit.>.The approach enables identification with high accuracy of the externalsources that trigger the spreading dynamics. Especially, a fullreconstruction of the stochastic and inhomogeneous interactions presentedin the real-world networks can be achieved from small amounts of polarized(binary) data, a virtue of compressive sensing. After the outbreak ofdiffusion, hidden sources outside the network, from which no direct routesof propagation are available, can be ascertained and located with highconfidence. This represents essentially a new paradigm for tracing, monitoring,and controlling epidemic invasion in complex networked systems, which willbe of value to defending and preserving the systems against disturbancesand attacks.Section <ref> presents a number of alternative methods for reconstructing complex and nonlinear dynamical networks. These are methods based on (1) response of the system to external driving,(2) synchronization (via system clone), (3) phase-space linearization,(4) noise induced dynamical correlation, and (5) automated reverseengineering. In particular, for the system response based method (1), the basic idea was to measure the collective response of the oscillator networkto an external driving signal. With repeated measurements of the dynamicalstates of the nodes under sufficiently independent driving realizations, the network topology can be recovered <cit.>. Since complex networks are generally sparse, the number of realizations of external driving can be much smaller than the network size. For the synchronization based method, the idea was to design a replicaor a clone system that is sufficiently close to the original network without requiring knowledge about network structure <cit.>. From the clonesystem, the connectivities and interactions among nodes can be obtaineddirectly, realizing network reconstruction. For the phase-space linearization method, L_1-minimization in the phase space of a networked system is employed to reconstruct the topology without knowledge of the self-dynamicsof nodes and without using any external perturbation to the state ofnodes <cit.>. For the method based on noise-induced dynamicalcorrelation, the principle was based on exploiting the rich interplay between nonlinear dynamics and stochasticfluctuations <cit.>. Under the condition that the influence of noise on the evolution of infinitesimal tangent vectors in the phase space of the underlying networked dynamical system is dominant, it can be argued <cit.> that the dynamical correlation matrix that can be computed readily from the available nodal time series approximates the network adjacency matrix, fully unveiling the network topology. For the automated reverse engineering method,the approach was based on problem solving using partitioning, automatedprobing and snipping <cit.>, a process that is akin to genetic algorithm. Each of the five methods will be described inreasonable details. Network reconstruction was pioneered in biological sciences. Section <ref> is devoted to a concise survey of the approaches to reconstructing biological networks. Those include methodsbased on correlation, causality, information-theoretic measures,Bayesian inference, regression and resampling, supervision andsemi-supervision, transfer and joint entropies, and distinguishing between direct and indirect interactions.Finally, in Sec. <ref>, we offer a general discussion of the field of data-based reconstruction of complex networks and speculate on a number of open problems. These include: localization of diffusion sources in complex networks, reconstruction of complex networks withbinary state dynamics, possibility of developing a universal framework of structural estimator and dynamics approximator for complex networks, and a framework of control and controllability for complex nonlineardynamical networks.§ COMPRESSIVE SENSING BASED NONLINEAR DYNAMICAL SYSTEMSIDENTIFICATION §.§ Introduction to the compressive sensing algorithm Compressive sensing is a paradigm developed in recent years by appliedmathematicians and electrical engineers to reconstruct sparse signalsusing only limited data <cit.>. The observations are measured by linear projectionsof the original data on a few predetermined, random vectors. Since the requirement for the observations is considerably less comparing toconventional signal reconstruction schemes, compressive sensing has beendeemed as a powerful technique to obtain high-fidelity signalespecially in cases where sufficient observations are not available. Compressive sensing has broad applications ranging from imagecompression/reconstruction to the analysis of large-scale sensornetworks <cit.>. Mathematically, the problem of compressive sensing is to reconstruct a vector 𝐚∈ℝ^N from linear measurements 𝐗 about 𝐚 in the form:𝐗 = 𝒢·𝐚where 𝐗∈ℝ^M and 𝒢 is an M× Nmatrix. By definition, the number of measurements is much less than that ofthe unknown signal, i.e, M≪ N. Accurate recovery of the originalsignal is possible through the solution of the following convexoptimization problem<cit.>:min𝐚_1 𝒢·𝐚 = 𝐗, where𝐚_1=∑_i=1^N|𝐚_i| is the l_1 norm of vector 𝐚.The general principle of solving the convex optimization problem Eq. (<ref>) can be described briefly, as follows. By introducing a new variable vector 𝐮∈ℝ^N, Eq. (<ref>)can be recast into a linear constraint minimization problem <cit.>:min∑_i=1^Nu_ia_i-u_i≤ 0 - a_i-u_i≤ 0 𝒢·𝐚 = 𝐗. Defining 𝐳=[𝐚^T,𝐮^T]^T, we can rewriteEq. (<ref>) as⟨ c_0, z⟩f_i(z)≤ 0 f'_i(z)≤ 0 𝒢_0·𝐳 = 𝐗, where f_i(z)=⟨𝐜_i, 𝐳⟩and f'_i(z)=⟨𝐜'_i, 𝐳⟩(⟨·⟩ denotes the inner product of the two vectors). Here, 𝐜_0, 𝐜_i,𝐜'_i∈ℝ^2N, 𝒢_0 is a M× 2N matrix, (𝐜_0)_j=0 for j≤ Nand (𝐜_0)_j=1 for j> N; (𝐜_i)_j=1for j=i, (𝐜_i)_j=-1 for j=N+i; (𝐜'_i)_j=1 for j=i, (𝐜'_i)_j=-1 for j=N+i; 𝒢_0=[0^M× N,𝒢]. To solve thelinear constraint minimization problem in Eq. (<ref>), onecan use the Karush-Kuhn-Tucker conditions <cit.>, i.e., at theoptimal point 𝐳^*, there exist vectors 𝐯^*∈ℝ^M, 𝐪^*∈ℝ^N, 𝐪'^*∈ℝ^N, where (𝐪^*)_j ≥ 0 and (𝐪^'*)_j≥ 0for j = 1, …, N, such that the following hold:𝐜_0 + 𝒢_0^T ·𝐯^* +∑_iq_i^* 𝐜_i+∑_i q_i^'*𝐜'_i=0, q_i^*f_i(z^*) = 0,i=1,...,N, q_i^'*f'_i(z^*) = 0,i=1,...,N,𝒢_0 ·𝐳^* =𝐗, f_i(z^*) ≤0, f'_i(z^*)≤0.Equation (<ref>) can be solved by the classical Newton methodin the valid solution set determined by the inequality constraints{q_i≥0,q_i^'≥ 0,f_i(z)≤ 0,f'_i(z)≤ 0}, where a point (𝐳,𝐯,𝐪,𝐪') in this set iscalled as an interior point. Define a residual vector for all the equality conditions in Eq. (<ref>) as𝐫=[𝐫_1^T,𝐫_2^T,𝐫_3^T,𝐫_4^T]^T,with 𝐫_i (i=1,…,4) given by where𝐫_1 =𝐜_0 + 𝒢_0^T ·𝐯+∑_i q_ic_i+∑_i q_i^'𝐜'_i,𝐫_2 = - λ·𝐟,𝐫_3 = - λ'·𝐟',𝐫_4 =𝒢_0 ·𝐳 - 𝐗,where 𝐟=[f_1(z),f_2(z),...,f_N(z)]^T, 𝐟'=[f'_1(z),f'_2(z),...,f'_N(z)]^T, λand λ' are diagonal matrices with(λ)_ii=q_i and (λ')_ii=q'_i.To find the solution to Eq. (<ref>), one linearizes theresidual vector r using the Taylor expansion about the point(𝐳,𝐯,𝐪,𝐪'), which gives𝐫(𝐳+Δ𝐳, 𝐯+Δ𝐯,𝐪 +Δ𝐪, 𝐪'+Δ𝐪') = 𝐫(𝐳, 𝐯, 𝐪, 𝐪') + 𝒥(𝐳,𝐯,𝐪,𝐪') ·( [Δ𝐳;Δ𝐯;Δ𝐪; Δ𝐪'; ]), where 𝒥 is the Jacobian matrix of 𝐫 given by𝒥 = ( [0𝒢_0^T𝒞^T 𝒞'^T; -λ·𝒞0 -ℱ0; -λ'·𝒞'00-ℱ';𝒢_0000;]), the N× 2N matrices 𝒞 and 𝒞' have 𝐜_iand 𝐜'_i as rows, respectively, ℱ and ℱ'are diagonal matrices with (ℱ)_ii=f_i(𝐳) and(ℱ')_ii=f'_i(𝐳). The steepest descent directioncan be obtained by setting zero the left-hand side of Eq. (<ref>), which gives ( [Δ𝐳;Δ𝐯;Δ𝐪; Δ𝐪'; ])=-𝒥^-1·𝐫. With the descent direction so determined, to solve Eq. (<ref>),one can update the solution by𝐳 = 𝐳+sΔ𝐳, 𝐯=𝐯+s Δ𝐯,𝐪=𝐪+s Δ𝐪, 𝐪'=𝐪'+s Δ𝐪'with step length s, where s should be chosen to guarantee that(𝐳+sΔ𝐳,𝐯+sΔ𝐯, 𝐪 + sΔ𝐪,𝐪'+sΔ𝐪') is aninterior point of the valid solution set Eq. (<ref>). Iteratingthis procedure gives the reconstructed sparse signal 𝐚. §.§ Mathematical formulation of systems identification based oncompressive sensingThe inverse problem of identifying nonlinear dynamical systems can be cast in the framework of compressive sensing so that optimal solutions can be obtained even when the number of base coefficients to be estimated is large and/or the amount of available data is small. Consider systems described byẋ =F( x),where x∈ R^m represents the set of externally accessible dynamical variables and F is a smooth vector function in R^m. The jth component of F( x) can be represented as a power series:[ F( x)]_j = ∑_l_1=0^n∑_l_2=0^n⋯∑_l_m=0^n (a_j)_l_1,⋯,l_m· x_1^l_1 x_2^l_2⋯ x_m^l_m,where x_k (k=1,⋯, m) is the kth component of the dynamical variable, and the scalar coefficient of each product term (a_j)_l_1,⋯,l_m∈ R is to be determined from measurements. Note that the terms in Eq. (<ref>) are allpossible products of different components with different powers, andthere are (1+n)^m terms in total.Without loss of generality, one can examine one dynamical variable of the system. (Procedures for other variables are similar.) For example, to construct the measurement vector X and the matrix G for the case of m = 3 (dynamical variables x, y, and z) and n = 3, one has the following explicit equation for the first dynamical variable: [F(x)]_1 ≡(a_1)_0,0,0x^0y^0z^0 + (a_1)_1,0,0x^1y^0z^0 + ⋯+ (a_1)_3,3,3x^3y^3z^3. Denote the coefficients of [ F( x)]_1 by a_1 = [(a_1)_0,0,0,(a_1)_1,0,0,⋯ ,(a_1)_3,3,3]^T. Assuming that measurements of x(t) at a set of time t_1,t_2,…,t_w are available, one can writeg(t) = [ x(t)^0y(t)^0z(t)^0, x(t)^0y(t)^0z(t)^1, ⋯, x(t)^3y(t)^3z(t)^3 ] such that [ F( x(t))]_1 =g(t) · a_1. Fromthe expression of [ F( x)]_1, one can choose themeasurement vector as 𝐗=[ẋ(t_1),ẋ(t_2), ⋯,ẋ(t_w)]^T,which can be calculated from time series. Finally, one obtains the following equation in the form𝐗 = 𝒢·𝐚_1:( [ ẋ(t_1); ẋ(t_2);⋮; ẋ(t_w);] )= ( [ g(t_1); g(t_2);⋮; g(t_w);] ) ( [ a_1 ]).To ensure the restricted isometry property <cit.>, onenormalizes 𝒢 by dividing elements in each column by the L_2 norm of that column: (𝒢')_ij = (𝒢)_ij/L_2(j) with L_2(j) =√(∑_i=1^M [(𝒢)_ij]^2), so that 𝐗 = 𝒢'·𝐚_1'. After the normalization, a_1'= a_1· L_2 can be determined via some standard compressive-sensing algorithm <cit.>. As a result, the coefficients a_1 are given by a_1'/L_2. To determine the set of power-series coefficients corresponding to a different dynamical variable, say y, one simply replaces the measurement vector by X =[ẏ(t_1),ẏ(t_2), ⋯,ẏ(t_w)]^T and use the same matrix 𝒢. This way all coefficients a_1, a_2, and a_3 of three dimensions can be estimated. §.§ Reconstruction and identification of chaotic systemsThe problem of predicting dynamical systems based on time series has been outstanding in nonlinear dynamics because, despite previousefforts <cit.> in exploiting the delay-coordinate embeddingmethod <cit.> to decode the topological propertiesof the underlying system, how to accurately infer the underlying nonlinear system equations remains largely an unsolved problem. In principle, a nonlinear system can be approximated by a large collection of linear equations in different regions of the phase space, which can indeed be accomplished through reconstructing the Jacobian matrices on a proper grid that covers the phase-space region of interest <cit.>. However, the accuracy and robustness of the procedure are challenging issues, including the difficulty with the required computations. For example, in order to be able to predict catastrophic bifurcations, local reconstruction of a large set of linearized dynamics is not sufficient but rather, an accurate prediction of the underlying nonlinear equations themselves is needed.In 2011 it was proposed <cit.> that compressive sensingprovides a powerful method for data based nonlinear systemsidentification, based on the principle that it is possible to fullyreconstruct dynamical systems from time series because the dynamics ofmany natural and man-made systems are determined by functions that can be approximated by series expansions in a suitable base. The task is then to estimate the coefficients in the series representation. In general, the number of coefficients to be estimated can be large but many of them may be zero (the sparsity condition). According to the conventional wisdom this would be a difficult problem as a large amount of data is required and the computations involved can be extremely demanding. However, compressive sensing <cit.>.provides a viable solution to the problem, where the basic principleis to reconstruct a sparse signal from small amount of observations, as measured by linear projections of the original signal on a fewpredetermined vectors. In Ref. <cit.>, a number of classic chaotic systems were used to demonstrate the compressive sensing based approach. Here we quote one example, the Hénon map <cit.>, a classicalmodel that has been used to address many fundamental issues in chaoticdynamics. The prediction of the map equations resembles that of a vectorfield. The map is given by: (x_n+1,y_n+1) = (1 - a x_n^2 + y_n, bx_n),where a and b are parameters. For b = 0.3, the map exhibits periodic and chaotic attractors for a < a_c ≈ 1.42625, where a_cis the critical parameter value for a boundarycrisis <cit.>, above which almost alltrajectories diverge. Assuming power-series expansions up to order 3in the map equations, the authors were able to identify the mapcoefficients with high accuracy using only a fewdata points. Figure <ref> shows the distributions ofthe estimated power-series coefficients, where extremely narrow peaksabout zero indicate that a large number of the coefficients are effectivelyzero, which correspond to nonexistent terms in the map equations. Coefficients that are not included in the zero peak correspond thento the existent terms and they determine the predicted map equations. Note that, to predict correctly the map equations, the number of requireddata points is extremely low. Similar results were obtained <cit.>for the classic standard map <cit.>, the chaotic Lorenzsystem <cit.>, and the chaotic Rössleroscillator <cit.>. To quantify the performance of the compressive sensing based systemsidentification method with respect to the amount of required data, the prediction errors were calculated <cit.>, which aredefined separately for nonzero (existing) and zero terms in the dynamical equations. The relative error of a nonzero term is defined as the ratio to the true value of the absolute difference between the predicted and true values. The average over the errors of all terms in a component is the prediction error E_nz of nonzero terms for the component. In contrast, the absolute error E_z is used for zero terms. Figures <ref>(a) and<ref>(b) show E_nz as a function of the ratioof the number n_m of measurements to the total number n_nz + n_zof terms to be predicted, for the standard map and the Lorenz system, respectively. Note that, for the standard map, it is necessary to explore alternative bases of expansion so that the sparsity condition can be satisfied. A practical strategy is that, assuming that a rough idea about the basic physics of the underlying dynamical system is available, one can choose a base that is compatible with the knowledge. In the case of the standard map, a base includingthe trigonometric functions can be chosen. The results inFigs. <ref>(a) and <ref>(b) indicate that, when the number n_m of measurements exceeds a thresholdn_t, E_nz becomes effectively zero. For convenience, one can define n_t by using the small threshold value E_nz = 10^-3 so that n_t is the minimum number of required measurements for an accurate prediction. Figures <ref>(a) and <ref>(b) show that n_t is much less than n_nz + n_z if n_nz, the number of nonzero terms is small. The performance of the method can thus be quantified by the threshold with respect to the numbers of measurements and terms to be predicted. As shown in Figs. <ref>(c) and <ref>(d) for the standard map and the Lorenz system, respectively, as the nonzero terms become sparser among all terms to be predicted (characterized by a decrease in n_nz/(n_nz+n_z) when n_nz+n_z is increased), the ratio of the threshold n_t to the total number of terms n_nz+n_z becomes smaller. These results demonstrated the advantage of the compressive-sensing based method to predict dynamical systems, i.e., high accuracy and extremely low required measurements. In general, to predict anonlinear dynamical system as accurately as possible, many reasonable terms should be assumed in the expansions, insofar as the percentage of nonzero terms is small so that the sparsity condition of compressive sensing is met.There are situations where the system is high-dimensional or stochastic. A possible solution is to employ the Bayesian inferenceto determine the system equations. In general the computationalchallenge associated with the approach can be formidable, but thepower-series or more general expansion based compressive-sensing methodmay present an effective strategy to overcome the difficulty. §.§ Predicting catastrophe in nonlinear dynamical systems§.§.§ Predicting catastrophic bifurcations based on compressive sensingNonlinear dynamical systems, in their parameter space, can often exhibitcatastrophic bifurcations that ruin the desirable or “normal”state of operation. Consider, for example, the phenomenon ofcrisis <cit.> where, as a system parameter is changed,a chaotic attractor collides with its own basin boundary and issuddenly destroyed. After the crisis, the state of the system iscompletely different from that associated with the attractorbefore the crisis. Suppose that the state beforethe crisis is normal and desirable and the state after the crisisis undesirable or destructive. The crisis can thus be regarded as acatastrophe that one strives to avoid at all cost. Catastrophic events, of course, can occur in different forms in all kinds of natural and man-made systems. A question of paramount importance is how to predict catastrophes in advance of their possible occurrences.This is especially challenging when the equations of the underlyingdynamical system are unknown and one must then rely on measured timeseries or data to predict any potential catastrophe.Compressive sensing based nonlinear systems identificationprovides an approach to forecasting potential catastrophicbifurcations <cit.>. Assume that an accurate modelof the system is not available, i.e., the system equations areunknown, but the time evolutions of the key variables of thesystem can be accessed through monitoring or measurements. Themethod <cit.> thus consists of three steps: (i) predicting the dynamical system based on time series, (ii) identifying the parameters of the system, and (iii) performing a computational bifurcation analysis using the predicted system equations to locate potential catastrophic events in the parameter space so as to determine the likelihood of system's drifting into a catastrophe regime. In particular, if the system operates at a parameter setting close to such a critical bifurcation, catastrophe is imminent as a small parameter change or a random perturbation can push the system beyond the bifurcation point. Consider the concrete case of crises in nonlinear dynamical systems. Once a complete set of system equations has been predicted and the parameters have been identified, one proceeds to examine the available parameter space. It should be noted that, to explore the multi-parameter space of a dynamical system can be challenging, but this can often lead to the discovery of new phenomena in dynamics. Examples are the phenomenon of double crises in systems with two bifurcationparameters <cit.> and the hierarchical structures in theparameter space <cit.>. In Ref. <cit.>, a number of examples were given in which the bifurcation diagrams computed from the predicted system equations agree well with those from the original systems, so all possiblecritical bifurcation points can be predicted accurately based on timeseries only. Figures <ref>(a) and<ref>(b) illustrate a predicted bifurcationdiagram from the chaotic Lorenz and Rössler systems, respectively.§.§.§ Using compressive sensing to predict tipping points incomplex systemsIt is increasingly recognized that a variety of complex dynamical systems ranging from ecosystems and the climate to economical, social, and infrastructure systems can exhibit tipping points at which irreversible transition from a normal functioning state to a catastrophic state occurs <cit.>. Examples of such transitions are blackouts in the power grids, sudden emergence of massive jamming in urban traffic systems, the shutdown of the thermohaline circulation in the North Atlantic <cit.>, sudden extinction of species in ecosystems <cit.>, and the occasional switches of shallow lakes from clear to turbid waters <cit.>. In fact, the seemingly abrupt transitions are the consequence of gradual changes in the system which can, for example, be caused by slow drifts in the external conditions. To understand the dynamical properties of the system near a tipping point, to predict the tendency for the system to drift toward the tipping point, to issue early warnings, and finally to apply control to reverse or to slow down the trend, are significant but extremely challenging research problems. Compounded with the difficulty is the fact the complex systems are often interdependent and non-stationary. For example, the evolution of an ecosystem depends on human behaviors, which in turn affects the well being of the human society. Infrastructure systems such as the power grids and communication networks are interdependent upon each other <cit.>, both beingaffected by human activities (social system). The concept of interdependenceis prevalent in many disciplines. At the present there is little understandingof tipping points in interdependent complex systems in terms of theiremergence and dynamical properties.In a dynamical system, the existence of one or several tipping points is intimately related to the concept of resilience <cit.>,which can be understood heuristically by resorting to the intuitive picture of a ball moving in a valley under gravity, as shown in Fig. <ref>. To the right of the valley there is a hill, or a potential barrier in terms of classical mechanics. The downhill side to the right of the barrier corresponds to a catastrophic behavior. Normal functioning of the system is represented as the confinement of ball's motion within the valley. If the valley is sufficiently deep (or the height of the barrier is sufficiently large), as shown in Fig. <ref>(a), there will be little probability for the ball to move across the top of the barrier towards catastrophic behavior, implying that the system is more resilient to random noise or external disturbances. However, if the barrier height is small, as shown in Fig. <ref>(b), the system is less resilient as small perturbation can push the ball over to the left side of the barrier. The top of the barrier thus corresponds to a tipping point, across which from the left the system will essentially collapse. In mechanics, the situation can be formulated using a potential function so that, mathematically, the motion of the ball can be described by the Hamilton's equations <cit.>. Given a dynamical system, if the potential landscape can be determined, it will be possible to locate the tipping points. In systems biology, the potential function is called the Waddington landscape <cit.>, which essentially determines the biological paths for cell development and differentiation <cit.> - the landscape metaphor. In the past few years, a quantitative approach to mapping out the potential landscape for gene circuits or generegulatory networks has been developed <cit.>. In nonlinear dynamical systems, a similar concept exists - quasipotential <cit.>.Because of non-stationarity, for a complex system in the real world,Fig. <ref> in fact represents only a “snapshot” of the potential landscape. For complex dynamical systems, the potential landscape must necessarily be time-dependent or non-stationary, which completely governs the emergence of tipping points and the global dynamics of the system. There are two types of time-varying disturbances to the landscape: (1) slow but eventually large changes in parameters and system equations and (2) fast but small random perturbation. The physical origins of these disturbances can be argued, as follows.Take, for example, the challenge of adding distributed renewable energy generation to the power grid. The time scale for changes is years. Over that time period, significant new renewables can be added,and yet the precise timing, location, and amount of distributedrenewable energy generation is unpredictable, because it is not possible to know how social decisions (in terms of regulation, business planning, and consumer choice) will play out. (A similar situation occurs for climate change: slow, long-term, secular, and nonlinear changes in climatic averages would occur, which will impact generation via water availability and temperature, etc. At the same time, weather remains a high frequency pattern on top of these slow changes.) Clearly the current grid is largely stable vis-a-vis existing distributions of weather variables, but how will that change in the future? For example, in the future, charging networks may comprise mainly slow charging stations <cit.> as the widespread use of fast charging stations would raise the power demand in the electrical grid <cit.>, causing tremendous difficulties for the managers to operate the grid. There can also be changing regulatory incentives or management structures. All these can lead to social non-stationarity. Mathematically, the social factors can be modeled by adiabatic changes to the system, which affect the potential landscape on time scales that are slower than that of the intrinsic dynamics.At short time scales, precise hourly patterns of electricity generationmay fluctuate due to changes in wind and cloud activity - a type of environmental non-stationarity. There can also be technological non-stationarity such as shifting demand patterns associated with consumer electronics, plug-in hybrid and electric vehicles, etc. These occur on short time scale, which can be modeled as random perturbationor noise to the system.Referring to Fig. <ref>, the topographic landscape metaphor of resilience, one can see that, if the system is stationary, there is a fixed threshold across which the system will collapse. In this case, the system resilience can simply be characterized by the distance to the threshold. However, for a non-stationary system, it is not possible to measure thethreshold distance by establishing the absolute positions of the ball (system) and the tipping point and then computing the difference between them. Instead, one must attempt to estimate the differences directly in real time. Two open questions are: is it possible to determine the non-stationarylandscape so as to predict the emergence and the locations of thetime-dependent tipping points? Can human intervention or control strategiesbe developed to prevent or significantly slow down the system's evolutiontoward a catastrophic tipping point? At the present there are no answers to these questions of the grandchallenge nature, but we wish to argue that compressive sensingbased nonlinear systems identification can provide insights into thefundamental issues pertinent to these questions. Consider,for example, a complex power grid system, in which the time seriessuch as the voltage and power at all generator nodes are available,as well as social interaction data that are typically polarized (e.g.,binary). There were recent efforts in reconstructing complex networks of nonlinear oscillators based on continuous time series <cit.>and in uncovering epidemic propagation networks using binarydata <cit.> (to be discussed inSec. <ref>). In principle, these approaches can be combined to deal withnonstationary complex systems. Specifically, large but slowly varyingphysical non-stationarity can be modeled through the appearance ofadditional, concrete mathematical terms involving voltage, phase,and current variables, or through the disappearance of certainterms. Social non-stationarity can be modeled by functions of Booleanvariables that generate polarized data or can be reconstructed usingthese data. It would then be possible to establish amathematical framework combining reconstruction methods for continuoustime series and polarized data. Potentially, this could represent an innovative and concrete approach to incorporating social data into a complex physical/technological system and assessing, quantitatively, the effect of social non-stationarity on the dynamical evolution of the system. §.§ Forecasting future states (attractors) of nonlinear dynamical systemsA dynamical system in the physical world is constantly subject torandom disturbance or adiabatic perturbation. Broadly speaking, thereare two types of perturbations: stochastic or deterministic. Stochasticdisturbances (or noise) can typically be described by random processesand they do not alter the intrinsic structure of the underlyingequations of the system. Deterministic perturbations, however, can cause the system equations or parameters to vary with time. Suppose theperturbations are adiabatic, i.e., T_i, the time scale of the intrinsicdynamics of the system, is much smaller than T_e, the time scale of theexternal perturbation. In this case, some “asymptotic states” or“attractors” of the system can still be approximately defined in a timescale that is much larger than T_i but smaller than T_e. When the dynamicsin such a time interval is examined in a long run, the attractor of thesystem will depend on time. Often, one is interested in forecasting the“future” asymptotic states of the system. Take the climate system as anexample. The system is under random disturbances, but adiabatic perturbationsare also present, such as CO_2 injected into the atmosphere due to humanactivities, the level of which tends to increase with time. The timescale for appreciable increase in the CO_2 level to occur (e.g.,months or years) is much larger than the intrinsic time scales of thesystem (e.g., days). The climate system is thus an adiabaticallytime-varying, nonlinear dynamical system. It is of interest to forecast what the future attractors of the system might be in order to determine whether it will behave as desired or sustainably. The issue of sustainability is, of course, critical to many other natural and man-made systems as well. To be able to forecast the future states of suchsystems is essential to assessing their sustainability.It was demonstrated that compressive sensing can be exploited forpredicting the future states (attractors) of adiabatically time varying dynamical systems <cit.>. The general problemstatement is: given a nonlinear dynamical system whose equationsor parameters vary adiabatically with time, but otherwise are completelyunknown, can one predict, based solely on measured time series, thefuture asymptotic attractors of the system? To be concrete, considerthe following dynamical system:d x/dt =F[ x,p(t)],where x is the dynamical variable of the system in thed-dimensional phase space and p(t) = [p_1(t),…,p_K(t)]denotes K independent, time-varying parameters of the system. Assume that both the form of F and p(t) are unknown but at timet_M, the end of the time interval during which measurements are taken,the time series x(t) for t_M - T_M ≤ t ≤ t_M are available,where T_M denotes the measurement time window. The idea was to predict, using compressive sensing, the precise mathematical forms of F andp(t) based on the available time series at t_M so that theevolution and the likely attractors of the system for t > t_M can becomputationally assessed and anticipated <cit.>. The predictedforms of F and p(t) at time t_M would contain errorsthat in general will increase with time. In addition, for t > t_M newperturbations can occur to the system so that the forms of F andp(t) may be further changed. It is thus necessary to execute the prediction algorithm frequently using time series available at the time. In particular, the system could be monitored at all times so that time series can be collected, and predictions should be carried out at t_i's, where … > t_i > … > t_M+2 > t_M+1 > t_M. For any t_i, the prediction algorithm is to be performed based on available time series in a suitable window prior to t_i.To formulate the problem of predicting time-varying dynamical systems in the framework of compressive sensing, one can expand allcomponents of the time-dependent vector field F[ x,p(t)]in Eq. (<ref>) into a power series in terms of both dynamical variables x and time t. The ith component F[ x,p(t)]_i of the vector field can be written as∑_l_1,⋯,l_m=1^n [(α_i)_l_1,⋯,l_mx_1^l_1⋯x_m^l_m·∑_w=0^v(β_i)_w t^w] ≡∑_l_1,⋯,l_m=1^n∑_w=0^v (c_i)_l_1,⋯,l_m;wx_1^l_1⋯ x_m^l_m· t^w,where x_k (k=1,⋯, m) is the kth component of the dynamicalvariable and c_i is the ith component of the coefficient vector to bedetermined. Assume that the time evolution of each term can be approximatedby the power series expansion in time, i.e.,∑_w=0^v(β_i)_w t^w. The power-series expansion allows us to cast Eq. (<ref>) into the standard form of compressive sensing, Eq. (<ref>). In principle,if every combined scalar coefficient (c_i)_l_1,⋯,l_m;wassociated with the corresponding term in Eq. (<ref>)can be determined from time series for t ≤ t_M, the vector fieldcomponent [ F( x, p(t))]_i becomes known. Repeating the procedure for all components, the entire vector fieldfor t > t_M can be found.To explain the compressive sensing based method in an intuitive way, one canconsider the special case where the number of components of the dynamical variables is m = 3 (x, y, and z), the order of the power series is l_1+l_2+l_3 ≤ 2, and the maximum power of time t inEq. (<ref>) is v=1, i.e., only the t^0 and t^1 termsare included. Focusing on one dynamical variable, say x, the totalnumber of terms in the power-series expansion is 20, as specified inFig. <ref>(a). Let the measurementsx(t), y(t), and z(t) be taken at times t_1,t_2,…,t_M,as shown in Fig. <ref>(b). The values of all20 power-series terms at these time instants can then be obtained, asshown in Fig. <ref>(c), where all the termsare divided into two blocks according to the distinct powers of the timevariable t: t^0 and t^1. The projection matrix 𝒢 inEq. (<ref>) thus consists of these two blocks. (In the general case where higher powers of the time variable is involved,𝒢 would contain a corresponding number of blocks.) Thecomponents of vector 𝐗 in Eq. (<ref>) are thefirst derivatives dx/dt evaluated at t_1,t_2,…,t_M,which can also be approximated by the measured time series x(t) atthese times. As shown in Fig. <ref>(c),Eq. (<ref>) for this simple example can bewritten in the form of Eq. (<ref>). To ensure sparsity,one can assume many terms in the power-series expansion up to somehigh order n so that the total number of terms inEq. (<ref>), N, will be quite large. As a result, 𝐚 is high-dimensional but most of its components are zero. The number of measurements, M, needs not be as large asN: M ≪ N. Another requirement of compressive sensing is therestricted isometric property that can be guaranteed by normalizing thematrix 𝒢 and by using linear-programming based signal-recoveryalgorithms <cit.>. To determine the set of power-series coefficientscorresponding to a different dynamical variable, say y, one simplyreplaces the measurement vector by 𝐗 =[ẏ(t_1),ẏ(t_2),…,ẏ(t_M)]^T. The matrix 𝒢, however, remains the same. The problem offorecasting time-varying nonlinear dynamical systems then fitsperfectly into the compressive-sensing paradigm. As a proof of principle, the authors of Ref. <cit.> used theclassical Lorenz chaotic system <cit.> as an example by incorporating explicit time dependence in a number of additional terms. The modified Lorenz system is given byẋ=-10(x - y) + k_1(t)y,ẏ= 28x - y - xz + k_2(t)z,ż= -(8/3)z+xy+[k_3(t) + k_4(t)]y,where k_1(t) = -t^2, k_2(t) = 0.5 t, k_3(t) = t, and k_4(t) = -0.5 t^2. Suppose that the system equations are unknown but only measured time series x(t), y(t), and z(t) in a finite time interval are available. The number of dynamical variables is m = 3 and we choose the orders of the power-series expansions in the three variables according to l_1+l_2+l_3 ≤ 3. The maximum power in the time dependence is chosen to be v = 2 so that explicit time-dependent terms t^0, t^1 and t^2 are considered. The total number of coefficients to be predicted is then (v+1)∑_i=1^3(i+1)(i+2)/2=57. (Note that, using low-order power-series expansions in both the dynamical variables and time is solely for facilitating explanation and presentation of results, while the forecasting principle is the same for realistic dynamical systems where much higher orders may be needed.)Figure <ref> shows the predicted coefficient values versus the term index for all three dynamical variables, where in each panel, solid triangles and open circles denote the predicted non-zero and zero coefficients, respectively, and the red dashed dividing lines indicate the terms associated with different powers of the time variable, i.e., t^0, t^1 and t^2 (from left to right). The meaning of these results can be explained by using any one of the dynamical variables. For example, for the x-component of the vector field, the prediction algorithm gives only 3 nonzero coefficients. By identifying the corresponding values of the term index, one can see that they correspond to the two terms without explicit time dependence: y, x, and the term that contains explicit such dependence: t^2y, respectively. A comparison of the predicted nonzero coefficient values with the actual ones in the original Eq. (<ref>) indicates that the method works remarkably well. Similar results were obtained for the y and z components of the vector field. When the vector field F[ x, p(t)] of the underlying dynamical system has been predicted, one can solve Eq. (<ref>) numerically to assess the state variables at any future time and the asymptotic attractors.Figures <ref>(a,b) present one example, wherea forecasted time series calculated from the predicted vector fieldis shown, together with the values of the corresponding dynamicalvariable from the actual Lorenz system at a number of time instants.The two cases shown are where the parameter functions k_i(t)(i=1,…,4) are all zero and time-varying, respectively. Excellentagreement was again obtained, indicating the power of the method topredict the future states and attractors of time-varying dynamicalsystems. The interpretation and implication ofFigs. <ref>(a,b) are the following. Note that t = 0 and t = 10 correspond to the beginning and the end of the measurement time window [t_1,t_M], respectively. For the original classical Lorenz system without time-varying parameters, the asymptotic attractor is chaotic, as can be seen from Fig. <ref>(a). However, as external perturbations are turned on at t = 0, there are four time-varying parameters in the system for t > 0. In this case, the asymptotic attractor becomes a fixed-point, as can be seen from the asymptotic behavior z → in Fig. <ref>(b). In both cases, by using limited amount of measurements, namely, available time series in the window [t_1,t_M], one obtains quite accurate forecasting results. The result exemplified in Fig. <ref>(b) is especially significant, as it indicates that the future state and attractors of time-varying dynamical systems can be accurately predicted based on limited data availability.An error and performance analysis was carried out in Ref. <cit.> by using the indicators E_nz and E_z, the prediction error forexistent and non-existent terms, respectively. Figure <ref>(a) shows, for the time-varying Lorenzsystem, E_nz versus the ratio of the number M of measurements tothe total number N of terms to be predicted. For all dynamicalvariables, one observes that, as M exceeds a threshold value M_t,E_nz becomes effectively zero, where M_t can be defined quite arbitrarily, e.g., the minimum number of measurements required to achieve E_nz=10^-3. The data requirement for accurate prediction can then be assessed by examining how M_t depends on the sparsity of the coefficient vector to be predicted, which can be defined as the ratio of the number N_nz of the nonzero terms to the total number N of terms to be predicted. Note that, N or the ratio N_nz/N can be adjusted by varying the order of the assumed power series. FromFig. <ref>(b), one can see that, as N_nz/N isdecreased (e.g., by increasing N) so that the vector to be predictedbecomes more sparse, the ratio M_t/N also decreases. In particular,for the smallest value of N_nz/N examined, where N = 357, onlyabout 5% of the data points are needed for accurate prediction,despite the time-varying nature of the underlying dynamical system. Figure <ref>(c) shows the prediction errors with respect to different length of the measurement window for a fixed number of datapoints. It can be seen that, when the length exceeds a certain (small)value so that the time series extends to the whole attractor in thephase space, E_nz approaches zero rapidly.Dynamical systems are often driven by time-periodic forces, such asthe classical Duffing system <cit.>. In such a case, it isnecessary to explore alternative bases of expansion with respect tothe time variable other than power series to ensure the sparsitycondition. A realistic strategy to choose a suitable expansion baseis to make use of the basic physics underlying the dynamical systemof interest. Insofar as an appropriate base can be chosen so thatthe coefficient vector to be predicted is sparse, the compressive sensing based methodology is applicable. § COMPRESSIVE SENSING BASED RECONSTRUCTION OF COMPLEXNETWORKED SYSTEMSCompressive sensing has recently been introduced to the field of networkreconstruction for continuous time coupled oscillatornetworks <cit.>, for evolutionary gamedynamics on networks <cit.>, for detecting hiddennodes <cit.>, for predicting and controllingsynchronization dynamics <cit.>,for reconstructing spreading dynamics based on binary data <cit.>,and for reconstructing complex geospatial networks through estimatingthe time delays <cit.>. In this Section we shall themethodologies and the main results. §.§ Reconstruction of coupled oscillator networksWe describe a compressive sensing based framework that enables a full reconstruction of coupled oscillator networks whose vector fields consist of a limited number of terms in some suitable base of expansion <cit.>. There are two facts that justify the use of compressive sensing. First, complex networks in the real world are typically sparse <cit.>. Second,the mathematical functions determining the dynamical couplings in aphysical network can be expressed by power-series expansions. The taskis then to estimate all the nonzero coefficients. Since the underlyingcoupling functions are unknown, the power series can contain high-order terms. The number of coefficients to be estimated can therefore be quite large. However, the number of nonzero coefficients may be onlya few so that the vector of coefficients is effectively sparse. Because the network structure as well as the dynamical and coupling functions are sparse, compressive sensing stands out as a feasible framework for full reconstruction of the network topology and dynamics.A complex oscillator network can be viewed as a high-dimensional dynamicalsystem that generates oscillatory time series at various nodes. The dynamics of a node can be written asẋ_i =F_i( x_i) + ∑_j=1,j≠ i^N𝒞_ij· ( x_j -x_i), (i=1,⋯ ,N),where x_i ∈ R^m represents the set of externally accessible dynamical variables of node i, N is the number of accessible nodes, and 𝒞_ij is the coupling matrix between the dynamical variables at nodes i and j denoted by𝒞_ij =( [ c_ij^1,1 c_ij^1,2⋯ c_ij^1,m; c_ij^2,1 c_ij^2,2⋯ c_ij^2,m;⋯⋯⋯⋯; c_ij^m,1 c_ij^m,2⋯ c_ij^m,m;]).In 𝒞_ij, the superscripts kl (k,l=1,2,...,m) stand for the coupling from the kth component of the dynamical variable at node i to the lth component of the dynamical variable at node j. For any two nodes, the number of possible coupling terms is m^2. If there is at least one nonzero element in the matrix 𝒞_ij, nodes i and j are coupled and, as a result, there is a link (or an edge) between them in the network. Generally, more than one element in 𝒞_ij can be nonzero. Likewise, if all the elements of 𝒞_ij are zero, there is no coupling between nodes i and j. The connecting topology and the interaction strengths among various nodes of the network can be predicted if we can reconstruct all the coupling matrices 𝒞_ij from time-series measurements.Generally, the compressive sensing based method consists of thefollowing two steps. First, one rewrites Eq. (<ref>) asẋ_i = [ F_i( x_i) - ∑_j=1,j≠ i^N𝒞_ij· x_i ]+∑_j=1,j≠ i^N𝒞_ij· x_j,where the first term in the right-hand side is exclusively a function of x_i, while the second term is a function of variables of other nodes (couplings). We define the first term to be f_i( x_i), which is unknown. In general, the kth component of f_i( x_i) can be represented by a power series of order up to n:[f_i( x_i)]_k ≡[ F_i( x_i) - ∑_j=1,j≠ i^N𝒞_ij· x_i ]_k = ∑_l_1=0^n∑_l_2=0^n⋯∑_l_m=0^n [(α_i)_k]_l_1,⋯,l_m [( x_i)_1]^l_1 [( x_i)_2]^l_2⋯ [( x_i)_m]^l_m,where ( x_i)_k (k=1,⋯, m) is the kth component of the dynamical variable at node i, the total number of products is(1+n)^m, and [(α_i)_k]_l_1,⋯,l_m∈ R^m is thecoefficient scalar of each product term, which is to be determined frommeasurements as well. Note that the terms in Eq. (<ref>) are allpossible products of different components with different power of exponents. As an example, for m=2 (the components are x and y) and n=2, the power series expansion is α_0,0 + α_1,0x + α_0,1y + α_2,0x^2 + α_0,2y^2 + α_1,1xy +α_2,1x^2y + α_1,2 xy^2 + α_2,2x^2y^2.Second, one rewrites Eq. (<ref>) asẋ_i =f_i( x_i) + 𝒞_i1· x_1 + 𝒞_i2· x_2 + ⋯ + 𝒞_iN· x_NThe goal is to estimate the various coupling matrices C_ij(j=1,⋯, i-1,i+1,⋯,N) and the coefficients of f_i( x_i) from sparse time-series measurements. According to the compressive sensing theory, to reconstruct the coefficients of Eq. (<ref>) from a small number of measurements, most coefficients should be zero - the sparse signal requirement. To include as many coupling forms as possible,one expands each term 𝒞_ij x_j in Eq. (<ref>)as a power series in the same form of f_i( x_i) butwith different coefficients:ẋ_i =f_1( x_1) +f_2( x_2)+ ⋯ +f_N( x_N).This setting not only includes many possible coupling forms but alsoensures that the sparsity condition is satisfied so that the predictionproblem can be formulated in the compressive-sensing framework. For an arbitrary node i, information about node-to-node coupling, or about the network connectivity, is contained completely inf_j(j≠ i). For example, if in the equation of i, a term in f_j(j≠ i) is not zero, there then exists coupling between i and j with the strength given by the coefficient of the term. Subtracting the coupling terms -∑_j=1,j≠ i^N𝒞_ij· x_i fromf_i in Eq. (<ref>), which is the sum of coupling coefficients of all f_j (j≠ i), the nodal dynamics F_i( x_i) can be obtained. That is, once the coefficients of Eq. (<ref>) have been determined, the node dynamics and couplings among the nodes are all known.The formulation of the method can be understood in a more detailed andconcrete manner by focusing on one component of the dynamical variableat all nodes in the network, say component 1. (Procedures for othercomponents are similar.) For each node, one first expands the corresponding component of the vector field into a power series up to power n. For a given node, due to the interaction between this component and other (m-1) components of the vector field, there are (n+1)^m terms in the power series. The number of coefficients to be determined for each individual nodal dynamics is thus (n+1)^m. Now consider a specific node, say node i. For every other node in the network, possible couplings from node i indicates the need to estimate another set of (n+1)^m power-series coefficients in the functions of 𝐟_j(𝐱_j).There are in total N(n+1)^m coefficients that need to be determined.The vector 𝐚 to be determined in the compressive sensingframework contains then N(n+1)^m components. For example, to constructthe measurement vector 𝐗 and the matrix 𝒢 for the case of m = 3 (dynamical variables x, y, and z) and n = 3, one obtains the following explicit dynamical equation for the first component of the dynamical variable of node i:Γ_i(x_i) = (a_i)_000· x_i^0y_i^0z_i^0 + ⋯ + (a_i)_003· x_i^0y_i^0z_i^3 + (a_i)_010· x_i^0y_i^1z_i^0 + ⋯ +(a_i)_100· x_i^1y_i^0z_i^0 + ⋯+(a_i)_333· x_i^3y_i^3z_i^3.We can denote the coefficients of Γ_i(x_i) by a_i = [(a_i)_000,(a_i)_001,⋯ ,(a_i)_333]^T. Assuming that measurements x_i(t) (i=1,…,N) at a set of time t_1,t_2,…,t_M are available, one denotesg_i(t) = [ x_i(t)^0y_i(t)^0z_i(t)^0, x_i(t)^0y_i(t)^0z_i(t)^1, ⋯, x_i(t)^3y_i(t)^3z_i(t)^3 ],such that Γ_i[x_i(t)]=g_i(t) · a_i. According to Eq. (<ref>), the measurement vector can be chosen as 𝐗 =[ẋ_i(t_1),ẋ_i(t_2), ⋯,ẋ_i(t_M)]^T, which can be calculated from time series. Finally, one obtains the following equation in the form 𝐗 = 𝒢·𝐚:( [ ẋ_i(t_1); ẋ_i(t_2);⋮; ẋ_i(t_M);] ) =( [ g_1(t_1) g_2(t_1)⋯ g_N(t_1); g_1(t_2) g_2(t_2)⋯ g_N(t_2);⋮⋮⋮⋮; g_1(t_M) g_2(t_M)⋯ g_N(t_M);] ) ·( [ a_1; a_2; ⋮; a_N; ]).To ensure the restricted isometry property <cit.>, one can normalize the coefficient vector by dividing the elements in each column by the L_2 norm of that column: (𝒢)_ij = (𝒢)_ij/L_2(j) with L_2(j) =√(∑_i=1^M [(𝒢)_ij]^2). After 𝐚 is determined via some standard compressive-sensing algorithm, the coefficients are given by 𝐚/L_2. To determine the set of power-series coefficients corresponding to a different component of the dynamical variable, say component 2, one simply replaces the measurement vector by 𝐗 = [ẏ_̇i̇(t_1),ẏ_̇i̇(t_2), ⋯,ẏ_̇i̇(t_M)]^T and use the same matrix 𝒢. This way all coefficients can be estimated. After the equations of all components of i are determined, one can repeat this process for all other nodes to reconstruct the whole system.The working of the method was illustrated by using networks ofcoupled chaotic Lorenz and Rössler oscillators asexamples <cit.>. The classical Lorenz and Rössler systems are given by [ẋ,ẏ,ż] = 10 (y-x), x(28 - z)-y, xy - (8/3)z] and [ẋ,ẏ,ż] = -y-z, x + 0.2y, 0.2 + z(x - 5.7)], respectively. Since m = 3, the power series of x, y and z can be chosen such that l_1 + l_2 + l_3 ≤ 3. The total number of the coefficients to be estimated is then N∑_i=1^3 (i+1)(i+2)/2+1=19N+1, where i=l_1 + l_2 + l_3 ranges from 1 to 3.Random and scale-free network topologies were studied <cit.>.In particular, the Lorenz oscillator network was chosen to be a Erdős-Rényi type of homogeneous random network <cit.>, generated by assuming a small probability of link for any pair of nodes. The coupling between nodal dynamics wasassumed to occur between the y and the z variables in the Lorenz equations, leading to the following coupling matrix: c_ij^3,2 = 1 if nodes i and j are connected and c_ij^3,2 = 0 otherwise. The Rössler oscillator network wasassumed to be a Barabási-Albert type of scale-free network <cit.>with a heterogeneous degree distribution. The coupling scheme is c_ij^1,3 = 1 for link between i and j. Both types of network structures are illustrated schematically in Fig. <ref>. Time series were generated by integrating the whole networked system with time step h = 10^-4 for 6× 10^6 steps. However, the number of “measured” data points required for the method to be successful can be orders of magnitude less than 6× 10^6, a fundamental advantage of compressive-sensing method. Specifically, random measurements were collected from the integrated time series and the number of elements in each row of the matrix 𝐆 is given by N(n+1)^m.Figure <ref> shows some representative results. For the random Lorenz network, the inferred coefficients were shown of node #2 associated with both the couplings with other nodes [Fig. <ref>(a)] and those with its own dynamics [Fig. <ref>(a)]. The term index is arranged from low to high values, corresponding to the order from low to high node number. The predicted coupling strengths between node #2 and others are shown in Fig. <ref>(a), where each term according to its index corresponds to a specific node. Nonzero terms belonging to nodes other than node #2 indicate inter-node couplings. The predicted interactions with nonzero coefficients (the value is essentially unity) are in agreement with the neighbors of node #2 in the sample random network in Fig. <ref>(b). The term 32 related to -6y is the coupling strength from node #2, which equals the sum of the coupling strengths from the other connected nodes. Figure <ref>(c) displays the inferred coefficients for both nodal dynamics and coupling terms in the three components x, y and z. All predicted terms with nonzero coefficients are in agreement with those in the equations of the dynamics of node #2, together with the inter-node coupling terms c^3,2.Figure <ref>(d) shows the predicted links between node #48 and others in a Rössler oscillator network with a scale-free structure. All existing couplings have been accurately inferred, as compared to the structure presented in Fig. <ref>(e), even though the interaction patterns among nodes are heterogeneous. Both the detected local dynamical and coupling terms associated with node #48 are indicated in Fig. <ref>(f), where in the x component, the term -8z is the combination of the local-dynamical term -z and the coupling of node #48 with 7 neighboring nodes. Since all the couplings have been successfully detected, the local-dynamical term -z in the x component can be separated from the combination so that all terms of node #48 are predicted. Similar results were obtained for all other nodes, leading to a complete andaccurate reconstruction of the underlying complex networkedsystem <cit.>.A performance analysis was carried out <cit.>, with the result that the number of required data points is much smaller thanthe number of terms in the power series function, a main advantage of the compressive-sensing technique. Insofar as the number of data points exceeds a critical value, the prediction errors are effectively zero, indicating the robustness of the reconstruction. One empirical observation was that, if thesampling frequency is high, the number of data points is not able tocover the dynamics in the whole phase space. In order to obtain a faithful prediction of the whole system, the sampling frequency must be sufficientlylow. Another important question was how the structural properties of the network affect the prediction precision. A calculation <cit.>of the dependence of the prediction error on the average degree⟨ k⟩ and the network size N indicated that, regardless of the network size, insofar as the network connections are sparse, the prediction errors remain to be quite small, providingfurther support for the robustness of the compressive-sensing based method.With respect to reconstruction of the network topology, computationsdemonstrated <cit.> that, despite the small prediction errors, all existing links in the original network can be predicted extremely reliably. The performance of the method forpredicting network structures can be quantified through the success rates for existing links (SREL) and nonexistent links (SRNL), defined to be the ratio between the number of successfully predicted links and total number of links and the ratio of the number of correctly predicted nonexistent links to the total number of nonexistent links, respectively. Figure <ref> shows the success rates versus the number of data points and t for both random Lorenz and Rössler oscillator networks. It can be seen that, when the number of data points and t are sufficiently large, both SREL and SRNL reach 100%. The inset in the upper-right panel shows the distribution of the coupling strengths in the Rössler network. For all tested numbers of data points (>54), there exist two sharp peaks centered at c=0 and c=1, corresponding to the absence of coupling and the existing coupling of strength 1.0, respectively. The narrowness of the two peaks in the distribution makes it feasible to distinguish existing links (with nonzero coupling strength) from nonexistent links (effectively with zero coupling). This provides an explanation for the 100% success rates shown in Fig. <ref>.The method was also tested <cit.> on a number of real-worldnetworks, ranging from social to biological and technological networks, where the nodal dynamics were assumed to be of the Lorenz and Rössler types. For five real-world networks, the prediction errors are shown in Table <ref>. It can be seen that all errors are small. §.§ Reconstruction of complex networks with evolutionary-game dynamicsMany complex dynamical systems in biology, social science, and economics can be mathematically modeled by evolutionary games <cit.>. For example, in recent years evolutionary game models played an important role in addressing thebiodiversity problem through microscopic modeling and the mechanism of species competition and coexistence at the level of individual interactions <cit.>. It was demonstrated <cit.> that compressive sensing can be exploited to reconstruct the full topology of the underlyingnetwork based on evolutionary game data. In particular, in a typicalgame, agents use different strategies in order to gain the maximumpayoff. Generally, the strategies can be divided into two types:cooperation and defection. It was shown <cit.> that, even whenthe available information about each agent's strategy and payoff islimited, the compressive-sensing based method can yield precise knowledgeabout the node-to-node interaction patterns in a highly efficient manner. The basic principle was further demonstrated by using an actual social experiment in which participants forming a friendship network played atypical game to generate short sequences of strategy and payoff data.In an evolutionary game, at any time a player can choose one of two strategies S: cooperation (C) or defection (D), which can be expressed as 𝐒(C) = (1, 0)^T and 𝐒(D) = (0, 1)^T. The payoffs of the two players in a game are determined by their strategies and the payoff matrix of the specific game. For example, for the prisoner's dilemma game (PDG) <cit.> and the snowdrift games (SG) <cit.>, the payoff matrices are𝒫_PDG =( [ 1 0; b 0; ]) or 𝒫_SG =( [ 1 1-r; 1+r 0; ]),respectively, where b (1<b<2) and r (0<r<1) are parameters characterizing the temptation to defect. When a defector encounters a cooperator, the defector gains payoff b in the PDG and payoff 1+r in the SG, but the cooperator gains the sucker payoff 0 in the PDG and payoff 1-r in the SG. At each time step, all agents play the game with their neighbors and gain payoffs. For agent i, the payoff isP_i = ∑_j ∈Γ_i𝐒_i^T ·𝒫·𝐒_j,where 𝐒_i and 𝐒_j denote the strategies of agents i and j at the time and the sum is over the neighboring set Γ_i of i. After obtaining its payoff, an agent updates its strategy according to its own and neighbors' payoffs, attempting to maximize its payoff at the next round. Possible mathematical rules to capture an agent's decision making process include the best-take-over rule <cit.>, the Fermiequation <cit.>, and payoff-difference-determined updating probability <cit.>. In the computational study of evolutionary game dynamics, the Fermi rule has been commonly used,which is defined, as follows. After a player i randomly chooses a neighbor j, i adopts j's status 𝐒_j with the probability <cit.>:W(𝐒_i←𝐒_j)=1/1+exp[(P_i-P_j)/κ],where κ characterizes the stochastic uncertainties in the game dynamics. For example, κ=0 corresponds to absolute rationality where the probability is zero if P_j < P_i and one if P_i < P_j, and κ→∞ corresponds to completely random decision. The probability W thus characterizes the bounded rationality of agents in society and the natural selection based on relative fitness in evolution.The key to solving the network-reconstruction problem lies in the relationship between agents' payoffs and strategies. The interactions among agents in the network can be characterized by an N × N adjacency matrix 𝒜 with elements a_ij=1 if agents i and j are connected and a_ij=0 otherwise. The payoff of agent x can be expressed byP_x(t) = a_x1𝐒_x^T(t) ·𝒫·𝐒_1(t) + ⋯+ a_x,x-1𝐒_x^T(t)·𝒫·𝐒_x-1(t) + a_x,x+1𝐒_x^T(t) ·𝒫·𝐒_x+1(t)+ ⋯ + a_xN𝐒_x^T(t) ·𝒫·𝐒_N(t),where a_xi (i=1, ⋯, x-1, x+1, ⋯, N) represents a possible connection between agent x and its neighbor i, a_xi𝐒_x^T(t) ·𝒫·𝐒_i(t) (i=1, ⋯, x-1, x+1, ⋯, N) stands for the possible payoff of agent x from the game with i (if there is no connection between x and i, the payoff is zero because a_xi=0), and t=1, ⋯, m is the number of round that all agents play the game with their neighbors. This relation provides a base to construct the vector 𝐏_x and matrix 𝒢_x in a proper compressive-sensing framework to obtain solution of the neighboring vector 𝐀_x of agent x. In particular, one can write𝐏_x=(P_x(t_1),P_x(t_2),⋯,P_x(t_m))^T,𝐀_x=(a_x1,⋯ , a_x,x-1,a_x,x+1, ⋯ , a_xN)^T,and 𝒢_x =( [F_x1(t_1)⋯ F_x,x-1(t_1) F_x,x+1(t_1)⋯F_xN(t_1);F_x1(t_2)⋯ F_x,x-1(t_2) F_x,x+1(t_2)⋯F_xN(t_2);⋮⋯⋮⋮⋮⋮;F_x1(t_m)⋯ F_x,x-1(t_m) F_x,x+1(t_m)⋯F_xN(t_m);]),where F_xy(t_i) = 𝐒_x^T(t_i) ·𝒫·𝐒_y(t_i). The vectors 𝐏_x, 𝐀_x and matrix 𝒢_x satisfy𝐏_x = 𝒢_x ·𝐀_x,where 𝐀_x is sparse due to the sparsity of the underlying complex network, making the compressive-sensing framework applicable. Since 𝐒_x^T(t_i) and 𝐒_y(t_i) in F_xy(t_i) come from data and 𝒫 is known, the vector 𝐏_x can be obtained directly while the matrix 𝒢_x can be calculated from the strategy and payoff data. The vector 𝐀_x can thus be predicted based solely on the time series. Since the self-interaction term a_xx is not included in the vector 𝐀_x and the self-column [F_xx(t_1),⋯ , F_xx(t_m)]^T is excluded from the matrix 𝒢_x, the computation required for compressive sensing can be reduced. In a similar fashion, the neighboring vectors of all other agents can be predicted, yielding the network adjacency matrix 𝒜 = (𝐀_1,𝐀_2,⋯ , 𝐀_N).The method was first tested <cit.> by implementing PDGand SG on three types of standard complex networks: random <cit.>,small-world <cit.> and scale-free <cit.>. In particular, time series of strategies and payoffs were recorded during the system'sevolution towards the steady state, which were used for uncovering theinteraction network topology. The performance of the method can bequantified in terms of the amount of required measurements fordifferent game types and network structures through the success ratesof existent links (SREL) and non-existent links (SRNL). If thepredicted value of an element of the adjacency matrix 𝒜 is closeto 1, the corresponding link can be deemed to exist. If the valueis close to zero, the prediction is that there is no link. In practice,a small threshold can be assigned, e.g., 0.1, so that the ranges ofthe existent and non-existent links are 1± 0.1 and 0± 0.1,respectively. Any value outside the two intervals is regarded as aprediction failure. For a single player, SREL is defined as theratio of the number of successfully predicted neighboring links tothe number of actual neighbors, and SRNL is similarly defined. Averagingover all nodes leads to the values of SREL and SRNL for the entire network. The reason for treating the success rates for existent and non-existent links separately lies in the sparsity of the underlying complex network, where the number of non-existent links is usually much larger than the number of existent links. The choice of the threshold does not affect the values of the success rates, insofar as it is not too close to one, nor too close to zero.The success rates of prediction for two types of games and three types of network topologies are shown in Fig. <ref>. The length of the time series is represented by the number of measurements collected during the temporal evolution normalized by the number N of agents, e.g., the value of one means that the number of used measurements equals N. For all combinations of game dynamics and network topologies examined, perfect success rate can be achieved with extremely low amount of data. For example, for random and small-world networks, the length of data required for achieving 100% success rate is between 0.3 and 0.4. This value is slightly larger (about 0.5) for scale-free networks, due to the presence of hubs whose connections are much denser than most nodes, although their neighboring vectors are still sparse. Figure <ref> thus demonstrates that the method is both accurate and efficient. The exceptionally low data requirement is particularly important for situations where only rare information is available. From this standpoint, evolutionary games are suitable to simulate such situations as meaningful data can be collected only during the transient phase before the system reaches its steady state, and game dynamics are typically fast convergent so that the transients are short. In addition, the robustness of the method was tested in situations wherethe time series are contaminated by noise <cit.>. For example, the case was studied where random noise of amplitude up to 30% is added to the payoffs of PDG. When the size of the data exceeds 0.4, the success rate approaches 100% for random networks. Similar performance was achieved for small-world and scale-free networks. The noise immunity embedded in the method is not surprising, as compressive sensing represents an optimization scheme that is fundamentally resilient to noise. In contrast, another type of noise, noise κ in the strategy updating process, plays a positive role in network reconstruction, because of the fact that this kind of noise can increase the relaxation time towards one of the absorbing states (all C or all D), thereby providing more information for successful reconstruction.The method can be generalized straightforwardly to weighted networkswith inhomogeneous node-to-node interactions. Using weights tocharacterize various interaction strengths, the elements of the weightedadjacency matrix 𝒲 can be defined asW_ij={ [ w_0 > 1, if i connects to j; 0, otherwise.;] .In the context of evolutionary games on networks, the weight W_ij characterizes the situation of aggregate investment. In particular, for both players, more investments in general will lead to more payoffs. Given the link weights, the weighted payoff P_i^w of an arbitrary individual is given byP_i^w = ∑_j ∈Γ_iw_ij𝐒_i^T ·𝒫·𝐒_j,where Γ_i denotes the neighboring set of i. With the evolutionary-game dynamics, the weighted network structure is taken into account by the weighted payoff P_i^w. To uncover such a network from data, the weighted payoff vector 𝐏_x^w, matrix 𝒢_x, and the weighted neighboring vector 𝐖_x for an arbitrary individual x are needed. The vectors 𝐏_x^w and 𝐖_x are given by𝐏_x^w = (P_x^w(t_1),P_x^w(t_2),⋯,P_x^w(t_m))^T,𝐖_x = (W_x1,⋯,W_x,x-1,W_x,x+1,⋯,W_xN)^T.Similar to unweighted networks, one has𝐏_x^w = 𝒢_x ·𝐖_x,where 𝐖_x can be calculated from the strategy and payoff data. The prediction accuracy can be conveniently characterized by various prediction errors, which are defined separately for link weights and non-existent links with zero weight. In particular, the relative error of a link weight is defined as the ratio of the absolute difference between the predicted weight and the true weight to the latter. The average error over all link weights is the prediction error E_w. However, a relative error for a zero (non-existent) weighted link cannot be defined, so it is necessary to use the absolute error E_z.Figure <ref> shows the prediction errors for PDG dynamics on a scale-free network with random link weights chosen uniformly from the interval [1.0,6.0]. It can be seen that the prediction errors decrease fast as the number of measurements is increased. As the relative data size exceeds about 0.4, the two types of prediction errors approach essentially zero, indicating that all link weights have been successfully predicted without failure and redundancy, despite that the link weights are random. Random andsmall-world networks were also tested with the finding that, to achieve the same level of accuracy, the requirement for data can be somewhat relaxed as compared with scale-free networks.The compressive sensing based reconstruction framework can be applied to real social networks. Especially, an experimental was reported in Ref. <cit.>, where 22 participants from Arizona State University played PDG together iteratively and, at each round each player was allowed to change his/her strategies to optimize the payoff. The payoff parameter is set (arbitrarily) to be b = 1.2. The player who had the highest normalized payoff (original payoff divided by the number of neighbors) summed over time was the winner and rewarded. During the experiment, each player was allowed to communicate only with his/her direct neighbors for strategy updating. Prior to experiment, there was a social tie (link) between two players if they had already been acquainted to each other; otherwise there was no link. Among the 22 players, two withdrew before the experiment was completed, so they were treated as isolated nodes. The network structure is illustrated in Fig. <ref>(a). It exhibits typical features of social networks, such as the appearance of a large density of triangles and a core consisting of 4 players (nodes 5, 11, 13, and 16), which is fully connected within and has more links than other nodes in the network. The core essentially consists of players who were responsible for recruiting other players to participate in the experiment. Each of the 20 players who completed the experiment played 31 rounds of games, and he/she recorded his/her own strategy and payoff at each time, which represented the available data base for prediction. The data used for each prediction run was randomly picked from this data base. The pre-existed friendship ties among the participants tend to favor cooperation and preclude the system from being trapped in the social dilemma, due to the relatively short data streams. However, for a long run, a full defection state may occur. In this sense, the recorded data were taken during the transient dynamical phase and were thus suitable for network reconstruction. The results are shown in Fig. <ref>(b). It can be seen that thesocial network was successfully uncovered, despite the complicated process of individual's decision making during the experiment. Compared to the simulation results, larger data set (about 0.6) is needed for a perfect prediction of social ties. This can be attributed to the relative smaller size and denser connections in the social network than in model networks.An interesting phenomenon is that the winner picked in terms of the normalized payoff had only two neighbors, in contrast to the players with the largest node degree, whose normalized payoffs were approximately at the average level, as shown in Fig. <ref>(c). In addition, the payoffs of players of smaller degrees were highly non-uniform, while those of higher degrees showed smaller difference. This suggests that players of high degree may not act as leaders due to their average normalized payoffs. This experimental finding was in striking agreement with numerical predictions in literature about the relationship between individuals' normalized payoffs and their node degrees <cit.>.It was also observed from experimental data that a typical player with a large number of neighbors failed to stimulate their neighbors to follow his/her strategies, suggesting that hubs may not be as influential in social networks. However, this finding should not be interpreted as a counter-example to the leader's role in evolutionary games <cit.>, since the network based on friendship may violate the absolute selfish assumption of players who tend to be reciprocal with each other.The method, besides being fully applicable to complex networks governed by evolutionary-game type of interactions, can be applied to other contexts where the dynamical processes are discrete in time and the amount of available data is small. For example, inferring gene regulatory networks from sparse experimental data is a problem of paramount importance in systems biology <cit.>. For such an application, Eq. (<ref>) shouldbe replaced by the Hill equation, which models generic interactions amonggenes. In an expansion using base functions specifically suited for generegulatory interactions, a compressive-sensing framework may beestablished. The underlying reverse-engineering problem can then besolved. A challenge that must be overcome is to represent the Hillfunction by an appropriate mathematical expansion so that the sparsityrequirement for compressive sensing can be met. §.§ Detection of hidden nodes in complex networks §.§.§ Principle of detecting hidden nodes based on compressive sensingWhen dealing with an unknown complex networked system that has a largenumber of interacting components organized hierarchically, curiosity demands that we ask the following question: are there hidden objects that are not accessible from the external world? The problem of inferring the existence of hidden objects from observations is quite challenging but it has significant applications in many disciplines of science and engineering. Here the meaning of “hidden” is that no direct observation of or information about the object is available so that it appears to the outside world as a black box. However, due to the interactions between the hidden object and other observable components in the system, it may be possible to utilize “indirect” information to infer the existence of the hidden object and to locate its position with respect to objects that can be observed. The difficulty to develop effective solutions is compounded by the fact that the indirect information on which any method of detecting hidden objects relies can be subtle and sensitive to changes in the system or in the environment. In particular, in realistic situations noise and random disturbances are present. It is conceivable that the “indirect” information can be mixed up with that due to noise or be severely contaminated. The presence of noise thus poses a serious challenge to detecting hidden nodes, and some effective “noise-mitigation” method must be developed.One approach to addressing the problem of detecting a hiddennode <cit.> was based on compressivesensing. The basic principle principle is that the existenceof a hidden node typically leads to “anomalies”in the quantities that can be calculated or deduced from observation.Simultaneously, noise, especially local random disturbances applied atthe nodal level, can also lead to large variance in these quantities. This is so because, a hidden node is typically connected to a few nodes in the network that are accessible to the external world, and a noise source acting on a particular node in the network may also be regarded as some kind of hidden object. Thus, the key to any detection methodology is to identify and distinguish the effects of hidden nodes on detection measure from those due to local noise sources.The key issues associated with detection of hidden nodes can be explained in a concrete manner by using the network setting shown schematically in Fig. <ref>, where there are 20 nodes, the couplings among the nodes are weighted, and the entire network is in a noisy environment, but a number of nodes also receive relatively strong random driving, e.g., nodes 7, 11, and 14.Assume an oscillator network so that the nodal dynamics are described by nonlinear differential equations, and that time series can be measured simultaneously from all nodes in the network except one, labeled as #20, which is a hidden node. The task of ascertaining the presence and locating the position of the hidden node are equivalent to identifying its immediate neighbors, which are nodes #3 and #7 in Fig. <ref>. Note that, in order to be able to detect the hidden node based on information from its neighboring nodes, the interactions between the hidden node and its neighbors must be directional from the former to the latter or be bidirectional. Otherwise, if the coupling is solely from the neighbors to the hidden node, the dynamics of the neighboring nodes will not be affected by the hidden node and, consequently, time series from the neighboring nodes will contain absolutely no information about the hidden node, rendering it undetectable. The action of local noise source on a node is naturally directional, i.e., from the source to the node.In Ref. <cit.>, it was demonstrated that, when the compressive-sensing paradigm is applied to uncovering the network topology <cit.>, the predicted linkages associated with nodes #3 and #7 are typically anomalously dense, and this piece of information is basically what is needed to identify them as the neighboring nodes of the hidden node. In addition, when differentsegments of the measurement data are used to reconstruct the couplingweights for these two nodes, the reconstructedexhibit significantly larger variances than thoseassociated with other nodes. However, the predicted linkages associatedwith the nodes driven by local noise sources can exhibit behaviorssimilar to those due to the hidden nodes, leading to uncertainty inthe detection of the hidden node. This issue is critical to developingalgorithms for real-world applications. One possible solution <cit.> was to exploit the principle of differential signal to investigatethe behavior of the predicted link weights as a function of the dataused in the reconstruction. Due to the advantage of compressive sensing,the required data amount can be quite small and, hence, even if themethod requires systematic increase in the data amount, it would stillbe reasonably small. It was argued that demonstrated <cit.>that, when the various ratios of the predicted weights associated withall pairs of links between the possible neighboring nodes and the hiddennode are examined, those associated with the hidden nodes and nodes understrong local noise show characteristically distinct behaviors, renderingunambiguous identification of the neighboring nodes of the hidden node.Any such ratio is essentially a kind of differential signal, becauseit is defined with respect to a pair of edges.§.§.§ Mathematical formulation of compressive sensing based detection of a hidden node *Compressive-sensing based method to uncover weightednetwork dynamics and topology. Consider the typical setting of a complex network of N coupled oscillators in a noisy environment. The dynamics of each individual node, when it is isolated from other nodes, can be described as 𝐱̇_i=𝐅_i(𝐱_i) + ξη_i, where 𝐱_i∈ℝ^m is the vector of state variables, η_i are an m-dimensional vector whose entries are independent Gaussian random variables of zero mean and unit variance, and ξ denotes the noise amplitude. A weighted network can be described by𝐱̇_i=𝐅_i(𝐱_i)+ ∑_j=1,j≠ i^N𝒲_ij· [𝐇(𝐱_j) -𝐇(𝐱_i)] + ξη_i,where 𝒲_ij∈ℝ^m × m is the coupling matrix between node i and node j, and 𝐇 is the coupling function.Defining𝐅_i^'(𝐱_i) ≡𝐅_i(𝐱_i)- 𝐇(𝐱_i) ·∑_j=1,j≠i^N𝒲_ij,one has𝐱̇_i=𝐅_i^'(𝐱_i)+ ∑_j=1,j≠ i^N𝒲_ij·𝐇(𝐱_j) +ξη_i,i.e., all terms directly associated with node i have been grouped into 𝐅_i^'(𝐱_i). One can then expand 𝐅^'(𝐱_i) into the following form:𝐅_i^'(𝐱_i)=∑_γ𝐚̃_i^(γ)·𝐠̃_i^(γ)(𝐱_i),where 𝐠̃_i^(γ)(𝐱_i) are a set of orthogonal and complete base functions chosen such that the coefficients 𝐚̃_i^(γ) are sparse. While the coupling function 𝐇(𝐱_i) can be expanded in a similar manner, for simplicity we assume that they arelinear: 𝐇(𝐱_i)=𝐱_i. This leads toẋ_i=∑_γ𝐚̃_i^(γ)·𝐠̃_i^(γ)(𝐱_i)+ ∑_j=1,j≠ i^N𝒲_ij·𝐱_j+ξη_i,where all the coefficients 𝐚̃_i^(γ) and weights 𝒲_ij need to be determined from time series 𝐱_i. In particular, the coefficient vector 𝐚̃_i^(γ) determines the nodal dynamics and the weighted matrices 𝒲_ij's give the full topology and coupling strength of the entire network.Suppose simultaneous measurements are available of all state variables 𝐱_i(t) and 𝐱_i(t+δ t) at M different uniform instants of time at interval Δ t apart, whereδ t ≪Δ t so that the derivative vector ẋ_ican be estimated at each time instant. Equation (<ref>)for all M time instants can then be written in a matrix form with thefollowing measurement matrix:𝒢_i=( [ 𝐠̃_i(t_1)𝐱_1(t_1) ⋯𝐱_k(t_1) ⋯𝐱_N(t_1); 𝐠̃_i(t_2)𝐱_1(t_2) ⋯𝐱_k(t_2) ⋯𝐱_N(t_2); ⋮ ⋮ ⋯ ⋮ ⋯ ⋮; 𝐠̃_i(t_M)𝐱_1(t_M) ⋯𝐱_k(t_M) ⋯𝐱_N(t_M) ]),where the index k in 𝐱_k(t) runs from 1 to N, k ≠ i, and each row of the matrix is determined by the available time series at one instant of time. The derivatives at different time can be written in a vector form as 𝐗_i= [𝐱̇_i(t_1), ⋯, 𝐱̇_i(t_M)]^T, and the coefficients from the functional expansion and the weights associated with all links in the network can be combined concisely into a vector𝐚_i as𝐚_i=[𝐚̃_i,𝐖_1i,⋯, 𝐖_i-1,i,𝐖_i+1,i,⋯,𝐖_N,i]^T,where [·]^T denotes the transpose. For a properly chosen expansionbase and a general complex network whose connections are sparse,the vector 𝐚_i to be determined is sparse as well. Finally, Eq. (<ref>) can be written as𝐗_i = 𝒢_i·𝐚_i + ξη_i.In the absence of noise or if the noise amplitude is negligibly small, Eq. (<ref>) represents a linear equation but the dimension of the unknown coefficient vector 𝐚_i can be much larger than that of 𝐗_i, and the measurement matrix will have many more columns than rows. In order to fully recover the network of N nodes with each node having m components, it is necessary to solve N × m such equations. Since 𝐚_i is sparse, insofar as its number of non-zero coefficients is smaller than the dimension of 𝐗_i, the vector 𝐚_i can be uniquely and efficiently determined through compressive sensing. *Detection of hidden node. A meaningful solution of Eq. (<ref>) based on compressive sensingrequires that the derivative vector 𝐗_i and the measurementmatrix 𝒢_i be entirely known which, in turn, requires time series from all nodes. In this case, the information available forreconstruction of the complex networked system is deemed to be complete <cit.>. In the presence of a hiddennode, for its immediate neighbors, the available information will not be complete in the sense that some entries of the vector 𝐗_i and the matrix 𝒢_i are unknown. Let h denote the hidden node. For any neighboring node of h, the vector 𝐗_i and the matrix 𝒢_i in Eq. (<ref>) now contain unknown entries at the locations specified by the index h. For any other node not in the immediate neighborhood of h, Eq. (<ref>) is unaffected. When compressive-sensing algorithm is used to solve Eq. (<ref>), there will then be large errors in the solution of the coefficient vector 𝐚_i associated the neighboring nodes of h, regardless of the amount of data used. In general, the so-obtained coefficient vector 𝐚_i will not appear sparse. Instead, most of its entries will not be zero, a manifestation of which is that the node would appear to have links with almost every other node in the network. In contrast, for nodes not in the neighborhood of h, the corresponding errors will be small and can be reduced by increasing the data amount, and the corresponding coefficient vector will be sparse. It is this observation which makes identification of the neighboring nodes of the hidden node possible in a noiseless or weak-noise situation <cit.>.The need and the importance to distinguish the effects of hidden nodefrom these of noise can be better seen by separating the term associated with h in Eq. (<ref>) from those with other accessible nodes in the network. Letting l denote a node in the immediate neighborhood of the hidden node h, we have𝐗_l = 𝒢^'_l·𝐚^'_l + (𝒲_lh·𝐱_h +ξη_l),where 𝒢^'_l is the new measurement matrix that can be constructed from all available time series. While background noise may be weak, the term 𝒲_lh·𝐱_h can in general be large in the sense that it is comparable in magnitude with other similar terms in Eq. (<ref>). Thus, when the network is under strong noise, especially for those nodes that are connected to the neighboring nodes of the hidden node, the effects of hidden node on the solution can be entangled with those due to noise. In addition, if the coupling strength from the hidden node is weak, it would be harder to identify the neighboring nodes. For example, ahidden node in a network with Gaussian weight distribution will be harder to detect, due to the finite probability of the occurrence of near zero weights. When the coupling strength is comparable or smaller than the background noise amplitude, the corresponding link cannot be detected.*Method to distinguish hidden nodes from local noise sources. The basic idea to distinguish the effects of hidden node and of localnoise sources <cit.> is based on the following consideration. Take two neighboring nodes of the hidden node, labeled as i and j. Because the hidden node is a common neighbor of nodes i and j, the couplings from the hidden node should be approximately proportional to each other, with the proportional constant determined by the ratio of their link weights with the hidden node. When the dynamical equations of nodes i and j are properly normalized, the terms due to the hidden node tend to cancel each other, leaving the normalization constant as a single unknown parameter that can be estimated subsequently.This parameter is the cancellation ratio, denoted as Ω_ij. As the data amount is increased, Ω_ij tends to its true value. Practically one then expects to observe a systematic change in theestimated value of the ratio as data used in the compressive-sensingalgorithm is increased from some small to relatively large amount. If only local noise sources are present, the ratio should show no systematic change with the data amount. Thus the distinct behaviors of Ω_ij as the amount of data is increased provides a way to distinguish the hidden node from noise and, at the same time, to ascertain the existence of the hidden node. For simplicity, assume that all coupled oscillators share the same localcoupling configuration and that each oscillator is coupled to any of itsneighbors through one component of the state vector only. Thus, eachrow in the coupling matrix 𝒲_ih associated with a linkbetween node i and h has only one non-zero element. Let p denotethe component of the hidden node coupled to the first component of nodei, the dynamical equation of which can then be written as[ẋ_i]_1 =[∑_γ𝐚̃_i^(γ)·𝐠̃_i^(γ)(𝐱_i)]_1 + [∑_k≠ i, h^N𝒲_ij·𝐱_j]_1+ w^1p_ih· [𝐱_h]_p + ξη_i,where [𝐱_h]_p denotes the time series of the pth component of the hidden node, which is unavailable, and w^1p_ih is the coupling strength between the hidden node and node i. The dynamical equation of the first component of neighboring node j of the hidden node has a similar form. LettingΩ_ij=w_ih^1p/w_jh^1p,be the cancellation ratio, multiplying Ω_ij to the equation ofnode j, and subtracting from it the equation for node i, one obtains[ẋ_i]_1= Ω_ij[ẋ_j]_1+ ∑_γ𝐚̃_i^(γ)·𝐠̃_i^(γ)(𝐱_i)+∑_k≠ i, h w_ik^1p [𝐱_k]_p- Ω_ij∑_γ𝐚̃_j^(γ)·𝐠̃_j^(γ)(𝐱_j) - Ω_ij∑_k≠ j, h w_jk^1p [𝐱_k]_p+(w_ih^1p -Ω_ijw_jh^1p) · [𝐱_h]_p +ξη_i - Ω_ijξη_j.It can be seen that the terms associate with [𝐱_h]_p vanishand all deterministic terms on the left-hand side ofEq. (<ref>) are known, which can then be solved by the compressive-sensing method. From the coefficient vectorso estimated, one can identify the coupling of nodes i and j toother nodes, except for the coupling between themselves since such termshave been absorbed into the nodal dynamics, and the couplings to theircommon neighborhood are degenerate in Eq. (<ref>) andcannot be separated from each other. Effectively, one has combinedthe two nodes together by introducing the cancellation ratio Ω_ij.As a concrete example, consider the situation where each oscillator hasthree independent dynamical variables, named as x, y and z. Forthe nodal and coupling dynamics polynomial expansions of order up ton can be chosen. The x component of the nodal dynamics[𝐅_i^'(𝐱_i)]_x for node i is:[𝐅_i^'(𝐱_i)]_x= ∑_l_x=0^n∑_l_y=0^n∑_l_z=0^n [a_l_xl_yl_z]_x· x_i^l_xy_i^l_yz_i^l_z,and the coupling from other node k to the x component can be written asC_ik^x=w_ik^xx·x_k+w_ik^xy·y_k +w_ik^xz·z_k,where w_ik^xy denotes the coupling weight from the y componentof node k to the x component of node i, and so on. The nodaldynamical terms in the matrix 𝒢_i are[𝐠̃_i]_x=[x^0_iy^0_iz^0_i,x^1_iy^0_iz^0_i, ⋯,x^n_iy^n_iz^n_i],and the corresponding coefficients are [a_l_xl_yl_z]_x. Thevector of coupling weights is [𝒲_ij]_x=[w_ij^xx,w_ij^xy,w_ij^xz].Equation (<ref>) becomes([ẋ_i(t_1);ẋ_i(t_2); ⋮; ẋ_̇i̇(̇ṫ_̇Ṁ)̇ ]) ≈([ẋ_j(t_1) 1 [𝐠̃_i(t_1)]_x [𝐠̃_j(t_1)]_xx_1(t_1) ⋯z_N(t_1);ẋ_j(t_2) 1 [𝐠̃_i(t_2)]_x [𝐠̃_j(t_2)]_xx_1(t_2) ⋯z_N(t_2); ⋮ ⋮ ⋮ ⋮ ⋮ ⋯ ⋮;ẋ_j(t_M) 1 [𝐠̃_i(t_M)]_x [𝐠̃_j(t_M)]_xx_1(t_M) ⋯z_N(t_M); ]) ·[ Ω_ij;c;ã^'_i;-Ω_ij·ã^'_j; w_i1^xx - Ω_ij w_j1^xx;⋮; w_iN^xz - Ω_ij w_jN^xz ], where c is the sum of constant terms from the dynamical equationsof nodes i and j, and ã^'_i is thecoefficient vector to be estimated which excludes all the constants.Using compressive sensing to solve this equation, one can recoverthe cancellation ratio Ω_ij and the equations of node i. When Ω_ij is known the dynamics of node j can be recovered from the coefficient vector -Ω_ij·ã^'_j.In Ref. <cit.>, an analysis and discussions were provided about the possible extension of the method to systems of characteristically different nodal dynamics and/or with multiple hidden nodes. In particular, it was shown that the method can be readily adopted to network systems whose nodal dynamics are not described by continuous-time differential equations but by discrete-time processes such as evolutionary-game dynamics. In such a case, the derivatives used for continuous-time systems can be replaced by the agent payoffs. The cancellation factors can then be calculated from data to differentiate the hidden nodes from local noise sources. It was also shown that, under certain conditions with respect to the coupling patterns between the hidden nodes and their neighboring nodes, the cancellation factors can be estimated even when there are multiple, entangled hidden nodes in the network. §.§.§ Examples of hidden node detection in the presence of noiseThe methodology of hidden node detection in the presence of noise can beillustrated using coupled oscillator networks. (Results fromevolutionary-game dynamical networks can be found in Ref. <cit.>.) As discussed in Sec. <ref>, given such a networked system, one can use compressive sensing to uncoverall the nodal dynamical equations and coupling functions <cit.>.The expansion base needs to be chosen properly so that the number ofnon-zero coefficients is small as compared with the total number N_tof unknown coefficients. All N_t coefficients constitute a coefficientvector to be estimated. The amount of data used can be convenientlycharacterized by R_m, the ratio of the number M of data points usedin the reconstruction, to N_t. The method to differentiate hidden nodes and noise wastested <cit.> using random networks of nonlinear/chaoticoscillators, where the nodal dynamics were chosen to be those fromthe Rössler oscillator <cit.>,[ẋ_i,ẏ_i,ż_i]=[ -y_i-z_i, x_i + 0.2y_i,0.2+z_i(x_i-5.7)],which exhibits a chaotic attractor. The size of the network wasvaried from 20 to 100, and the probability of connection between anytwo nodes is 0.04. The network link weights are equally distributed in[0.1, 0.5] (arbitrary). Background noise of amplitude ξ wasapplied (independently) to every oscillator in the network, withamplitude varying from 10^-4 to 5×10^-3. The noiseamplitude is thus smaller than the average coupling strength of the network. The tolerance parameter ε in thecompressive sensing algorithm can be adjusted in accordance with thenoise amplitude <cit.>. Time series are generated by using the standard Heun's algorithm <cit.> to integrate thestochastic differential equations. To approximate the velocity field,a third-order polynomial expansion was used in the compressive-sensingformulation. (In Ref. <cit.>, more examples can be foundusing network systems of varying sizes, different weight distributionsand topologies, and alternative nodal dynamics.) *Illustration of hidden node detection. Consider the network inFig. <ref>, where only background noise is present andthere are no local noise sources. Linear coupling between any pair of connected nodes is from the z-component to the x-component in the Rössler system. From the available time series (nodes #1-19),the coefficient vector can be solved using compressive sensing. In particular, for node i, the terms associated with couplings from the z-components of other nodes appear in the ith row of the coupling matrix. As shown in Fig. <ref>(a), when the data amount is R_m=0.7, the network's coupling matrix can be predicted. The predicted links and the associated weights are sparse for all nodes except for nodes #3 and #7, the neighbors of the hidden node. While there are small errors in the predicted weights due to background noise, the predicted couplings for the two neighbors of the hidden node, which correspond to the 3rd and the 7th row in the coupling matrix, appear to be from almost all other nodes in the network and some coupling strength is even negative. Such anomalies associated with the predicted coupling patterns of the neighboring nodes of the hidden node cannot be removed by increasing the data amount. Nonetheless, it is precisely these anomalies which hint at the likelihood that these two “abnormal” nodes are connected with a hidden node.While abnormally high connectivity predicted for a node is likelyindication that it belongs to the neighborhood of the hidden node,in complex networks there are hub nodes with abnormally large degrees,especially for scale-free networks <cit.>. In order todistinguish a hidden node's neighboring node from some hub node, one can exploit the variance of the predicted coupling constants, which can be calculated from different segments of the available data sets.Due to the intrinsically low-data requirement associated withcompressive sensing, the calculation of the variance is feasiblebecause any reasonable time series can be broken into a number ofsegments, and prediction can be made from each data segment. For nodesnot in the neighborhood of the hidden node, the variances are smallas the predicted results hardly change when different segments of thetime series are used. However, for the neighboring nodes of the hiddennode, due to lack of complete information needed to construct the measurement matrix, the variance values can be much larger. Figure <ref>(b) shows the variance σ^2 in thepredicted coupling strength for all 19 accessible nodes. It can beseen that the values of the variance for the neighboring nodes of thehidden node, nodes #3 and #7, are at or above the upper dashed lineand are significantly larger than those associated with allother nodes that all fall below the lower dashed line. This indicatesstrongly that they are indeed the neighboring nodes of the hidden node.The gap between the two dashed lines can be taken as a quantitativemeasure of the detectability of the hidden node. The larger the gap,the more reliable it is to distinguish the neighbors of the hidden nodefrom the nodes that not in the neighborhood. The results inFig. <ref> thus indicate that the location of the hiddennode in the network can be reliably inferred in the presenceof weak background noise. The size of the gap, or the hidden-node detectability depends on the system details. A systematic analysis of the detectability measure was done <cit.>, where it was foundthat the variance due to the hidden node is mainly determined by thestrength of its coupling with the accessible nodes in the network.It was also found that system size and network topology have littleeffect on the hidden-node detectability. It is worth emphasizing that the detectability relies also on successful reconstruction of all nodes that are not in the neighborhood of the hidden node, which determine the lower dashed line in Fig. <ref>.The reliability of the reconstruction results can be quantified by investigating how the prediction errors in the link weights of allaccessible nodes, except the predicted neighbors of the hidden node,change with the data amount. For an existent link, one can use thenormalized absolute error E_nz, the error in the estimated weightwith respect to the true one, normalized by the value of the true linkweight. Figure <ref> shows the results for N = 100. Thelink weights are uniformly distributed in the interval [0.1, 0.5] andthe background noise amplitude is ξ=10^-3. The tolerance parameterin the compressive-sensing algorithm was set to be ε=0.5,which is optimal for this noise amplitude. (Details of determining theoptimal tolerance parameter for different values of the background noiseamplitude can be found in Ref. <cit.>.) It can be seen thatfor R_m>0.4, E_nz decreases to the small value of about 0.01,which is determined by the background noise level. As R_m is increasedfurther, the error is bounded by a small value determined by the noiseamplitude, indicating that the reconstruction is robust. Although thevalue of E_nz does not decrease further toward zero due to noise,the prediction results are reliable in the sense that the predictedweights and the real values agree with each other, as shown in the inset of Fig. <ref>, a comparison of the actual and the predicted weights for all existent links. All the predicted resultsare in the vicinities of the corresponding actual values, as indicatedby a heavy concentration of the dots along the diagonal line. The centralregion in the dot distribution has brighter color than the marginal regions, confirming that vast majority of the predicted results are accurate.In Ref. <cit.>, it was further shown that robust reconstructioncan be achieved regardless of the network size, connection topology andweight distributions, insofar as sufficient data are available.The error measure E_nz to characterize the accuracy of the reconstruction is similar to z-scores, or the standard score in statistics, with the minor difference being that the z-scores use the standard derivatives of the distribution to normalize the raw scores, whilethe exact values were used <cit.> in the examples. In realisticapplications the exact values are usually not available, so it is necessaryto use the z-score measure.It should be emphasized that there are two types of “dense” connections:one from reconstruction and another intrinsic to the network. In particular, in the two-dimensional representation of the reconstruction results [e.g.,Fig. <ref>(A)], the neighboring nodes of the hidden node typically appear densely linked to many other nodes in the network. Thesecan be a result of lack of incomplete information (i.e., time series) due to the hidden node (in this case, there is indeed a hidden node), or the intrinsic dense connection pattern associated with, for example, a hub node in a scale-free network. The purpose of examining the variancesof the reconstructed connections from independent data segments is fordistinguishing these two possibilities. Extensivecomputations <cit.> indicated that a combination of the denseconnection and large variance can ascertain the existence of hidden nodewith confidence. *Differentiating hidden node from local noise sources. When strong noise sources are present at certain nodes, the predictedcoupling patterns of the neighboring nodes of these nodes will showanomalies. (Here the meaning of the term “strong” is that theamplitudes of the random disturbances are order-of-magnitude largerthan that of background noise.) The method based on the cancellationratio was demonstrated <cit.> to be effective at distinguishinghidden nodes from local noise sources, insofar as the hidden node has at least two neighboring nodes not subject to such disturbances. To be concrete, consider a network of N = 61 coupled chaotic Rössleroscillators, which has 60 accessible nodes and one hidden node (#61)that is coupled to two neighbors: nodes #14 and #20, as illustratedschematically in Fig. <ref>. Assume a strongnoise source is present at node #54. It was found <cit.>that the reconstructed weights match their true values to high accuracy. It was also found that the reconstructed coefficients including the ratio Ω_ij are all constant and invariant with respect todifferent data segments, a strong indication that the pair of nodes are the neighboring nodes of the same hidden node, thereby confirming its existence.When there are at least two accessible nodes in the neighborhood of the hidden node that are not subject to strong noisy disturbance, such as nodes #14 and #20, as the data amount R_m is increasedtowards 100%, the cancellation ratio should also increase and approachunity. This behavior is shown as the open circles inFig. <ref>(a). However, when a node is driven by a local noisesource, regardless of whether it is in the neighborhood of the hidden node,the cancellation ratio calculated from this node and any other accessiblenode in the network exhibits a characteristically different behavior.Consider, for example, nodes #14 and #54. The reconstructedconnection patterns of these two nodes both show anomalies, as theyappear to be coupled with all other nodes in the network. In contrastto the case where the pair of nodes are influenced by the hidden nodeonly, here the cancellation ratio does not exhibit any appreciable increaseas the data amount is increased, as shown with the crosses inFig. <ref>(a). In addition, the average variance values ofthe predicted coefficient vectors of the two nodes exhibitcharacteristically different behaviors, depending on whether any onenode in the pair is driven by strong noise or not. In particular, forthe node pair #14 and #20, since neither is under strong noise,the average variance will decrease toward zero as R_m approaches unity,as shown in Fig. <ref>(b) (open circles). In contrast, forthe node pair #14 and #54, the average variance will increase withR_m, as shown in Fig. <ref>(b) (crosses). This is because,when one node is under strong random driving, the input to thecompressive-sensing algorithm will be noisy, so its performance willdeteriorate. However, compressive sensing can perform reliably when theinput data are “clean,” even when they are sparse. Increasing the dataamount beyond a threshold is not necessarily helpful, but longer andnoisier data sets can degrade significantly the performance. The resultsin Figs. <ref>(a,b) thus demonstrate that the cancellation ratio between a pair of nodes, in combination with the average variance of the predicted coefficient vectors associated with the two nodes, can effectively distinguish a hidden node from a local noise source. If there are more than one hidden node or there is a cluster of hidden nodes, the procedure to estimate the cancellation factors is similar but requires additional information about the neighboring nodes of the hidden nodes. The cancellation-factor based method can be extended to network systems with nodal dynamics not of the continuous-time type, such as evolutionary-game dynamics <cit.>. *Multiple entangled hidden nodes. When there are multiple hidden nodes in a network, there is a possibility that an identified node is connected to more than one hidden node. For example, node b in all three panels of Fig. <ref> is affected by hidden nodes H1 and H2. The cancellation factor(s) can still be estimated if each hidden node in the network has at least two nodes (otherwise it can be treated as a local noise source).For simplicity, first consider the situation of two entangled hidden nodes in that they have overlapping neighborhoods. Some possible coupling patterns are shown in Fig. <ref>. In panel (A), the two hidden nodes share three NH nodes but with different coupling strength. In panels (B) and (C), the hidden nodes share two or one common node(s). If the two hidden nodes do not share any nodes, the cancellation factors can be estimated independently using the same method as for the situation of one hidden node. Anther extreme case is that the two hidden nodes have and only have two identical nodes, which is equivalent to the case of one hidden node.A procedure was developed <cit.> to estimate the cancellationfactors for the situation in Fig. <ref>(A). Theprocedure can be extended to the other two cases in a straightforwardmanner by setting zero the weights from the hidden nodes to nodes aand (or) b. For any of the three neighboring nodes i ∈ [a, b, c],its dynamical equation can be expanded as[ẋ_i]_1 =[∑_γ𝐚̃_i^(γ)·𝐠̃_i^(γ)(𝐱_i)]_1 + [∑_k≠ i, H_1, H_2^N𝒲_ij·𝐱_j]_1 + w^1p_i,H_1· [𝐱_H_1]_p + w^1p_i,H_2·[𝐱_H_2]_p + ξη_i.To cancel the hidden-node effect in one node, e.g., node b, oneneeds the time series of the two other non-hidden (NH) nodes, aand c, so as to cancel the coupling terms from the two hidden nodes.Let the corresponding cancellation factors be Ω_ba andΩ_bc. A new dynamical equation without the interferences fromthe hidden nodes can then be obtained:[ẋ_a]_1 - Ω_ba [ẋ_b]_1 - Ω_ca[ẋ_c]_1=[∑_γ𝐚̃_a^(γ)·𝐠̃_a^(γ)(𝐱_a)]_1 - Ω_ba·[∑_γ𝐚̃_b^(γ)·𝐠̃_b^(γ)(𝐱_b)]_1 - Ω_ca·[∑_γ𝐚̃_c^(γ)·𝐠̃_c^(γ)(𝐱_c)]_1+[ w^1p_a,H_1 - Ω_ba w^1p_b,H_1 - Ω_ca w^1p_c,H_1] · [𝐱_H_1]_p + [ w^1p_a,H_2 - Ω_ba w^1p_b,H_2 - Ω_ca w^1p_c,H_2] · [𝐱_H_2]_p+[∑_j≠ i, H_1, H_2^N (𝒲_aj - Ω_ba𝒲_bj -Ω_ca𝒲_cj )·𝐱_j]_1 + ξ (η_a -Ω_baη_b - Ω_caη_c),where it is assumed that the coefficients associated with the couplingterms from the hidden nodes are zero. This can be achieved when the equation([ w_b,H_1 w_c,H_2; w_b,H_2 w_c,H_2 ]) ·([ Ω_ba; Ω_ca ]) ≡ℳ_w·([ Ω_ba; Ω_ca ]) = ([ w_a,H_1; w_a,H_2 ])holds and has only one trivial solution. The couplings to the hiddennodes should thus satisfy the condition (ℳ_w)=2.One can then estimate the cancellation factors using Eq. (<ref>), which is free of influence from the hidden nodes.To demonstrate the procedure, a small chaotic Rössler oscillator network of five nodes was tested, asillustrated in Fig. <ref>(A). A reconstructionprocedure was carried out for #b, utilizing the time series from all three NH nodes. Since the unknown coefficients are highly correlated and dense in this small system, the least squares method can be chosen for reconstruction with relative data amount R_m=1.2.The predicted and the actual results are shown in the Fig. <ref>.In panel (A), terms #1 and #2 denote the two cancellation factors. Thenodal dynamics for node #b, #a and node #c are listed in order.Panel (B) shows the entangled couplings in the network. All predictedterms match well with the actual ones.When an NH node is coupled with K (K≥3) hidden nodes, their successful detection requires that every hidden node be connected with two or more NH nodes. Then K cancellation factors can be estimated when the coupling weights satisfy the condition (ℳ_w)=K, where the elements m_ih in ℳ_w correspond to the coupling terms from the hth hidden node to the ith NH node. §.§ Identifying chaotic elements in neuronal networksCompressive sensing can be exploited to identify a subset of chaoticelements embedded in a network of nonlinear oscillators from timeseries. The oscillators, when isolated, are not identical in that theirparameters are different, so dynamically they can be in distinct regimes.For example, all oscillators can be described by differential equationsof the same mathematical form, but with different parameters. Consider the situation where only a small subset of the oscillators are chaotic, andthe remaining oscillators are in dynamical regimes of regular oscillations.Due to the mutual couplings among the oscillators, the measured time seriesfrom most oscillators would appear random. The challenge is to identifythe small subset of originally (“truly”) chaotic oscillators.The problem of identifying chaotic elements from a network of coupled oscillators arises in biological systems and biomedical applications. For example, for a network of coupled neurons that exhibit regular oscillations in a normal state,the parameters of each isolated neuron are in regular regime. Under external perturbation or slow environmental influences the parameters of some neurons can drift into a chaotic regime. When this occurs the whole network would appear to behave chaotically, which may correspond to certain disease. The coupling and nonlinearity stipulate that irregular oscillations at the network level can emerge even if only a few oscillators have gone “bad.” It is thus desirable to be able to pin down the origin of the ill-behaved oscillators - the few chaotic neurons among a large number of healthy ones.One might attempt to use the traditional approach of time-delayed coordinate embedding to reconstruct the phase space of the underlying dynamical system <cit.> and then to compute the Lyapunov exponents <cit.>. However, for a network of nonlinear oscillators, the phase-space dimension is high and an estimate of the largest Lyapunov exponent would only indicate if the whole coupled system is chaotic or nonchaotic, depending on the sign of the estimated exponent. In principle, using time series from any specific oscillator(s) would give qualitatively the same result. Thus, the traditional approach cannot give an answer as to which oscillators are chaotic when isolated.Recently, a compressive sensing based method was developed <cit.> to address the problem of identifying a subset of ill-behaved chaoticelements from a network of nonlinear oscillators, majority of thembeing regular. In particular, for a network of coupled, mixed nonchaoticand chaotic neurons, it was demonstrated that, by formulating thereconstruction task as a compressive sensing problem, the systemequations and the coupling functions as well as all the parameters canbe obtained accurately from sparse time series. Using the reconstructedsystem equations and parameters for each and every neuron in the networkand setting all the coupling parameters to zero, a routine calculationof the largest Lyapunov exponent can unequivocally distinguish thechaotic neurons from the nonchaotic ones.§.§.§ Basic procedure of identifying chaotic neuronsFigure <ref>(a) shows schematically arepresentative coupled neuronal network. Consider a pair of neurons,one chaotic and another nonchaotic when isolated (say #1 and #10,respectively). When they are embedded in a network, due to coupling, thetime series collected from both will appear random and qualitatively similar, as shown in Figs. <ref>(b) and<ref>(c). It is visually quite difficult todistinguish the time series and to ascertain which node is originallychaotic and which is regular. The difficulty is compounded by the fact that the detailed coupling scheme is not known a priori. Suppose thatthe chaotic behavior leads to undesirable function of the network andis to be suppressed. A viable and efficient method is to apply smallpinning controls <cit.> to therelatively few chaotic neurons to drive them into some regular regime.(An implicit assumption is that, when all neurons are regular, thecollective dynamics is regular. That is, the uncommon but not unlikelysituation that a network systems of coupled regular oscillators wouldexhibit chaotic behaviors is excluded.) Accurate identification of thechaotic neurons is thus key to implementing the pinning control strategy.Given a neuronal network, the task is thus to locate all neurons that are originally chaotic and neurons that are potentially likely to enter into a chaotic regime when they are isolated from the other neurons or when the couplings among the neurons are weakened. The compressive sensing based approach <cit.> consists of two steps. Firstly, theframework is employed to estimate, from measured time series only,the parameters in the FHN equation for each neuron, as well as thenetwork topology and various coupling functions and weights. This canbe done by expanding the nodal dynamical equations and the coupling functions into some suitable mathematical base as determined by thespecific knowledge about the actual neuronal dynamical system, and thencasting the problem into that of determining the sparse coefficientsassociated with various terms in the expansion. The nonlinear systemsidentification problem can then be solved using the standard compressive sensing algorithm. Secondly, all coupling parameters are set to zero so that the dynamical behaviors of each and every individual neuron can be analyzed by calculating the Lyapunov exponents. These witha positive largest exponent are identified as chaotic.A typical time series from a neuronal network consists of a sequence of spikes in the time evolution of the cell membrane potential. It was demonstrated <cit.> that the compressive sensing basedreconstruction method works well even for such spiky time series. Thedependence of reconstruction accuracy on data amount was analyzedto verify that only limited data are required to achieve high accuracyin reconstruction.§.§.§ Example: identifying chaotic neurons in theFitzHugh-Nagumo (FHN) networkThe FHN model, a simplified version of the biophysically detailed Hodgkin-Huxley model <cit.>, is a mathematical paradigm for gaining significant insights into a variety of dynamical behaviors in real neuronal systems <cit.>. For a single, isolated neuron, thecorresponding dynamical system is described by the followingtwo-dimensional, nonlinear ordinary differential equations:d V/dt = 1/δ [ V( V - a)(1 - V) - W],d W/dt =V - W - b + S(t),where V is the membrane potential, W is the recover variable, S(t) is the driving signal (e.g., periodic signal), a, b, and δ are parameters. The parameter δ is chosen to be infinitesimal so that V(t) and W(t) are “fast” and “slow” variables, respectively. Because of the explicitly time-dependent driving signal S(t), Eq. (<ref>) is effectively a three-dimensionaldynamical system, in which chaos can arise <cit.>. For anetwork of FHN neurons, the equations ared V_i/dt = 1/δ [ V_i ( V_i -a) (1- V_i) - W_i] + ∑_i=1^N c_ij(V_j - V_i)d W_i/dt =V_i - W_i -b + S(t),where c_ij is the coupling strength (weight) between the ith andthe jth neurons (nodes). For c_ij = c_ji, the interactions between any pair of neurons are symmetric, leading to a symmetric adjacency matrix for the network. For c_ij c_ji, the network is asymmetrically weighted.Consider the FHN model with sinusoidal driving: S(t) = r sinω_0 t. The model parameters are r=0.32,ω_0=15.0, δ = 0.005, and b=0.15. For a=0.42, anindividual neuron exhibits chaos. Representative chaotic time series andthe corresponding dynamical trajectory are shown in Fig. <ref>. Reconstruction of an isolated neuron can be done by setting zero allcoupling terms in network. For this purpose power series of order 4can be used as the expansion base <cit.> so that there are17 unknown coefficients to be determined. Three consecutive measurements aresampled at time interval τ=0.05 apart and a standard two-point formulacan be used to extrapolate the derivatives. From a random starting point, 12 data points were generated. Results of reconstruction are shown inFigs. <ref>(a) and <ref>(b)for variables V and W, respectively. The last two coefficientsassociated with each variable represent the strength of the drivingsignal. Since only the variable W receives a sinusoidal input, the lastcoefficient in W is nonzero. By comparing the positions of the nonzeroterms and the previously assumed vector form 𝐠_i(t), onecan fully reconstruct the dynamical equations of any isolated neuron.In particular, from Figs. <ref>(a) and<ref>(b) it can be seen that all estimatedcoefficients agree with their respective true values.Figure <ref>(c) shows how the estimated coefficients converge to the true values as the number of data points is increased. It can be seen that, for over 10 data points, all theparameters associated with a single FHN neuron can be faithfullyidentified.Next consider the network of coupled FHN neurons as schematicallyshown in Fig. <ref>(a), where the couplingweights among various pairs of nodes are uniformly distributed in theinterval [0.3, 0.4]. The network topologyis random with connectionprobability p=0.04. From time series the compressive sensing matrix foreach variable of all nodes can be reconstructed. Since the couplings occuramong the variables V of different neurons, the strengths of all incominglinks can be found in the unknown coefficients associated with different V variables. Extracting all coupling terms from the estimated coefficients gives all off-diagonal terms in the weighted adjacency matrix. Figure <ref> shows the reconstructed adjacency matrix as compared with the real one for R_m = 0.7, where R_m is the relative number of data points normalized by the total number of unknown coefficients. It can be seen that the compressive sensing based method can predict all links correctly, in spite of thesmall errors in the predicted weight values. The errors are mainly dueto the fact that there are large coefficients in the system equationsbut the coupling weights are small.Using the weighted adjacency matrix, one can identify the coupling terms in the network vector function so as to extract the terms associated witheach isolated nodal velocity field. The value of parameter a can then be identified and the largest Lyapunov exponent can be calculated foreach individual neuron. The results are shown inFigs. <ref>(a) and <ref>(b). It can be seen that, for this example, neuron #1 has a positive largest exponentwhile the largest exponents for all others are negative, so #1 isidentified as the only chaotic neuron among all neurons in the network.§.§ Data based reconstruction of complex geospatial networks and nodal positioningComplex geospatial networks with components distributed in the realgeophysical space are an important part of the modern infrastructure.Examples include large scale sensor networks and various subnetworksembedded in the Internet. For such a network, often the set of activenodes depends on time: the network can be regarded as static only inrelatively short time scale. For example, in response to certainbreaking news event, a social communication network within the Internet mayemerge, but the network will dissolve itself after the event and itsimpacts fade away. The connection topologies of such networks areusually unknown but in certain applications it is desirable to uncoverthe network topology and to determine the physical locations ofvarious nodes in the network. Suppose time series or signals can becollected from the nodes. Due to the distributed physical locationsof the nodes, the signals are time delayed. An illustrative example of a complex geospatial network is shown in Fig. <ref>, where there is a monitoring center that collects data from nodes atvarious locations, but their precise geospatial coordinates are unknown.The normal nodes are colored in green. There are also hidden nodesthat can potentially be the sources of threats (e.g., those representedby dark circles). Is it possible to uncover the network topology,estimate the time delays embedded in the signals from different nodes,and then determine their physical locations? Can the existence of ahidden node be ascertained and its actual geophysical location bedetermined? A recent work <cit.> showed that these questions can beaddressed by using the compressive sensing based reconstructionparadigm. In particular, the time delays of the dynamics at variousnodes can be estimated using time series collected from a singlelocation. Note that there were previous methods of finding time delaysin complex dynamical systems, e.g., those based on synchronization <cit.>, Bayesian estimation <cit.>, and correlation between noisy signals <cit.>. The compressivesensing based method provides an alternative approach that has the advantages of generality, high efficiency, low data requirement, and applicability to large networks. It was demonstrated <cit.>that the method can yield estimates of the nodal time delays withreasonable accuracy. After the time delays are obtained, the actualgeospatial locations of various nodes can be determined by using, e.g.,a standard triangular localization method <cit.>,given that the locations of a small subset of nodes are known. Hiddennodes can also be detected. These results can potentially be usefulfor applications such as locating sensors in wireless networks andidentifying/detecting/anticipating potential geospatialthreats <cit.>, an area of importance and broad interest.*Reconstruction of time delays using compressive sensing.Consider a continuous-time oscillator network with time delayedcouplings <cit.>, where for every link,the amount of delay is proportional to the physical distance of this link. The time delayed oscillator network model has been widely used instudying neuronal activities <cit.> and food webs inecology <cit.>. Mathematically, the system can be written as 𝐱̇_i=𝐅_i[𝐱_i(t)] + ∑_j=1, j≠ i^N𝒲_ij· [𝐱_j(t- τ_ij)- 𝐱_i(t)],for i = 1, …, N, where 𝐱_i∈ℝ^m is the m-dimensional state variable of node i and 𝐅_i[𝐱_i(t)] is the vector field for its isolated nonlinear nodal dynamics. For a link l_ij connecting nodes i and j,the interaction weight is given by the m× m weight matrix 𝒲_ij∈ℝ^m × m with its element w_ij^p,q representing the coupling from the qth component of node j to the pth component of node i. For simplicity, we assume only one component of w_ij^p,q is non-zero and denotes it as w_ij. The associated time delay is denoted as τ_ij. For a modern geospatial network, the speed of signal propagation is quite high in a proper medium (e.g., optical fiber). The time delay can thus be assumed to be small and the Taylor expansioncan be used to express the delay coupling termsto the first order, e.g., x_i(t- τ_ji) ≈ x_i(t) - τ_jiẋ_i, where ẋ_i is the time derivative. When the coupling function between any pair of nodes is linear, in a suitable mathematical basis constructed from the timeseries data, 𝐠̃^(γ) [𝐱_i (t)],the coupling and time delayed terms, together with the nodaldynamical equations, can be expanded into a series. The task isto estimate all the expansion coefficients. Assume linear couplingfunctions and causality so that all τ_ij (i,j=1,…,N)are positive (for simplicity). All terms directly associated withnode i can be regrouped into 𝐅_i^' [𝐱_i(t)], where𝐅_i^'[𝐱_i(t)] ≡𝐅_i [𝐱_i(t)] - 𝐱_i(t)·∑_j=1, j≠ i^N𝒲_ij,and 𝐅_i^' [𝐱_i(t)] has been expanded into the following series form:𝐅_i^' [𝐱_i(t)] = ∑_γα̃^(γ)·𝐠̃^(γ) [𝐱_i (t) ],with 𝐠̃^(γ) [𝐱_i (t)] representinga suitably chosen set of orthogonal and complete base functions sothat the coefficients α̃^(γ) are sparse. The time delayed variable 𝐱_j(t - τ_ij) can beexpanded as𝐱_j( t - τ_ij)≈𝐱_j(t)-τ_ij𝐱̇_j(t).All the coupling terms with inhomogeneous time delays associated withnode i can then be written as[∑_j=1, j≠ i^N𝒲_ij𝐱_j(t - τ_ij)]_p≡∑_j=1, j≠ i^N[ℬ_ij·𝐱_j(t) + 𝒞_ij·𝐱_j(t)],where ℬ_ij = 𝒲_ij and 𝒞_ij= -𝒲_ijτ_ij. In the compressive sensing framework, Eq. (<ref>) can thenbe written in the following compact form:𝐱̇_i(t) = ∑_γα̃^(γ)·𝐠̃^(γ) [𝐱_i (t) ]+ ∑_j=1, j≠ i^N[ℬ_ij·𝐱_j(t) + 𝒞_ij·𝐱_j(t)],which is a set of linear equations for data collected at differenttime t, where α̃^(γ), ℬ_ijand 𝒞_ij are to be determined. If the unknown coefficientvectors can be reconstructed accurately, one has complete informationabout the nodal dynamics as represented by𝐅^'[𝐱(t)], the topology and interacting weights of the underlying network as represented by 𝒲_ij, as well as the time delays associated with the nonzero links because of therelations 𝒲_ij= ℬ_ij. Note that, if thecoupling form is nonlinear, the relationship between the delay term𝐜_ij and the time delays τ_ij would be hardto interpret, especially when the exact coupling form is not known. *Nodal positioning and reconstruction of geospatial network.After obtaining the time delays, one can proceed to determining the actual positions of all nodes. If time series are collected simultaneously from all nodes at the data collecting node, the estimated coupling delay τ_ij associated with the link l_ij is proportional to the physical distance d_ij = d_ji of the link. However, in reality strictly synchronous data collection is not possible. For example, if the signals are collected, e.g., at a location s outside the network with varying time delays τ_si, the estimated delays associated with various links in the network are no longer proportional to the actual distances. The varying delays due to asynchronous data collectioncan be canceled and the distances can still be estimated asd_ij = (c/2)( τ_ij + τ_ji), where τ_ij is the signaldelay associated with node j from the reconstruction of node i, viceversa for τ_ji, and c is the signal propagation speed.When the mutual distances between the nodes have been estimated, onecan determine their actual locations, e.g., by using the standard triangular localization algorithm <cit.>. This method requires that the positions of N_B reference nodes be known, the so-called beacon nodes. Starting from the beacon nodes, the triangulation algorithm makes use of the distances to these referencenodes to calculate the Cartesian coordinates of the detected nodes. Thebeacon node set can then be expanded with the newly located nodes. Nodes that are connected to the new beacon set, each with more than three linksin the two-dimensional space, can be located. The process continues untilthe locations of all nodes have been determined, or no new nodes can belocated. The choice of the proper initial beacon set to fully reconstructthe network depends on the network topology. For example, for ascale-free network, one can choose nodes that were firstly added duringthe process of network generation as the initial beacon set, therebyguaranteeing that all nodes can be located using the procedure. Anempirical rule is then to designate the largest degree nodes as theinitial beacon node set.More specifically, given the positions of k reference nodes (or beaconnodes) (x_k, y_k), and their distances d_i,1, d_i,2, ⋯ d_i,kto the target node i, one can calculate the position of node i using the triangular localization method <cit.>, for k largerthan the space dimension. In general, it is necessary to solve the leastsquares optimization problem ℋ·𝐱_i =𝐛, where 𝐱_i=[x_i, y_i ]^T is the position of node i, and ℋ = [ 𝐱_1,𝐱_2, ⋯, 𝐱_k]^T is the position matrix corresponding to the set of beacon nodes, where 𝐛=0.5×[ D_1, D_2, ⋯, D_k]^T and D_k=d_ik^2- y_k^2 + x_k^2. To locate the positions of all nodesin the network, one starts with a small set of beacon nodes whose actual positions are known. Initially one can locate the nodes that are connectedto at least three nodes in the set of beacon nodes, insofar as the threereference nodes are not located on a straight line. When this is done,the newly located nodes can be added into the set of reference nodesand the neighboring nodes can be located through the new set of beaconnodes. This process can be iterated until the positions of all nodesare determined or no more qualified neighboring nodes can be found.For a general network, such an initial beacon set may not be easilyfound. A special case is scale-free networks, for which the initialbeacon set can be chosen as the nodes with the largest degrees. For arandom network, one can also choose the nodes of the largest degree as the initial beacon node set, and use a larger beacon set to locate most ofthe nodes in the network. For an arbitrary network topology, the followingsimple method was proposed <cit.> to select the set ofbeacon nodes: estimate the distances from one node to all otherunconnected nodes using the weighted shortest distance and then proceedwith the triangular localization algorithm. There are alternativelocalization algorithms based on given distances, e.g, themultidimensional scaling method <cit.>.*An example of reconstruction of a complex geospatial network.A numerical example was presented in Ref. <cit.> to demonstrate the reconstruction of complex geospatial networks, in which all nodes were assumed to be distributed in a two-dimensional square, or athree-dimensional cube of unit length. The network topology wasscale free <cit.> or random <cit.>, and its sizewas varied. A nonlinear oscillator was placed at each node, e.g.,the Rössler oscillator ([ẋ,ẏ,ż] =[-y-z,x+0.2y,0.2x+z(x-0.2)]). The coupling weights were asymmetricand uniformly distributed in the interval [0.1, 0.5]. A small thresholdwas assigned to the estimated weight as w_0=0.05 (somewhat arbitrary),where if the estimated weight is larger (smaller) than w_0, thecorresponding link is regarded as existent (nonexistent). As a result, the following holds: d_ij=c ·τ_ij= c ·τ_ji, and the parameter c was chosen to be 100 (arbitrarily). Linearcoupling functions were chosen for any pair of connected nodes, where the interaction occurs between the z-variable of one node and thex-variable of another. The time series used to reconstruct thewhole network system were acquired by integrating the coupled delayeddifferential-equation system <cit.> with step size 5× 10^-5.The vector fields of the nodal dynamics were expanded into a powerseries of order l_x + l_y + l_z ≤ 3. The derivatives required for the compressive sensing formulation were approximated from time series by the standard first order Gaussian method. The data requirement was characterized by R_m, the ratio of the number of data points usedto the total number of unknown coefficients to be estimated. The beaconnodes were chosen to be those having the largest degrees in the network,and their positions were assumed to be known.Figure <ref> summarizes the major stepsrequired for reconstructing a complex geospatial network usingcompressive sensing, where N = 30 nodes connecting with eachother in a scale-free manner are randomly distributed in atwo-dimensional square. Oscillatory time series are collected fromeach node, from which the compressive sensing equations can be obtained,as shown in Figs. <ref>(A) and <ref>(B). The reconstructed coefficients for the nodal dynamical equations contain the coupling weights B_ij=w_ijand the delay terms C_ij=-w_ij×τ_ij. The links withreconstructed weights larger than the threshold w_0 are regarded asactual (existent) links, for which the time delays τ_ij can beestimated as τ_ij = - C_ij/w_ij. Repeating this procedurefor all nodes, the weighted adjacency matrix (which defines the networktopology) and the time delay matrix can be determined. The estimatedadjacency matrix and the time delays are displayed inFigs. <ref>(C) and <ref>(D),respectively, which match well with those of the actual network. Notethat the reconstructed time delays are symmetric with respect to thelink directions, as shown in Fig. <ref>(D),which is correct as they depend only on the corresponding physicaldistances. With the estimated time delays, the four largest degree n odes, node #1 ∼#4, are chosen as the beacon nodes, so that thelocations of all remaining nodes can be determined. The fully reconstructedgeospatial network is shown in Fig. <ref>(E), wherethe red rectangles indicate the locations of the beacon nodes. The blackcircles denote the actual locations of the remaining nodes and the headsof the blue arrows indicate their estimated positions (shorter arrowsmean higher estimation accuracy). The amount of data used is relativelysmall: R_m=0.5.A detailed performance analysis was provided in Ref. <cit.> where the issue of positioning accuracy was addressed. Specifically, to locate all nodes in a two-dimensional space requires knowledge of the positions of at least three nodes (minimally four nodes in the three-dimensional space). Due to noise, the required number of beacon nodes will generally be larger. Since node positioning is based on time delays estimated from compressive sensing, which contain errors, the number of required beacon nodes is larger than three even in two dimensions. The positioning accuracy canbe quantified <cit.> by using the normalized error M_r, defined as the medium distance error between the estimated and actual locations for all nodes (except the beacon nodes), normalized by the distributed length L. Figure <ref> shows M_r versus the fraction R_B of the beacon nodes. The reconstruction parameters are chosen such that the errors in the time delay estimation is D_nz≈ 0.12. For small values of R_B, the positioning errors are large. Reasonable positioning errors are obtained when R_B exceeds, say, 0.2.The compressive sensing based approach can be used to ascertain theexistence of a hidden node and to estimate its physical location ina complex geospatial network <cit.>.To detect a hidden node, it is necessary toidentify its neighboring nodes <cit.>. For anexternally accessible node, if there is a hidden node in its neighborhood, the corresponding entry in the reconstructed adjacency matrix will exhibit an abnormally dense pattern or contain meaningless values. In addition, the estimated coefficients for the dynamical and coupling functions of such an abnormal node typically exhibit much larger variations when different data segments are used, in comparison with those associated with normal nodes that do not have hidden nodes in their neighborhoods. The mathematical formulation of the method to uncover a hidden node in complex geospatial networks can be foundin Ref. <cit.>. For the network of size N = 30 inFig. <ref>, initially, there are only 29 timeseries, one from each of the normal node, and it is not knowna priori that there would be a hidden node in the network.The network was reconstructed to obtain the estimated weightsand time delays, as shown in Fig. <ref>(A).One can see that the connection patterns of some nodes are relatively dense and the values of the weights and time delays are meaningless (e.g., negative values), giving the first clue that these nodes may be the neighboring nodes of a hidden node. To confirm that this is indeed the case, the available time series are divided into a number of segments based on the criterion that the data requirement for reconstruction is satisfied for each segment. As shown in Fig. <ref>(B), extraordinarily large variancesin the estimated coefficients associated with the abnormal nodes arise. Combining results from Figs. <ref>(A) and <ref>(B), one can claim with confidence that the four nodes are indeed in the immediate neighborhood of a hidden node,ascertaining its existence in the network. The method also works ifthere are more than one hidden node, given that they do not sharecommon neighboring nodes.§.§ Reconstruction of complex spreading networks from binary dataAn important class of collective dynamics is virusspreading and information diffusion in social and computernetworks <cit.>. We discuss here the problem of reconstructing thenetwork hosting the spreading process and identifying the source ofspreading using limited measurements <cit.>. This is achallenging problem due to (1) the difficulty in predicting and monitoring mutations of deadly virus and (2) absence of epidemic threshold inheterogeneous networks <cit.>.The problem is directly relevant to affairs of significant current interest such as rumor propagation in the online virtual communities, which can causefinancial loss or even social instabilities (an example being the 2011irrational and panicked acquisition of salt in southeast Asian countriescaused by the nuclear leak in Japan). In such a case, identifying thepropagation network for controlling the dynamics is of great interest.A significant challenge in reconstructing a spreading network liesin the nature of the available time series: they are polarized,despite stochastic spreading among the nodes. Indeed, the link patternand the probability of infection are encrypted in the binary statusof the individuals, infected or not.There were recent efforts in addressing the inverse problem of some special types of complex propagation networks <cit.>. For example, for diffusion process originated from a single source, the routes of diffusion from the source constitute a tree-like structure. If information about the early stage of the spreading dynamics is available, it would be feasible to decode all branches that reveal the connections from the source to its neighbors, and then to their neighbors, and so on. Taking into account the time delays in the diffusion process enables a straightforward inference of the source in a complex network through enumerating all possible hierarchical trees <cit.>.However, if no immediate information about the diffusion process is available,the tree-structure based inference method is inapplicable, and the problemof network reconstruction and locating the source becomes intractable, hindering control of diffusion and delivery of immunization.The loss of knowledge about the source is common in real situations. For example, passengers on an international flight can carry a highly contagious disease, making certain airports the immediate neighbors of the hidden source, which would be difficult to trace. In another example, the source could be migratory birds coming from other countries or continents. A general data-driven approach, applicable in such scenarios, is an outstanding problem in network science and engineering.Recently, a compressive sensing based framework was developed to reconstruct complex spreading networks based on binarydata <cit.>. Since the dynamics of epidemic propagationare typically highly stochastic with binary time series, the standard power series expansion method to fit the problem into the compressive sensing paradigm (as discussed in preceding sections) is not applicable, notwithstanding the methods of alternative sparsity enforcing regularizersand convex optimization used in Ref. <cit.> to infer networks. The idea in Ref. <cit.> was then to develop a scheme to implementthe highly nontrivial transformation associated with the spreadingdynamics in the paradigm of compressive sensing. Two prototypical modelsof epidemic spreading on model and real-world (empirical) networks were studied: the classicsusceptible-infected-susceptible (SIS) dynamics <cit.> and the contact process (CP) <cit.>. Inhomogeneous infectionand recovery rates as representative characteristics of the naturaldiversity were incorporated into the diffusion dynamics to bettermimic the real-world situation.The basic assumption is then that only binary time series can be measured,which characterize the status of any node, infected or susceptible, at anytime after the outbreak of the epidemic. The source that triggers thespreading process is assumed to be externally inaccessible (hidden). Infact, one may not even realize its existence from the available time series.The method developed in Ref. <cit.> enables, based onrelatively small amounts of data, a full reconstruction of the epidemicspreading network with nodal diversity and successful identificationof the immediate neighboring nodes of the hidden source (therebyascertaining its existence and uniquely specifying its connections tonodes in the network). The framework was validated with respect todifferent amounts of data generated from various combinations of the network structures and dynamical processes. High accuracy, high efficiency and applicability in a strongly stochastic environment with measurement noise and missing information are the most striking characteristics of the framework <cit.>. As a result, broad applications can beexpected in addressing significant problems such as targeted controlof virus spreading in computer networks and rumor propagation on socialnetworks.§.§.§ Mathematical formulation *Spreading processes. The SIS model <cit.> is a classic epidemic model to study avariety of spreading behaviors in social and computer networks.Each node of the network represents an individual and links are connectionsalong which the infection can propagate to others with certain probability. At each time step, a susceptible node i in state 0 is infected withrate λ_i if it is connected to an infected node in state 1. [Ifi connects to more than one infected neighbor, the infection probabilityP^01 is given by Eq. (<ref>) below.] At the sametime, infected nodes are continuously recovered to be susceptible at therate δ_i. The CP model <cit.> describes, e.g.,the spreading of infection and competition of animals over a territory.The main difference between SIS and CP dynamics lies in the influence ona node's state from its vicinity. In both SIS and CP dynamics, λ_iand δ_i depend on the individuals' immune systems and are selectedfrom a Gaussian distribution characterizing the natural diversity. Moreover,a hidden source is regarded as infected at all time. *Mathematical formulation of reconstruction from binary data based on compressive sensing. Assume thatthe disease starts to propagate from a fraction of the infected nodes. The task is to locate any hidden source based solely on binary time series after the outbreak of infection. The state of an arbitrary nodei is denoted as S_i, whereS_i={[ 0, susceptible;; 1,infected. ].Due to the characteristic difference between the SIS dynamics and CP, it is useful to treat them separately.SIS dynamics. The probability P_i^01(t) of an arbitrary node i being infected by its neighbors at time t isP_i^01(t) =1-(1-λ_i)^∑_j=1,j i^N a_ijS_j(t),where λ_i is the infection rate of i, a_ij stands for the elements of the adjacency matrix (a_ij=1 if i connects to j and a_ij=0 otherwise), S_j(t) is the state of node j at t, and thesuperscript 01 denotes the change from the susceptible state (0) to the infected state (1). At the same time, the recovery probability of i isP_i^10(t)=δ_i, where δ_i is the recovery rate of nodei and the superscript 10 denotes the transition from the infected stateto the susceptible state. Equation (<ref>) can berewritten asln[1-P_i^01(t)]=ln(1-λ_i) ·∑_j=1,j i^N a_ijS_j(t).Suppose measurements at a sequence of times t= t_1, t_2, ⋯, t_m are available. Equation (<ref>) leads to the following matrix form 𝐗_m× 1 = 𝒢_m× (N-1)·𝐚_(N-1)× 1: [ ln[1-P_i^01(t_1)]; ln[1-P_i^01(t_2)]; ⋮; ln[1-P_i^01(t_m)] ] = [ S_1(t_1)⋯ S_i-1(t_1) S_i+1(t_1)⋯ S_N(t_1); S_1(t_2)⋯ S_i-1(t_2) S_i+1(t_2)⋯ S_N(t_2);⋮⋮⋮⋮⋮⋮; S_1(t_m)⋯ S_i-1(t_m) S_i+1(t_m)⋯ S_N(t_m) ][ln(1-λ_i)a_i1;⋮; ln(1-λ_i)a_i,i-1; ln(1-λ_i)a_i,i+1;⋮;ln(1-λ_i)a_iN ], between node i and all other nodes, and it is sparse for a generalcomplex network. It can be seen that, if the vector 𝐗_m× 1and the matrix 𝒢_m× (N-1) can be constructed from timeseries, 𝐚_(N-1)× 1 can then be solved by usingcompressive sensing. The main challenge here is that the infection probabilities P_i^01(t) atdifferent times are not given directly by the time series of the nodalstates. A heuristic method to estimate the probabilities can bedevised <cit.> by assuming that the neighboring set Γ_iof the node i is known. The number of such neighboring nodes is givenby k_i, the degree of node i, and their states at time t can bedenoted asS_Γ_i(t) ≡{ S_1(t), S_2(t), ⋯, S_k_i(t) }.In order to approximate the infection probability, one can use S_i(t)=0so that at t+1, the node i can be infected with certain probability.In contrast, if S_i(t)=1, S_i(t+1) is only related with the recoveryprobability δ_i. Hence, it is insightful to focus on the S_i(t)=0case to derive P_i^01(t). If one can find two time instants:t_1,t_2∈ T (T is the length of time series), such thatS_i(t_1)=0 and S_i(t_2)=0, one can calculate thenormalized Hamming distance H[S_Γ_i(t_1),S_Γ_i(t_2)]between S_Γ_i(t_1) and S_Γ_i(t_2), defined as the ratio of the number of positions with different symbols between them andthe length of string. If H[S_Γ_i(t_1),S_Γ_i(t_2)]=0,the states at the next time step, S_i(t_1+1) and S_i(t_2+1),can be regarded as as i.i.d Bernoulli trials. In this case, using thelaw of large numbers, one haslim_l→∞1l∑_ν=1^lS_i(t_ν+1) → P_i^01(t̂_α), ∀t_ν, S_i(t_ν)=0, H[S_Γ_i(t̂_α),S_Γ_i(t_ν)]=0.A more intuitive understanding of Eq. (<ref>) is that,if the states of i's neighbors are unchanged, the fraction of times ofi being infected by its neighbors over the entire time period willapproach the actual infection probability P_i^01. Note, however,that the neighboring set of i is unknown and to be inferred. Astrategy is then to artificially enlarge the neighboring setS_Γ_i(t) to include all nodes in the network except i. DenoteS_-i(t)≡{S_1(t),S_2(t),…,S_i-1(t),S_i+1(t), …,S_N(t)}.If H[S_-i(t_1),S_-i(t_2)]=0, the condition H[S_Γ_i(t_1),S_Γ_i(t_2)]=0 can be ensured.Consequently, due to the nature of the i.i.d Bernoulli trials, application of the law of large numbers leads tolim_l→∞1l∑_ν=1^lS_i(t_ν+1) → P_i^01(t̂_α), ∀t_ν, S_i(t_ν)=0,H[S_-i(t̂_α), S_-i(t_ν)]=0.Hence, the infection probability P_i^01(t̂_α) of anode at t̂_α can be evaluated by averaging over itsstates associated with zero normalized Hamming distance between the strings of other nodes at some time associated with t̂_α.In practice, to find two strings with absolute zero normalized Hammingdistance is unlikely. It is then necessary to set a threshold Δso as to pick the suitable strings to approximate the law of largenumbers, that is1l∑_ν=1^l≫ 1S_i(t_ν+1)≃1l∑_ν=1^l≫ 1P_i^01(t_ν), ∀t_ν, S_i(t_ν)=0,H[S_-i(t̂_α), S_-i(t_ν)]<Δ,where S_-i(t̂_α) serves as a base for comparison with S_-i(t) at all other times and (1/l)∑_ν=1^l≫ 1P_i^01(t_ν) ≃ P_i^01(t̂_α). Since H[S_-i(t̂_α),S_-i(t_ν)] is not exactly zero, there is a small difference between P_i^01(t̂_α) and P_i^01(t_ν) (ν=1,⋯,l). It is thus useful toconsider the average of P_i^01(t_ν) for all t_ν to obtain P_i^01(t̂_α), leading to the right-hand side of Eq. (<ref>). Let ⟨ S_i(t̂_α+1)⟩= (1/l)∑_ν=1^l≫ 1S_i(t_ν+1)⟨ P_i^01(t̂_α) ⟩=(1/l)∑_ν=1^l≫ 1P_i^01(t_ν).In order to reduce the error in the estimation, one can implementthe average on S_-i(t) over all selected strings using Eq. (<ref>). The averaging process is with respect tothe nodal states S_j,j≠ i(t) on the right-hand side of themodified dynamical Eq. (<ref>). Specifically, averaging overtime t restricted by Eq. (<ref>) on both sides of Eq. (<ref>) yields⟨ln[1-P_i^01(t)]⟩=ln(1-λ_i)∑_j=1,ji^N a_ij⟨S_j(t)⟩.For λ_i small compared with insignificant fluctuations, thefollowing approximation holds: ln[1-⟨P_i^01(t)⟩]≃⟨ln[1-P_i^01(t)]⟩,which leads toln[1-⟨P_i^01(t)⟩]≃ln(1-λ_i) ∑_j=1,ji^N a_ij⟨S_j(t)⟩.Substituting ⟨ P_i^01(t̂_α)⟩ by ⟨S_i(t̂_α+1)⟩, one finally getsln[1-⟨ S_i(t̂_α+1)⟩ ]≃ln(1-λ_i) ·∑_j=1,j i^N a_ij⟨ S_j(t̂_α)⟩.While the above procedure yields an equation that bridges the links of an arbitrary node i with the observable states of the nodes, a single equation does not contain sufficient structural information about the network. The second step is then to derive a sufficient number of linearly independent equations required by compressive sensing to reconstructthe local connection structure. To achieve this, one can choose a series of base strings at a number of time instants from a set denoted by T_base,in which each pair of strings satisfyH[S_-i(t̂_β),S_-i(t̂_α)]> Θ,∀t̂_α,t̂_β∈ T_base,where t̂_α and t̂_β correspond to the time instants of two base strings in the time series and Θ is a threshold. For each string, the process of establishing the relationship betweenthe nodal states and connections can be repeated, leading to a set ofequations at different values of t̂_α inEq. (<ref>). This process finally gives rise to a setof reconstruction equations in the matrix form: [ [ ln[1-⟨ S_i(t̂_1+1)⟩]; ln[1-⟨ S_i(t̂_2+1)⟩];⋮; ln[1-⟨ S_i(t̂_m+1)⟩] ]]= [ [ ⟨ S_1(t̂_1)⟩⋯ ⟨ S_i-1(t̂_1)⟩ ⟨ S_i+1(t̂_1)⟩⋯ ⟨ S_N(t̂_1)⟩; ⟨ S_1(t̂_2)⟩⋯ ⟨ S_i-1(t̂_2)⟩ ⟨ S_i+1(t̂_2)⟩⋯ ⟨ S_N(t̂_2)⟩;⋮⋮⋮⋮⋮⋮; ⟨ S_1(t̂_m)⟩⋯ ⟨ S_i-1(t̂_m)⟩ ⟨ S_i+1(t̂_m)⟩⋯ ⟨ S_N(t̂_m)⟩ ]] [ [ln(1-λ_i)a_i1;⋮; ln(1-λ_i)a_i,i-1; ln(1-λ_i)a_i,i+1;⋮;ln(1-λ_i)a_iN ]],where t̂_1, t̂_2,⋯ , t̂_m correspond to the timeassociated with m base strings and ⟨·⟩ denotethe average over all satisfied t. The vector 𝐗_m× 1and the matrix 𝒢_m× (N-1) can then be obtained basedsolely on time series of nodal states and the vector𝐚_(N-1)× 1 to be reconstructed is sparse, renderingapplicable the compressive sensing framework. As a result, exact reconstruction of all neighbors of node i from relatively small amounts of observation can be achieved. In a similarfashion the neighboring vectors of all other nodes can be uncovered fromtime series, enabling a full reconstruction of the whole network bymatching the neighboring sets of all nodes.CP dynamics. The infection probability of an arbitrary node i is given byP_i^01(t)=λ_i∑_j=1,j i^N a_ijS_j(t)/k_i,where k_i is the degree of the node i, and the recovery probabilityis P_i^10(t)=δ_i. In close analogy to the SIS dynamics,one has ⟨S_i(t̂_α+1)⟩≃⟨P_i^01(t̂_α)⟩ =λ_i∑ a_ij⟨S_j(t̂_α)⟩/ k_i.One can then choose a series of base strings using a proper thresholdΘ to establish a set of equations, expressed in the matrix form 𝐗_m× 1 = 𝒢_m× (N-1)·𝐚_(N-1)× 1, where 𝒢 has the same form as inEq. (<ref>), but 𝐗 and 𝐚 aregiven by𝐗 = [⟨ S_i(t̂_1 +1)⟩, ⟨ S_i(t̂_2 +1)⟩, ⋯, ⟨ S_i(t̂_m +1)⟩]^T, 𝐚 = [ λ_i/k_ia_i1, ⋯, λ_i/k_ia_i,i-1, λ_i/k_ia_i,i+1, ⋯, λ_i/k_ia_iN]^T.The reconstruction framework based on building up the vector𝐗 and the matrix 𝒢 is schematically illustrated inFig. <ref>. It is noteworthy that the framework canbe extended to directed networks in a straightforward fashion due tothe feature that the neighboring set of each node can be independentlyreconstructed. For instance, the neighboring vector 𝐚 canbe defined to represent a unique link direction, e.g., incoming links.Inference of the directed links of all nodes yields the full topologyof the entire directed network. *Inferring inhomogeneous infection rates. The values of the infection rate λ_i of nodes can be inferred after the neighborhood of each node has been successfully reconstructed. The idea roots in the fact that the infection probability of a node approximated by the frequency of being infected calculated from time series is determined both by its infection rate and by the number of infected nodes in its neighborhood. An intuitive picture can be obtained by considering the following simple scenario in which the number of infected neighbors of node i does not change with time. In this case, the probability of i being infected at each time step is fixed. One can thus count the frequency of the 01 and 00 pairs embedded in the time series of i. The ratio of the number of 01 pairs over the total number of 01 and 00 pairs gives approximately the infection probability. The infection rate can then be calculated by using Eqs. (<ref>) and (<ref>) for the SIS andCP dynamics, respectively. In a real-world situation, however, the numberof infected neighbors varies with time. The time-varying factor can betaken into account by sorting out the time instants corresponding todifferent numbers of the infected neighbors, and the infection probabilitycan be obtained at the corresponding time instants, leading to a set of values for the infection rate whose average represents an accurate estimate of the true infection rate for each node.To be concrete, considering all the time instants t_ν associatedwith k_I infected neighbors, one can denote S_i^(k_I)= (1/l)∑_ν=1^lS_i (t_ν+1), t_ν,∑_j∈Γ_i S_j(t_ν) = k_I, S_i(t_ν) =0,where Γ_i is the neighboring set of node i, k_I is thenumber of infected neighbors, and S_i^(k_I) represents theaverage infected fraction of node i with k_I infected neighbors.Given S_i^(k_I), one can rewrite Eq. (<ref>) by substituting S_i^(k_I) for P_i^01(t) and λ_i^(k_I)for λ_i, which yields λ_i^(k_I) = 1-exp[ln( 1 - S_i^(k_I)) /k_I].The estimation error can be reduced by averaging λ_i^(k_I)with respect to different values of k_I, as follows:λ_i^true(SIS) ≈⟨λ_i^(k_I)⟩ = 1/N_λ_i∑_k_I∈λ_iλ_i^(k_I),where λ_i denotes the set of all possible infected neighborsduring the epidemic process and N_λ_i denotes the number ofdifferent values of k_I in the set. Analogously, for CP, one can evaluate λ_i^true from Eq. (<ref>) as λ_i^true(CP) ≈⟨λ_i^(k_I)⟩ = 1/N_λ_i∑_k_I∈λ_iS_i^(k_I) k_i/k_Iwhere k_i = ∑_j=1^N a_ij is the node degree of i. After all the links of i have been successfully reconstructed, S_i^(k_I)can be obtained from the time series in terms of the satisfiedS_i(t_ν +1), allowing one to infer λ_i^true viaEqs. (<ref>) and (<ref>).The method so described for estimating the infection rates isapplicable to any type of networks insofar as the network structurehas been successfully reconstructed <cit.>.§.§.§ Reconstructing complex spreading networks: examples *Reconstructing networks and inhomogeneous infection andrecovery rates. A key performance indicator of the binary data based reconstructionframework is the number of base strings (equations) for a varietyof diffusion dynamics and network structures. It is necessary tocalculate the success rates for existent links (SREL) and null connections (SRNC), corresponding to non-zero and zero element values in the adjacency matrix, respectively, in terms of the number of base strings. The binary nature of the network dynamical process and datarequires that the strict criterion be imposed <cit.>, i.e.,a network is regarded to have been fully reconstructed if and only if both success rates reach 100%. The sparsity of links makes itnecessary to define SREL and SRNC separately. Since the reconstructionmethod is implemented for each node in the network, SREL and SRNCcan be defined with respect to each individual node and, the twosuccess rates for the entire network are the respective averagedvalues over all nodes. The issue of trade-off can also be considered in terms of the true positive rate (TPR - for correctly inferred links)and the false positive rate (FPR - for incorrectly inferred links).In Ref. <cit.>, a large number of examples of reconstruction were presented. Take an example where there is no hidden source, andbinary time series can be obtained by initiating the spreading processfrom a fraction of infected nodes. Figure <ref>(a) showsthe reconstructed values of the components of the neighboring vector𝐗 of all nodes. Let n_t̂ be the number of basestrings normalized by the total number of strings. For small valuesof n_t̂, e.g., n_t̂=0.1, the values of elementsassociated with links and those associated with null connections (actual zeros in the adjacency matrix) overlap, leading to ambiguitiesin the identification of links. In contrast, for larger values ofn_t̂, e.g., n_t̂=0.4, an explicit gap emerges betweenthe two groups of element values, enabling correct identification ofall links by simply setting a threshold within the gap. The successrates (SREL and SRNC) as a function of n_t̂ for SIS and CPon both homogeneous and heterogeneous networks are shown inFigs. <ref>(b,c), where nearly perfect reconstructionof links are obtained, insofar as n_t̂ exceeds a relativelysmall value - an advantage of compressed sensing. The exact reconstructionis robust in the sense that a wide range of n_t̂ values canyield nearly 100% success rates. The reconstruction method iseffective for tackling real networks in the absence of any a priori knowledge about its topology.Since the network is to be reconstructed through the union of allneighborhoods, one may encounter “conflicts” with respect to the presence/absence of a link between two nodes as generated by the reconstruction results centered at the two nodes, respectively. Such conflictswill reduce the accuracy in the reconstruction of the entire network.The effects of edge conflicts can be characterized by analyzing the consistency of mutual assessment of the presence or absence of linkbetween each pair of nodes, as shown in Figs. <ref>(b,c).It can be seen that inconsistency arises for small values of n_t̂but vanishes completely when the success rates reach 100%,indicating perfect consistency among the mutual inferences of nodesand consequently guaranteeing accurate reconstruction of the entirenetwork.While the number of base strings is relatively small compared with the network size, it is necessary to have a set of strings atdifferent time with respect to a base string to formulate themathematical reconstruction framework. How the length of the time series affects the accuracy of reconstruction wasstudied <cit.>. Figures <ref>(a,b) show the success rates as a function of the relative length n_t of time series for SIS and CP dynamics on both homogeneous and heterogeneous networks, respectively, where n_t is the total length of time series from thebeginning of the spreading process divided by the network size N. The results demonstrate that, even for very small values of n_t, most links can already be identified, as reflected by the high values of the success rate shown. Figures <ref>(c,d) show the minimumlength n_t^min required to achieve at least 95% successrate for different network size. For both SIS and CP dynamics onnetworks, n_t^min decreases considerably as Nis increased. This seemingly counterintuitive result is due to thefact that different base strings can share strings at different timesto enable reconstruction. In general, as N is increased, n_t̂will increase accordingly. However, a particular string can belong todifferent base strings with respect to the threshold Δ, accountingfor the slight increase in the absolute length of the time series andthe reduction in n_t^min. The dependence of the success rateon the average node degree ⟨ k⟩ for SIS and CP on differentnetworks was investigated as well <cit.>. The results inFigs. <ref> and <ref> demonstrate the highaccuracy and efficiency of the compressive sensing based reconstruction method based on small amounts of binary data.In practice, noise is present and it is also common for time seriesfrom certain nodes to be missing, so it is necessary to test theapplicability of the method under these circumstances.Figures <ref>(a,b) show the dependence of the successrate on the fraction n_f of states in the time series thatflip due to noise for SIS and CP dynamics on two types of networks,respectively. It can be seen that the success rates are hardlyaffected, providing strong support for the applicability of thereconstruction method. For example, even when 25% of the nodalstates flip, about 80% success rates can still be achieved forboth dynamical processes and different network topologies.Figures <ref>(c,d) present the success rate versus thefraction n_m of unobservable nodes, the states of which are externally inaccessible. It can be seen that the high successrate remains mostly unchanged as n_m is increased fromzero to 25%, a somewhat counterintuitive but striking result. It was found that <cit.>, in general, missing informationcan affect the reconstruction of the neighboring vector, as reflectedby the reduction in the gap between the success rates associated withthe actual links and null connections. However, even for high valuesof n_f, e.g., n_f=0.3, there is still a clear gap,indicating that a full recovery of all links is achievable. Takentogether, the high accuracy, efficiency and robustness against noise and missing information provide strong credence for the validity andpower of the framework for binary time-series based network reconstruction.Having reconstructed the network structure, one can estimate the infection and recovery rates of individuals to uncover their diversity in immunity. This is an essential step to implement target vaccination strategy in a population or on a computer network to effectively suppress/prevent the spreading of virus at low cost, as a large body of literature indicates that knowledge about the network structure and individual characteristics is sufficient for controlling the spreading dynamics <cit.>. An effective method wasproposed <cit.> to infer the individuals' infection rates λ_i based solely on the binary time series of the nodal statesafter an outbreak of contamination. In particular, after all linkshave been successfully predicted, λ_i can be deduced fromthe infection probabilities that can be approximated by the correspondinginfection frequencies. These probabilities depend on both λ_i and the number of infected neighbors. The reproduced infection rates λ_i of individuals for both SIS and CP dynamics on different networks were in quite good agreement with the true values with small prediction errors. An error analysis revealed uniformly highaccuracy of the method <cit.>. The inhomogeneous recovery ratesδ_i of nodes can be predicted from the binary time series in amore straightforward way, because δ_i's do not depend on thenodal connections. Thus the framework is capable of predictingcharacteristics of nodal diversity in terms of degrees and infectionand recovery rates based solely on binary time series of nodal states. *Locating the source of spreading. Assume that a hidden source exists outside the network but there are connections between it and some nodes in the network. In practice, the source can be modeled as a special node that is always infected. Starting from the neighborhood of the source, the infection originates from the source and spreads all over the network. One first collects a set oftime series of the nodal states except the hidden source.As discussed in detail in Sec. <ref>, the basic idea of ascertaining and locating the hidden source is based onmissing information from the hidden source when attempting toreconstruct the network <cit.>. In particular, in order to reconstruct the connections belonging to the immediate neighborhood of the source accurately, time series from the source are needed to generate the matrix 𝒢 and the vector 𝐗. But since the source is hidden, no time series from it are available,leading to reconstruction inaccuracy and, consequently, anomalies inthe predicted link patterns of the neighboring nodes. It is thenpossible to detect the neighborhood of the hidden source by identifyingany abnormal connection patterns <cit.>, which canbe accomplished by using different data segments. If the inferred linksof a node are stable with respect to different data segments, thenode can be deemed to have no connection with the hidden source;otherwise, if the result of inferring a node's links variessignificantly with respect to different data segments, the node is likely to be connected to the hidden source. The standard deviationof the predicted results with respect to different data segments canbe used as a quantitative criterion for the anomaly. Once theneighboring set of the source is determined, the source can beprecisely located topologically.An example <cit.> is shown in Fig. <ref>,where a hidden source is connected with four nodes in the network[Fig. <ref>(a)], as can be seen from the network adjacencymatrix [Fig. <ref>(b)]. The reconstruction framework was implemented on each accessible node by using different sets of datain the time series. For each data set, the neighbors of all nodeswere predicted, generating the underlying adjacency matrix. Averaging over the elements corresponding to each location in all the reconstructed adjacency matrices leads to Fig. <ref>(c), in whicheach row corresponds to the mean number of links in a node'sneighborhood. The inferred links of the immediate neighbors of thehidden source exhibit anomalies. To quantify the anomalies, thestructural standard deviation σ was calculated from differentdata segments, where σ associated with node i is definedthrough the ith row in the adjacency matrix as σ_i = 1/N∑_j=1^N√(1/g∑_k=1^g(a_ij^(k) - ⟨ a_ij⟩)^2),where j denotes the column, a_ij^(k) represents the elementvalue in the adjacency matrix inferred from the kth group of the data, ⟨ a_ij⟩ = (1/g)∑_k=1^g a_ij^k is the mean value of a_ij, and g is the number of data segments. Applying Eq. (<ref>) to the reconstructed adjacency matricesgives the results in Fig. <ref>(d), where the valuesof σ associated with the immediate neighboring nodes of thehidden source are much larger than those from others (which areessentially zero). A threshold can be set in the distribution ofσ_i to identify the immediate neighbors of the hidden source.§ ALTERNATIVE METHODS FOR RECONSTRUCTING COMPLEX, NONLINEAR DYNAMICAL NETWORKS §.§ Reconstructing complex networks from response dynamics Probing into a dynamical system in terms of its response to external drivingsignals is commonly practiced in biological sciences. This approach isparticularly useful for studying systems with complex interactions and dynamical behaviors. The basic idea of response dynamics was previously exploited forreconstructing complex networks of coupled phase oscillators <cit.>.Through measuring the collective response of the oscillator network toan external driving signal, the network topology can be recovered through repeated measurement of the dynamical states of the nodes, provided that thedriving realizations are sufficiently independent of each other. Sincecomplex networks are generally sparse, the number of realizations of external driving can be much smaller than the network size.A network of coupled phase oscillators with a complex interacting topology isdescribed byϕ̇_i=ω_i + ∑_j=1^N J_ijf_ij(ϕ_j - ϕ_i) + I_i,m,where ϕ̇_i(t) is the phase of oscillator i at time t, ω_iis its natural frequency, J_ij is the coupling strength from oscillator jto oscillator i (weighted adjacency matrix, where J_ij=0 indicates theabsence of a link from j to i), and f_ij's are the pairwisecoupling functions among the connected oscillators. The nodes on which the external driving signals are imposed are specified as I_i,m, where m=0indicates no driving signal. When driving is present, the collective frequencyisΩ_m = ω_i + ∑_j=1^N J_ijf_ij(ϕ_j,m- ϕ_i,m) + I_i,m.The frequency difference between the driven and non-driven systems isD_i,m= ∑_j=1^N J_ij[f_ij(ϕ_j,m -ϕ_i,m)-f_ij(ϕ_j,0 - ϕ_i,0)].For sufficiently small signal strength, the phase dynamics of the oscillatorsare approximately those of the original. We can approximatef_ij(x)=f_ij'(Δ_ji,0) + 𝒪(x^2) and let θ_j,mdenote ϕ_j,m-ϕ_j,0. After a number of experiments, the networkconfiguration converges to𝒟 = 𝒥̂·θ,where Ĵ_ij is the Laplacian matrix withĴ_ij={[ J_ijf_ij'(Δ_ji,0),for i≠ j,; -∑_k,k≠ i J_ikf_ik'(Δ_ki,0), i=j. ].Note that the phase and frequency difference matrices θand 𝒟 are measurable. Thus, the networkstructure characterized by the matrix 𝒥̂can be solved by 𝒥̂ = 𝒟·θ^-1.The reconstruction method basedon the response dynamics not only can reveal links among the nodes but alsocan provide a quantitative estimate of the interaction strengths represented inthe matrix 𝒥̂. In general the number of experimentalrealizations required is M=N to ensure a unique solution of thematrix 𝒥̂ to represent the network structure.However, most real networks are sparsein the sense that the degree of a node is usually much less than thenetwork size, i.e., K≪ N. To obtain the matrix 𝒥̂with the least number of links is effectively a constraint for reducingthe required data amount through some optimization algorithms. Specifically, constraint (<ref>) can be used toparameterize the family of admissible matrices through (N-M)N parameters,P_ij, in terms of a singular value decomposition ofθ^ T=𝒰 in a standard way, where thesingular matrix 𝒮 of dimension M× N contains thesingular values on the diagonal. The coupling matrices can be reformulatedto be 𝒥̂ = 𝒟·𝒰·𝒮̃·𝒱^ T + 𝒫·𝒱, where P_ij is set to bezero for j≤ M, S̃_ij=δ_ij/σ_i ifσ_i>10^-4 and S̃_ii=0 if σ_i ≤ 10^-4. Finally,with respect to the parameter matrix 𝒫,the incoming links of any node i can beinferred by minimizing the 1-norm of the ith row vector of Ĵ asĴ_i _1 := ∑_j=1;j≠ i^N |ĵ_ij|.Numerical tests <cit.> showed that the number M of therequired experimental realizations can be substantially smaller than N toyield reasonable reconstruction results.A detailed description of the response dynamics based approach to network reconstruction can be found in Ref. <cit.>. In general, thereconstruction method is applicable to networked systems whose behaviors are dominated by the linearized dynamics about some stable state. An issue concerns how the reconstruction method can be extended to networked systemscharacterized by more than one variable, i.e., systems beyond phase coupled oscillators. Another issue is whether it would be possible to reconstructa network from time series without any perturbation to the nodal dynamics.Addressing these issues is of both theoretical and practical importance.§.§ Reconstructing complex networks via system clone Exploiting synchronization between a driver and a response system throughfeedback control led to a method for reconstructing complexnetworks <cit.>. The basic idea was to design a replica or aclone system that is sufficiently close to the original network withoutrequiring knowledge about the network structure. From the clone system, the connectivities and interactions among the nodes can be obtained directly,realizing the goal of network reconstruction.The conventional method to deal with driver-response systemsis to design proper feedback control to synchronize the state of theresponse system with that of the driver system. To achieve reconstruction, the elements in the adjacency matrix characterizing the network topologyof the clone system are treated as the variables to be synchronized withthose of the original system. With an appropriate feedback control, theadjacency matrix of the clone system can be “forced” to converge to theunknown adjacency matrix in the original system due to synchronization.For this approach to be effective, the local dynamics ofeach node needs to be known. In addition, the local dynamical andthe coupling functions need to be Lipschitz continuous <cit.>.More details of the synchronization approach can be described, as follows. Consider a networked system described by a set of differential equations𝐱̇_i= 𝐟(𝐱_i) + ∑_j=1^N a_ij𝐡_j (𝐱_j),where 𝐱_i denotes the state of node i (i=1,⋯, N), a_ijis the ijth element of the adjacency matrix 𝒜,𝐟(𝐱_i) represents the local dynamics of node i, and𝐡_j is the coupling function. In order to realize stablesynchronization, the functions 𝐟_i and 𝐡_i arerequired to be Lipschitzian for all nodes. To design a clone system requires that the functions 𝐟_i and 𝐡_i be known a priori.The clone system under feedback control can be written as 𝐲̇=𝐟_i(𝐲_i)+ ∑_j=1^N b_ij𝐡_j(𝐲_j) + 𝐪_i + 𝐮_i, ḃ_ij= -γ_ij𝐡_j(𝐲_j)·𝐞_i,where γ_ij are positive coefficients used to control the evolutionof b_ij so as to “copy” the dynamics of the original system,𝐪_i represents the modeling errors, and𝐞_i ≡𝐲_i - 𝐱_i denotes the differencebetween the states of the clone and the original systems. To ensure that theclone system can approach the original system, the following Lyapunovfunction can be exploited:Ω̇=∑_i=1^N e_i^2 +∑_i=1^N ∑_j=1^N(1/γ_ij)(b_ij-a_ij)^2.The Lipschitzian and stability constraints give rise to a negative Lyapunovexponent and the feedback control as𝐮_i = -k_1 𝐞_i-1/4ε_1δ^2𝐞_i,where k_1>L_1 + NL_2, L_1 and L_2 are the positive constants in theLipschitzian constraint. It can be proven <cit.> that feedbackcontrol in the form of Eq. (<ref>) in combination with the Lipschitzian constraint can guarantee thatthe Lyapunov exponent is zero or negative so that the clonesystem converges to the original system with small errors. Numerical tests in <cit.> demonstrated the working of the synchronization-basedreconstruction method. In an alternative approach <cit.>, it was argued that if the localdynamics of the nodes are available, a direct solution of the networktopology is possible without the need of any clone system. The basic ideacame from the fact that, if the local dynamics and the coupling functionsare available, then the only unknown parameters are the coupling strengthsassociated with the adjacency matrix. With data collected at different times,a set of equations for the coupling strengths can be obtained, which can be solved by using the standard Euclidean L_2-norm minimization. Numericaltests showed that this reconstruction method is effective for both transientand attracting dynamics <cit.>. §.§ Network reconstruction via phase-space linearizationFor nonlinear dynamical networks, there was a method <cit.> based on chaotic time-series analysis, where time series were assumed to be available from some or all nodes of the network. The nodal dynamics were assumed to be described by autonomous systems with a few coupling terms. That is, the network is sparse. The delay-coordinate embedding method <cit.> was then used to reconstruct the phase space of the underlying networked dynamical system. The principal idea was to estimate the Jacobian matrix of the underlying dynamics, which governs the evolutions of infinitesimal vectors in the tangent space along a typical trajectory of the system. Mathematically, the entries of the Jacobian matrix are mutual partial derivatives of the dynamical variables on different nodes in the network. A statistically significant entry in the matrix implies a connection between the two nodes specified by the row and the column indices of that entry. Because of the mathematical nature of the Jacobian matrix, i.e., it is meaningful only for infinitesimal tangent vectors, linearization of the dynamics in the neighborhoods of the reconstructed phase-space points is needed, for which constrained optimization techniques <cit.> were found to be effective <cit.>. The approach <cit.> was based on usingL_1-minimization in the phase space of a networked system to reconstructthe topology without knowledge of the self-dynamics of the nodes and withoutusing any external perturbation to the state of nodes. In particular, onedata point x_*^t is chosen from the phase space at time t and fromthe neighborhood of N nearest data points from the time series. Fornode i (the ith coordinate), the dynamics at t around x_*^t canbe expressed by using a Taylor series expansion, yielding𝐱_i^t+1=𝐅_i(𝐱_*^t) +D𝐅_i(𝐱_*^t)· (𝐱_i^t - 𝐱_*^t)+ O(𝐱_i^t - 𝐱_*^t ^2),where the higher-order terms can be omitted when implementing thereconstruction. A key parameter affecting the reconstruction accuracy is size of the neighborhood in the phase space. If the size is relatively large,more data points will be available but the equations will have largererrors. If the size is small, the amount of available data will bereduced. There is thus a trade-off for choosing the neighborhood size.Selecting a different time t, a set of equations in the form ofEq. (<ref>) can be established for reconstruction. The directneighbors of node i (the ith column of the adjacency matrix 𝒜)can be recovered by estimating the coefficients for 𝐅_i in thelocal neighborhoods in the phase space. The coefficients can be solvedfrom the set of equations by using a standard L_1-minimization procedure.After the coefficients have been estimated, it is necessary to set athreshold to discern the neighbors of node i, where true and falsepositive rates can be used to quantify the reconstruction performance.Discrete dynamical systems were used to demonstrate the working of the reconstruction method <cit.>. §.§ Reconstruction of oscillator networks based on noise induceddynamical correlationThe effect of noise on the dynamics of nonlinear and complex system hasbeen intensively investigated in the field of nonlinear science.For example, the interplay between nonlinearity and stochasticity can lead to interesting phenomenon such as stochasticresonance <cit.>, where a suitable amount of noise can counter-intuitively optimize thecharacteristics of the system output such as the signal-to-noise ratio.Previous studies also established the remarkable phenomenon ofnoise-induced frequency <cit.> or coherenceresonance <cit.>. Specifically, when a nonlinear oscillator is under stochastic driving, a dominant Fourier frequency in its oscillations can emerge, resulting in a signal that can be much more temporarily regular than that without noise <cit.>. A closely related phenomenon is noise-induced collective oscillation or stochastic resonance in the absence of an external periodic driving in excitable dynamical systems <cit.>. In complex networks,there was a study of the effect of noise on the fluctuation of nodalstates about the synchronization manifold <cit.>, with a number of scaling properties uncovered. The rich interplay between nonlinear dynamics and stochastic fluctuationspointed at the possibility that noise may be exploited for networkreconstruction, leading to the development of a number ofmethods <cit.>. In general, these methods do not assume any a priori knowledge about the nodal dynamics and there is no need to impose any external perturbation. Under the condition that the influence of noise on the evolution of infinitesimal tangentvectors in the phase space of the underlying networked dynamical system is dominant, it can be argued <cit.> that the dynamicalcorrelation matrix that can be computed readily from the available nodaltime series approximates the network adjacency matrix, fully unveilingthe network topology. In a general sense, noise is beneficial for network reconstruction. Supposethat all nodes in an oscillator network are in a synchronous state. Without external perturbation, the coupled oscillators behave as a single oscillator so that the effective interactions among the oscillators vanish, rendering impossible to extract the interaction pattern from measurements. However,noise can induce desynchronization so that the time series would containinformation about the underlying interaction patterns. It was demonstrated <cit.> that noise can bridge dynamicsand the network topology in that nodal interactions can be inferred from the noise-induced correlations. Consider N nonidentical oscillators, each of which satisfies 𝐱̇_i = 𝐅_i(𝐱_i) in the absence of coupling, where 𝐱_i denotes the d-dimensional state variable of the ith oscillator. Under noise, the dynamics of the whole coupled-oscillator system can be expressed as:𝐱̇_i = 𝐅_i(𝐱_i) - c ∑^N_j=1 L_ij𝐇(𝐱_j) + η_i,where c is the coupling strength, 𝐇: ℝ^d →ℝ^d denotes the coupling function of oscillators, η_i is the noise term, L_ij = -1 if j connects to i (otherwise 0) for i ≠ j and L_ii = -∑_j=1,j≠ i^NL_ij. Due to nonidentical oscillators and noise, an invariant synchronization manifold does not exist. Let 𝐱̅_i be the counterpart of 𝐱_i in the absence of noise, and assume a small perturbation ξ_i, we can write 𝐱_i = 𝐱̅_i + ξ_i. Substituting this into Eq. (<ref>), we obtain:ξ̇ = [ Dℱ̂(𝐱̅) -cℒ̂⊗ DĤ(𝐱̅) ] ·ξ + η,where ξ = [ξ_1,ξ_2, … , ξ_N ]^T denotes the deviation vector, η = [η_1,η_2,…, η_N]^T is the noise vector, ℒ̂ names the Laplacian matrix of elements L_ij (i,j=1,…,N), Dℱ̂(𝐱̅) = diag [Dℱ̂_1(𝐱̅_1), Dℱ̂_2(𝐱̅_2),⋯,Dℱ̂_N(𝐱̅_N)] and Dℱ̂_i are d× d Jacobian matrices of𝐅_i, ⊗ denotes direct product, andDĤ is the Jacobian matrix of the coupling function𝐇.Let Ĉ denote the dynamical correlation matrix of oscillators⟨ξξ^T ⟩, wherein C_ij = ⟨ξ_i ξ_j ⟩and ⟨·⟩ is the time average. One obtains0 =⟨ d(ξξ^T)/dt⟩ = -·Ĉ- Ĉ·Â^T + ⟨ηξ^T ⟩ + ⟨ξη^T ⟩,where  = -Dℱ̂(𝐱̅) + c ℒ̂⊗ DĤ(𝐱̅). ξ(t) can be solved from Eq. (<ref>), yielding the expression of ⟨ηξ^T ⟩ and ⟨ξη^T ⟩: ⟨ξη^T ⟩= ⟨ηξ^T ⟩= 𝒟̂/2. As a result, Eq. (<ref>) can be simplified to:·Ĉ + Ĉ·Â^T = 𝒟̂.The general solution of Ĉ can be written as (Ĉ) =(𝒟̂)/(Î⊗ +  ⊗Î),where (𝒳̂) is a vector containing all columns of matrix 𝒳̂.For one-dimensional state variable and linear coupling, with Gaussian whitenoise 𝒟̂=σ^2 Î, and negligible intrinsicdynamics Dℱ̂, Eq. (<ref>) can be simplified to ℒ̂·Ĉ +Ĉ·ℒ̂^T = σ^2Î/c. For undirected networks, the solution of Ĉ becomesĈ = σ^2/2cℒ̂^†,where ℒ̂^† denotesthe pseudo inverse of matrix ℒ. SinceĈ can be calculated from time series, and σ and care constant, setting a threshold in the element values of matrixĈ for identifying realnonzero elements can lead to full reconstruction of all the connections(Fig. <ref>). A key issue is how to determinethe threshold.In Refs. <cit.>, it was argued that the threshold canbe set based on the diagonal elements (autocorrelation) in Ĉ:C_ii≃σ^2/2ck_i(1+ 1/⟨ k⟩).The formula of C_ii obtained from a second-orderapproximation is consistent with the finding of noise-inducedalgebraic scaling law in Ref. <cit.>.Specifically, from Eq. (<ref>), let S ≡∑_i=1^N 1/C_ii = 2cl^2/[σ^2 (N+l)], where l=∑_i=1^Nk_i =N⟨ k⟩ is twice the total number oflinks. The integral part of l can be identified via l=(Sσ^2 + √(S^2 σ^4 + 8cNSσ^2 ))/4c. The threshold C_M^† (or [σ^2/(2c)]C_M^†) is chosen such that ∑^M_m=1Φ(C_m^†) = l, where Φ(C_m^†) is theunnormalized distribution of C_m^†. Then the connection matrix ℒ can be obtained (see Fig. <ref>). Figure <ref> shows the scaling property of autocorrelation versusthe node degree k for different nodal dynamics and different types ofnetworks, as predicted by Eq. (<ref>). The numerical results arein good agreement with the theoretical prediction.The reconstruction method by virtue of noise was improved <cit.>,based on reconstruction formula (<ref>). It was argued that,to use the formula, the coupling strength c and the noise varianceσ should be known a priori, a condition that may be difficult to meet in realistic situations. A simple method to eliminate both c andσ without requiring any a priori parameters was thenintroduced <cit.>. Specifically, note thatσ^2/2cC_ii^+=k_i,σ^2/2cC_ij^+=-A_ij,i≠ j,where σ and c can be simultaneously eliminated by the ratior_ij of the diagonal and off-diagonal elements of Ĉ^+in Eq. (<ref>):r_ij={[0, i and j are disconnected,; -1/k_i,i and j are connected. ].A threshold is still necessary to fully separate links and zero elementsin the adjacency matrix. A heuristic method was developed <cit.>for determining the threshold. While based on the same principle, theimproved reconstruction method appears indeed more practical.An alternative method to reconstruct the network topology of a dynamicalsystem contaminated by white noise was developed <cit.> throughmeasurement of both the nodal state x and its velocity ẋ, with the following formula: Â= ℬ̂·Ĉ^-1,where Ĉ = ⟨𝐱𝐱^ T⟩ isthe state-state correlation matrix, andℬ̂ = ⟨ẋ x^ T⟩ is thevelocity-state correlation matrix. An interesting feature is that, bymeasuring the velocity, the white noise variance is not requiredto be known and the formula applies (in principle) to any strength of noise.However, measuring the instantaneous velocity of the state variables canbe difficult and the errors would affect the reconstruction performance dramatically. There is in fact a trade-off between the generalityof the reconstruction method and the measurement accuracy. Nevertheless,the work <cit.> is theoretically valuable, and it was validatedby using a linear system and a nonlinear cell cycle dynamics model. Inthe nonlinear model, there are both active andinhibitive links, corresponding to positiveand negative elements in the adjacency matrix. It was demonstrated that bothclasses of links can be successfully reconstructed <cit.>. §.§ Reverse engineering of complex systems *Automated reverse engineering of nonlinear dynamical systems. Reconstructing nonlinear biological and chemical systems is of greatimportance. An automated reverse engineering approach was developedto solve the problem by using partitioning, automated probing andsnipping <cit.>. Firstly, partitioning allows thealgorithm to characterize the variables in a nonlinear system separately bydecoupling the interdependent variables. It was argued that Bayesian networkscannot model mutual dependencies among the variables, a common patternin biological and other regulation networks with feedback. It was alsoarticulated that the partitioning procedure is able to reveal the underlyingstructure of the system to a higher degree as compared with alternativemethods. Secondly, instead of passively accepting data to model a system,automated probing uses automatically synthesized models to create a testcriterion to eliminate unsuitable models. Because the test cannot beanalytically derived, candidate tests are optimized to maximize theagreement between data and model predictions. Thirdly, snippingautomatically simplifies and optimizes models. For all models obtained byautomated probing, one calculates their prediction errors against systemdata, perturbs the existent models randomly, evaluate the newly createdmodels, and compare the performance of the modified models with thatof the corresponding original models. In this way, models are constantlyevolving, with those with better performance replacing the inferior ones. Thisprocess tends to yield more accurate and simpler models. The process isakin to a genetic algorithm, but in the former, models evolve and areconstantly improved, whereas in the latter, some adaptive functions evolve.Four synthetic systems and two physical systems wereused <cit.> to test the automated reverse engineeringapproach. The method was demonstrated to be robust against noise and scalable to interdependent and nonlinear systems with many variables. The limitations of the method were discussed as well. Especially, the method isrestricted to systems in which all variables are observable. In addition,discrete time series as input to the method can lead to inconsistence with the synthesized models. Without including any fuzzy effect, the processmay not yield explicit models. Potential solutions to these problems weresuggested <cit.>.*Constructing minimal models for complex system dynamics. In Ref. <cit.>, the authors introduced a method of reverseengineering to recover a class of complex networked system described byordinary differential equations of the formdx_i/dt= M_0(x_i(t))+ ∑_j=1^N A_ij M_1(x_i(t))M_2(x_j(t)),where the adjacency matrix 𝒜 defines the interacting components, M_0(x_i(t)) describes the self-dynamics of each node,M_1(x_i(t))M_2(x_j(t)) characterizes the interaction between nodes iand j. Equation (<ref>) was argued to be suitable formany social, technological and social systems.The system dynamics is uniquely characterized by three independentfunctions and the aim is to construct the model in the form m=(M_0(x),M_1(x), M_2(x)),corresponding to a point in the model space 𝕄. The goal wasto develop a general method to infer the subspace 𝕄(𝒳)by relying on minimal a priori knowledge of the structure of M_0(x),M_1(x), and M_2(x). The key lies in using the system's response toexternal perturbations, a common technique used in biological experimentssuch as genetic perturbation, which is feasible for technological andsocial networked systems as well. The link between the observed systemresponse and the leading terms of m was established, enablingthe formulation of 𝒳 into a dynamical equation. Contrary totraditional reverse engineering, the approach <cit.> givesthe boundaries of 𝕄(𝒳) rather than a specific modelm.To infer m, the three functions were expressed <cit.> interms of a Hahn series <cit.>M_0(x) =∑_n=0^∞ A_n(x_0 - x)^Π_0(n)M_1(x) =∑_n=0^∞ B_n(x_0 - x)^Π_1(n)M_2(x) =∑_n=0^∞ C_n(x_0 - x)^Π_2(n).The challenge was to discern the coefficients A_n, B_n and C_n foruncovering the functional form (<ref>), which can be overcomeby exploiting the system's response to external perturbation.Under perturbation, the temporal dynamics is characterizedby the time-dependent relaxation from the original steady state, x_i,to the perturbed steady-state x_i(t →∞) =x_i + dx_i.System (<ref>) can then be linearized about the steadystate. After relaxation, the system's new state is characterized by aresponse matrix. Based on the permanent perturbation, m can beestimated viaM_0(x)∼ 1/(R(x))^2∫[(R(x))^2+θ+ 𝒪((R(x))^ϕ+(2+θ)) ]dx M_1(x)∼M_0(x)R(x) M_2(x)∼ {[ (R(x))^δ+1-φ +𝒪( (R(x)^ϕ + (δ +1 -φ))),β =0; y_0-(R(x))^β +𝒪((R(x))^ϕ + β ),β >0 ].whereR(x)∼{[x^-1/ξ +𝒪(x^ϕ(-1/ξ)) , δ=0; (x_0-x)^1/δ +𝒪((x_0-x)^ϕ(1/δ)),δ >0 ].and x_0 and y_0 are arbitrary constants. A set of parameters need to be determined, which exhibit certain scalingproperties. Specifically, the relaxation time scales with θ, i.e.,τ_i ∼ k_i^θ, the steady-state activity either scales withξ as x_i ∼ k_i^ξ or has the relation x_i ∼ k_i^ξ,node i's impact on neighbors obeys I_i ∼ k_i^φ; and i's stability against perturbation in its vicinity leads toS_i ∼ k_i^δ (see Fig. <ref>).The determination of the parameters gives only the leading terms ofM_0(x), M_1(x) and M_2(x). Additional terms can be usedto better model complex systems. The reconstructed model is calleda minimal model because it captures the essential features of the underlyingmechanism without any additional constraint. The subspace given by thereconstruction method is robust to parameter selection. Models subject tothe subspace can produce consistent data with observations <cit.>.§ INFERENCE APPROACHES TO RECONSTRUCTION OF BIOLOGICAL NETWORKSHigh-throughput technologies such as microarrays and RNA sequencing producea large amount of experimental data, making genome-scale inference oftranscriptional gene regulation possible. The reconstruction of biologicalinteractions among genes is of paramount importance to biological sciences at different levels. A variety of approaches aiming to reconstruct geneco-expression networks or regulation networks have been developed.In general, there are three types of experimental gene data: geneco-expression data, gene knockout data, and transcriptional factors. Gene interaction networks can be classified into two categories:(1) co-expression networks, in which the nodes represent genes and theedges represent the degree of similarity in the expression profiles ofthe genes, and (2) transcription-regulatory networks, in which the nodesrepresent either transcription factors or target genes, and edgescharacterize the causal regulatory relationships. Reconstructing neuronal networks and brain functional networks fromobservable data has also been an active area of research. Typicalexperimental data include blood oxygen level dependent (BOLD) signals,electroencephalography (EEG) and stereoelectroencephalography (SEEG) data,magnetoencephalography (MEG) data, spike data, calcium image data, etc.To uncover the interaction structure among the neurons or distinct braindomains from the signals, a number of reconstruction approaches have beendeveloped and utilized in neuroscience. §.§ Correlation based methodsCorrelation based methods are widely used for inferring the associations between two variables.§.§.§ Value based methods Pearson's correlation coefficient measuring the strength of thelinear relationship between two random variables is widely used in manyfields <cit.>. The correlation is defined ascorr=∑^n_i=1(x_i - x̅)(y_i-y̅)/(n-1)S_xS_y,where x̅ and y̅ are the sample means and S_x and S_yare the standard deviations of x and y, respectively. The Pearson'scorrelation assumes that data is normally distributed and is sensitiveto outliers.Distance Covariance (dCov) provides a nonparametric test to examinethe statistical dependence of two variables <cit.>. Forsome given pairs of measurement (x_i, y_i), i=1, 2, …, n forvariables X and Y, let 𝒜 denote the pairwiseEuclidean distance matrix of X with a_ij=|x_i - x_j| and ℬbe the corresponding matrix for Y,where |.| denote the Euclidean norm. Define the doubly centered distancematrix 𝒜^ c whose elements are given by a^c_ij = a_ij-a̅_i. - a̅_.j+a̅_..,where a̅_i. is i^th row mean, a̅_.j is the j^thcolumn mean, and a̅_.. is the grand mean of 𝒜.Centered distance matrix ℬ^ ccan be defined similarly. The squared sample distancecovariance is defined to be the arithmetic average of the products of𝒜^ c and ℬ^ c:dcov^2(X,Y)=1/n^2∑_ija^c_ijb^c_ij. The Theil-Sen estimator, proposed inRefs. <cit.>, is defined asβ̂_1 =median{m_ij=y_i- y_j/x_i - x_j: x_ix_j, 1≤ i ≤ j ≤ n},where the median can characterize the relationship between the two variables. This estimate is robust, unbiased, and less sensitive to outliers. Partial correlation and information theory(PCIT) <cit.> extracts all possible interactiontriangles and applies Data Processing Inequality (DPI) to filter indirectinteractions using partial correlation. The partial correlation coefficientc^ p_ij between two genes i and jwithin an interactiontriangle (i,j, k) is defined asc^p_ij= corr(x_i, x_j)- corr(x_i, x_k) corr(x_j, x_k)/√((1- corr(x_i, x_k))^2(1- corr(x_j, x_k))^2),where corr(.,.) is Pearson's correlation coefficient.§.§.§ Rank based methods Compared with value based correlation measures, rank based correlationmeasures are more robust and insensitive to outliers. TheSpearman and Kendall measures are the most commonly used. In particular,Spearman's correlation is simply the Pearson's correlation coefficientincorporated with ranked expression <cit.>, and Kendall's τcoefficient <cit.> is defined asτ(x_i, x_j) =con(x^r_i, x^r_j)- dis(x^r_i, x^r_j)/1/2n(n-1),where x^r_i and x^r_j are the ranked expression profiles of genes iand j, con(.,.) and dis(.,.) represent the numbers ofconcordant and disconcordant pairs, respectively.Inner Composition Alignment (ICA) <cit.> was proposed toinfer directed networks from short time series by extending the Kendall'sτ measure. Given time series x^l and x^k of length n from subsystems l and k over the same time intervals, let π^l be thepermutation that arranges x^l in a nondecreasing order. The seriesg^k,l=x^k(π^l) is the reordering of the time series x^k withrespect to π^k. The ICA is formulated asτ^l → k= 1 - ∑^n-2_i=1∑^n-1_j=i+1w_ijΘ[(g^k,l_j+1-g^k,l_i)(g^k,l_i - g^k,l_j)]/1/2(n-1)(n-2),where w_ij denotes the weight between points i and j, and Θ[x]is the Heaviside step function. Compared with Kendall's τ measure,ICA can infer direct interactions and eliminate indirect interaction by usingthe partial version of ICA.Hoeffding's D coefficient is a rank-based nonparametric measure ofassociation <cit.>. The statistic coefficient D isdefined asD = (n-2)(n-3)D_1+D_2-2(n-2)D_3/n(n-1)(n-2)(n-3)(n-4),whereD_1 = ∑^n_i=1Q_i(Q_i-1),D_2 = ∑^n_i=1(R_i-1)(R_i-2)(S_i-1)(S_i-2),D_3 = ∑^n_i=1(R_i-2)(S_i-2)Q_i,R_i is the rank of X_i, S_i is the rank of Y_i, and the bivariaterank Q_i is the number of both X and Y values less than thei^th point.It was pointed out <cit.> that real gene interactions may change asthe intrinsic cellular state varies or may exist only under a specificcondition. That is, for a long time series of the co-expression data,perhaps only partof the data are meaningful for revealing interactions. In this regard, twonew co-expression measures based on the matching patterns of local expressionranks were proposed. Specifically, when dealing with time-course data, themeasure W_1 is defined asW_1 = ∑^n-k+1_i=1I(ϕ(x_i,…,x_i+k-1)=ϕ(y_i,…,y_i+k-1)) +I(ϕ(x_i,…,x_i+k-1)=ϕ(-y_i,…,-y_i+k-1)),where I(.) is an indicator function, and ϕ is the rank function thatreturns the indices of the elements after they have been sorted in anincreasing order, and W_1 counts the number of continuous subsequences oflength k with matching and reverse rank patterns. For non-time-series data,where the order is not meaningful, a more general measure, W_2 can bedefined:W_2 = ∑_1≤ i_1<… < i_k ≤ nI(ϕ(x_i_1,…,x_i_k) =ϕ(y_i_1,…,y_i_k))+I(ϕ(x_i_1,…,x_i_k) =ϕ(-y_i_1,…,-y_i_k)).§.§ Causality based measures Wiener-Granger Causality (WGC), pioneered byWiener <cit.> and Granger <cit.>,is a commonly used measure to infer the causal influence betweentwo variables, e.g., the expression time series ofgenes and the spiking neural time series. The basic idea of WGC isstraightforward. Given two time series {X_t} and {Y_t}, if using thehistory of both X and Y is more successful to predict X_t+1 thanexclusively using the history of X, Y is said to be G-cause X. Theidea of WGC is similar to that of the transfer entropy, but WGC uses somecorrelation measure instead of mutual information. WGC is widely used inmeasuring the functional connectivity among subdomains of brain based onEEG, MEG, SEEG data <cit.>. Fundamentally, Granger test is a linear method operated on the hypothesis that the underlyingsystem can be described as a multivariate stochastic process. Thus, inprinciple, there is no guarantee that the method would be effective fornonlinear systems, in spite of efforts to extend the methodology to strongly coupled systems <cit.>.In the traditional Granger framework, measurement noise is generally detrimental in the sense that, as its amplitude is increased the value of the detected causal influence measure decreases monotonically, leading to spurious detection outcomes <cit.>. The transfer entropy framework is applicable <cit.> to both linear and nonlinear systems, but often the required data amount is prohibitively large. In the special case of Gaussian dynamical variables, the two methods, one of the autoregressive nature (Granger test) and another based on information theoretic concepts (transfer entropy, to be discussed below), are in fact equivalent to each other <cit.>. An alternative information theoretic measure, the causation entropy, was also proposed <cit.>.Convergent cross mapping framework (CCM) is based on delay coordinateembedding, the paradigm of nonlinear time seriesanalysis <cit.>. The CCM method can deal with both linear and nonlinear systems with small data sets, and it has been applied to data from different contexts, such as EEG data <cit.>, FMRI <cit.>, fishery data <cit.>, economic data <cit.>, and cerebral auto-regulation data <cit.>. It was also found that properly applied noise can enhance the CCM performance in inferring causalrelations <cit.>.The nonlinear dynamics based CCM method was proposed <cit.> todetect and quantify causal influence between a pair of dynamical variables through the corresponding time series. The starting point is to reconstruct a phase space, for each variable, based on the delay-coordinate embedding method <cit.>. Specifically, for time series x(t), the reconstructed vector is X(t)=[x(t),x(t-τ),...,x(t-(E_x-1)τ)], where τ is the delay time and E_x is the embedding dimension. For variable y, a similar vector can be constructed in the E_y dimensional space. Let M_X and M_Y denote the attractor manifolds in the E_x- and E_y-dimensional space, respectively. If x and y are dynamically coupled, there is a mapping relation between M_X and M_Y. The CCM method measures how well the local neighborhoods in M_X correspond to those in M_Y. In particular, the cross-mapping estimate of a given Y(t), denoted as Ŷ(t)|M_X, is based on a simplex projection <cit.> that is essentially a nearest-neighbor algorithm involving E+1 nearest neighbors of X(t) in M_X. (Note that E+1 is the minimum number of points required for a bounding simplex in the E-dimensional space.) The time indices of the E+1 nearest neighbors are denoted as t_1, t_2 , ..., t_E+1 in the order of distances to X(t) from the nearest to the farthest, i.e., point X(t_1) is the nearest-neighboring point of X(t) in M_X. These time indices are used to identify the points (putative neighborhoods) in M_Y, namely, to find the points at the corresponding instants: Y(t_1), Y(t_2), ..., and Y(t_E+1), which are used to estimate Ŷ(t) through the weighted averageŶ(t)|M_X = ∑_i=1^E+1 w_i(t)· Y(t_i),wherew_i(t) = μ_i(t)/∑_j μ_j(t)is the weight of the vector Y(t_i),μ_i(t) =exp{-d[X(t),X(t_i)]/d[X(t),X(t_1)]},and d[X(t), X(t_i)] is the Euclidean distance between the two vector points X(t) and X(t_i) in M_X. An estimated time series ŷ(t) can then be obtained from Ŷ(t)|M_X. Likewise, the cross mapping from Y to X can be defined analogously so that the time series of x(t) can be predicted from the cross-mapping estimate X̂(t)|M_Y.The correlation coefficient between the original time series y(t) and the predicted time series ŷ(t) from M_X, denoted as ρ_Y|M_X, is a measure of CCM causal influence from y to x. Larger value of ρ_Y|M_X implies that y is a stronger cause of x, while ρ_Y|M_X≤ 0 indicates that y has no influence on x. The relative strength of causal influence can be defined as R=ρ_X|M_Y-ρ_Y|M_X, which is a quantitative measure of the casual relationship between x and y. A positive value of R indicates that x is the CCM cause of y. §.§ Information-theoretic based methods §.§.§ Mutual information Mutual Information (MI) <cit.> is used widelyto quantify the pairwise mutual dependency between two variables. For twovariables X and Y, the mutual information is defined in term ofMI(X;Y)=∑_x∈ X∑_y∈ Y p(x,y)log (p(x,y)/p(x)p(y)),where p(x,y) is the joint probability distribution functions of X and Y,and p(x) and p(y) are the marginal probability distribution function ofX and Y, respectively. Note that MI is nonnegative and symmetric:MI(X;Y)= MI(Y;X). The larger the value of MI, the more highlythe two variable are correlated. The key to applying the mutual information forquantifying associations in continuous data is to estimate the probabilitydistribution from the data. A difficult and open issue is how to obtainunbiased mutual information value. There are four commonly used MI-based network reconstruction methods:relevance network approach (RELNET) <cit.>, contextlikelihood of relatedness (CLR) <cit.>, maximum relevance/minimumredundancy feature selection (MRNET) <cit.>, and thealgorithm for accurate reconstruction of cellular networks(ARACNE) <cit.>. In RELNET, a threshold τ isused to distinguish actual links from null connections. However,this method has a limitation, i.e., the indirectrelationship may lead to high mutual information values and failures todistinguish direct from indirect relationships.The CLR method integrates z-score to measure the significance of thecalculated MI. For node i, the mean μ_i and the standard deviationδ_i of the empirical distribution of the mutual informationMI(x_i; x_k), (k=1,…, n) are calculated. The z-scorez_ij of MI(x_i;x_j) is defined asz_ij=max (0, MI(x_i;x_j)-μ_i/δ_i).The value of z_ji can be defined analogously. A combined score, whichquantifies the relatedness between i and j, is expressed asẑ_ij = √(z_ij^2 + z_ji^2).The MRNET method can be used to infer a network by repeating the maximumrelevance, minimum redundancy (MRMR) <cit.> feature selectionmethod for all variables. The key to the MRMR algorithm lies in selectinga set of variables with a high value of the mutual information with the targetvariable (maximum relevance) but meanwhile the selected variables aremutually maximally independent (minimum redundancy). The aim of thismethod is to associate direct interactions with high rank values, but associateindirect interactions with low rank values.The MRMR algorithm is essentially a greedy algorithm. Firstly, the MRMRselects the variable x_i with the highest MI value regarding the target y.Next, given a set of selected variables, a new variable x_k is chosento maximize the value of s^y_k = u^y_k - r^y_k, where u^y_k is therelevance term and r^y_k is the redundancy term. More precisely, one hasu^y_k = MI(x_k; y), r^y_k =1/|S|∑_x_l ∈ SMI(x_k; x_l).By setting each gene to be the target, one can calculate all the valuesof s^x_i_x_j, and the relatedness value between any pair of x_iand x_j is the maximum of (s^x_i_x_j, s^x_j_x_i). A modificationof MRNET can be found in Ref. <cit.>.ARACNE is the extension of RELNET. In the ARACEN, Data ProcessingInequality (DPI) <cit.>, a well-known information theoreticproperty, is used to overcome the limitation in the RELNET. Specifically,the DPI stipulates that if variables x_i and x_j interact only through athird variable x_k, one has MI(x_i; x_j)≤min(MI(x_i, x_k), MI(x_j, x_k)),which is used to eliminate indirect interactions. ARACNE starts from anetwork in which the value of MI for each edge is larger than τ. Theweakest edge in each triplet, for example the edge between i and j, isregarded as an indirect interaction and is removed ifMI(x_i; x_j)≤min( MI(x_i, x_k),MI(x_j, x_k))-ϵ ,where ϵ is a tolerance parameter.§.§.§ Maximal information coefficient Maximal Information Coefficient (MIC) is a recently proposedinformation based method, belonging to the larger class of maximalinformation-based, non-parametric exploration statistics <cit.>.The basic idea of MIC is that, if a relationship exists between twovariables, a grid can be drawn on the scatterplot of the two variablesto partition the data so that the relationship between the two variablesin the two-dimensional diagram can be uncovered.The MIC of a given pair of data D is calculated, as follows. First,one partitions the x-values of D into x bins and the y-values intoy bins according to a uniform partition grid G(x,y). One thencalculates the mutual information with respect to this partition,which is denoted as I(D|_G). For fixed D with a different grid G,even with the same x bins and y bins, the value of the mutualinformation will be different. A characteristic matrix ℳ(D) canbe defined asℳ(D_x,y)=I^*(D, x, y)/log min(x,y),where I^*(D, x, y)=max{I(D|_G)} is the maximum over all grids G withx and y bins. The value of MIC of the variable pair D is given byMIC(D)= max{M(D)_x,y}.There were claims <cit.> that the MIC does notoutperform MI, suggesting that the mutual information may be a morenatural and practical way to quantify statistical associations.§.§.§ Maximum entropy One way to model the spiking activity of a population of neurons isthrough the classic Ising model <cit.>, in terms of the following maximum entropy distribution:P(σ_1,σ_2,…,σ_N)=1/Zexp[∑_ih_iσ_i+1/2∑_i jJ_ijσ_iσ_j ],where σ_i=± 1 is the spiking activity of cell i and thenormalized factor is the partition function Z in statistical physics.The effective coupling strengths J_ij are then chosen so that theaverages (⟨σ_i ⟩,⟨σ_i σ_j ⟩)in this distribution agree with the results from numerical or actualexperiments. §.§ Bayesian network The Bayesian networks (BNs) <cit.> area commonly used probabilistic graphical model, represented as a directacyclic graph (DAG), in which nodes represents a set of random variablesand the direct links signify their conditional dependencies. For example, inthe case of inferring a gene regulatory network, a Bayesian network couldrepresent the probabilistic relationships between transcription factorsand their target genes. The BN method is associated with, however, highcomputational cost (especially for large networks). Unlike the alternative pairwise association estimation methods discussed above, the BN methodrequires that all possible DAGs be exhaustively or heuristically searched, scored, and kept either as a best-scoring network or a network constructedby averaging the scores of all the networks. In order to optimize theposterior probabilities in BN, a variety of heuristic search algorithmswere developed, e.g., simulated annealing, max-min parent and childrenalgorithm <cit.>, Markov blanketalgorithm <cit.>, Markov boundary inductionalgorithm <cit.>. The optimization algorithmsintegrated into BN notwithstanding, the method is practically applicablebut only to biological networks of relatively small size. §.§ Regression and resampling Interring gene regulatory networks can be treated as a feature selectionproblem. The expression level of a target gene can be predicted by itsdirect regulate transcription factors. It is often assumed that the links among the genes are sparse. For example, the lasso <cit.>is a widely used regression method for network inference, and it is ashrinkage and selection method for linear regression. Specifically,it minimizes the usual sum of squared errors, with a bound on the L_1 normof the coefficients, which is used to regularize the model. Given theexpression data of genes, in a steady state experiment, the featureselection can be formulated asmin1/2n ||y - Xw||^2_2 + λ ||w||_1,where λ is an adjustable parameter for controlling the sparsityof the coefficients. A direct use of Lasso to infer networks has twoshortcomings: (a) it is known to be an unstable procedure in terms ofthe selected features, and (b) it does not provide confidence scores forthe selected features. As a result, a stability selection procedure isoften integrated into Lasso <cit.>. One first resamples thedata into several sub-data based on bootstrap, then apples the Lassoto solve these sub-data regression problems, and aggregates the finalscore for each feature to select more confident features.Beside the steady-state data, time series data can also be tackled usingthe Lasso in the sense that the current expression level of thetranscription factors is the predictors of the change in the expression ofthe target gene. A group Lasso method using both steady-state and timeseries data was proposed <cit.>, in which the paircoefficients of a single transcription factor across both steady-stateand time-series data are either both zero or both non-zero.The gene regulation can also be modeled with nonlinear models, e.g.,polynomial regression models <cit.> and sigmoidfunctions <cit.>. §.§ Supervised and semi-supervised methods Supervised and semi-supervised methods treat reconstruction asa classification problem. A number of methods were developed to accomplishthis goal, with some works showing that the performance of the supervisedmethods is better than that of theunsupervised methods <cit.>. Generally, prior to applying a supervised method, two types of training datasets are required: the expression profile of each gene and a list of priorinformation about whether the known transcriptional factors and genes are regulated. Support Vector Machine (SVM) is a prevailing supervised classificationmethod that has been successfully used in inferring gene regulatory networks.The basic idea of SVM is to find an apparent gap that divide the data points ofthe separate categories as widely as possible. By integrating thekernel function, SVM can be extended to nonlinear classification. Gene Network Inference with Ensemble of Trees (GENIE), similar to MRNET,identifies the best subset of regulator genes with random forest andextra-trees for regression and feature selection instead of MI and MRMR. The method of Supervised inference of regulatory networks(SIRENE) <cit.> uses SVM as a classifier to learn the decisionboundary given the training dataset, and then generates the labels(whether links exist or do not) for the prediction dataset. Through asemi-supervised method, the unlabeled data are also included into thetraining data <cit.>. §.§ Transfer and joint entropies Transfer Entropy (TE), as an information theoretic method, isoften used for reconstructing neuron networks <cit.>. Compared with the mutualinformation, TE can be employed to reveal the causal relationships from thehistorical time series of two variables. Given the time series {x_t}and {y_t} for x and y respectively, TE can be expressed as:TE_y->x=H(x_t|x_t-1:x_t-m) - H(x_t|y_t-1:y_t-l,x_t-1:x_t-m),where H(x) is the Shannon entropy of x, and l,m are the length ofthe historical time series. In the case of inferring neuronal networks, x_tcan be the number of spikes in a specific time window, and l and mare often set to be 1.Joint Entropy (JE) <cit.>. In calculating the MI orcorrelations for neural networks, the number of spikes in each time windowis used, but the temporal patterns between spikes are ignored. For JE, thecross-inter-spike-intervals (cISI), defined as cISI = t_y - t_xbetween two neurons, is used. The JE is calculated based on cISI asJE(X,Y) = H( cISI),where H is the entropy of cISI. In Ref <cit.>, it was shownthat TE and JE perform better than other methods. In Ref. <cit.>,the authors demonstrated that, by integrating high-order historical timeseries (l > 1 and m>1) and multiple time delays, TE can be improvedmarkedly. §.§ Distinguishing between direct and indirect interactions Based on a motif analysis, it was found <cit.> that threetypes of generic and systematic errors exist for inferring networks:(a) fan-out error, (b) fan-in error, and (c) cascade error. The cascadeerror originates from the tendency for incorrectly predicted “shortcuts”to cascades, also known as an error to misinterpret indirect links asdirect links. For example, if nodes 1 and 2, and nodes 2 and 3 in thetrue (direct) network are strongly dependent upon each other,then high correlations willalso be visible between nodes 1 and 3 in the observed (direct andindirect) network. Such errors are always present in the pairwisecorrelation, mutual information or other similarity metrics between apair of nodes.It was proposed <cit.> that a deconvolution methodcan be used to distinguish direct dependencies in networks. In particular, the weights of an observed network 𝒢_obs to be inferred bythe correlation or mutual information can be modeled as a sum of bothdirect weights 𝒢_dir in the real network andindirect weights 𝒢^2_dir,𝒢^3_dir, etc. due to indirect paths: 𝒢_obs=𝒢_dir + 𝒢^2_dir +𝒢^3_dir+… =𝒢_dir· (ℐ - 𝒢_dir)^-1.The direct network can then be obtained as𝒢_dir=𝒢_obs· (ℐ+𝒢_obs)^-1Through a matrix similarity transformation, one obtains𝒢_obs=𝒰·𝒮_obs·𝒰^-1, where 𝒮_obs is a diagonal matrixwhose elements are λ_obs, which holds similarly for𝒢_dir: 𝒢_dir=𝒰·𝒮_dir·𝒰^-1, andλ_dir=λ_obs/(1+λ_obs). The direct networkcan then be inferred.The problem of extracting local response from global response toperturbation was first addressed using a modular response analysis(MRA) <cit.>. Since then many works haveappeared <cit.>). A local response coefficient r_ijis defined asr_ij=lim_Δ x_j (∂lnx_i/∂lnx_j), i j,which quantifies the sensitivity of module i to the change in modulej with the states of other modules unchanged. However, in a realsituation, an external intervention to perturb a parameter p_jintrinsic to module j can cause a local change in x_j and thenpropagates to the whole system. After the network has relaxed into anew steady state, the global response coefficient, an often measuredquantity in practice, can be obtained asR_i p_j=dlnx_i/dp_j. The relationbetween the local and global responses can be established.The idea of extracting local from global responses using statisticalsimilarity measure based methods and a global silence method were articulated and developed <cit.>. Given the observed globalresponse matrix 𝒢, measured by the correlation or themutual information, the local response matrix 𝒮 can beobtained as𝒮 =[𝒢 - ℐ + 𝒟·(𝒢-ℐ)·𝒢]·𝒢^-1,where ℐ is the identity matrix and the function D(ℳ)sets the off-diagonal elements of matrix ℳ to be zero.Through this method, theindirect links become silenced so that the direct and indirect interactionscan be distinguished.§ DISCUSSIONS AND FUTURE PERSPECTIVESNearly two decades of intense research in complex networks have resulted in a large body of knowledge about network structures and their effectson various dynamical processes on networks. The types of processes thathave been investigated include synchronization <cit.>, virus spreading <cit.>, traffic flow <cit.>, and cascading failures <cit.>. A typical approach in the field is toimplement a particular dynamical process of interest on networks whoseconnecting topologies are completely specified. Often, real-world networkssuch as the Internet, the power grids, transportation networks, and variousbiological and social networks are used as examples to demonstrate therelevance of the dynamical phenomena found from model networks. While thisline of research is necessary for discovering and understanding variousfundamental phenomena in complex networks, the “inverse” or “reverse engineering” problem of predicting network structure and dynamics fromdata is also extremely important. The basic hypothesis underlying theinverse problem is that the detailed structure of the network and the nodal dynamics are totally unknown, but only a limited set of signals or time series measured from the network is available. The question is whether the intrinsic structure of the network and the dynamical processes can beinferred solely from the set of measured time series. Compared with the“direct” network dynamics problem, there has been less effort in theinverse problem. Data-based reverse engineering of complex networks, withgreat application potential, is important not only for advancing networkscience and engineering at a fundamental level but also for meeting theneed to address an array of applications where large-scale, complex networksarise. In this Review, we attempt to provide a review, as comprehensive aspossible, of the recent advances in the network inverse problem. A focus of this Review is on exploiting compressive sensing for addressing various inverse problems in complex networks (Secs. <ref> and <ref>). The basic principle of compressive sensingis to reconstruct a signal from sparse observations, which are obtainedthrough linear projections of the original signal on a few predetermined vectors. Since the requirements for observations can be relaxed considerably as compared with those associated with conventional signal reconstructionschemes, compressive sensing has received much recent attention and it isbecoming a powerful technique to obtain high-fidelity signal for applicationswhere sufficient observations are not available. The key idea rendering possible exploiting compressive sensing for reverseengineering of complex network structures and dynamics lies in the fact that any nonlinear system equation or coupling function can be approximated bya power-series expansion. The task of prediction becomes then that ofestimating all the coefficients in the power-series representations of thevector fields governing the nodal dynamics and the interactions. Since theunderlying vector fields are unknown, the power series can containhigh-order terms. The number of coefficients to be estimated can thereforebe quite large. Conventional wisdom would count this as a difficultproblem since a large amount of data is required and the computationsinvolved can be extremely demanding. However, compressive sensing is ideallysuited for this task. In this Review we have described in detail how compressive sensing can be used for uncovering the full topology andnodal dynamical processes of complex, nonlinear oscillator networks, for revealing the interaction patterns in social networks hostingevolutionary game dynamics, for detecting hidden nodes, for identifying chaotic elements in neuronal networks, for reconstructing complex geospatial networks and nodal positioning, and for mapping out spreading dynamics on complex networks based on binary data. In addition to the compressive sensing based paradigm, we have alsoreviewed a number of alternative methods for reconstruction of complexnetworks. These include methods based on response to external driving signals, system clone (synchronization), phase-space linearization,noise induced dynamical correlation, and reverse engineering of complexsystems. Representative methodologies to reconstruct biological networks have also been briefly reviewed: methods based on correlation, causality, information-theoretic measures, Bayesian inference, regression andresampling, supervision and semi-supervision, transfer and jointentropies, anddistinguishing between direct and indirect interactions. Data-based reconstruction of complex networks belongs to the broad fieldof nonlinear and complex systems identification, prediction, and control. There are many outstanding issues and problems. In the following we listseveral open problems which, in our opinion, are at the forefront of this area of research. §.§ Localization of diffusion sources in complex networksDynamical processes taking place in complex networks are ubiquitous in nature and in engineering systems, examples of which include disease or epidemic spreading in the human society <cit.>,virus invasion in computer networks <cit.>, andrumor propagation in online social networks <cit.>.Once an epidemic emerges, it is often of great interest to be able to identify its source within the network accurately and quickly so that proper control strategies can be devised to contain or even to eliminate the spreading process. In general, various types of spreading dynamics can be regarded as diffusion processes in complex networks, and it is of fundamental interest to be able to locate the sources of diffusion. A straightforward, brute-force search for the sources requires accessibility of global information about the dynamical states of the network. However, for large networks, a practical challenge is that our ability to obtain and process global information can often be quite limited, making brute-force search impractical with undesired or even disastrous consequences. For example, the standard breadth-first search algorithm for finding the shortest paths, when implemented in online social networks, can induce information explosion even for a small number of searching steps <cit.>. Recently, in order to locate the source of the outbreak of Ebola virus in Africa, five medical practitioners lost their lives <cit.>. All these call for the development of efficient methodologies to locate diffusion sources based only on limited, practically available information without the need of acquiring global information about the dynamical states of the entire network.There have been pioneering efforts in addressing the source localization problem in complex networks, such as those based on the maximum likelihood estimation <cit.>, belief propagation <cit.>,the phenomena of hidden geometry of contagion <cit.>, particlediffusion and coloration <cit.>, and inversespreading <cit.>. In addition, some approaches have been developed for identifying super spreaders that promote spreading processes stemming from sources <cit.>.In spite of these efforts,achieving accurate source localization from asmall amount of observation remains challenging. A systematic frameworkdealing with the localization of diffusion sources for arbitrary networkstructures and interaction strength has been missing.A potential approach <cit.>to addressing the problem of network source localization is to investigate thefundamental issue of locatability: given a complex network and limited(sparse) observation, are diffusion sources locatable? Answering this question can then lead to a potential solution to the practical and challenging problemof actual localization: given a network, can a minimum set of nodes beidentified which produce sufficient observation so that sources at arbitrarylocations in the network can actually be located? A two-step solution strategy was recently suggested <cit.>. Firstly, one develops aminimum output analysis to identify the minimum number of messenger/sensornodes, denoted as N_m, to fully locate any number of sources in anefficient way. The ratio of N_m to the network size N,n_m≡ N_m/N, thus characterizes the source locatabilityof the network in the sense that networks requiring smaller values ofn_m are deemed to have a stronger locatability of sources. Theminimum output analysis can be carried out by taking advantage of the dualrelation between the controllability theory <cit.> and the canonical observability theory <cit.>. Secondly, given N_m messenger nodes, one can formulate the source localization problem as a sparse signal reconstruction problem, which can be solved by using compressive sensing. The basic properties of compressive sensing allow one to accurately locate sources from a small amount of measurement from themessenger nodes, much less than that required by the conventional observabilitytheory. Testing these ideas using variety of model and real-world networks,the authors found that the connection density and degree distribution play asignificant role in source locatability, and sources in a homogeneous anddenser network are more readily to be located. A striking and counterintuitivefinding <cit.> was that, for an undirected network with one connectedcomponent and random link weights, a single messenger node is sufficient tolocate any number of sources.Theoretically, the combination of the minimum output analysis (derived from the controllability and observability theories for complex networks) and the compressive sensing based localization methodconstitutes a general framework for locating diffusion sources in complex networks. It represents a powerful paradigm to quantify the source locatability of a network exactly and to actually locate the sources efficiently and accurately. Because of the compressive sensing based methodology,the framework is robust against noise, paving the way to practical implementation in a noisy environment. The framework also provides significant insights into the open problem ofdeveloping source localization methods for time variant complex networkshosting nonlinear diffusion processes. §.§ Data based reconstruction of complex networks with binary-state dynamics Complex networked systems consisting of units with binary-state dynamics are common in nature, technology, and society <cit.>. In such a system, each unit can be in one of the two possible states, e.g.,being active or inactive in neuronal and gene regulatorynetworks <cit.>, cooperation or defection in networks hosting evolutionary game dynamics <cit.>, being susceptible or infected in epidemic spreading on social and technological networks <cit.>, two competing opinions in social communities <cit.>, etc. The interactions among the units are complex and a state change can be triggered either deterministically (e.g., depending on the states of their neighbors) or randomly. Indeed, deterministic and stochastic state changes can account for a variety of emergent phenomena, such as the outbreak of epidemic spreading <cit.>, cooperation among selfish individuals <cit.>, oscillations in biological systems <cit.>, power blackout <cit.>, financial crisis <cit.>, and phase transitions in natural systems <cit.>. A variety of models were introduced to gain insights into binary-state dynamics on complex networks <cit.>, such as the voter models for competition of two opinions <cit.>, stochastic propagation models for epidemic spreading <cit.>, models of rumor diffusion and adoption of new technologies <cit.>, cascading failure models <cit.>, Ising spin models for ferromagnetic phase transition <cit.>, and evolutionary games for cooperation and altruism <cit.>. A general theoretical approach to dealing with networks hosting binary state dynamics was developed <cit.>based on the pair approximation and the master equations, providing agood understanding of the effect of the network structure on the emergentphenomena.Compressive sensing can be exploited to address the inverse problem ofbinary-state dynamics on complex networks, i.e., the problem of reconstructingthe network structure and binary dynamics from data. Of particular relevance to this problem is spreading dynamics on complex networks, where the available data are binary: a node is either infected or healthy. In such cases, a recent work <cit.> demonstrated that the propagation network structure can be reconstructed and the sources of spreading can be detected by exploiting compressive sensing (Sec. <ref>).However, for binary state network dynamics, a general reconstruction framework has been missing. The problem of reconstructing complex networks with binary-state dynamics is extremely challenging, for the following reasons. (i) The switching probability of a node depends on the states of its neighbors according to a variety of functions for different systems, which can be linear, nonlinear, piecewise, or stochastic. If the function that governs the switching probability is unknown, difficulties may arise in obtaining a solution of the reconstruction problem. (ii) Structural information is often hidden in the binary states of the nodes in an unknown manner and the dimension of the solution space can be extremely high, rendering impractical (computationally prohibitive) brute-force enumeration of all possible network configurations. (iii) The presence of measurement noise, missing data, and stochastic effects in the switching probability make the reconstruction task even more challenging, calling for the development of effective methods that are robust against internal and external random effects.A potential approach <cit.> to developing a general and robustframework to reconstruct complex networks based solely on the binary states of the nodes without any knowledge about the switching functionsis based on the idea of linearizing theswitching functions from binary data. This allows one to convert the network reconstruction into a sparse signal reconstructionproblem for local structures associated with each node. Exploiting thenatural sparsity of complex networks, one can employ the lasso <cit.>,an L_1 constrained fitting method for statistics and data mining, toidentify the neighbors of each node in the network from sparse binary datacontaminated by noise. The underlying mechanism that justifies the linearization procedure by conducting tests using a number of linear, nonlinear and piecewise binary-state dynamics can be established on a large number ofmodel and real complex networks <cit.>. Universally highreconstruction accuracy was found even for small data amount with noise.Because of its high accuracy, efficiency and robustness against noise andmissing data, the framework is promising as a general solution to the inverse problem of network reconstruction from binary-state time series.The data-based linearization method can be useful for dealing with generalnonlinear systems with a wide range of applications. §.§ Universal structural estimator and dynamics approximator for complex networks Is it possible to develop a universal and completely data-driven framework for extracting network topology and identifying the dynamical processes, without the need to know a priori the specific types of network dynamics? An answer to this question would be of significant value not only to complexity science and engineering but also to modern data science where the goal is to unearth the hidden structural information and to predict the future evolution of the system. A partial solutionto this problem emerged recently, where the concept of universalstructural estimator and dynamics approximator for complex networkedsystems was proposed <cit.>, and it was demonstrated that such aframework or “machine” can indeed be developed for a large number ofdistinct types of network dynamical processes. While it remains an open issue to develop a rigorous mathematical framework for the universalmachine, the preliminary work <cit.> can be regarded as aninitial attempt towards the development of a universal framework fornetwork reconstruction and dynamics prediction.The key principle is the following. In spite of the dramatic difference inthe types of dynamics in terms of, e.g., the interaction setting and stateupdating rules, there are two common features shared by many dynamicalprocesses on complex networks: (1) they are stochastic, first-order Markovianprocesses, i.e., only the current states of the systems determine theirimmediate future; and (2) the nodal interactions are local. The two featuresare characteristic of a Markov network (or a Markov random field) <cit.>. In particular, a Markov network is an undirected and weighted probabilistic graphical model that is effective at determining the complex probabilistic interdependencies in situations where the directionality in the interaction between connected nodes cannot be naturally assigned, in contrast to the directed Bayesian networks <cit.>. A Markov network has twotypes of parameters: a node bias parameter that controls its preference of state choice, and a weight parameter characterizing the interaction strength of each undirected link. The joint probability distribution of the state variables 𝐗 = (x_1, x_2, …, x_N)^T is given by P(𝐗) = ∏_Cϕ(𝐗_C)/∑_𝐗∏_Cϕ(𝐗_C), where ϕ(𝐗_C) is the potential function for a well-connected network clique C, and the summation in the denominator is over all possible system state 𝐗. If this joint probability distribution is available, literally all conditional probability interdependencies can be obtained. The way to define a clique and to determine its potential function plays a key role in the Markov network's representation power of modeling the interdependencies within the system. In Ref. <cit.>, the possibility was pursued of modeling the conditional probability interdependence of a variety of dynamical processes on complex networked systems via a binary Ising Markov network with its potential function in the form of the Boltzmann factor, exp(-E), where E is the energy determined by the local states and their interactions along with the network parameters (the link weights and node biases) in a log-linear form <cit.>. This is effectively a sparse Boltzmann machine <cit.> adopted to complex network topologies without hidden units. (Note that hidden units can play a critical role in ordinary Boltzmann machines <cit.>). A temporalevolutionary mechanism was introduced <cit.> as a persistent samplingprocess for such a machine based on the conditional probabilities obtainedvia the joint probability, and generate a Markov chain of persistently sampled state configurations to form state transition time series for each node. The model was named a sparse dynamical Boltzmann machine(SDBM) <cit.>.In dynamical processes on complex networks, such as epidemic spreading or evolutionary game dynamics, the state of each node at the next time step is determined by the probability conditioned on the current states of its neighbors (and its own state in some cases). There is freedom to manipulate the conditional probabilities that dictate the system behavior in the immediate futurethrough change in the parameters such as weights and biases. A basic question is then, for an SDBM, is it possible to properly choose these parameters so that the conditional probabilities produced are identical or nearly identical to those of a typical dynamical process with each given observed system state configuration? If the answer is affirmative, the SDBM can serve as a dynamics approximator of the original system, and the approximated conditional probabilities possess predictive power for the system state at the next time step. When such an SDBM is found for many types of dynamical process on complex networks, it effectively serves as a universal dynamics approximator. Moreover, if the detailed statistical properties of the state configurations can be reproduced in the long time limit, that is, if the time series generated by this SDBM are statistically identical or nearly identical to those from the original system,the SDBM is a generative model of the observed data (in the language of machine learning), which is potentially capable of long term prediction.When an approximator exists for each type of dynamics on a complex network, the topology of the SDBM is nothing but that of the original network, providing a simultaneous solution to the problem of network structure reconstruction. Previous works on the inverse static or kinetic Ising problems led to methods of reconstruction for Ising dynamics through maximization of the datalikelihood (or pseudo-likelihood) function via the gradient descent approaches <cit.>.Instead of adopting these approaches, compressive sensing can be exploited. By incorporating a K-means clustering algorithm into the sparse solutionobtained from compressive sensing, it was demonstrated <cit.> thatnearly perfect reconstruction of the complex network topology can be achieved.In particular, using 14 different types of dynamical processes on complexnetworks, it was found that, if the time series data generated by thesedynamical processes are assumed to be from its equivalent SDBMs, the universalreconstruction framework is capable of recovering the underlining networkstructure of each original dynamics with almost zero error. This representssolid and concrete evidence that SDBM is capable of serving as a universalstructural estimator for complex networks. In addition to being able toprecisely reconstruct the network topologies, the SDBM also allows the linkweights and the nodal biases to be calculated with high accuracy. In fact, the the universal SDBM is fully automated and does not require any subjectiveparameter choice <cit.>.§.§ Controlling nonlinear and complex dynamical networks An ultimate goal of systems identification and prediction is to control. The coupling between nonlinear dynamics and complex network structure presentstremendous challenges to our ability to formulate effective control methodologies.In spite of the rapid development of network science and engineering towardunderstanding, analyzing and predicting the dynamics of large complex networksystems in the past fifteen years, the problem of controlling nonlineardynamical networks has remained largely open.The field of controlling chaos in nonlinear dynamical systems has been active for more than two decades since the seminal work of Ott, Grebogi, and Yorke <cit.>. The basic idea was that chaos, while signifying random or irregular behavior, possesses an intrinsically sensitive dependence on initial conditions that can be exploited for controlling the system using only small perturbation. This feature, in combination with the fact that a chaotic system possesses an infinite set of unstable periodic orbits, each leading to different system performance, implies that a chaotic system can be stabilized about some desired state with optimal performance. Controlling chaos has sincebeen studied extensively and examples of successful experimental implementationabound in physical, chemical, biological, and engineeringsystems <cit.>. The vast literature on controlling chaos, however,has been mostly limited to low-dimensional systems, systems that possessone or a very few unstable directions (i.e., one or a very few positiveLyapunov exponents <cit.>). Complex networks with nonlinear dynamics are generally high dimensional, rendering inapplicable existing methodologiesof chaos control.In the past several years, a framework for determining the linear controllability of network based on the traditional control and graph theories emerged <cit.>, leading to a quantitative understanding of the effect of network structure on its controllability. In particular, a structural controllability framework was proposed <cit.>, revealing that the ability to steer a complex network toward any desired state, as measured by the minimum number of driver nodes, is determined by the set of maximum matching, which is the maximum set of links that do not share starting or ending nodes. A main result was that the number of driver nodes required for full control is determined by the network's degree distribution <cit.>. The framework was established for weighted and directed networks. An alternative framework, the exact controllability framework, was subsequently formulated <cit.>, which was based on the principle of maximum multiplicity to identify the minimum set of driver nodes required to achieve full control of networks with arbitrary structures and link weight distributions. Generally, a limitation of such rigorous mathematical frameworks of controllability is that the nodal dynamical processes must be assumed to be linear. For nonlinear dynamical networks, to establish a mathematical controllability framework similar to those based on the classic Kalman's rank condition <cit.> for linear networks is an unrealistically broad objective. Traditionally, controllability for nonlinear control can be formulated based on Lie brackets <cit.>, but to extend the abstract framework to complex networks may be difficult. A recent work extended the linear controllability and observability theory to nonlinear networks with symmetry <cit.>. In spite of the previous works, at the present there is no known general framework for controlling nonlinear dynamics on complex networks.Due to the high dimensionality of nonlinear dynamical networks and the rich variety of behaviors that they can exhibit, it may be prohibitively difficult to develop a control framework that is universally applicable to different kinds of network dynamics. In particular, the classic definition of linear controllability, i.e., a network system is controllable if it can be driven from an arbitrary initial state to an arbitrary final state in finite time, is generally not applicable to nonlinear dynamical networks. Instead, controlling collective dynamical behaviors may be more pertinent and realistic <cit.>. Our viewpoint is that, for nonlinear dynamical networks, control strategies may need to be specific and system-dependent. Recently, a control framework for systems exhibitingmultistability was formulated <cit.>. A definingcharacteristic of such systems is that, for a realistic parameter setting,there are multiple coexisting attractors in the phasespace <cit.>. The goal is to drive the system from one attractor to another usingphysically meaningful, temporary and finite parameter perturbation, assumingthat the system is likely to evolve into an undesired state (attractor) orthe system is already in such a state and one wishes to implement control tobring the system out of the undesired state and steer it into a desired one. It should be noted that dynamical systems with multistability are ubiquitous inthe real world ranging from biological and ecological to physical systems <cit.>.In biology, nonlinear dynamical networks with multiple attractors have been employed to understand fundamental phenomena such as cancer mechanisms <cit.>, cell fate differentiation <cit.>, and cell cycle control <cit.>. For example, Boolean network models were used to study gene evolution <cit.>, attractor number variation with asynchronous stochastic updating <cit.>, gene expression in the state space <cit.>, and organism system growth rate improvement <cit.>. Another approach is to abstract key regulation genetic networks <cit.> (or motifs) from all associated interactions, and to employ synthetic biology to modify, control and finally understand the biological mechanisms within these complicated systems <cit.>. An earlier application of this approach led to a good understanding of the ubiquitous phenomenon of bistability in biological systems <cit.>, where there are typically limit cycle attractors and, during cell cycle control, noise can trigger a differentiation process by driving the system from a limit circle to another steady state attractor <cit.>. In general, there are two candidate mechanisms for transition or switching between different attractors <cit.>: through signals transmitted within cells and through noise, which were demonstrated recently using synthetic genetic circuits <cit.>. More recently, a detailed numerical study was carried out of how signal-induced bifurcations in a tri-stable genetic circuit can lead to transitions among different cell types <cit.>.The control and controllability framework for nonlinear dynamical networkswith multistability can be formulated <cit.> based on the conceptof attractor networks <cit.>. An attractor network is definedin the phase space of the underlying nonlinear system, in which each noderepresents an attractor and a directed edge from one node to another indicatesthat the system can be driven from the former to the latter using experimentally feasible, temporary, and finite parameter changes. A well connected attractor network implies a strong feasibility that the system can be controlled to reach a desired attractor. The connectivity of the attractor network can then be used to characterize the controllability of the nonlinear dynamical network. More specifically, for a given pair ofattractors, the relative weight of the shortest path is the number ofaccessible control parameters whose adjustments can lead to the attractortransition as specified by the path. Gene regulatory networks (GRNs) wereused <cit.> to demonstrate the practicality of the control framework, which includes low-dimensional, experimentally realizable synthetic gene circuits and a realistic T-cell cancer network of 60 nodes. A finding was that noise can facilitate control by reducing therequired amplitude of the control signal. In fact, the development of thenonlinear control framework <cit.> was based entirely on physical considerations, rendering feasible experimental validation.The framework can be adopted to controlling nonlinear dynamical networks other than the GRNs. For example, for the Northern European power grid network recently studied by Menck et al. <cit.>, a rewiring method was proposed and demonstrated to be able to enhance the system stability through the addition of extra transmission lines. For a power grid network, the synchronous states are desired while other states, e.g., limit cycles, detrimental. Treating the link density (or number) as a tunable parameter, the minimum transfer capacity required for extra lines to realize the control can be estimated through our method. Another example is Boolean networks with discrete dynamics, for which a perturbation method wasproposed based on modifying the update rules to rescue the system from the undesired states <cit.>.An attractor network can be constructed based on perturbation to multiple parameters to drive the system out of the undesired, damaged states toward a normal (desired) state. For biological systems, an epigeneticstate network (ESN) approach was proposed <cit.> to analyze the transitions among different phenotypic processes. In an ESN, nodes represent attractors and edges represent pathways between a pair of attractors. By construction, different parameter values would result in a different ESN. This should be contrasted to an attractor network, in which nodes are attractors butedges are directed and represent controllable paths (through parameterperturbation) to drive the system from one attractor to another.§ ACKNOWLEDGEMENTWe thank Dr. R. Yang (formerly at ASU), Dr. R.-Q. 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"authors": [
"Wenxu Wang",
"Ying-Cheng Lai",
"Celso Grebogi"
],
"categories": [
"physics.data-an",
"nlin.CD",
"physics.soc-ph"
],
"primary_category": "physics.data-an",
"published": "20170427223103",
"title": "Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems"
} |
Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands;[email protected] Institute of Astrophysics, Universiteit Leuven, Celestijnenlaan 200 D, 3001 Leuven, Belgium European Space Astronomy Centre (ESA/ESAC), Operations Department, Villanueva de la Cañada(Madrid), Spain Department of Astronomy, Stockholm University, Oskar Klein Center, SE-106 91 Stockholm, Sweden Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD, USA UK Astronomy Technology Centre, Royal Observatory Edinburgh, Blackford Hill, Edinburgh, EH9 3HJ, UKThe formation process of massive stars is still poorly understood. Massive young stellar objects (mYSOs) are deeply embedded in their parental clouds, they are rare and thus typically distant, and their reddened spectra usually preclude the determination of their photospheric parameters. M17 is one of the best studied Hii regions in the sky, is relatively nearby, and hosts a young stellar population. With X-shooter on the ESO Very Large Telescope we have obtained optical to near-infrared spectra of candidate mYSOs, identified by <cit.>, and a few OB stars in this region. The large wavelength coverage enables a detailed spectroscopic analysis of their photospheres and circumstellar disks. We confirm the pre-main sequence (PMS) nature of six of the stars and characterise the O stars. The PMS stars have radii consistent with being contracting towards the main sequence and are surrounded by a remnant accretion disk. The observed infrared excess and the (double-peaked) emission lines provide the opportunity to measure structured velocity profiles in the disks. We compare the observed properties of this unique sample of young massive stars with evolutionary tracks of massive protostars by <cit.>, and propose that these mYSOs near the western edge of the Hii region are on their way to become main-sequence stars (∼ 6 - 20 M_⊙) after having undergone high mass-accretion rates (Ṁ_ acc∼ 10^-4 - 10^-3 M_⊙ yr^-1). Their spin distribution upon arrival at the zero age main sequence (ZAMS) is consistent with that observed for young B stars, assuming conservation of angular momentum and homologous contraction.Massive pre-main sequence stars in M17Based on observations collected at the European Southern Observatory at Paranal, Chile (ESO programmes 60.A-9404(A), 085.D-0741, 089.C-0874(A), and 091.C-0934(B)). M.C. Ramírez-Tannus 1 L. Kaper1 A. de Koter1,2F. Tramper3 A.Bik4 L.E. Ellerbroek1 B.B. Ochsendorf5 O.H. Ramírez-Agudelo6 H. Sana2 December 30, 2023 =========================================================================================================================================================================================================================================§ INTRODUCTIONIn the past decades significant progress has been made regarding the understanding of star formation. However, the formation of massive stars – the only mode of star formation observable in external galaxies – remains a key unsolved problem <cit.>. One of the main reasons is that observations of the earliest phases in the life of a massive star are scarce. This is explained by the expected short formation timescale (∼ 10^4 - 10^5 yr) of massive stars, and the severe extinction (A_V∼ 10 - 100 mag) by which the surrounding gas and dust obscure their birth places from view. Additionally, massive stars are rare and, as a consequence, located at relatively large distances.Observational and theoretical evidence is accumulating that the formation process of massive stars is through disk accretion, similar to low-mass stars. This despite the physical processes acting near hot, massive stars (e.g., radiation pressure, ionisation) that counteract the accretion flow onto the forming star <cit.>. In the accretion scenario, one expects that soon after the gravitational collapse of the parental cloud the pre-stellar core is surrounded by an extended accretion disk <cit.>. Apparently this is not a very efficient process, as most of these high-mass protostars seem to drive outflows <cit.>.Already before arriving on the main sequence, the young massive star is expected to produce a strong ultraviolet (UV) radiation field ionising its surroundings <cit.>. At this stage the object will become detectable at radio and infrared wavelengths through the heated dust and recombination of hydrogen in an expanding hyper- or ultra-compact Hii (UCHii) region <cit.>. Unfortunately, only very little is known about the physical properties of the central (massive) stars at this stage of formation <cit.>.Given the short formation time of massive stars, the accretion rate must be high <cit.>. At these accretion rates, a massive young stellar object (mYSO) is expected to bloat up to about 100 R_⊙, resulting in a relatively low effective temperature and modest UV luminosity. The accretion process is unlikely to be constant with time; the blob structure observed in Herbig Haro outflows indicates that strong variations occur in the mass in- and outflow rate of young intermediate-mass stars <cit.>. Simulations of different accretion models applied to mYSOs show that the accretion rate increases as the mYSO grows in mass <cit.>. Once the accretion rate diminishes, the “bloated” pre-main-sequence (PMS) star contracts to the main sequence on the Kelvin-Helmholtz timescale. Recently, candidates for such bloated, massive PMS stars have been spectroscopically confirmed <cit.>.The environment (multiplicity, clustering) and the large distances to massive (proto)stars make their circumstellar disks very difficult to resolve <cit.>. In contrast, disks around intermediate-mass stars (2 < M_⋆ < 8 M_⊙) have been characterised using (sub)millimeter as well as optical and near-infrared spectroscopy <cit.>. The rotational structure of the disk provides information about the role of magnetic fields that can slow down the rotation rate below pure Keplerian. Super-Keplerian rotation of the disk could indicate that the inner disk contributes significantly to the gravitational potential of the system <cit.>. In the mass range from 20–30 M_⊙ circumstellar disks have been detected, but there is no consensus about their general properties.M17, located in the Carina-Sagittarius spiral arm of the Galaxy is one of the best studied giant Hii regions <cit.>. Its distance has been accurately determined by measuring the parallax of the CH_3OH maser source G15.03-0.68: d = 1.98^+0.14_-0.12 kpc <cit.>. The bright blister Hii region is embedded in a giant molecular cloud complex, and divides the molecular cloud into two components: M17 South and M17 North, containing a total gas mass (molecular and atomic) of about 6 × 10^4 M_⊙ <cit.>. The centre of the Hii region hosts the cluster NGC 6618 including 16 O stars and over 100 B stars <cit.>, providing the ionising power for the Hii region. Many of the OB stars are suspected to be binaries <cit.>, which would explain why they are more luminous than expected from their spectral type. NGC 6618 is a young cluster, its age is estimated at about 1 Myr <cit.>, while the surrounding molecular cloud hosts pre-main-sequence stars. The photo-dissociation region to the southwest of NGC 6618 includes several candidatemYSOs: the hyper-compact Hii region M17-UC1 surrounded by a circumstellar disk <cit.>, and IRS5, a young possibly quadruple system of which the primary star, IRS5A (B3–B7 V/III), is a high-mass protostellar object <cit.>. <cit.> discovered a sample of high-mass (5–20 M_⊙) young stellar objects in the same area. Their SEDs display a near-IR excess, some show CO bandhead and (double-peaked) Pa δ emission, likely due to the presence of a circumstellar disk. These objects are the subject of study in this paper. We are now on the verge of being able to quantitatively test the predictions of massive star formation models. Several samples of mYSOs have been presented <cit.>. However, so far only for a handful of mYSOs the photospheric spectrum has been detected. We have obtained high-quality spectra with X-shooter on the ESO Very Large Telescope (VLT) of candidate mYSOs, first identified by , and a few other O and B starsin M17. The large wavelength coverage of the X-shooter spectra (and additional infrared photometry) allows for a detailed analysis of the spectral energy distribution, including an infrared excess produced by the dust component in the accretion disk. The observed (double-peaked) emission lines provide the opportunity to study the dynamical structure of the gaseous component of the disk.This paper is organised as follows. In Sect. <ref> we present our target sample, observations and data reduction procedure. In Sect. <ref> we provide the spectral classification of the (pre-) main-sequence OB stars in our sample. After that (Sect. <ref>) we present their spectral energy distributions. To further constrain the properties of the mYSOs we modeled the optical spectra using . These results are shown in Sect. <ref>, where we also place the PMS objects in the Hertzsprung-Russell diagram and compare their position with PMS tracks and isochrones. Additional evidence for the presence of circumstellar disks is obtained from near-infrared spectroscopy (Sect. <ref>). We discuss the age distribution of the massive PMS population in M17, the extinction towards this region, the presence of circumstellar disks, and the spin properties of the young massive stars in Sect. <ref>. We summarise our conclusions in Sect. <ref>. § VLT/X-SHOOTER SPECTROSCOPYWe obtained optical to near-infrared (300–2500 nm) spectra of candidate mYSOs, and some OB stars in M17. We used the X-shooter spectrograph mounted on UT2 of the VLT <cit.>. The spectra of B275 and B358 were obtained as part of the X-shooter science verification program; B275 has been subject of study in <cit.>. For some objects multiple spectra have been taken. A log of the observations is listed in Table <ref>. §.§ Target sampleWe selected our targets based on the study ofwho obtained K-band spectra of the young massive star population of M17, complemented by optical and 1 μm spectroscopy for some of the moderately reddened objects. Several of the targets have also been studied by <cit.> and some are included in the studies of <cit.>, <cit.>, and <cit.>.reported that B111, B164, B253, and B311 do not present any NIR excess nor emission lines in their spectra and they are likely OB-star members of the central cluster NGC 6618. It is likely that all our targets belong to NGC 6681, at least the (massive) main sequence stars. Based on the NIR excess derived from their position in the NIR colour-colour diagram (CCD, see also Sect. <ref>) and displayed by the spectral energy distribution (SED, see also Sect. <ref>), and the apparent presence of CO bandhead and/or double-peaked emission lines, the candidate mYSOs according toare B163, B243, B268, B275, B289, B331, and B337. We also observed B215 (IRS15): this mYSO candidate has been studied by <cit.>. The spectral types fromand <cit.> are listed in Table <ref>. We obtained an X-shooter spectrum of B358, but according to <cit.> andthe spectrum is that of an early-/mid-G supergiant. It has not been detected in X-rays <cit.> and it is likely a background post-AGB star. From examining the X-shooter spectrum we support this conclusion.Figure <ref> (left panel) shows a colour composite image of M17 based on 2MASS J (blue), H (green) and K (red) observations <cit.>. The stars for which we collected X-shooter spectra are labeled. The area includes the bright nebular emission produced by the Hii region and the central cluster NGC 6618. The mYSO candidates are predominantly located near the excited rim of the surrounding molecular cloud. This observation suggests that those objects could be the product of a later phase in the star formation process. In the right panel a colour composite is shown based on images taken with Spitzer <cit.> and Herschel <cit.>. This image provides insight into the (dusty) surroundings of the Hii region. The 3.6 μm image traces the illuminated edges of the cloud irradiated by the massive stars, whereas the 8 μm emission is usually associated with emission features produced by polycyclic aromatic hydrocarbons (PAHs) and/or warm (100-200 K) dust that emits at these wavelengths when heated by ultraviolet radiation <cit.>. Throughout this paper we have color-coded the labels to present our results: the objects whose PMS nature is confirmed are plotted in blue, the OB stars in grey (black in figure <ref>), the objects for which we cannot confirm the PMS nature in green and the post-AGB star in red.§.§ Colour-colour and colour-magnitude diagram of X-shooter targets in M17The (K, J-K) colour-magnitude diagram is shown in Fig <ref>; the J, H, and K-band magnitudes are from the 2MASS database <cit.>. Similar diagrams have been presented byand <cit.>, but not using the recent distance estimate of 1.98 kpc <cit.>. The zero-age main sequence (ZAMS) is marked with a drawn line; the dashed line represents the reddening vector. The ZAMS properties for spectral types later than B9 were taken from <cit.> and for the earlier type stars from . We over-plot the properties for the O dwarf stars from <cit.> for reference. We used the <cit.> extinction law in order to plot the reddening lines. The candidate mYSOs have a large (J-K) colour and a bright K-band magnitude.Figure <ref> presents the near-infrared colour-colour diagram including the OB-type stars for which an X-shooter spectrum has been obtained. In the colour-colour diagram a near-infrared excess becomes very apparent, and thus the possible presence of circumstellar material. The solid line represents the unreddened main sequence. The dashed lines indicate the reddening lines for an O3 dwarf and an M0 giant. Most of the mYSO candidates are located to the right of the reddening line of an O3 V star, consistent with the expectation that these objects host a circumstellar disk. B163, B289, and B331 are candidate mYSOs according to ; however, they do not exhibit a near-infrared excess in the colour-colour diagram. Nevertheless, B331 and B163 are mYSO candidates according to our criteria: they present double-peaked emission lines, CO bandhead emission and/or infrared excess beyond the K-band. We will discuss the spectral energy distribution and the possible presence of an infrared excess in Sect. <ref>, also at mid-infrared wavelengths.§.§ Reduction VLT/X-shooter spectra The X-shooter spectra were obtained under good weather conditions,with seeing ranging from 05 and 1 and clear sky. With the exception of the 2012 B289 spectrum and the 2009 B275 science verification spectrum, the spectrograph slit widths used were 1 (UVB, 300 – 590 nm), 09 (VIS, 550 – 1020 nm), and 04 (NIR, 1000 – 2480 nm), resulting in a spectral resolving power of 5100, 8800, and 11300, respectively. The slit widths for the 2010 B275 observations were 16, 09, and 09 resulting in a resolving power of 3300, 8800, and 5600, respectively. For the 2012 B289 observations we used the 08, 07, and 04 slits corresponding to a resolving power of 6200, 11000, and 11300 for the UVB, VIS and NIR arms, respectively. The spectra were taken in nodding mode and reduced using the X-shooter pipeline <cit.> version 2.7.1 running under the ESO Reflex environment <cit.> version 2.8.4. The flux calibration was obtained using spectrophotometric standards from the ESO database. We then scaled the NIR flux to match the absolutely calibrated VIS spectrum. The telluric correction was performed using the software tool v1.2.0[<http://www.eso.org/sci/software/pipelines/skytools/molecfit>] <cit.>. Parts of the spectra are shown in Figs. <ref> to <ref> and the full spectrum is available in the on-line material of this paper. We used therecipe to reduce the data, meaning that the sky subtraction is performed by subtracting the two different nodding positions. We note that due to the spatial variation of the nebular lines along the slit, residuals from the sky subtraction are present in some of the the reduced spectra (e.g., [Nii] 6548,6584 Å, [Sii] 6716,6731 Å and [Siii] 9069,9532 Å, and the H lines).§ SPECTRA AND SPECTRAL CLASSIFICATIONTraditionally, the blue spectral region (400 – 500 nm) is used to perform the spectral classification of OB-type stars. In the case of the M17 sources studied in this paper the blue region is severely affected by extinction, but is for most of our objects detected in the X-shooter spectrum (Fig. <ref>). Many targets show the full hydrogen Balmer, Paschen and Brackett (up to Brγ) series,Hei and Heii lines (the latter in the O-type stars), the Ca triplet lines, and diffuse interstellar bands. Figs. <ref> and <ref> provide an overview of the Ca triplet region (820–880 nm) and the K-band (2100–2400 nm, including Brγ and the CO bandheads) for all targets.With the exception of B163, B331 and B337 which are strongly affected by interstellar absorption in the blue (A_V > 10), we classified the stars based on the strength of the H, He and metal lines in the UVB spectrum. For the O stars we visually compared our spectra with the previously catalogued and published spectra by the Galactic O Star Spectroscopic Survey <cit.>. For the B stars we used the criteria and spectra published by <cit.>.We looked for the presence of [Oi] 6300. This forbidden line is associated with bi-polar outflows/jets, disk winds, and the disk surface in Herbig Ae/Be stars.The [Oi] line is thought to originate from the region where the UV radiation from the star impinges on the disk surface <cit.>. We also paid special attention to the luminosity subclass Vz, which refers to objects with a substantially stronger Heii4686 line. These objects are hypothesised to be the youngest optically observable O type stars <cit.>. Given the young age of M17, one might expect such stars to be present. Following this classification scheme we are able to determine the spectral types with an accuracy of one subtype, unless stated otherwise in Sect. <ref>. We constrained the luminosity class of the stars using the surface gravity derived in Sect. <ref>. An overview of the spectral features described in this section is presented in Table <ref> and the results of the spectral classification are listed in Table <ref>. In most cases the spectral classification agrees well with the one provided in the literature byand <cit.>.§.§ Individual sourcesIn this section we discuss the X-shooter spectra of the individual targets. In parentheses we list the updated spectral types corresponding to this work. The spectra are displayed in full in Appendix <ref>. B111 (O4.5 V): This clearly is an early O-type spectrum: the Heii lines at 4200, 4541, and 4686 Å are strong, consistent with an O4.5 V type star. The spectrum does not show Hei 4387, and Hei 4471 is weak. The hydrogen Balmer (including Hα) and Paschen series lines are strongly in absorption. Using its NIR spectrum,classified B111 as a kO3-O4 based on the Heii, Nii lines and the lack of C iv. According to our K-band spectrum B111 is a kO5-O6 type star, based on the presence of Heii lines and weak, if any, C iv emission. The spectrum does not include any (nebular) emission lines.B163 (kA5): Due to severe interstellar extinction it is not possible to identify any photospheric features in the UVB arm of the X-shooter spectrum. Longward of ∼8500 Åthe first Paschen line becomes detectable. The NIR spectrum also shows the Brackett series in absorption, no elements other than H are seen. Based on this spectrum we assign a mid-A spectral type. Two of the CO first overtone transitions are detected, indicating the presence of a circumstellar disk. The spectrum includes nebular emission lines in the hydrogen series, Hei lines (e.g., 10830 Å), and [Niii] 9069,9532 Å. This points to an energetic ionising radiation field as expected given the location of B163 near the core of NGC 6618 (Fig. <ref>). According to our criteria B163 is a mYSO. B164 (O6 Vz): The presence of Heii 4541, 4686, and the fact that these lines are stronger than Hei 4387 and 4471 demonstrate that B164 is an O-type star. The ratio Heii 4200 / Hei (+ii) 4026 Å is near unity which indicates a spectral type O6 V. Based on the ratio EW(Heii 4686) / EW(Heii 4542) > 1.10 we conclude that it belongs to the Vz class. From the K-band classification we conclude that B164 is a kO5-O6 type star. classified B164 as kO7-O8 based on the Niii and Hei lines. No emission lines are detected.B215 (IRS15; B0-B1 V): No Heii lines are detected, which indicates that B215 must be of later spectral type than B0. The hydrogen lines of the Balmer, Paschen and Brackett series are in absorption, and the Hei absorption lines are weak (e.g., 4922). The absence of Mgii 4481 suggests that the spectral type is earlier than B1; therefore, we conclude that this star is of spectral type B0-B1. The K-band classification is consistent with kO9-B1. <cit.>classified this object as an extreme class I source, and <cit.> pointed out that it is a ∼26 M_⊙ star surrounded by an extensive remnant disk. The spectrum contains nebular emission lines in the hydrogen series, Hei lines and forbidden emission lines [Nii] 6548,6584, [Sii] 6716,6731 and [Siii] 9069,9532. We also identify [Oi] 6300 emission: a broad component (∼100 km/s) on top of a nebular component. By inspecting the 2D frames we conclude that the emission seen in Hα has a nebular origin. From the X-shooter spectrum alone we cannot conclude that this is a mYSO.B243 (B8 V): The hydrogen lines of the Balmer series are in absorption. The strongest Balmer lines show a central emission component; Hα is strongly in emission. The Paschen series absorption lines are filled in with double-peaked emission lines. The Brackett series lines mainly exhibit a double-peaked emission component. Furthermore, the `auroral' Oi 7774 and 8446 lines show prominent double-peaked emission. The Caii triplet is not present. The spectrum includes Hei absorption lines (e.g. 4471, 5876, 10830); the ratio Hei 4026 / Hei 4009 is around two. The Hei 4471 and Mgii 4481 line ratio is close to unity which points towards spectral type B8. The JHK CCD indicates NIR excess, weak CO-bandhead emission is detected, and it presents [Oi] 6300 emission. Therefore we conclude that B243 is a mYSO.B253 (B3-B5): The spectrum displays strong and broad hydrogen absorption lines, with a central (nebular) emission component. Hei absorption lines are present (e.g., 4026, 4471, 4922, 10830 Å), also with a central (nebular) emission component; the Hei lines are most prominent in the blue part of the spectrum. No Heii lines are detected, nor the Cii 4267 Å line; therefore, we conclude that B253 is a B3-B5 type star. The NIR CMD and CCD show no evidence for a NIR excess. Its K-band spectrum is kB5 or later. Many forbidden emission lines are present (e.g., [Oiii], [Nii], [Sii]) that, together with the Hei emission, indicate a high degree of ionisation of the surrounding Hii region.B268 (B9-A0): The hydrogen Balmer and Paschen series are prominently in absorption and include a central nebular emission component. The Hα and Hβ profiles are filled in by a circumstellar emission component. The Hei lines are very weakly present, the nebular emission component dominates the line profiles. The fact that Hei 4471 and Mgii 4481 have almost the same strength indicates that the spectral type is late-B to early-A. The Ca triplet lines show double-peaked emission and a red-shifted absorption component that reaches below the continuum and could indicate the presence of an accretion flow <cit.>. We confirm the findings ofwho classified B268 as a mYSO, based on the presence of CO bandhead emission and the Paδ line.B275 (B7 III):<cit.> performed a detailed spectral classification, resulting in spectral type B7 III. They concluded that B275 is a pre-main-sequence star contracting towards the main sequence. TheHei 4009 Å and Cii 4267 Å lines are very weak and when considering the Si ii 4128 Å to Mg ii 4481 Å ratio the spectral type becomes B6-B7. The Oi 8446 Å and Ca triplet lines show pronounced double-peaked emission, as do several of the hydrogen lines on top of a photospheric absorption profile. Both first- and second-overtone CO emission is detected. The [Oi] 6300 emission line is present. Together with double-peaked emission features and NIR excess, this points to the presence of a rotating circumstellar disk. B275 is a massive YSO.B289 (O9.7 V): The spectrum includes Heii 4686 and Heii 5411 absorption lines, but the presence of Heii 4200 is hard to confirm given the low signal-to-noise ratio. The Hei (+ii) 4026 line is present, Hei 4144 and 4387 are weak and Hei 4471 and 5876 are strong. In addition, it is possible to identify the C iii 4647/50/51 line complex with a similar strength as the Heii 4686 line, so that the spectral type is O9.7 V. From the K-band we obtain a kO9-B1 spectral type. According to , B289 might have a NIR excess and be a late O star. The JHK CCD does not display evidence for a NIR excess and there are no emission lines, therefore we cannot conclude that it is a mYSO. The1.5 and 2.0 μm photometry by <cit.> using MANIAC mounted at the ESO La Silla 2.2 m telescope shows IR excess. It is one of the sources surrounded by an IR-bright dusty disk as reported by <cit.>. The spectrum shows some weak nebular emission lines. B311 (O8.5 Vz): The Hei (+ii) 4026 andHei 4471 lines are almost equally strong and sharp.Hei λλ 4144, 4387, and Heii λλ 4541, 4686 also show up in the spectrum. The Ciii 4068/69/70 complex is in absorption together with Siiv 4089 and 4116. We classify this star as O8.5 Vz because the ratio EW(Heii 4686) /EW(Hei 4471) > 1.10. From the NIR spectrum,classified it as later than O9-B2 because it has Hei in absorption and lacks Niii and Heii. <cit.> and <cit.> detected an IR-bright dusty disk in the N and Q-bands. However, <cit.> resolved a bow shock associated with it, which is responsible for the NIR emission. B311 is a main-sequence star according to our criteria. The result of the K-band classification agrees well with the visual spectral type (kO9-B1). Nebular emission is weakly present. B331 (late-B): The UVB part of the spectrum is not detected due to the severe interstellar extinction towards this source. The red and near-infrared spectrum is dominated by strong emission lines. In the case of the hydrogen series lines these are superposed on a broad and shallow photospheric profile. According to <cit.>, B331 is a B2 V type star based on its visual spectrum. No helium lines are detected in the NIR part of the spectrum, indicating that B331 is a late-B or early-A-type star. It exhibits Brγ, Oi, CO bandhead emission, and several double-peaked emission lines indicating the presence of a rotating circumstellar disk. The SED includes a NIR excess, making it a bonafide mYSO. The spectrum contains some weak nebular emission lines.B337 (late-B): This object is deeply embedded so that it is not possible to detect the blue part of the spectrum. It does show the Paschen series in absorption with a narrow central nebular emission component. The Ca triplet lines show pronounced double-peaked emission and a blue absorption component remnant of the sky subtraction. The SED presented byis consistent with a B5 V type star when correcting for the distance. No Hei lines can be identified in the X-shooter spectrum pointing towards a late-B or early-A-type star. The NIR CCD indicates a NIR excess and the Caii triplet lines are in emission and double-peaked; we do not detect CO bandhead emission. B337 is a mYSO. § SPECTRAL ENERGY DISTRIBUTIONWe construct the spectral energy distribution (SED) of the star by dereddening the X-shooter spectrum, as well as the available photometric data (extending into the mid-infrared wavelength range). We fit the slope of the SED in the photospheric domain (400 – 820 nm) to Castelli & Kurucz models <cit.>. As an initial guess, we took the associated T_ eff and log g from the Castelli & Kurucz[Table 2 in <http://www.stsci.edu/hst/observatory/crds/castelli_kurucz_atlas.html>] model corresponding to the spectral type reported in Sect. <ref>. We then cross-check our choice of T_ eff and log g with the results obtained in Sect. <ref> and perform the necessary iterations. The final values of T_ eff and log g, corresponding to the best fit Castelli & Kurucz models used, are displayed in Figure <ref>. We thereby constrain the visual extinction A_V and the total to selective extinction R_V = A_V/E(B-V) towards the objects, adopting the parametrisation of <cit.>. From comparing these SEDs to observed spectra a near-infrared excess should become apparent (Figure <ref>).The flux calibrated X-shooter spectra and the photometric data points are shown by the black lines and squares, respectively. The photometric data in the UBVRI bands were taken from <cit.> who used the 123 cm telescope on Calar Alto, and the 50 cm telescope on the Gamsberg in South West Africa. These measurements have an accuracy of ±0.5 mag. We highlight that the apertures used for this study are quite large, therefore the measurements might be contaminated by close neighbours. For B215 and B331 we obtained the V-band magnitude from <cit.>, and for B111 from the AAVSO Photometric all sky survey (APASS)[<https://www.aavso.org/apass>]. The JHK magnitudes weretaken from the 2MASS survey <cit.> which provides photometry for our objects with an accuracy better than 0.3 mag, with the exception of B215 whose JHK photometry we obtained from <cit.>. The mid-IR magnitudes were measured with the Infrared Array Camera (IRAC) on the Spitzer Space Telescope <cit.> as part of the Spitzer Legacy Science Program GLIMPSE <cit.>; the photometric accuracy achieved for our targets is better than 0.2 mag. The N and Q-band magnitudes (10.5 and 20 μm) were obtained (when available) from <cit.> who observed M17 with the ground-based infrared camera MANIAC. The aperture sizes were 2" for B337, 3" for B331 and B275, 5" for B311, 7" for B215, and 10" for B289; the accuracy in their measurements varies from 10 to 30%. The dereddened spectra and photometry are shown in blue. The dashed grey line represents the Castelli & Kurucz model corresponding to the spectral type of the star. The temperature and logg corresponding to the best Castelli & Kurucz model is indicated in the bottom-right corner; note that this is not identical to the temperature and logg of the star, but it corresponds to the adopted Castelli & Kurucz model. The resulting extinction and stellar parameters are labeled as well.§.§ Extinction parametersIn order to obtain independent values of the total extinction A_V and the total to selective extinction R_V, we implemented a χ^2 fitting algorithm. We first dereddened the X-shooter spectrum varying R_V from 2 to 5.5 in steps of 0.1 and then fitted the slope of the Castelli & Kurucz model to that of the dereddened X-shooter spectrum in the photospheric domain to determine the total V-band extinction, A_V. This allowed us to constrain R_V for all sources, except for B163. For this star we lack spectral coverage at λ < 850 nm. As the infrared excess starts at 1000 nm, a too limited spectral range is available for constraining R_V.The obtained values for R_V range from 3.3 to 4.7 and A_V varies from ∼6 to ∼14 mag (Table <ref>).observe a similar range, while <cit.> found R_V = 3.9 ± 0.2 for their sample. The latter authors argue that the extinction to M17 is best described by a contribution of foreground extinction (A_V = 2 mag with R_V = 3.1, the average Galactic value) plus an additional contribution produced by local ISM dust. We derive R_V and A_V for each individual sight-line. The A_V values that we obtained agree within the errors with those calculated by <cit.> from the total hydrogen column density N(H).The X-shooter spectra include strong diffuse interstellar bands (DIBs).reported that their strength did not vary with A_V (or E(B-V)) as seen in other Galactic sightlines. Massive star forming regions are known to exhibit anomalous extinction properties <cit.>. We will report the DIB behaviour towards sightlines in M17 in a separate paper. §.§ Stellar radiusTo estimate the radius of the stars we scaled the observed flux to the flux produced by the stellar surface given by the Castelli & Kurucz model, using the distance to M17 (d=1.98 kpc). The difference between the measured flux F_λ and the flux from the model F_Kur can be corrected for by multiplying by a factor (R_⋆/d)^2, where R_⋆ is the stellar radius, and d is the distance to the Sun. We calculated the luminosity logL/L_ using the V-band magnitude from <cit.>, the distance to M17, the A_V values from our fit (Sect. <ref>), and the bolometric correction corresponding to the spectral type from <cit.>.To assess the uncertainties in the parameters, we calculate the probability P that its χ^2 value differs from the best-fit χ^2 due to random fluctuations: P=1-Γ(χ^2 /2,ν /2), where Γ is the incomplete gamma function and ν the number of degrees of freedom. We normalise the χ^2 such that the reduced χ^2 corresponding to the best fitting model has a value of unity and we select all models with P ≥ 0.32 as acceptable fits representing the 68% confidence interval. The finite exploration of the parameter space may result in an underestimation of the confidence interval near the borders of P(χ^2,ν) = 0.32. To avoid an underestimation of the errors we select as boundaries of the 68% confidence interval the first combination of parameters that do not satisfy P(χ^2,ν) ≥ 0.32 <cit.>. Like in the case of B275 <cit.> the obtained stellar radius of the mYSOs does not correspond to the value expected for a main-sequence star. This corroborates our earlier finding that these mYSOs indeed are massive pre-main-sequence stars that are still contracting towards the main sequence. In Sect. <ref> we will show that this is consistent with the logg values independently measured from the broadening of the diagnostic lines. § MODELING THE PHOTOSPHERIC SPECTRUMTo further constrain the temperature, luminosity, projected rotational velocity, and surface gravity of the stars we used an automatic fitting algorithm developed by <cit.> and <cit.>, which compares the H i, Hei, Heii, and N absorption lines with model profiles produced by the non-LTE stellar atmosphere model<cit.>. For our PMS stars we did not fit the He and N abundances but fixed them to be consistent with the solar values. This is in agreement with what we would expect for such a young stellar population. This method applies the genetic fitting algorithm<cit.> which allows us to explore the parameter space in an extensive way <cit.>.calculates non-LTE line-blanketed stellar atmospheres and accounts for a spherically symmetric stellar wind. It can be used to examine the dependence of H, Hei, and Heii photospheric lines on T_ eff and logg. Its application to B stars has been successfully tested by <cit.>.Inputs to the fit are the absolute V-band magnitude (M_V) and the radial velocity (RV) of the star. We calculated M_V using the apparent magnitude reported by <cit.> and the extinction coefficient A_V found in Sect. <ref> adopting a distance of 1.98 kpc. RV was measured following the procedure described by <cit.> where we simultaneously adjust the spectral lines for a given object, taking into account variations in the signal-to-noise ratio. The RV values vary from -11 to 20 km s^-1 and are listed in Tab. <ref>; for B243, B268, and B289 we list the RV values obtained from the 2013, 2013, and 2012 observations, respectively. We calculated the RV dispersion using the standard deviation of a gaussian distribution (σ_1D) and we find the strikingly low value of ∼5 km s^-1. This is in contrast with the expectation that the massive star population in M17 includes many (close) binaries <cit.>. This points to a lack of short period binaries or a low binary fraction. A quantitative investigation of the low σ_1D is presented in <cit.>.Several atmospheric parameters are obtained from the fitting procedure. The effective temperature (T_ eff) can be constrained using the relative strength of the H i, Hei, and Heii lines. The surface gravity (logg) is obtained from fitting the wings of the Stark-broadened H i lines. This parameter allows us to constrain the luminosity class <cit.>. The projected rotational velocity (vsini) is a natural outcome of the fitting procedure as the models are convolved with a rotational profile to reproduce the observed spectrum. The mass-loss rate is mainly determined by fitting the Hα line. Only for B111 (logṀ = -6.00 ± 0.1) and B164 (logṀ = -6.35 ± 3.65), the two hottest stars in our sample, we obtain a reliable measurement. The parameter describing the rate of acceleration of the outflow (β) cannot be constrained by our data and was therefore fixed to the theoretical value predicted by <cit.>, i.e. β = 0.8 for main-sequence stars. Given the low mass-loss rates, the value adopted for β does not affect the model results. Another parameter in the fitting procedure is the microturbulent velocity: we allowed this parameter to vary from 5 to 50 km s^-1, but due to the lack of Si lines in our spectra this parameter remains poorly constrained. This does not affect the determination of the other stellar parameters.The bolometric luminosity (logL/L_⊙) is obtained by applying the bolometric correction to the absolute magnitude used as input. The luminosity and effective temperature are used to calculate the radius of the star R_⋆. Using logg we can calculate the spectroscopic mass of the star M_ spec=gR^2/G, where G is the gravitational constant. Given the uncertainty in logg it is very difficult to constrain M_ spec as is evident from Tab. <ref>. To calculate the errors in the parameters we follow the procedure described by <cit.> and <cit.>. The best-fitting model is selected based on the χ^2 value in the same way as described in Sect. <ref>. The normalisation of the χ^2 in this approach is only valid if the best fit is a good representation of the data, which we checked visually for each case (see Appendix <ref>). We select all models with P(χ^2, ν) ≥ 0.05 as acceptable fits, representing the 95% confidence interval to the fitted parameters.The results from the fitting procedure are listed in Tab. <ref>; the temperatures obtained withare consistent with the spectral types derived in Sect. <ref>. The best fitting model, and other acceptable fits (5% significance level or higher models) are shown in Appendix <ref>. The lines used for each fit are displayed in figures <ref> to <ref>; we have given the same weight to all the lines in the fitting procedure.It is important to point out that the confidence intervals cited in this paper represent the validity of the models as well as the errors on the fit. They do not include the contribution from systematic errors due to the model assumptions, continuum placement biases, etc. For a detailed analysis of the systematic errors and their impact on the parameter determination the reader is referred to <cit.>. As can be seen in Tab. <ref> the error bars differ significantly from star to star. Large error bars are obtained in two cases: (i) when the parameter space is poorly explored near the border of the confidence interval and, therefore, the first model that does not satisfy P(χ^2, ν) = 0.05 is considerably outside the confidence interval; (ii) when the parameter is poorly constrained due to e.g., a low signal-to-noise ratio.For B275 a good fit of the spectrum could not be obtained while leaving all parameters free. To mitigate this, we first constrained vsin i using a fit to only the helium lines. The obtained range of valid values was then used in a fit including the full set of diagnostic lines. Hβ was not included in this fit because of the presence of a very strong 4880 Å DIB blending the red wing of the line. This approach results in an acceptable fit to the spectrum, consistent with the results presented in <cit.>; we note that the red wings of Hei 6678 and 5875 are not well represented. The morphology of these lines and other Hei lines could be an indication that this star is in a binary system; follow-up observations of this source are required to confirm or reject this hypothesis.§.§ Comparison of the radius estimated by two methods The radius of the star is estimated in two different ways following the procedures described in the previous sections. In Fig. <ref> we show R_⋆ obtained by fitting the SEDs to Castelli & Kurucz models versus the one obtained via the genetic algorithm fitting. The diagonal line represents the one to one correlation and each symbol corresponds to one of our targets. B163, B331, and B337 were left out because for these stars it was not possible to identify any absorption lines, and therefore we did not include them in the GA fitting. The conclusion is that the two methods yield radii that are consistent within the errors. §.§ Hertzsprung-Russell diagramFigure <ref> shows the theoretical Hertzsprung-Russell diagram (HRD) based on the values of T_ eff and L obtained in the previous sections. We plotted the PMS tracks from <cit.> with the ZAMS mass labeled and open symbols indicating lifetimes. We also present the birthline for accretion rates of 10^-3, 10^-4, and 10^-5 M_⊙yr^-1. Assuming that the accretion is constant and that the Hosokawa tracks provide a correct description of the PMS evolution we conclude that 80% of our sample must have experienced an on average high accretion rate (10^-3 - 10^-4 M_⊙ yr^-1) and the remaining stars an accretion rate of at least 10^-5 M_⊙ yr^-1.All our confirmed PMS stars (blue circles) are located far away from the ZAMS and their positions can be compared with PMS tracks. The O stars (B111, B164, and B311; grey squares) and the stars without disk signatures (B289 and B215; green triangles) in the X-shooter spectra are located at or near the ZAMS. The position of the B-type star B253 is consistent with it being a PMS star, but its spectrum shows no signatures for the presence of a circumstellar disk.§ EVIDENCE FOR THE PRESENCE OF CIRCUMSTELLAR DISKSIn this section we present an overview of the emission-line features thought to be produced by the circumstellar disks. We discuss the nature of the disks of the PMS objects identified in the previous sections and measure the disk rotational velocity, V_ disk, using several hydrogen lines. In some of the objects we identify CO bandhead emission. The infrared excess observed in the SEDs (Sect. <ref>) provides information on the dust component of the disk.We observe several double-peaked emission lines and/or CO bandhead emission in the visual to near-infrared spectrum in six of our targets: B163, B243, B268, B275, B331, and B337. A selection of double-peaked lines along the X-shooter wavelength range is shown in Figs. <ref> to <ref>.The fact that we see double-peaked emission is indicative of a rotating circumstellar disk. For all of these objects we observe an infrared excess in their SEDs (Sect. <ref>). The atypical morphologies observed in the Balmer and Brackett series of B243 and B268 might be an indication of active accretion; nevertheless, further higher resolution observations are needed to confirm this scenario. §.§ Velocity structure of the gaseous disks To measure the characteristic projected rotational velocity properties of the gaseous component of the circumstellar disks, we selected a sample of double-peaked emission lines for each of the sources and fitted two Gaussian functions to measure the peak to peak separation. The projected rotational velocity measured in this way corresponds to half of the peak to peak separation. The Gaussian functions fitted to the lines are shown with the solid red lines in Figs. <ref> to <ref> and the results from these measurements are listed in Tab. <ref>. The Balmer, Paschen, and Brackett series and Caii triplet lines are plotted in each of the columns (from left to right) and lines are labeled in the left part of the plot. The measured peak to peak separation is indicated to the right.Figure <ref> shows the disk projected velocities measured from each of the hydrogen lines against logλ f_lu, where f_lu is the oscillator strength. We uselogλ f_lu as a measure of the relative strength of a given line within a line series. The oscillator strengths for the hydrogen lines were obtained from <cit.>. For B163 and B337 we do not have sufficient velocity measurements to draw any conclusions. For four out of the six gaseous disks detected (B243, B268, B275, and B331) we see a clear trend of the projected disk rotational velocity from the hydrogen recombination lines with line strength. This suggests a structured velocity profile of the gaseous disks, in qualitative agreement with the prediction that the high excitation lines form in the inner region (dense and hot) of the disk while the low excitation lines form over a larger area (and on average more slowly rotating part) of the disk. B268 and B275 also exhibit Caii triplet and Oi double-peaked emission lines. The velocities measured from these lines are higher than those of the hydrogen lines. This indicates that these lines are formed closer to the star than the hydrogen lines.Assuming that the disks have a Keplerian velocity structure and adopting the mass of the central object, it is possible to roughly calculate the distance from the star at which the lines are formed. This results in a range from a few hundred to one thousand R_⊙ (tens to a few hundreds R_⋆).§.§ CO bandhead emission CO overtone emission is produced in high-density (10^10 – 10^11 cm^-3) and high-temperature (2500 – 5000 K) environments <cit.>. CO is easily dissociated, and therefore must be shielded from the strong UV radiation coming from the star. These conditions are expected in the inner regions of (accretion) disks, which makes the CO bandheads an ideal tool to trace the disk structure around mYSOs <cit.>. The shape of the CO lines can be modelled by a circumstellar disk in Keplerian rotation <cit.>. The blue shoulder in the bandheads is a measure of the inclination of the disk: an extended blue shoulder indicates a high inclination angle (i.e., near "edge-on" view). We detect CO-bandhead emission in B163, B243, B268, B275, and B331 (see Fig. <ref>).Although CO-bandhead emission is rare, it is also seen in some B[e] supergiants <cit.>. However, these evolved B[e] stars show numerous forbidden emission lines in their optical spectra, while we only observe the well known [Oi] 6300 in some of the sources. Hence, the observed spectra support our hypothesis that the observed disks are related with the accretion process in contrast to a possible origin such as in (more evolved) B[e] stars <cit.>. Modelling of the bandheads and the double-peaked hydrogen emission lines will be the subject of a forthcoming paper.§ DISCUSSION§.§ Age distribution of the massive PMS population in M17Age estimates of M17 show that its stellar population is not older than 1 Myr. There is no evidence for a supernova explosion in that area <cit.>, consistent with such a young age. We estimated the age of the main-sequence stars by comparing their position in the HRD with Milky Way evolutionary tracks from <cit.>. We compare the position of our stars with evolutionary tracks that account for the effects of rotation considering a flat distribution of spin rates starting at the measured vsini. To do so, we used the bayesian tool [Theweb-service is available at <www.astro.uni-bonn.de/stars/bonnsai>.] from <cit.>. Within the uncertainties of the models, the ages obtained for B111, B164, and B311 are 1.60, 0.98, and 0.82 Myr, in fair agreement with the age estimates for the region (Tab. <ref>). As a way of testing our classification of the PMS stars, we estimated their ages using main sequence tracks through . We obtained main sequence ages ranging from 10 to 90 Myr, which would be inconsistent with M17 being a young region. Assuming that our classified PMS stars are part of M17, we conclude that these stars cannot be post-, but must be pre-main sequence objects.We estimated the age of the PMS stars by comparing their effective temperatures and luminosities with PMS tracks <cit.>. For these stars we find an age span from tens of thousands of years to a few hundred thousand years. If we compare the position of the (presumable) main-sequence stars B215, B253, and B289 to isochrones, we obtain age estimates of ∼9, ∼50, and ∼4 Myr, respectively. B253 does not present emission lines nor IR excess in its SED, and B215 and B289 do not present emission lines in the X-shooter spectrum but have IR excess longward of 2.3 μm. If these sources are PMS stars this suggests that their circumstellar disks have already (at least partially) disappeared (see also Sect. <ref>). Given the size of our sample and the uncertainties in the models we cannot draw a firm conclusion about the age of M17 nor about the possible presence of two distinct populations in this region <cit.>. Nevertheless we observe a trend of age with luminosity: the less luminous objects are further away from the ZAMS than their more luminous counterparts. This is in line with the idea <cit.> that more massive stars form faster and therefore spend less time on the PMS tracks than lower-mass stars. §.§ Extinction towards the PMS stars in M17Table <ref> lists the extinction properties of our targets. We find that the extinction in the V-band varies from ∼5 to ∼14 mag and that the total to selective extinction is 3.3 < R_V < 5. The two sources with the highest extinction are situated near or in the irradiated molecular cloud (see Fig.<ref>), but there is no general trend with location in the H ii region. The sources with higher A_V tend to have larger values of R_V, although the correlation is weak. The overall conclusion is that the extinction is quite patchy, with substantial variation on a spatial scale of 50 arcsec (corresponding to a geometrical scale of 0.5 pc at the distance of M17), similar as to the findings of . A dust disk local to the star, for example, may dominate the line-of-sight extinction towards the sources <cit.>. Studies of individual sources therefore should not rely on average properties of the region, but should be based on a detailed investigation of the extinction in the line of sight. §.§ Presence of circumstellar disks around massive PMS stars We detect signatures of circumstellar disks in six of our sources: B163, B243, B268, B275, B331, and B337. The full sample includes two O-type stars, B111 and B164, that are not classified as potential YSO sources by , but as main sequence objects, and do not reveal disk signatures. Two stars, B215 (IRS15) and B289, do not show evidence for gaseous disk material in their spectral lines but do feature excess infrared continuum emission indicative of dust in the circumstellar environment. We thus find that 60% of our massive YSO candidates show clear evidence for circumstellar disks in their spectrum and another 20% likely also feature disks based on NIR excess emission. In Fig. <ref> we have labeled the sources with a gaseous disk detectable in the X-shooter spectrum with a blue dot, the ones with only NIR excess in the SED with a green triangle and the O and B stars with a grey square. The stars with gaseous disks are located further away from the ZAMS than the other sources. However, we cannot link this in a straightforward way to an evolutionary effect. Having noted this, and taking into account that our sources with non-detectable disks are younger than 10^5 yrs, we can conclude that massive stars up to ∼20 M_⊙ retain disks up to less than 10^5 years upon arrival on the ZAMS. We identify the presence of [Oi] 6330 emission in three sources (B215, B243, and B275). InHerbig Ae/Be this line is mostly formed in stars with a flaring disk <cit.>.The spectral profiles of some lines in B268 and B243 might be an indication that this star is actively accreting. For the other sources, as we do not find strong evidence for infall or the presence of jets, we do not know whether these sources are actively accreting or that the observed disks are structures remnant of the formation process. §.§ Spin properties of the massive PMS stars in M17 The projected rotational velocity (vsini) distribution for 216 O stars in 30 Dor has been published by <cit.>. Their distribution shows a two-component structure consisting of a peak at 80 km s^-1 with 80% of the stars having 0< vsini < 300 km s^-1 and a high velocity tail (containing 20% of the stars) extending up to 600 km s^-1. <cit.> studied 300 stars spanning spectral types from O9.5 to B3 in 30 Dor; they find a bimodal distribution with 25% of the stars with 0 < vsini < 100 km s^-1 and a high velocity tail between 200 < vsini < 350 km s^-1.They estimated vsini using a Fourier transform method, which allows them to separate the rotational broadening from other broadening mechanisms. <cit.> measured vsini for 97 OB stars in the Milky Way. They find that 80% of their sample rotate slower than 200 km s^-1, and that the remaining 20% has 200 < vsini < 400 km s^-1. Tab. <ref> shows the vsini measured for the stars in our sample. We find that around 30% of our sample is rotating relatively fast (around 200 km s^-1).We show the cumulative vsini distribution functions of these works together with our results in Fig. <ref>. To quantitatively compare these distributions we performed a Kuiper test, which allows to test the null hypothesis that two observed distributions are drawn from the same parent distribution. The significance level of the Kuiper statistic, p_K, is a percentage that indicates how similar the compared distributions are. Small values of p_K show that our cumulative distribution is significantly different from the one it is compared to. The p_K values obtained from comparing our distribution with the ones of <cit.>, <cit.>, and <cit.> are p_K=13%, 99%, and 18%, respectively. As these values are not lower than 10%, they do not allow us to reject the null hypothesis. Upon arrival on the ZAMS our sample will span a roughly similar range of masses as the B-star sample in 30 Dor <cit.>. Therefore, it is most appropriate to compare our findings to this sample. According to the models by <cit.> and <cit.> the vsini distribution of stars in our mass range will not change significantly during the first few Myrs of evolution. <cit.> point out that macroturbulent motions only need to be taken into account in cases of relatively slow spinning stars (vsini<80 km s^-1). We note that <cit.> discuss the possibility of their presumably single star sample to be polluted by (relatively rapidly spinning) post-interaction binaries. We estimated the vsini that our PMS stars will have upon arrival on the ZAMS using the ZAMS radii corresponding to the end of the PMS tracks from <cit.>. We applied angular momentum conservation and assumed that the stars are rigidly rotating (which is analogous to homologous contraction). When comparing this vsini distribution (dashed line in Fig. <ref>) with the B-star sample in 30 Dor we find that p_k = 87%.Assuming that our stars represent a progenitor population of the LMC B-star sample studied by <cit.> (neglecting metallicity effects and pollution from post-interaction systems in the B stars in 30 Dor sample) we find that the contraction of the PMS stars during the main sequence is consistent with being homologous. Of course, it is premature to firmly state this given the small sample size and the caveats mentioned above.§ CONCLUSIONSWe performed VLT/X-shooter observations of young OB stars in M17. We classified and modelled the photospheric spectra usingin order to derive their stellar parameters. We identified the presence of gaseous and dusty disks in some of them based on the emission lines in the spectrum and on the IR excess observed in the spectral energy distribution. We confirm the PMS nature of six objects in this region and conclude that they are on their way of becoming B main sequence stars with masses ranging from 6 to 20 M_⊙. This constitutes a unique sample of PMS stars that allows us to test theoretical star formation models. Our findings can be summarised as follows:* We confirm the PMS nature of six of the mYSO candidates presented by . We conclude that most of our objects must have experienced, on average, high accretion rates. * The age of the PMS objects has been obtained by comparing their position in the HRD with pre-main sequence tracks of <cit.>, whereas the age of the OB stars was estimated using the tool<cit.> and comparing to tracks of <cit.>. For the O stars we obtained ages younger than 2 Myr. The pre-main sequence stars have estimated PMS lifetimes of a few hundred thousand years. Given the uncertainty in the age and the fact that we have only a few stars, we cannot conclude that the PMS population corresponds to a second generation of star formation in M17 but found no indication in favour either. * We measured the visual and total to selective extinction towards our objects by fitting Castelli & Kurucz models corresponding to the spectral types. We confirm that the extinction towards M17 is highly variable, as usually observed in star forming regions. We point out that a dust disk local to the star may dominate the line-of-sight extinction towards the sources. Based on the NIR excess (> 2 μm) observed in the spectral energy distributions we found dusty disks in eight of our targets. * Via (double-peaked) emission lines we found evidence for gaseous circumstellar disks in six of our targets, all of which also include a dust component. We measured the projected rotational velocity of the disks from each of the double-peaked lines and found a structured velocity profile among the hydrogen recombination lines for four out of the six disks. For two out of the six disks we were able to identify Caii triplet and Oi double-peaked emission lines. The velocities measured from these lines are larger than from the ones measured from hydrogen suggesting that they are formed closer to the star than the hydrogen lines. * We measured the projected rotational velocities, vsini, of our stars. About 30% of our sample rotate at around 200 km s^-1 or faster. The PMS objects are expected to contract and therefore have spun up upon arrival on the main-sequence. Assuming homologous contraction and the absence of processes causing angular momentum loss in their remaining PMS evolution, the vsini distribution of our sample appears consistent with that of the B stars in 30 Dor once they ignite hydrogen. We note that the contraction in PMS stars is not well understood. Two of the objects will have a vsini >300 km s^-1 upon arrival on the MS.With this unique, though still small sample we show the potential for constraining models of star formation. A larger sample is needed in order to robustly assess the validity of different theories. We thank the anonymous referee for for carefully reading the manuscript and many helpful and insightful comments and suggestions. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This research made use of Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration, 2013). MCRT is funded by the Nederlandse Onderzoekschool Voor Astronomie (NOVA). § FULL X-SHOOTER SPECTRA. Normalised VLT/X-shooter spectra of the objects studied in this paper (Table 1), combining the UVB, VIS and NIR arm. Line identifications are indicated above the spectrum. Note the presence of telluric absorption features centred at 690, 720, 820, 920, 1400, 2010 and 2060 nm. The attenuated region between the H- and the K band (1830 - 1980 nm) has been omitted. The blue part of the spectrum for B163, B331, and B337 is not visible due to severe interstellar extinction. We have included the spectra of B111, B275, and B311 as an example, the version of the appendix for the full sample will be available on A&A. §FITTING RESULTSIn this appendix we show in detail the model fits for our targets and the spectral lines used in each of the cases. We have included the models for B111 and B164 as an example, the version of the appendix for the full sample will be available on A&A. | http://arxiv.org/abs/1704.08216v1 | {
"authors": [
"María Claudia Ramírez-Tannus",
"Lex Kaper",
"Alex de Koter",
"Frank Tramper",
"Arjan Bik",
"Lucas E. Ellerbroek",
"Bram B. Ochsendorf",
"Oscar H. Ramírez-Agudelo",
"Hugues Sana"
],
"categories": [
"astro-ph.SR",
"astro-ph.GA"
],
"primary_category": "astro-ph.SR",
"published": "20170426170442",
"title": "Massive pre-main sequence stars in M17"
} |
December 30, 2023 1Department of Physics and Astronomy, The University of Western Ontario, London, ON Canada N6A 3K7 2Department of Physics, Central Michigan University, Mount Pleasant, MI 48859 USA 3US Naval Observatory, Flagstaff Station, 10391 W. Naval Observatory Rd, Flagstaff, AZ 86001 USA We utilize a multi-step modelling process to produce synthetic interferometric and spectroscopic observables, which are then compared to their observed counterparts.Our extensive set of interferometric observations of the Be star 48 Per, totaling 291 data points, were obtained at the Navy Precision Optical Interferometer from 2006 Nov 07 to 2006 Nov 23. Our models were further constrained by comparison with contemporaneous Hα line spectroscopy obtained at the John S. Hall Telescope at Lowell Observatory on 2006 Nov 1. Theoretical spectral energy distributions, SEDs, for 48 Per were confirmed by comparison with observations over a wavelength regime of 0.4 to 60 microns from <cit.> and <cit.>. Our best-fitting combined model from Hα spectroscopy, Hα interferometry and SED fitting has a power law density fall off, n, of 2.3 and an initial density at the stellar surface of ρ_0 = 1.0 × 10^-11with a inclination constrained by Hα spectroscopy and interferometry of 45^∘± 5^∘. The position angle for the system, measured east from north, is 121^∘ ± 1^∘. Our best-fit model shows that the disk emission originates in a moderately large disk with a radius of 25 R_* which is consistent with a disk mass of approximately 5 × 10^24 g or 3 × 10^-10 M_*. Finally, we compare our results with previous studies of 48 Per by <cit.> with agreement but find, with our much larger data set, that our disk size contradicts the findings of <cit.>. § INTRODUCTION Classical Be stars are distinguished by the presence of Balmer emission lines in their spectra.As first proposed by <cit.>, the Balmer lines are attributed to an equatorial disk of material surrounding the star (; ). Other defining characteristics of Be stars include linearly polarized light, infrared and radio continuum excess due to radiative processes within the disk and rapid stellar rotation.As well, these systems often exhibit variability over a range of time scales (for details, see the recent review by <cit.>).The classical B-emission (Be) star 48 Per (HD 25940, HR 1273, spectral type B3V) is well studied and located at a distance of 146 pc[based on Hipparcos parallaxes <cit.>]. <cit.> originally classified this star as pole-on but the appearance of doubly peaked Hα profiles reported by <cit.> led <cit.> to suggest that it has an inclination of 34^∘ to 40^∘. Since then, the value of the inclination for this system has remained contentious. The reported changes in the spectral line shape and in brightness (see <cit.> and references therein) point to periods of variability exhibited by 48 Per. However, we note that 48 Per was particularly stable over the time our observations were acquired (see the next Section for more detail).Studies by <cit.> and <cit.> combined interferometry with other observables for 48 Per, such as polarimetry and spectroscopy, and their work is ideally suited to detailed comparison with the results presented here. The <cit.> study, hereafter `Q97', obtained interferometric observations with the Mark III Interferometer <cit.> as well as spectropolarimetric observations.Although Mark III has since been decommissioned, it was a predecessor to the instrument used for this study, the Navy Precision Optical Interferometer, and the two instruments share some characteristics. 48 Per was observed with six distinct interferometric baselines resulting in a set of 46 observations in the study by Q97. Through modeling they were able to place bounds on the size and inclination of the Hα emitting region. They conclusively demonstrated, for the first time, that Be star disks could not be both geometrically and optically thick. More recently, <cit.> or `D11', obtained data from the Center for High Angular Resolution Astronomy (CHARA) interferometer <cit.> to constrain estimates for the size of the Hα emission region for 48 Per. Q97 confirm the nearly pole-on orientation found by <cit.> by determining a minimum inclination of 27^∘ which is also consistent with an inclination of 30± 10^∘ more recently reported by D11.The overall progression of this study is as follows; in Section <ref> we detail our observational program, and Section <ref> provides an overview of the code used to calculate the theoretical disk models, along with the data pipeline we developed to analyze the model and observational data.The results of this analysis are presented in Section <ref>.Finally, Section <ref> discusses our findings along with a comparison to other work and implications. § OBSERVATION PROGRAM Our observations of 48 Per were obtained at the Navy Precision Optical Interferometer (NPOI), located near Flagstaff, AZ, USA.The NPOI has an unvignetted aperture of 35 cm with an effective aperture for the observations of 12.5 cm set by the diameter of the feed system optics. See <cit.> and <cit.> for additional technical descriptions of this facility. Typically, observations from up to five unique baselines are obtained simultaneously; for this study, baseline lengths ranged from 18.9 to 64.4 m.A total of 291 squared visibility measurements from a 150 Å wide spectral channel containing the Hα emission line (i.e., centered at 6563 Å) of 48 Per were obtained at NPOI in the autumn of 2006. The specific dates of observation are listed in Table <ref>, and Table <ref> provides details such as time, (u,v)-space coordinates, and baseline specifier for each individual observation. Figure <ref> shows the (u,v) plane coverage based on five unique baselines, where individual observations are represented by open circles and the arcs illustrate the possible coverage from the meridian to 6^ h east (dotted lines) and from the meridian to 6^ h west (solid lines).The squared visibilities from the Hα channel were calibrated with respect to the continuum channels following the methodology outlined in <cit.>.This approach allows for a more effective removal of instrumental and atmospheric effects since the Hα and continuum channels are recorded simultaneously and correspond to the same direction on the sky (i.e., come from the same source).Because the photosphere of the central star is not expected to be spatially resolved by our baselines at any significant level, the continuum channels are assumed to follow a uniform disk (UD) model with an angular diameter of 0.306 mas (based on the spectral type adopted stellar parameters, including Hipparcos distance, from Table <ref>).Figure <ref> shows the calibrated squared visibilities from the Hα containing channel (V^2_Hα) for all five unique baselines as a function of radial spatial frequency, along with a UD model curve that represents the continuum channels.Although strictly speaking an external calibrator is not required to calibrate the interferometric squared visibilities from the Hα channel when the continuum channels are utilized as a calibration reference, we have utilized observations of an external reference star to correct for small higher order channel-to-channel instrumental variations following the procedure described by <cit.>.For the purpose of these higher order corrections we utilized a nearby non-Be type star, ϵ Per(HR 1220, HD 24760), for which the observations were interleaved with those of 48 Per.cr 2 0pc Interferometric NPOI ObservationsUT Date Number of V^2_Hαmeasurements 2006 Nov 07182006 Nov 08182006 Nov 09262006 Nov 10102006 Nov 11142006 Nov 1412006 Nov 15102006 Nov 16222006 Nov 17422006 Nov 18242006 Nov 20362006 Nov 21422006 Nov 22242006 Nov 234ccccc 5 0pc Interferometric ObservationsJD-2,450,000 u vV^2_Hα Baseline1 10^6 cycles/rad 10^6 cycles/rad ± 1 σ 4046.749 17.189 -21.882 0.834 ± 0.026 AC-AE 4046.749 -29.224 0.422 0.849 ± 0.033 AC-AW 4046.782 21.140 -18.991 0.854 ± 0.062 AC-AE 4046.782 -30.994 -4.121 0.893 ± 0.128 AC-AW 4046.815 24.338 -15.402 0.793 ± 0.040 AC-AE 1The baselines AC-AE, AC-AW, AW-W7, AC-W7, AE-W7 correspond to lengths of 18.9, 22.2, 29.5, 51.6, and 64.4 m, respectively. 0.8Table <ref> is published in its entirety in a machine readable format.A portion is shown here for guidance regarding its form and content. To complement our interferometric observations we have also obtained contemporaneousobservations with the Solar Stellar Spectrograph (SSS), an echelle spectrograph attached to the John S. Hall Telescope at Lowell Observatory (). The emission line is singly peaked, consistent with a disk viewed more pole-on to mid inclinations. Over the course of our observing program theemission line remained remarkably stable. Ourspectra acquired on 2006 Nov 1 and on 2006 Dec 9 that bracket the time frame of our interferometric data are indistinguishable, withequivalent widths of 28.2 and 28.1 Å, respectively. Based on a larger set of seven Hα profiles obtained over the time period from 2006 Apr 10 to 2006 Dec 30, the emission profile shows only 1.4% variation based on the standard deviation of the set. The observed Hα spectrum (blue circles) obtained on 2006 Nov 1 is shown in Figure <ref> along with a sample of our best-fitting models based on our figure-of-merit value, F, analysis discussed in the next section. § MODELLING§.§ Data Pipeline:Bedisk, Beray and 2dDFTwas developed by <cit.>. It is a non-local thermodynamic equilibrium (non-LTE) modelling code which calculates self-consistent temperature distributions based on the corresponding density distribution and level populations within the disk <cit.>. For the present study, the density structure within the circumstellar disk was described by a power law: ρ(R,Z) = ρ_0 (R_*/R)^n exp[-(Z/H)^2] where ρ_0 is the density at the disk-star boundary, Z is the distance from the plane of the disk measured normal to the disk and H is the vertical scale height of the disk measured perpendicular to the disk. We assume that H is in approximate vertical hydrostatic equilibrium with a temperature 0.6 × T_eff of the star, see <cit.> for details.The temperature and density distributions as well as level populations calculated withare used as inputs to<cit.>. The - sequence is used to obtain a model intensity image of the disk system on the plane of the sky.calculates a formal solution of the radiative transfer equation through the disk along approximately 10^5 rays from the observer's line of sight. The computational region extends from the photosphere to a distance (in terms of stellar radii) specified by the user. For this study, models were computed for disk sizes of 6.0, 12.5, 25.0 and 50.0 R_*.Interferometric data is in the form of a series of squared visibilities (V^2) as a function of spatial frequency, which is itself a representation of the interferometric baseline <cit.>.The 2D Discrete Fourier Transform (2dDFT) code <cit.> takes the 2-dimensional discrete Fourier transform of theimage and produces V^2 as a function of spatial frequency. The code then compares the model to a set of observations supplied by the user and estimates goodness-of-fit based on a reduced χ^2 test. This reduction pipeline was developed by <cit.> and first used to model the Be star o Aquarii. In addition to determining the density distribution within the disk, our Fourier analysis is used to calculate the system's angular dimensions on the sky, the position angle of the system, the disk mass and corresponding angular momentum. §.§ Model ParametersThe spectral type B3Ve was adopted for 48 Per, which is consistent across the two comparison studies of Q97 and D11, and with the Bright Star Catalog <cit.>.Further, this is in agreement with examples in the literature dating back over the past six decades (see, for example, <cit.> or <cit.>).The stellar parameters for the B3Ve type were determined by linear interpolation fromand are provided in Table <ref>.lr 2 0pc Adopted Stellar Parameters for 48 Per. Parameter Value M (M_) 1 7.6 R (R_) 1 4.8 L (L_) 2580 T_ eff (K) 2 18800 log g 4.0 Distance (pc) 3 146.2 ± 3.5 Angular Diameter (mas) 0.306 1adopted from Table 15.8 of <cit.> 2interpolated in Table 15.7 of <cit.> 3adopted from <cit.>§.§ Computational Grid The parameter space was chosen to be consistent with n and ρ_0 values that would be expected for Be star disks based on historical predictions <cit.> and on contemporary studies (see, for example, section 5.1.3 of <cit.> for a summary of recent results in the literature). As mentioned above, our models were computed for a range of disk sizes of 6.0, 12.0, 25.0 and 50.0 R_*. Other model parameters were varied as follows; 1.5 ≤ n ≤ 4.0 in steps of 0.25, 1.0× 10^-13≤ρ_0 ≤ 2.5 × 10^-10in increments of 2.5 over each order of magnitude, with inclinations ranging from 20^∘ to 65^∘ in steps of 5^∘.§ RESULTS§.§ Hα Spectroscopy Ourline profile models were compared directly to the observed spectrum obtained on 2006 Nov 1. Our model spectra were convolved with a Gaussian of FWHM of 0.656 Å to match the resolving power of 10^4 of the observed spectra. For each comparison, the percentage difference between the observed line and model prediction were averaged over the line from 6555 Å to 6570 Å to determine figure-of-merit value, F, computed by,ℱ=1/N∑_i w_i | F_i^obs-F_i^mod|/F_i^obs,withw_i = |F^obs_i/F^obs_c - 1 |, where the sum is over all of the points in the line. F^obs_i and F^mod_i are the observed flux and the model flux, respectively. F^obs_c is the observed continuum flux. Equation <ref> emphasizes the fit in the core and peak of the line while minimizing differences in the wings. Finally F was normalized by the best fit, i.e. F/F_min, so that in our analysis the model best fit has a value of 1. This technique of matching the core of the line was found to be useful in a previous study for o Aquarii <cit.>. Overall, our model spectra were too weak in the wings similar to the results of D11 and <cit.>. Figure <ref> shows the four best-fitting spectra within 20% of the best-fitting model along with the observed line. The density parameters for each model are listed in the legend in the upper right of the figure along with the value of F. The parameters in brackets in the legend correspond to ρ_0 in , n, disk size in R_*, and inclination angle. The parameters corresponding to our best fit are = 5.0× 10^-12 , 2.0, 50 R_*, 45^∘ (blue line on Figure <ref>). The average inclination for the four best-fit models is 46 ± 5^∘. The Hα line observed for 48 Per has an EW of 28.2 Å and exhibits the singly-peaked profile we expect to see from a disk system with low to moderate inclination. Q97 estimated the lower limit for the inclination angle of 48 Per to be 27^∘. D11 determined a best-fit inclination from their kinematic model of 30 ± 10^∘. Our model spectra for inclinations at 30^∘ and lower did not reproduce the observed line shape well. The modellines were too narrow and the wings of the line were too weak. Considering a slightly larger set of 16 best-fitting models, corresponding to F within ∼ 30 % of the best fit, there are 3 models with an inclination of 35^∘, and the remainder in this set have inclinations between 40^∘ and 55^∘ with only one model at this highest value. The average inclination of this set is 47 ± 7^∘. The models with the greater inclinations tended to fit the wings better since broader lines occur naturally with increasing inclination but the spectra corresponding to highest inclinations have a doubly-peaked shape unlike the observed profile. Figure <ref> shows the inclination versus F/F_min for all of our computed models for F/F_min≤ 2. The symbols in the legend in the lower left of the Figure correspond to the values of the density power law exponent, n. The values of the base density, ρ_0, vary with n. Generally, for small n, i.e. slower density fall-off with increasing distance from the central star, ρ_0 is also correspondingly reduced to obtain a similar amount of material in the disk to produce the Hα emission and vice versa. The horizontal dotted lines on Figure <ref> correspond to the inclination ± 1 σ obtained from Gaussian disk fits, GD, to the interferometry for ease of comparison. See Section <ref> for more details about the geometric fits.We see from Figure <ref>, and as discussed above, that our best-fitting models for F/F_min≤ 1.2 have inclinations with 46 ± 5^∘. However, with slight increases in the value of F/F_min to within ∼ 30 % we see a range in the inclination of ∼ 30^∘ to 55^∘. The lower limit of this range is consistent with the lower limit obtained by Q97 and the result of D11. Note all of our spectroscopic best-fitting models with F/F_min≤ 2.0 shown in Figure <ref> corresponded to models computed with a disk size of 50 R_*.§.§ Hα Interferometry The image file outputs fromwere fed into 2dDFT to obtain models of V^2 against spatial frequency, which were then compared directly to data obtained from interferometry by a reduced χ^2 calculation. The results of the best fit to V^2 data,= 3.0, 1.0× 10^-10 , 45^∘, are shown in Figure <ref>. The model V^2 symbols are plotted as green circles (181 points), red triangles (62 points) and blue plus signs (48 points). The colours represent the degree of agreement between the model visibilites and NPOI observations. The green points have V^2 that agree with the observed data within the errors. Given the the majority of the points are in this category (over 60%) and we conclude that our model represents the data reasonably well within ± 1 σ. The red and blue symbols represent model V^2 that have χ^2 too low (21%) and too high (16%), respectively. The reduced χ^2 corresponding to the best-fit model is 1.39 with a position angle, PA, of 121±1^o. In order to assess our model predictions, we compared our predicted Beray visibilities (shown in Figure <ref>) with our best-fitting model obtained by our interferometry analysis. In Figure <ref>, the visibilities are plotted as a function of the spatial frequency for the same best-fit model shown in Figure <ref>. The observed data are shown in black with the model in red. The dashed line corresponds to the star of a uniform disk of 0.306 mas. The residuals between the model and the data are shown in the bottom panel and demonstrate that our model visibilities fit the observations within ± 1 σ. Figure <ref> shows the PA for our models that correspond to χ^2/ν≤ 5. Horizontal lines correspond to the mean PA (solid blue line)and the mean ± 1 σ (blue dotted lines) obtained from model fits to the interferometry for χ^2/ν≤ 2.5. We note that for models with χ^2/ν≤ 2.5 that there is considerable scatter in the PA of about the mean of 140^∘ of ∼ 15^∘. However, the five best-fit models corresponding to χ^2/ν≤ 1.5 shown on Figure <ref> have a tight range of PAs of 121 ± 1^∘. This is good agreement with the PA determined from the best elliptical Gaussian fit to the interferometry described next.§.§ V^2 Geometric Fits It is also interesting to compare the size theemitting region and position angle of our models with geometric fits to the interferometry data. We note that geometric fits use simple functions but no physics to determine some basic physical characteristics of the disk. Here we follow the technique described in <cit.> as developed by <cit.>. Geometric fits were also computed by D11 and Q97 for 48 Per so we include a comparison with their work as well. Tables <ref> and <ref> compare the results of geometric fits to the visibilities using UD and GD fits, respectively. These tables also provide the axis ratios of the minor axis to major axis, the position angle of the major axis of the disk on the sky with respect to north, the fractional contribution from the photosphere of the central star to the Hα containing interferometric signal, c_*, the reduced χ^2 and the number of data points (N) used to obtain each result. The reader is referred to <cit.> and references therein for more details about our geometric models.lcccccr 7 Uniform Disk (UD) Geometric Fits based on Hα channels.Study Major Axis Axis Ratio Position Angle c_*χ^2/ν N[mas][degrees] [stellar[number contribution]of data] This study 5.70 ± 0.11 0.69±0.02119±30.876±0.002 1.480291D11 3.4 ± 0.2 0.77 ± 0.06110± 19—0.56 3lcccccr 7 Elliptical Gaussian Disk Geometric Fits based on Hα V^2 Data. Study Major Axis Axis Ratio Position Angle c_*χ^2/ν N[mas][degrees] [stellar[number contribution]of data]This study 3.24± 0.08 0.71 ± 0.03122±30.855± 0.0031.456 291Q97 Modified Fit2.77 ± 0.56 0.89 ± 0.13 680.271- 46D11 result 2.1 ± 0.2 0.76 ± 0.05 115± 33— 0.62 31Not a fitted parameter.The value is based on expected photon counts in 1 nm wide channel based on a star with the same V magnitude.For a wider spectral channel this value is expected to be closer to unity. Table <ref> shows good agreement with D11 for the axis ratio and position angle for the UD fits, however, we obtain a larger major axis than D11. There may be several reasons for this discrepancy. The fitting procedure is slightly different in each study with D11 determining the disk parameters by removing the stellar contribution. More significantly, we note that we have a substantially larger set of interferometry data consisting of 291 points providing a greater sky coverage in the (u,v) plane (see Figure <ref>). Table 1 and figure 1 in D11 shows the details of their observations and (u,v) plane coverage which is much less extensive compared to our data set. Q97 also fit their data for 48 Per with a UD and a ring-like model but the specific details about these geometric fits are not provided in their paper because they resulted in larger χ^2 than their GD fits. However, they mention that these models were not significantly different from the results for their GD models shown in Table <ref>.A comparison of the results for the GD fits are presented in Table <ref> and show good agreement with the major axis between Q97 and this study. D11 obtained a smaller major axis than we obtain however D11's result does agree with Q97 within the errors. The axis ratios point to a disk that is not viewed at large inclination angle that would result in large deviations from circular symmetry. There is agreement in the PA obtained except for the modified model (that takes the contribution of the star into account) by Q97 which gives a PA about half the other values presented in Table <ref>. We also note that our definition of c_* is the same as c_p used by Q97, however since the filters (or spectral channels) have different widths, the values are different. Having said that, Q97 used a 1 nm wide filter and did not fit for the parameter c_p, but instead determined the value of this parameter based on photometric counts and the expected values based on the V and (B - V) index. Our UD and GD fits to interferometry listed in Tables <ref> and <ref> give predicted axis ratios of 0.69±0.02 and 0.71±0.03, respectively. If we assume an infinitely thin disk, these ratios translate into inclination angles of ∼ 46^∘, nearly identical to our inclination of 45 ± 5^∘ from spectroscopy. Recall the horizontal dotted lines on Figure <ref> corresponding to the inclination ± 1 σ obtained from GD fits, plotted for convenience, with a range of predicted inclinations for F/F_min≤ 2 from our spectroscopic analysis. Clearly there is strong agreement for 48 Per's inclination obtained from geometric fits and our emission line modeling.A comparison of Tables <ref> and <ref> reveals that the major axis for the UD fits are always larger than the GD fits for each respective study. As a minor point of clarification, this is expected because the UD fits represent the major axis as the largest extent of the disk projected on the plane of the sky but for the GD fit, the size is proportional to the width of the Gaussian which only contains 68% of the light. Finally, we note that the best-fit elliptical Gaussian fit from our interferometry, shown in Figure <ref>, gives a PA of 121.65 ± 3.17^∘ in good agreement with our PA model results shown in Figure <ref> for our five best-fit models with PAs of 121 ± 1^∘ corresponding to χ^2/ν≤ 1.5. §.§ Spectral Energy Distributions Spectral energy distributions, SEDs, were also computed with beray for wavelengths between 0.4 and 60 microns for comparison to the reported observations of <cit.> (optical and near-IR) and <cit.> (IR). The model SEDs were computed for the same range of density parameters and disk sizes as described in Section <ref>. The observed fluxes were de-reddened for an E(B-V)=0.19 <cit.> following the extinction curve of <cit.> assuming a standard R_V of 3.1. This is a relatively large amount of reddening for a nearby star like 48 Per; however, it is close to the galactic plane with a galactic latitude of just b=-3.05^∘. We note, however, that the reddening is negligible for wavelengths greater than about 10 microns.Using the best-fit models from Hα and V^2, we found that the circumstellar contribution in the visual band to the reddening is negligible and therefore the E(B-V) must be completely of interstellar origin.Figure <ref> shows the best-fit model corresponding to a χ^2/ν=0.46 with parameters= 2.5× 10^-11 , 3.0, 25 R_*, and i=50^∘. We adopted an absolute error of ± 0.02 in the log for the fluxes of <cit.> and used the reported errors of <cit.> for the longer wavelengths. A second model computed with parameters simulating an essentially disk-less system, is also shown, illustrating the underlying stellar continuum and the magnitude of the IR excess due to the disk. The best-fit SED shown in Figure <ref> has a slightly steeper power-law index compared to the Hα fit, although it is consistent with power-law of the best-fit model to the interferometric visibilities. However, there are a number of additional models with χ^2/ν of the order of unity, and this point is further discussed in the next section. Finally, while the best-fit model SED inclination of i=50^∘ is consistent with the previous Hα and V^2 fits, we note that the SED is a poor constraint on disk inclination at infrared wavelengths <cit.>. §.§ Combined Results from Spectroscopy, Interferometry, and SED fits Table <ref> summarizes our model best-fit results based on Hα spectroscopy, Hα interferometry and SED fits. We note that the best models from spectroscopy, interferometry and SED fitting are reasonably consistent. Hα spectroscopy and Hα interferometry are consistently best-fit with models of a disk size of 50 R_* while the best-fit SED corresponds to a disk size of 25 R_* in our grid. Although we note that the set of best-fitting SEDs corresponding to χ^2/ν≤ 1 span a range of disk size from 6 to 50 R_disk.Figure <ref> summarizes the best-fitting models for Hα spectroscopy, Hα interferometric V^2 and SED fitting. The solid ellipses and filled symbols represent models within 20% of the best-fit from the Hα line profile (red ellipses, triangles) and Hα interferometric V^2 (blue ellipses, squares) and for χ^2/ν≤ 1 from the SED fitting (black ellipses, diamonds). The dotted ellipses and unfilled symbols, using the same colours and symbols for each diagnostic as before, represent models within 50% of the best-fit from the Hα line profile and Hα interferometric V^2 and for χ^2/ν≤ 1.4 from the SED fitting and demonstrate the robustness of our fitting procedure.Some of the symbols presented on Figure <ref> represent more than one model since the same value of ρ_0 and n may have been selected with different inclination and R_disk. For the Hα spectroscopy there are four models within 20%. The parameters corresponding to these models are shown in Figure <ref>. For the Hα interferometry there are 19 models within 20% corresponding to χ^2/ν from 1.39 to 1.67. These 19 models have an average n = 2.7 ± 0.3, ρ_0 = (5.5±4.2) × 10^-11 , and inclination of 38^∘ ± 12^∘. There are 5 models corresponding to χ^2/ν≤ 1 from the SED fitting. These models have a range of n from 2.0 to 3.0 while ρ_0 ranges from 7.5 × 10^-12 to 2.5 × 10^-11 . As mentioned above these models span a range of R_disk sizes and interestingly all have an inclination of 50^∘ with the exception of one model with an inclination of 30^∘. We emphasize as discussed in Section <ref> that the SED is not a good constraint on inclination.lccc 4 0pc Best-fit Hα, V^2 and SED model results.Fit n ρ_0 [g cm^-3] i [degrees] Hα Profile 2.0 5.0 × 10^-1245 ± 5V^2 3.0 1.0 × 10^-10 45 ± 12SED3.02.5 × 10^-11 (50) 11The inclination is not well constrained by the SED.The dashed lines in Figure <ref> show the model (n = 2.3 and logρ_0 = -11.0) corresponding to the intersection of Hα spectroscopy, Hα visibilities and optical/IR SED fitting that is most consistent with these three observational data sets. As discussed above the model fits to the Hα spectroscopy and interferometrycorresponded to disk sizes of 50 R_*. By taking our best-fit image computed withat i = 0^∘ (face-on) and constrained by Hα interferometry, we can integrate from the central star along a radial rays over distance to obtain a better estimate of the extent of the Hα emitting region. At large distances from central star, the Hα emission tends to originate from an increasingly diffuse disk. Therefore, we chose to integrate until 90% of the Hα flux is contained within the disk. Our best-fit model with n = 2.3 and logρ_0 = -11.0, corresponding to spectroscopy, the visibilities and SED fitting as shown by the dashed lines in Figure <ref> gives us R_90/R_* of 25, where R_90 represents the radial distance corresponding to 90% of the Hα emission. We note that this calculation includes more of the Hα flux compared to what is enclosed within the extent of the major axis of the geometric GD (defined as FWHM of the Gaussian) listed in Table <ref>. This integrated disk size corresponds to a mass for the Hα emitting region of 5 × 10^24 g or 3 × 10^-10 M_*. We note that this is likely a lower limit to the disk mass in the Hα emitting region since as mentioned above, our models produce Hα spectra that are too weak in the wings. Finally, following the prescription in <cit.>, we use our disk mass to determine the angular momentum, J_disk, in the disk compared to the central star's momentum, J_*. We determined the critical velocity consistent with the stellar parameters listed in Table <ref> and assuming 80% critical rotation this gives an equatorial velocity of 360 km s^-1 for this calculation. For the model corresponding to the best-fit fromall three diagnostics, with a disk mass of 5 × 10^24 g or 3 × 10^-10 M_*, we obtain a value for J_ disk/J_* of 4 × 10^-8. § DISCUSSION AND SUMMARY Our best-fit models corresponding to our Hα spectroscopy, interferometry, and SED fits are summarized in Table <ref>. Figure <ref> is a graphical representation of the best-fitting regions corresponding to the constraints based on different observational data sets. Table <ref> and Figure <ref> show that 48 Per has a moderately dense disk with values of n ∼ 2 to 3 and log ρ_0 ∼ -11.7 to -9.6 or ρ_0 ∼ 2.0 × 10^-12 to 2.5 × 10^-10 . The radial extent with all of the best-fitting models for the Hα and V^2 fits correspond to the largest disks (50 R_*) in our grid. The model fits for smaller disk sizes in our computational grid resulted in poorer fits for the Hα spectroscopy and V^2. The combined results from all three diagnostics overlap for a model with n = 2.3 and ρ_0 = 1.0 × 10^-11 corresponding to the dashed lines in Figure <ref>. From spectroscopic analysis, there are four models within 20% of F/F_min value, and 19 models within 20% with χ^2/ν ranging from 1.39 to 1.68 corresponding to the visibilities. These models point to a disk inclination of of 45^∘± 5^∘. From the SED fitting there are five models with χ^2/ν≤ 1. These models span a range of i ∼ 30^∘ to 50^∘. This is consistent with early studies on Be star disks using infrared continuum measurements that demonstrated, especially for low to moderate inclinations, that viewing angle is not well constrained by these measurements <cit.>.As discussed in Section <ref> our model spectral lines were too weak in the wings. Therefore, for line fitting purposes we used a core-weighted formula for our figure of merit, F, which places more emphasis on the central portion of the line (recall Figure <ref>). D11 also found that it was not possible to fit the broad wings in the Hα line for 48 Per. They adopted an ad hoc scheme to account for non-coherent electron scattering, a process which redistributes absorbed line photons resulting in broader lines. This process has been well studied in the literature (see, for example, <cit.> for a detailed treatment) but it is difficult to properly account for in models because it lacks an analytic solution. The fact that our models were weak in the wings may also be due to this process. Alternatively, the poorer fit in the wings could also be due to the fact that a single value of n for each model was adopted for this study. Finally, as briefly discussed in Section <ref>, we calculated the critical velocity of the star directly from the stellar parameters listed in Table <ref>. If the stellar parameters resulted in an under-estimate of the disk rotation, then potentially this could contribute to the fact that our model spectra were too narrow in the wings.As reported above in Section <ref> the best-fitting models for Hα spectroscopy and interferometry from our grid corresponded to 50 R_*. However, as previously discussed, by taking our best-fit model with parameters with n = 2.3 and logρ_0 = -11.0, obtained from interferometry, spectroscopyand SED fitting (see the dashed line on Figure <ref>), we calculated a better approximation for the radial extent of the Hα emitting region of R_90/R_* = 25. As mentioned in Section <ref> this model dependent calculation includes more Hα flux than a geometric fit would contain. Consequently, the calculated disk size is correspondingly bigger than some results in the literature. Other studies have determined the radial extent of the Hα emitting region for sets of Be stars and specific stars. For example, using Hα spectroscopy <cit.> find the emitting regions of 20 R_* for 24 bright Be stars, <cit.> determine sizes of 7 to 19for a sample of 41 Be stars, and more recently <cit.> find a concentration of disk sizes of 5 to 20 R_*. Size estimates based on interferometric GD models encompassing 80% of a star's brightness at FWHM for Hα emitting regions for 12 Be stars are shown in <cit.>.Estimated radii range from 3.24 R_* for β CMi <cit.> to 16.36 R_* for ψ Per (D11). Our model result of 25 R_* is greater, as expected, than some of the sample averages presented above but is in general agreement with other results in the literature.Finally, based on our estimated size of the emitting region of 48 Per we determine the mass and angular momentum content of the disk. For the model corresponding to the best fit from all three diagnostics we obtain a disk mass of 5 × 10^24 g or 3 × 10^-10 M_* and J_ disk/J_* of 4 × 10^-8. This value for the disk mass represents a lower limit since our models were too weak in the wings but is in reasonable agreement with the result presented by <cit.> for 48 Per of 11.2 × 10^-10 M_⊙ or 2.23 × 10^24 g. Using a similar technique as the work presented here, <cit.>, found the mass and angular momentum of disk of the late spectral type Be star, o Aqr, of 1.8 × 10^-10 M_* and 1.6 × 10^-8 J_*, respectively. <cit.> analyzed variability in theequivalent widths for a sample of 49 Be stars.They determined that over the time frame of their study, which overlaps our observations, 48 Per was remarkably stable. Given the apparent stability of this system, it is interesting to compare our best-fit values of n = 2.3 and log ρ_0 of -11.0 with other values presented in the literature. <cit.> modeled 48 Per using a power law fall-off for the density distribution constrained with data from the Infrared Astronomical Satellite (IRAS). Their model is described in terms of an opening angle and stellar parameters are slightly different from our work, but the results are similar. <cit.> obtain an n of 2.5 and a range of log ρ_0 of -11.8 to -11.5.<cit.> studied the continuum emission of this system using pseudo-photosphere model and obtained n of 2.5 and log ρ_0 of ∼ -11.48. Most recently, <cit.> used IRAS, Japanese Aerospace infrared satellite (AKARI) and Wide-field Infrared Survey Explorer (WISE) observations to constrain their viscous decretion disk model. <cit.> find n of 2.9, 2.8 and 2.7 and log ρ_0 of -11.4, -11.4 and -11.5, using IRAS, AKARI and WISE data, respectively. Despite the differences in the models and adopted stellar parameters, the values obtained for 48 Per with <cit.>, <cit.> and <cit.> are remarkably similar. <cit.> report that n ≤ 3.0 indicates a dissipating disk. During the time of our observations, the 48 Per is quite stable. However, <cit.> mention that disk dissipation seems to occur over much longer time scales so it would be interesting to follow this system over a more extended time frame. In future, we plan to extend our modeling technique to include a two-component power law for the density parameter, n, to account for changes in disk density with radial distance from the star.The authors would like to thank an anonymous referee for comments, suggestions and questions that helped to improve this paper. BJG extends thanks to Dave Stock, Anahí Granada, and Andy Pon for their mentorship as well as their helpful feedback on prior versions of this article.BJG also recognizes support from The University of Western Ontario. CEJ and TAAS wish to acknowledge support though the Natural Sciences and Engineering Research Council of Canada. CT acknowledges support from Central Michigan University and the National Science Foundation through grant AST-1614983, and would like to thank Bryan Demapan for assistance with the interferometric data processing. The Navy Precision Optical Interferometer is a joint project of the Naval Research Laboratory and the US Naval Observatory, in cooperation with Lowell Observatory and is funded by the Office of Naval Research and the Oceanographer of the Navy. We thank the Lowell Observatory for the telescope time used to obtain the Hα line spectra, and the US Naval Observatory for the NPOI data that were presented in this work. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. | http://arxiv.org/abs/1704.08733v1 | {
"authors": [
"C. E. Jones",
"T. A. A. Sigut",
"B. J. Grzenia",
"C. Tycner",
"R. T. Zavala"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170427202318",
"title": "The disk physical conditions of 48 Persei constrained by contemporaneous Hα spectroscopy and interferometry"
} |
Time-domain Brillouin scattering assisted by diffraction gratings Vitalyi Gusev ================================================================= [1]The first two authors contributed equally to this workSkip-Gram Negative Sampling (SGNS) word embedding model, well known by its implementation in “word2vec” software, is usually optimized by stochastic gradient descent. However, the optimization of SGNS objective can be viewed as a problem of searching for a good matrix with the low-rank constraint. The most standard way to solve this type of problems is to apply Riemannian optimization framework to optimize the SGNS objective over the manifold of required low-rank matrices. In this paper, we propose an algorithm that optimizes SGNS objective using Riemannian optimization and demonstrates its superiority over popular competitors, such as the original method to train SGNS and SVD over SPPMI matrix. § INTRODUCTIONIn this paper, we consider the problem of embedding words into a low-dimensional space in order to measure the semantic similarity between them. As an example, how to find whether the word “table” is semantically more similar to the word “stool” than to the word “sky”? That is achieved by constructing a low-dimensional vector representation for each word and measuring similarity between the words as the similarity between the corresponding vectors.One of the most popular word embedding models <cit.> is a discriminative neural network that optimizes Skip-Gram Negative Sampling (SGNS) objective (see Equation <ref>). It aims at predicting whether two words can be found close to each other within a text. As shown in Section <ref>, the process of word embeddings training using SGNS can be divided into two general steps with clear objectives: Step 1. Search for a low-rank matrix X that provides a good SGNS objective value;Step 2. Search for a good low-rank representation X=WC^⊤ in terms of linguistic metrics, where W is a matrix of word embeddings and C is a matrix of so-called context embeddings.Unfortunately, most previous approaches mixed these two steps into a single one, what entails a not completely correct formulation of the optimization problem. For example, popular approaches to train embeddings (including the original “word2vec” implementation) do not take into account that the objective from Step 1 depends only on the product X=WC^⊤: instead of straightforward computing of the derivative w.r.t. X, these methods are explicitly based on the derivatives w.r.t. W and C, what complicates the optimization procedure. Moreover, such approaches do not take into account that parametrization WC^⊤ of matrix X is non-unique and Step 2 is required. Indeed, for any invertible matrix S, we haveX=W_1C_1^⊤ = W_1SS^-1C_1^⊤ = W_2C_2^⊤,therefore, solutions W_1C_1^⊤ and W_2C_2^⊤ are equally good in terms of the SGNS objective but entail different cosine similarities between embeddings and, as a result, different performance in terms of linguistic metrics (see Section <ref> for details).A successful attempt to follow the above described steps, which outperforms the original SGNS optimization approach in terms of various linguistic tasks, was proposed in <cit.>. In order to obtain a low-rank matrix X on Step 1, the method reduces the dimensionality of Shifted Positive Pointwise Mutual Information (SPPMI) matrix via Singular Value Decomposition (SVD). On Step 2, it computes embeddings W and C via a simple formula that depends on the factors obtained by SVD. However, this method has one important limitation: SVD provides a solution to a surrogate optimization problem, which has no direct relation to the SGNS objective. In fact, SVD minimizes the Mean Squared Error (MSE) between X and SPPMI matrix, what does not lead to minimization of SGNS objective in general (see Section <ref> and Section 4.2 in <cit.> for details).These issues bring us to the main idea of our paper: while keeping the low-rank matrix search setup on Step 1, optimize the original SGNS objective directly. This leads to an optimization problem over matrix X with the low-rank constraint, which is often <cit.> solved by applying Riemannian optimization framework <cit.>. In our paper, we use the projector-splitting algorithm <cit.>, which is easy to implement and has low computational complexity. Of course, Step 2 may be improved as well, but we regard this as a direction of future work.As a result, our approach achieves the significant improvement in terms of SGNS optimization on Step 1 and, moreover, the improvement on Step 1 entails the improvement on Step 2 in terms of linguistic metrics. That is why, the proposed two-step decomposition of the problem makes sense, what, most importantly, opens the way to applying even more advanced approaches based on it (e.g., more advanced Riemannian optimization techniques for Step 1 or a more sophisticated treatment of Step 2).To summarize, the main contributions of our paper are: * We reformulated the problem of SGNS word embedding learning as a two-step procedure with clear objectives;* For Step 1, we developed an algorithm based on Riemannian optimization framework that optimizes SGNS objective over low-rank matrix X directly;* Our algorithm outperforms state-of-the-art competitors in terms of SGNS objective and the semantic similarity linguistic metric <cit.>. § PROBLEM SETTING §.§ Skip-Gram Negative Sampling In this paper, we consider the Skip-Gram Negative Sampling (SGNS) word embedding model <cit.>, which is a probabilistic discriminative model. Assume we have a text corpus given as a sequence of words w_1,…, w_n, where n may be larger than 10^12 and w_i ∈ V_W belongs to a vocabulary of words V_W. A context c∈ V_C of the word w_i is a word from set {w_i-L, ... , w_i-1, w_i+1, ..., w_i+L} for some fixed window size L. Let w⃗, c⃗∈ℝ^d be the word embeddings of word w and context c, respectively. Assume they are specified by the following mappings:𝒲: V_W →ℝ^d,𝒞: V_C →ℝ^d.The ultimate goal of SGNS word embedding training is to fit good mappings 𝒲 and 𝒞.Let D be a multiset of all word-context pairs observed in the corpus. In the SGNS model, the probability that word-context pair (w, c) is observed in the corpus is modeled as a following dsitribution:P(#(w,c)≠ 0 | w, c)==σ(⟨w⃗, c⃗⟩) = 1/1+exp(-⟨w⃗,c⃗⟩),where #(w,c) is the number of times the pair (w,c) appears in D and ⟨x⃗,y⃗⟩ is the scalar product of vectors x⃗ and y⃗. Number d is a hyperparameter that adjusts the flexibility of the model. It usually takes values from tens to hundreds.In order to collect a training set, we take all pairs (w,c) from D as positive examples and k randomly generated pairs (w,c) as negative ones. The number of times the word w and the context c appear in D can be computed as#(w)=∑_c ∈ V_c#(w,c), #(c)=∑_w ∈ V_w#(w,c)accordingly. Then negative examples are generated from the distribution defined by #(c) counters:P_D(c)=#(c)/|D|.In this way, we have a model maximizing the following logarithmic likelihood objective for all word-context pairs (w,c):l_wc = #(w,c)(logσ(⟨w⃗, c⃗⟩) + + k·𝔼_c' ∼ P_Dlogσ(-⟨w⃗, c⃗'⃗⟩)).In order to maximize the objective over all observations for each pair (w,c), we arrive at the following SGNS optimization problem over all possible mappings 𝒲 and 𝒞:l=∑_w∈ V_W∑_c∈ V_C(#(w,c)(logσ(⟨w⃗, c⃗⟩)+ +k·𝔼_c' ∼ P_Dlogσ(-⟨w⃗, c⃗'⃗⟩)) ) →max_𝒲,𝒞.Usually, this optimization is done via the stochastic gradient descent procedure that is performed during passing through the corpus <cit.>.§.§ Optimization over Low-Rank Matrices Relying on the prospect proposed in <cit.>, let us show that the optimization problem given by (<ref>) can be considered as a problem of searching for a matrix that maximizes a certain objective function and has the rank-d constraint (Step 1 in the scheme described in Section <ref>). §.§.§ SGNS Loss FunctionAs shown in <cit.>, the logarithmic likelihood (<ref>) can be represented as the sum of l_w,c(w⃗,c⃗) over all pairs (w,c), where l_w,c(w⃗,c⃗) has the following form:l_w,c(w⃗,c⃗)=#(w,c)logσ(⟨w⃗,c⃗⟩)+ +k#(w)#(c)/|D|logσ(-⟨w⃗,c⃗⟩).A crucial observation is that this loss function depends only on the scalar product ⟨w⃗,c⃗⟩ but not on embeddings w⃗ and c⃗ separately:l_w,c(w⃗,c⃗)=f_w,c(x_w,c),wheref_w,c(x_w,c)=a_w,clogσ(x_w,c)+b_w,clogσ(-x_w,c),and x_w,c is the scalar product ⟨w⃗,c⃗⟩, anda_w,c=#(w,c),b_w,c=k#(w)#(c)/|D|are constants. §.§.§ Matrix Notation Denote |V_W| as n and |V_C| as m. Let W ∈ℝ^n× d and C ∈ℝ^m× d be matrices, where each row w⃗∈ℝ^d of matrix W is the word embedding of the corresponding word w and each row c⃗∈ℝ^d of matrix C is the context embedding of the corresponding context c. Then the elements of the product of these matricesX=WC^⊤are the scalar products x_w,c of all pairs (w,c):X=(x_w,c), w ∈ V_W, c∈ V_C.Note that this matrix has rank d, because X equals to the product of two matrices with sizes (n× d) and (d× m). Now we can write SGNS objective given by (<ref>) as a function of X:F(X) = ∑_w ∈ V_W∑_c ∈ V_C f_w,c(x_w,c), F: ℝ^n× m→ℝ.This arrives us at the following proposition: theoremProposition SGNS optimization problem given by (<ref>) can be rewritten in the following constrained form:X∈ℝ^n× mmaximizeF(X),subject toX ∈ℳ_d,where ℳ_d is the manifold <cit.> of all matrices in ℝ^n× m with rank d:ℳ_d = {X∈ℝ^n× m: rank(X)=d}. The key idea of this paper is to solve the optimization problem given by (<ref>) via the framework of Riemannian optimization, which we introduce in Section <ref>.Important to note that this prospect does not suppose the optimization over parameters W and C directly. This entails the optimization in the space with ((n+m-d)· d) degrees of freedom <cit.> instead of ((n+m)· d), what simplifies the optimization process (see Section <ref> for the experimental results).§.§ Computing Embeddings from a Low-Rank Solution Once X is found, we need to recover W and C such that X = W C^⊤(Step 2 in the scheme described in Section <ref>). This problem does not have a unique solution, since if (W, C) satisfy this equation, then W S^-1 and C S^⊤ satisfy it as well for any non-singular matrix S. Moreover, different solutions may achieve different values of the linguistic metrics (see Section <ref> for details). While our paper focuses on Step 1, we use, for Step 2, a heuristic approach that was proposed in <cit.> and it shows good results in practice. We compute SVD of X in the formX = U Σ V^⊤,where U and V have orthonormal columns, and Σ is the diagonal matrix, and use W = U √(Σ),C = V √(Σ)as matrices of embeddings.A simple justification of this solution is the following: we need to map words into vectors in a way that similar words would have similar embeddings in terms of cosine similarities:cos(w⃗_1, w⃗_2) = ⟨w⃗_1, w⃗_2⟩/‖w⃗_1 ‖·‖w⃗_2 ‖.It is reasonable to assume that two words are similar, if they share contexts. Therefore, we can estimate the similarity of two words w_1, w_2 ass(w_1,w_2)=∑_c∈ V_C x_w_1, c· x_w_2, c,what is the element of the matrix XX^⊤ with indices (w_1, w_2). Note thatXX^⊤ = U Σ V^⊤ V Σ U^⊤ = U Σ^2 U^⊤.If we choose W=UΣ, we exactly obtain ⟨w⃗_⃗1⃗, w⃗_⃗2⃗⟩ = s(w_1,w_2), since WW^⊤=XX^⊤ in this case. That is, the cosine similarity of the embeddings w⃗_⃗1⃗, w⃗_⃗2⃗ coincides with the intuitive similarity s(w_1,w_2). However, scaling by √(Σ) instead of Σ was shown in <cit.> to be a better solution in experiments.§ PROPOSED METHOD§.§ Riemannian Optimization §.§.§ General SchemeThe main idea of Riemannian optimization <cit.> is to consider (<ref>) as a constrained optimization problem. Assume we have an approximated solution X_i on a current step of the optimization process, where i is the step number. In order to improve X_i, the next step of the standard gradient ascent outputs the pointX_i+∇ F(X_i),where ∇ F(X_i) is the gradient of objective F at the point X_i. Note that the gradient ∇ F(X_i) can be naturally considered as a matrix in ℝ^n× m. Point X_i+∇ F(X_i) leaves the manifold ℳ_d, because its rank is generally greater than d. That is why Riemannian optimization methods map point X_i+∇ F(X_i) back to manifold ℳ_d. The standard Riemannian gradient method first projects the gradient step onto the tangent space at the current point X_i and then retracts it back to the manifold:X_i+1 = R(P_𝒯_M(X_i + ∇ F(X_i))),where R is the retraction operator, and P_𝒯_M is the projection onto the tangent space.Although the optimization problem is non-convex, Riemannian optimization methods show good performance on it. Theoretical properties and convergence guarantees of such methods are discussed in <cit.> more thoroughly. §.§.§ Projector-Splitting AlgorithmIn our paper, we use a simplified version of such approach that retracts point X_i + ∇ F(X_i) directly to the manifold and does not require projection onto the tangent space P_𝒯_M as illustrated in Figure <ref>:X_i+1=R(X_i+∇ F(X_i)). Intuitively, retractor R finds a rank-d matrix on the manifold ℳ_d that is similar to high-rank matrix X_i+∇ F(X_i) in terms of Frobenius norm. How can we do it? The most straightforward way to reduce the rank of X_i+∇ F(X_i) is to perform the SVD, which keeps d largest singular values of it:1: U_i+1, S_i+1, V_i+1^⊤←SVD(X_i+∇ F(X_i)),2: X_i+1← U_i+1S_i+1V_i+1^⊤.However, it is computationally expensive. Instead of this approach, we use the projector-splitting method <cit.>, which is a second-order retraction onto the manifold (for details, see the review <cit.>). Its practical implementation is also quite intuitive: instead of computing the full SVD of X_i+∇ F(X_i) according to the gradient projection method, we use just one step of the block power numerical method <cit.> which computes the SVD, what reduces the computational complexity.Let us keep the current point in the following factorized form:X_i = U_i S_i V_i^⊤,where matrices U_i∈ℝ^n× d and V_i∈ℝ^m× d have d orthonormal columns and S_i∈ℝ^d× d. Then we need to perform two QR-decompositions to retract point X_i+∇ F(X_i) back to the manifold:1: U_i+1, S_i+1←QR((X_i+∇ F(X_i))V_i),2: V_i+1, S_i+1^⊤←QR((X_i+∇ F(X_i))^⊤ U_i+1),3: X_i+1← U_i+1S_i+1V_i+1^⊤.In this way, we always keep the solution X_i+1= U_i+1 S_i+1 V_i+1^⊤ on the manifold ℳ_d and in the form (<ref>).What is important, we only need to compute ∇ F(X_i), so the gradients with respect to U, S and V are never computed explicitly, thus avoiding the subtle case where S is close to singular (so-called singular (critical) point on the manifold). Indeed, the gradient with respect to U (while keeping the orthogonality constraints) can be written <cit.> as:∂ F/∂ U = ∂ F/∂ X V S^-1,which means that the gradient will be large if S is close to singular. The projector-splitting scheme is free from this problem.§.§ AlgorithmIn case of SGNS objective given by (<ref>), an element of gradient ∇ F has the form:(∇ F(X))_w,c=∂ f_w,c(x_w,c)/∂ x_w,c==#(w,c)·σ(-x_w,c)-k#(w)#(c)/|D|·σ(x_w,c).To make the method more flexible in terms of convergence properties, we additionally use λ∈ℝ, which is a step size parameter. In this case, retractor R returns X_i + λ∇ F(X_i) instead of X_i + ∇ F(X_i) onto the manifold.The whole optimization procedure is summarized in Algorithm <ref>.[h] Riemannian Optimization for SGNS § EXPERIMENTAL SETUP§.§ Training Models We compare our method (“RO-SGNS” in the tables) performance to two baselines: SGNS embeddings optimized via Stochastic Gradient Descent, implemented in the original “word2vec”, (“SGD-SGNS” in the tables) <cit.> and embeddings obtained by SVD over SPPMI matrix (“SVD-SPPMI” in the tables) <cit.>. We have also experimented with the blockwise alternating optimization over factors W and C, but the results are almost the same to SGD results, that is why we do not to include them into the paper. The source code of our experiments is available online[<https://github.com/AlexGrinch/ro_sgns>]. The models were trained on English Wikipedia “enwik9” corpus[<http://mattmahoney.net/dc/textdata>], which was previously used in most papers on this topic. Like in previous studies, we counted only the words which occur more than 200 times in the training corpus <cit.>. As a result, we obtained a vocabulary of 24292 unique tokens (set of words V_W and set of contexts V_C are equal). The size of the context window was set to 5 for all experiments, as it was done in <cit.>. We conduct three series of experiments: for dimensionality d=100, d=200, and d=500.Optimization step size is chosen to be small enough to avoid huge gradient values. However, thorough choice of λ does not result in a significant difference in performance (this parameter was tuned on the training data only, the exact values used in experiments are reported below).§.§ EvaluationWe evaluate word embeddings via the word similarity task. We use the following popular datasets for this purpose: “wordsim-353” (<cit.>; 3 datasets),“simlex-999” <cit.> and “men” <cit.>. Original “wordsim-353” dataset is a mixture of the word pairs for both word similarity and word relatedness tasks. This dataset was split <cit.> into two intersecting parts: “wordsim-sim” (“ws-sim” in the tables) and “wordsim-rel” (“ws-rel” in the tables) to separate the words from different tasks. In our experiments, we use both of them on a par with the full version of “wordsim-353” (“ws-full” in the tables). Each dataset contains word pairs together with assessor-assigned similarity scores for each pair. As a quality measure, we use Spearman's correlation between these human ratings and cosine similarities for each pair. We call this quality metric linguistic in our paper.§ RESULTS OF EXPERIMENTSFirst of all, we compare the value of SGNS objective obtained by the methods. The comparison is demonstrated in Table <ref>. We see that SGD-SGNS and SVD-SPPMI methods provide quite similar results, however, the proposed method obtains significantly better SGNS values, what proves the feasibility of using Riemannian optimization framework in SGNS optimization problem. It is interesting to note that SVD-SPPMI method, which does not optimize SGNS objective directly, obtains better results than SGD-SGNS method, which aims at optimizing SGNS. This fact additionally confirms the idea described in Section <ref> that the independent optimization over parameters W and C may decrease the performance. However, the target performance measure of embedding models is the correlation between semantic similarity and human assessment (Section <ref>). Table <ref> presents the comparison of the methods in terms of it. We see that our method outperforms the competitors on all datasets except for “men” dataset where it obtains slightly worse results. Moreover, it is important that the higher dimension entails higher performance gain of our method in comparison to the competitors.To understand how our model improves or degrades the performance in comparison to the baseline, we found several words, whose neighbors in terms of cosine distance change significantly. Table <ref> demonstrates neighbors of the words “five”, “he” and “main” for both SVD-SPPMI and RO-SGNS models. A neighbor is marked bold if we suppose that it has similar semantic meaning to the source word. First of all, we notice that our model produces much better neighbors of the words describing digits or numbers (see word “five” as an example). Similar situation happens for many other words, e.g. in case of “main” — the nearest neighbors contain 4 similar words for our model instead of 2 in case of SVD-SPPMI. The neighbourhood of “he” contains less semantically similar words in case of our model. However, it filters out irrelevant words, such as “promptly” and “dumbledore”.Table <ref> contains the nearest words to the word “usa” from 11th to 20th. We marked names of USA states bold and did not represent top-10 nearest words as they are exactly names of states for all three models. Some non-bold words are arguably relevant as they present large USA cities (“akron”, “burbank”, “madison”) or geographical regions of several states (“midwest”, “northeast”, “southwest”), but there are also some completely irrelevant words (“uk”, “cities”, “places”) presented by first two models.Our experiments show that the optimal number of iterations K in the optimization procedure and step size λ depend on the particular value of d. For d=100, we have K=7, λ = 5· 10^-5, for d=200, we have K=8, λ=5·10^-5, and for d=500, we have K=2, λ = 10^-4. Moreover, the best results were obtained when SVD-SPPMI embeddings were used as an initialization of Riemannian optimization process.Figure <ref> illustrates how the correlation between semantic similarity and human assessment scores changes through iterations of our method. Optimal value of K is the same for both whole testing set and its 10-fold subsets chosen for cross-validation. The idea to stop optimization procedure on some iteration is also discussed in <cit.>.Training of the same dimensional models (d=500) on English Wikipedia corpus using SGD-SGNS, SVD-SPPMI, RO-SGNS took 20 minutes, 10 minutes and 70 minutes respectively. Our method works slower, but not significantly. Moreover, since we were not focused on the code efficiency optimization, this time can be reduced.§ RELATED WORK §.§ Word Embeddings Skip-Gram Negative Sampling was introduced in <cit.>. The “negative sampling” approach is thoroughly described in <cit.>, and the learning method is explained in <cit.>. There are several open-source implementations of SGNS neural network, which is widely known as “word2vec”. [Original Google word2vec: <https://code.google.com/archive/p/word2vec/>][Gensim word2vec: <https://radimrehurek.com/gensim/models/word2vec.html>]As shown in Section <ref>, Skip-Gram Negative Sampling optimization can be reformulated as a problem of searching for a low-rank matrix. In order to be able to use out-of-the-box SVD for this task, the authors of <cit.> used the surrogate version of SGNS as the objective function. There are two general assumptions made in their algorithm that distinguish it from the SGNS optimization: * SVD optimizes Mean Squared Error (MSE) objective instead of SGNS loss function.* In order to avoid infinite elements in SPMI matrix, it is transformed in ad-hoc manner (SPPMI matrix) before applying SVD.This makes the objective not interpretable in terms of the original task (<ref>). As mentioned in <cit.>, SGNS objective weighs different (w, c) pairs differently, unlike the SVD, which works with the same weight for all pairs and may entail the performance fall. The comprehensive explanation of the relation between SGNS and SVD-SPPMI methods is provided in <cit.>. <cit.> give a good overview of highly practical methods to improve these word embedding models.§.§ Riemannian OptimizationAn introduction to optimization over Riemannian manifolds can be found in <cit.>. The overview of retractions of high rank matrices to low-rank manifolds is provided in <cit.>. The projector-splitting algorithm was introduced in <cit.>, and also was mentioned in <cit.> as “Lie-Trotter retraction”. Riemannian optimization is succesfully applied to various data science problems: for example, matrix completion <cit.>, large-scale recommender systems <cit.>, and tensor completion <cit.>.§ CONCLUSIONS In our paper, we proposed the general two-step scheme of training SGNS word embedding model and introduced the algorithm that performs the search of a solution in the low-rank form via Riemannian optimization framework. We also demonstrated the superiority of our method by providing experimental comparison to existing state-of-the-art approaches.Possible direction of future work is to apply more advanced optimization techniques to the Step 1 of the scheme proposed in Section <ref> and to explore the Step 2 — obtaining embeddings with a given low-rank matrix. § ACKNOWLEDGMENTSThis research was supported by the Ministry of Education and Science of the Russian Federation (grant 14.756.31.0001). natexlab#1#1[Absil and Oseledets(2015)]absil2015low P-A Absil and Ivan V Oseledets. 2015. Low-rank retractions: a survey and new results. Computational Optimization and Applications 62(1):5–29.[Agirre et al.(2009)Agirre, Alfonseca, Hall, Kravalova, Paşca, and Soroa]agirre2009study Eneko Agirre, Enrique Alfonseca, Keith Hall, Jana Kravalova, Marius Paşca, and Aitor Soroa. 2009. A study on similarity and relatedness using distributional and wordnet-based approaches. 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In EMNLP.[Tan et al.(2014)Tan, Tsang, Wang, Vandereycken, and Pan]tan2014riemannian Mingkui Tan, Ivor W Tsang, Li Wang, Bart Vandereycken, and Sinno Jialin Pan. 2014. Riemannian pursuit for big matrix recovery. In ICML. volume 32, pages 1539–1547.[Udriste(1994)]udriste1994convex Constantin Udriste. 1994. Convex functions and optimization methods on Riemannian manifolds, volume 297. Springer Science & Business Media.[Vandereycken(2013)]vandereycken2013low Bart Vandereycken. 2013. Low-rank matrix completion by riemannian optimization. SIAM Journal on Optimization 23(2):1214–1236.[Wei et al.(2016)Wei, Cai, Chan, and Leung]wei2016guarantees Ke Wei, Jian-Feng Cai, Tony F Chan, and Shingyu Leung. 2016. Guarantees of riemannian optimization for low rank matrix recovery. SIAM Journal on Matrix Analysis and Applications 37(3):1198–1222.acl_natbib | http://arxiv.org/abs/1704.08059v1 | {
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"title": "Riemannian Optimization for Skip-Gram Negative Sampling"
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Department of Physics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia The classical ground state magnetic response of the Heisenberg model when rotationally invariant exchange interactions of integer order q>1 are added is found to be discontinuous, even though the interactions lack magnetic anisotropy. This holds even in the case of bipartite lattices which are not frustrated, as well as for the frustrated triangular lattice. The total number of discontinuities is associated with even-odd effects as it depends on the parity of q via the relative strength of the bilinear and higher order exchange interactions, and increases with q. These results demonstrate that the precise form of the microscopic interactions is important for the ground state magnetization response. 71.70.Gm Exchange Interactions, 75.10.Hk Classical Spin Models, 75.10.Pq Spin Chain ModelsDiscontinuous classical ground state magnetic response as an even-odd effect in higher order rotationally invariant exchange interactions N. P. Konstantinidis December 30, 2023 ========================================================================================================================================= The Heisenberg model plays an extremely important role in the study of magnetism of strongly correlated electron systems <cit.>. Within its context discontinuities of the ground state magnetization were found, showing that it can be tuned between well-separated values with small changes of an external field. Such discontinuities naturally occur in the presence of magnetic anisotropy, where the field forces the spins to non-continuously change their orientation toward special directions in spin space along which the energy is more efficiently minimized <cit.>. Frustrated clusters are of special interest, as competing interactions can lead to discontinuous ground state magnetization response in the absence of anisotropy, with the ground state spin configuration completely changing its symmetry as the discontinuity is traversed <cit.>. A non-continuous magnetic response is therefore associated with preferential directions in spin space or non-trivial connectivity of the interacting spins.Here a new source of discontinuities is identified, rotationally invariant exchange interactions of integer order q>1, which require neither magnetic anisotropy nor frustration to generate magnetization jumps. Such discontinuities occur for lattices as simple as the bipartite, as well as for the frustrated triangular lattice that was further considered. These interactions are associated with even-odd effects with respect to the parity of q: for positive higher order exchange they generate ground state discontinuities only when q is odd, while for negative there are discontinuities for both even and odd q. In the latter case the total number of magnetization gaps also depends on the parity of q. For the same type of parity, the total number of discontinuities increases with q.It has been shown that higher order exchange terms are often important for the explanation of experimental data <cit.>. This has been the case with the biquadratic exchange interaction (q=2) <cit.>, which can even be quite stronger than the bilinear exchange <cit.>. Its importance can not be understated also from the theoretical point of view <cit.>, for example for the chain hosting spins of magnitude s=1 <cit.>, and it has also been considered along with biqubic terms for s=3/2 <cit.>. It was also investigated in higher dimensions <cit.>, and for the magnetism of iron-based supeconductors <cit.>. The classical ground state magnetic response with bilinear and biquadratic exchange has been calculated for short odd-numbered chains <cit.>, while in the case of the icosahedron it has been shown to generate many magnetization discontinuities <cit.>. A similar Hamiltonian in two dimensional spin space includes the standard bilinear exchange and generalized nematic interactions <cit.>.Here rotationally invariant exchange interactions with q ranging from 2 to 9 are considered. For spins s the highest order non-trivial exchange interaction term of this type is of order 2s. This means that for higher q the classical treatment of the problem where the bilinear competes with the 2s-order exchange interaction provides a very good description of the quantum problem, and that the even-odd effects in q can also be viewed as effects related to s being integer or half-integer. The underlying lattices are of bipartite form, chains and rectangles, and the frustrated triangular lattice.The Hamiltonian including bilinear and higher order exchange interactions as well as a magnetic field term isH = ∑_<ij> [ J s⃗_i ·s⃗_j + J' ( s⃗_i ·s⃗_j )^q ] -h ∑_i=1^N s_i^zThere are N spins s⃗_i which are classical unit vectors in three-dimensional spin space. <ij> indicates that interactions are limited to nearest neighbors i and j. The first term is the bilinear exchange interaction, scaled with J, and the second the higher order exchange interaction of integer order q>1, scaled with J'. The exchange interactions are taken to be isotropic in spin space. The magnetic field h⃗ in the Zeeman term points along the z direction without any loss of generality. The interactions are parametrized as J=cosω and J'=sinω. The bilinear exchange favors antiparallel nearest-neighbors for positive J and parallel for negative J. When J' is positive the second term in Hamiltonian (<ref>) favors antiparallel nearest-neighbor spins for odd q and perpendicular for even q. For negative J' it favors ferromagnetically coupled spins irrespectively of the parity of q. The situation is further complicated by the Zeeman term, through which the spins gain maximum magnetic energy when pointing in the direction of the field. The competition of these three terms determines the magnetic properties. In addition, for lattices such as the triangular their frustrated connectivity plays an important role. A ground state magnetization discontinuity is associated with a non-continuous change of the lowest energy spin configuration, which originates in a more efficient energy minimization as the field increases. Here the lowest energy configuration of Hamiltonian (<ref>) is numerically calculated as a function of h for ω ϵ [0,2π) <cit.>. The direction of each spin s⃗_i is defined by a polar θ_i and an azimuthal ϕ_i angle. All angles are randomly initialized and each one is moved opposite its gradient direction until the energy minimum is reached. The procedure is repeated for different initial configurations to ensure that the global lowest energy configuration is found.Firstly bipartite structures are considered, chains and rectangles with periodic boundary conditions, which lack any frustration.Minimization of Hamiltonian (<ref>) shows that for lower and higher ω, where the bilinear exchange is positive or weakly negative and the ground state is not ferromagnetic in zero field, the lowest energy configuration is the same for both types of structures for increasing N and different q, and consequently also the one for the corresponding infinite lattices. The configuration is planar with a unit cell of two spins with the same polar angle θ, and azimuthal angles that differ by π. Thus each spin's nearest-neighbors point in the same direction, while θ is given from the solution of the equation( 2 cos^2θ -1)^q-1 = 1/qJ' ( h/4 cosθ - J )(see App. <ref> for the case of zero field). The solution of Eq. (<ref>) can be discontinuous at a magnetic field h_d. Fig. <ref> plots the magnetization per spin M/N=cosθ as a function of h for q=3. When J'/J=0.41667 (or ω=0.12567 π) a discontinuity appears which originates in the higher order exchange interaction, as it is present when J=0 (ω=π/2), and survives up to ω=0.70483π.Fig. <ref>(a) plots the discontinuity field for J'>0. The jump occurs for odd but not for even q, bringing about an even-odd effect in the parity of higher order exchange interactions. An odd q favors antiparallel nearest-neighbors, while an even q perpendicular. The discontinuity exists for a specific ω range (see also App. <ref>), and is always present for J=0 (ω=π/2). It requires neither magnetic anisotropy nor frustration in the interactions. A higher q pushes it towards smaller magnetic fields and increases its width per spin Δ M/N (Fig. <ref>(c)).For J>0 and J'<0 another magnetization discontinuity appears, irrespectively of the parity of q. The higher order exchange interactions favor parallel spins for any q exactly like the magnetic field, and compete with the antiferromagnetic bilinear exchange. This competition is now the origin of the discontinuity. The lowest energy configuration changes abruptly from the one predicted by Eq. (<ref>) to the ferromagnetic one, with the corresponding discontinuity fields plotted in Fig. <ref>(b) (see also App. <ref>). There is another even-odd effect, with the discontinuity triggered by an infinitesimal positive J for even q, while a finite J value is needed for odd q. On the other hand only a small negative value of J' is required to generate the jump close to the bilinear exchange limit for any q. Then only for even q ≥ 6 and higher ω the discontinuity breaks up in two, with the one leading to saturation following the pattern of the odd q and lower even q discontinuities. This can also be seen in the plot of the width per spin of the jumps (Fig. <ref>(d)). These results show that the ground state magnetic response gets richer with q, demonstrating also the importance of the detailed form of the microscopic interactions and not only their symmetry for the precise determination of the magnetization curve <cit.>.Hamiltonian (<ref>) is frustrated in the case of the triangular lattice. In the absence of higher order exchange interactions and in finite field it has an accidental classical ground state degeneracy that is lifted by thermal <cit.> and quantum fluctuations <cit.>. This order by disorder effect is also generated by nonmagnetic impurities <cit.> and anisotropic terms <cit.>. When higher order exchange interactions are included minimization of Hamiltonian (<ref>) with periodic boundary conditions shows that they also break the degeneracy and induce order (see App. <ref> for the case of zero field). The ground state has a triangular unit cell, with spin configurations selected from the finite field ground state degenerate manifold of ω=0 and plotted in Fig. <ref> (see App. <ref>). The frustrated connectivity of the triangular lattice generates a ground state magnetization response with more discontinuities than the one of the bipartite lattices. Such discontinuities were first found for a finite version of the triangular lattice, the icosahedron, already for q=2 <cit.>. Again there is an even-odd effect for J'>0, with discontinuities occuring only for odd q. Figure <ref>(a) plots the corresponding fields for q=3. An infinitesimal positive J' is sufficient to generate two magnetization jumps, which merge for higher ω. The lower field discontinuity changes the configuration from the Y to the fan,while the higher field jump leads to the non-coplanar”umbrella” configuration. The higher order exchange interaction generates the discontinuities, with both occuring when the bilinear exchange J=0 (ω=π/2). This time the frustrated connectivity allows the jumps to survive down to infinitesimal J'. When J>0 and J'<0two discontinuities occur. The lower changes the spin configuration from the ”umbrella” to the V. This magnetization gap also requires an infinitesimal value of (negative) J' to occur. The higher field jump leads the spins directly to the ferromagnetic configuration, similarly to the case of the bipartite lattices. A jump to saturation has been found in the quantum case for the antiferromagnetic Heisenberg model in frustrated lattices and molecules <cit.>. It is stressed that infinitesimal deviations from the purely bilinear exchange case generate discontinuous ground state response irrespectively of the sign of J'.Figure <ref>(b) plots the discontinuity fields when q=9. Contrary to the case of the bipartite lattices the discontinuity diagram becomes richer for J'>0, with a maximum of five discontinuities for 0.067898 π≤ω≤ 0.10199 π. An infinitesimal J' generates now a total of three jumps, with the inverted Y configuration appearing for small fields. The jumps are now related not only to a change of the configuration type but also to discontinuous polar angles within the same type of configuration (discontinuities 4 and 5). These results show as in the case of the bipartite lattices that the precise value of q is important for the determination of the ground state magnetic response.Figure <ref>(c) shows the discontinuities for q=2. An infinitesimal deviation from ω=3π/2 generates a jump by changing the configuration from the UUD associated with the 1/3 magnetization plateau of the triangular lattice to the saturated one. The highest field discontinuities direct all spins to be parallel to the field, as in the bipartite lattices case. An infinitesimal (negative) J' generates the 1/3 magnetization plateau as indicated by the (red) dashed lines. This plateau is a feature of the antiferromagnetic Heisenberg model in the quantum case <cit.>. For small negative J' the field drives the system from the Y to the UUD and then to the V configuration, similarly to the effect of finite temperature in the J'=0 case <cit.>. In Fig. <ref>(d) it is shown that a higher value of q=8 again enriches the magnetization response generating more jumps, with the inverted Y configuration entering again for small fields and higher ω. Again the response for J>0 and J'<0 showcases an even-odd effect with respect to q. The discontinuity strengths corresponding to Fig. <ref> are plotted in Fig. <ref> (for the rest of the q values see App. <ref>).In conclusion, the classical ground state magnetization response has been calculated for bipartite lattices and the triangular lattice, when isotropic exchange interactions of integer order q>1 compete with the standard bilinear exchange interaction. These interactions generate magnetization discontinuities even though there is no magnetic anisotropy or necessarily frustration. The total number of discontinuities is associated with even-odd effects in q, and also increases with q. These results indicate that the precise form of the interactions and not only the symmetry of the Hamiltonian is important for the determination of zero-temperature properties, especially since a general interaction between spins can be expressed as a series expansion in powers of q. This is also expected for the thermodynamic properties where all the states are involved, as was shown in the case of nematic interactions <cit.>. Similar calculations can be performed for frustrated clusters where the addition of biquadratic exchange along with the special connectivity has led to a multitude of discontinuities already for a cluster as small as the icosahedron <cit.>, and more specifically for an even q and bilinear and biquadratic exchange both positive, something not possible for the bipartite and triangular lattices. § ZERO FIELD LOWEST ENERGY CONFIGURATION FOR BIPARTITE LATTICESThe lowest energy configuration in the absence of a field is given by the solution of Eq. (<ref>) for h=0. The range of ω for which the lowest energy configuration is antiferromagnetic (AFM) and ferromagnetic (FM) for 2 ≤ q ≤ 9 for a bipartite lattice is listed in Table <ref>.§ LOWEST ENERGY CONFIGURATION MAGNETIZATION IN A FIELD FOR BIPARTITE LATTICES Table <ref> lists the ω range of the two discontinuities for odd q for a bipartite lattice, while Table <ref> lists the ω range of the discontinuities for even q. The saturation field h_sat=4(J+qJ') except when the zero field ground state is ferromagnetic or a discontinuity leads directly to saturation.§ ZERO FIELD LOWEST ENERGY CONFIGURATION FOR THE TRIANGULAR LATTICEThe range of ω for which the lowest energy configuration in zero magnetic field has spins at 120^o degrees with each other, is FM or of the UUD form for 2 ≤ q ≤ 9 for the triangular lattice is listed in Tables <ref> and <ref> (results for q=2 have been presented in Ref. <cit.>).§ LOWEST ENERGY CONFIGURATION MAGNETIZATION IN A FIELD FOR THE TRIANGULAR LATTICETable <ref> lists the ω range of the discontinuities for odd q for the triangular lattice, while Table <ref> lists the ω range of the discontinuities for even q. Table <ref> lists the ranges of ω for the limits of the magnetization plateaus for the triangular lattice. The saturation field h_sat=9(J+qJ') except when the zero field ground state is ferromagnetic or a discontinuity leads directly to saturation. Figure <ref> shows the discontinuity magnetic fields for q=4, 5, 6, and 7, and Fig. <ref> the corresponding magnetization changes per spin. Fig. <ref> shows one of the q=5 discontinuties in greater detail. | http://arxiv.org/abs/1704.08210v2 | {
"authors": [
"N. P. Konstantinidis"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170426165146",
"title": "Discontinuous classical ground state magnetic response as an even-odd effect in higher order rotationally invariant exchange interactions"
} |
The disk physical conditions of 48 Persei constrained by contemporaneous Hα spectroscopy and interferometry C. E. Jones1, T. A. A. Sigut1, B. J. Grzenia1, C. Tycner2, R. T. Zavala3 Received: date / Revised version: date =========================================================================================================== A permutation class C is splittable if it is contained in a merge of two of its proper subclasses, and it is 1-amalgamable if given two permutations σ, τ∈ C, each with a marked element, we can find a permutation π∈ C containing both σ and τ such that the two marked elements coincide. It was previously shown that unsplittability implies 1-amalgamability. We prove that unsplittability and 1-amalgamability are not equivalent properties of permutation classes by showing that the class 1423, 1342 is both splittable and 1-amalgamable. Our construction is based on the concept of LR-inflations, which we introduce here and which may be of independent interest. § INTRODUCTION In the study of permutation classes, a notable interest has recently been directed towards the operation of merging. We say that a permutation π is a merge of σ and τ if the elements of π can be colored red and blue so that the red elements form a copy of σ and the blue elements form a copy of τ. For instance, Claesson, Jelínek and Steingrímsson <cit.> showed that every 1324-avoiding permutation can be merged from a 132-avoiding permutation and a 213-avoiding permutation, and used this fact to prove that there are at most 16^n 1324-avoiding permutations of length n.A general problem that follows naturally is how to identify when a permutation class C has proper subclasses A and B, such that every element of C can be obtained as a merge of an element of A and an element of B. We say that such a permutation class C is splittable. Jelínek and Valtr <cit.> showed that every inflation-closed class is unsplittable and the class of σ-avoiding permutations, where σ is a direct sum of two nonempty permutations and has length at least four, is splittable. Furthermore, they mentioned the connection of splittability to more general structural properties of classes of relational structures studied in the area of Ramsey theory, most notably the notion of 1-amalgamability. We say that a permutation class C is 1-amalgamable if given two permutations σ, τ∈ C, each with a marked element, we can find a permutation π∈ C containing both σ and τ such that the two marked elements coincide.Not much is known about 1-amalgamability of permutation classes. Jelínek and Valtr <cit.>, using a more general result from Ramsey theory, showed that unsplittability implies 1-amalgamability, and they raised the question whether there is a permutation class that is both splittable and 1-amalgamable. In this paper, we answer this question by showing that the class 1423, 1342 has both properties.For this task, we will introduce a slightly weaker property than being inflation-closed, that is being closed under inflating just the elements that are left-to-right minima. We say that an element of permutation π is a left-to-right minimum, or just LR-minimum, if it is smaller than all the elements preceding it. In Section <ref> we shall prove that certain properties of a permutation class C imply that its closure under inflating LR-minima is splittable and 1-amalgamable. And finally in Section <ref> we show that the class 1423, 1342 is actually equal to the class 123 closed under inflating left-to-right minima and that 123 has the desired properties.§ BASICSA permutation π of length n ≥ 1 is a sequence of all the n distinct numbers from the set [n] = { 1, 2, …, n }. We denote the i-th element of π as π_i. Note that we omit all punctuation when writing out short permutations, e.g., we write 123 instead of 1, 2, 3. The set of all permutations of length n is denoted S_n.We say that two sequences of distinct numbers a_1, …, a_n and b_1, …, b_n are order-isomorphic if for every two indices i < j we have a_i < a_j if and only if b_i < b_j. Given two permutations π∈ S_n and σ∈ S_k, we say that π contains σ if there is a k-tuple 1 ≤ i_1 < i_2 < ⋯ < i_k ≤ n such that the sequence π_i_1, π_i_2, …, π_i_k is order-isomorphic to σ and we say that such a sequence is an occurrence of σ in π. Furthermore, we say that the corresponding function f: [k] → [n] defined as f(j) = i_j is an embedding of σ into π. In the context of permutation containment, we often refer to the permutation σ as a pattern.A permutation that does not contain σ is σ-avoiding and we let σ denote the set of all σ-avoiding permutations. Similarly, for a set of permutations F , we let F denote the set of permutations that avoid all elements of F. Note that for small sets F we omit the curly braces, e.g., we simply write σ, ρ instead of {σ, ρ}.We say that a set of permutations C is a permutation class if for every π∈ C and σ contained in π, σ belongs to C as well. Observe that a set of permutations C is a permutation class if and only if there is a set F such that C = F. Moreover, for every permutation class C, there is a unique inclusionwise minimal set F such that C=F; this set F is known as the basis of C. A class is said to be principal if its basis has a single element, i.e., if the class has the form σ for a permutation σ.Suppose that π∈ S_n is a permutation, let σ_1, …, σ_n be an n-tuple of non-empty permutations, and let m_i be the length of σ_i for i ∈ [n]. The inflation of π by the sequence σ_1, …, σ_n, denoted by π[σ_1, …, σ_n], is the permutation of length m_1 +⋯+ m_n obtained by concatenating n sequences σ_1 σ_2 ⋯σ_n with these properties: * for each i ∈ [n], σ_i is order-isomorphic to σ_i, and* for each i, j ∈ [n], if π_i < π_j, then all the elements of σ_i are smaller than all the elements of σ_j. For two sets of permutations A and B, we let A[B] denote the set of all the permutations that can be obtained asan inflation of a permutation from A by a sequence of permutations from B. We say that a set of permutations A is ·[B]-closed if A[B] ⊆ A, and similarly a set of permutations B is A[·]-closed if A[B] ⊆ B. Finally, we say that a set of permutations C is inflation-closed if C[C] ⊆ C.There is a nice way to characterize an inflation-closed class through its basis. We say that a permutation π is simple if it cannot be obtained by inflation from smaller permutations, except for the trivial inflations π[1, …, 1] and 1[π]. Inflation-closed permutation classes are precisely the classes whose basis only contains simple permutations <cit.>.§ SPLITTABILITY AND 1-AMALGAMABILITY We now focus on the properties of splittability and 1-amalgamability of permutation classes. Mostly, we state or rephrase results that were already known. For more detailed overview, especially regarding splittability, see Jelínek and Valtr <cit.>.§.§ Splittability We say thata permutation π is a merge of permutations τ and σ, if it can be partitioned into two disjoint subsequences, one of which is an occurrence of σ and the other is an occurrence of τ. For two permutation classes A and B, we write A ⊙ B for the class of all merges of a (possibly empty) permutation from A with a (possibly empty) permutation from B. Trivially, A ⊙ B is again a permutation class.Conversely, we say that a multiset of permutation classes { P_1, …, P_m} forms a splitting of a permutation class C if C ⊆ P_1 ⊙⋯⊙ P_m. We call P_i the parts of the splitting. The splitting is nontrivial if none of its parts is a superset of C, and the splitting is irredundant if no proper submultiset of { P_1, …, P_m} forms a splitting of C. A permutation class C is then splittable if C admits a nontrivial splitting.The following simple lemma is due to Jelínek and Valtr <cit.>.For a class C of permutations, the following properties are equivalent: * C is splittable.* C has a nontrivial splitting into two parts.* C has a splitting into two parts, in which each part is a proper subclass of C.* C has a nontrivial splitting into two parts, in which each part is a principal class.Following the previous Lemma <ref>, we can characterize a splittable class C by the splittings of the form {π, σ}, where both π and σ are permutations from C. We want to identify permutations inside C that cannot define any such splitting.Let C be a permutation class. We say that a permutation π∈ C is unavoidable in C, if for any permutation τ∈ C, there is a permutation σ∈ C such that any red-blue coloring of σ has a red copy of τ or a blue copy of π. We let U_C denote the set of all unavoidable permutations in C. It is easy to see that a permutation π is unavoidable in C if and only if C has no nontrivial splitting into two parts with one part being π. A more detailed overview of the properties of unavoidable permutations was provided by Jelínek and Valtr <cit.>, we will mention only the observations needed for our results.Note that for a nonempty permutation class C, the set of unavoidable permutations U_C is in fact a nonempty permutation class contained in the class C. We can use the class of unavoidable permutations to characterize the unsplittable permutation classes.A permutation class C is unsplittable if and only if U_C = C. Furthermore, we can show that if C is closed under certain inflations then also U_C is closed under the same inflations. Again, the following result is due to Jelínek and Valtr <cit.>. Let C be a permutation class. If, for a set of permutations X, the class C is closed under ·[X], then U_C is also closed under ·[X], and if C is closed under X[·], then so is U_C. Consequently, if C is inflation-closed, then U_C = C and C is unsplittable.§.§ AmalgamabilityNow let us introduce the concept of amalgamation, which comes from the general study of relational structures.We say that a permutation class C is π-amalgamable if for any two permutations τ_1, τ_2 ∈ C and any two mappings f_1 and f_2, where f_i is an embedding of π into τ_i, there is a permutation σ∈ C and two mappings g_1 and g_2 such that g_i is an embedding of τ_i into σ and g_1 ∘ f_1 = g_2 ∘ f_2. We also say, for k ∈ℕ that a permutation class C is k-amalgamable if it is π-amalgamable for every π of order at most k. Furthermore, a permutation class C is amalgamable if it is k-amalgamable for every k.Note that k-amalgamability implies (k-1)-amalgamability, so we have an infinite number of increasingly stronger properties. However, the situation is quite simple in the case of the permutation classes. As shown by Cameron <cit.>, there are only five infinite amalgamable classes, the classes 12, 21, the class of all layered permutations 231, 312, the class of their complements 213, 132 and the class of all permutations. These are also the only permutation classes that are 3-amalgamable, implying that for any k≥ 3, a permutation class is k-amalgamable if and only if it is amalgamable.In contrast, very little is known about 1-amalgamable and 2-amalgamable permutation classes. In this paper, we are particularly interested in the 1-amalgamable permutation classes.Let C be a permutation class. We say that a permutation π∈ C is 1-amalgamable in C, if for every τ∈ C and every prescribed pair of embeddings f_1 and f_2 of the singleton permutation 1 into π and τ there is a permutation σ∈ C and embeddings g_1 and g_2 of π and τ into σ such that g_1 ∘ f_1 = g_2 ∘ f_2. We use A_C to denote the set of all 1-amalgamable permutations in C. Trivially, A_C is a permutation class contained in C. Moreover, the properties of A_C are largely analogous to those of U_C, as shown by the next several results. A permutation class C is 1-amalgamable if and only if A_C=C. Similarly to U_C, the set A_C is closed under the same inflations as the original class C. Let C be a permutation class. If, for a set of permutations X, the class C is closed under ·[X], then A_C is also closed under ·[X], and if C is closed under X[·], then so is A_C. Consequently, if C is inflation-closed, then A_C = C and C is 1-amalgamable. Suppose that C is closed under ·[X]. We can assume that X itself is inflation-closed since if C is closed under ·[X], it is also closed under ·[X[X]].Let π∈ A_C be a 1-amalgamable permutation of order k and let ρ_1, …, ρ_k be permutations from X. Our goal is to prove that π[ρ_1, …, ρ_k] also belongs to A_C. We can assume, without loss of generality, that all ρ_i are actually equal to a single permutation ρ. Otherwise, we could just take ρ∈ X that contains every ρ_i (this is possible since X is inflation-closed) and prove the stronger claim that π[ρ, …, ρ] belongs to A_C. Let us use π[ρ] as a shorthand notation for π[ρ, …, ρ].It is now sufficient to show that π[ρ] belongs to A_C for every π∈ A_C and ρ∈ X. Fix a permutation τ∈ C and two embeddings f_1 and f_2 of the singleton permutation into π[ρ] and τ. We aim to find a permutation σ∈ C and two embeddings g_1 and g_2 of π[ρ] and τ into σ such that g_1 ∘ f_1 = g_2 ∘ f_2. We can straightforwardly decompose f_1 into an embedding h_1 of the singleton permutation into π, by simply looking to which inflated block order-isomorphic to ρ the image of f_1 belongs, and an embedding h_2 of the singleton permutation into ρ, determined by restricting f_1 only to that copy of ρ. Since π belongs to A_C, there is a permutation σ' with embeddings g_1' and g_2' of π and τ such that g_1' ∘ h_1 = g_2' ∘ f_2.Define σ = σ'[ρ], and view σ as a concatenation of blocks, each a copy of ρ. Let us define mapping g_1 by simply using g_1' to map blocks of π[ρ] to the blocks of σ, each element in π[ρ] gets mapped to the same element of the corresponding copy of ρ in σ. Then define mapping g_2 by using g_2' to map its elements to the blocks of σ and then within the copy of ρ to the single element in the image of h_2. It is easy to see that g_1 and g_2 are in fact embeddings of π[ρ] and τ into σ. Also the images of g_1 ∘ f_1 and g_2 ∘ f_2 must lie in the same block of σ. And finally these images must be equal since we used h_2 to place the single element from the image of g_2 inside each block of σ.We now show that if C is closed under X[·] then so is A_C. Fix a permutation ρ∈ X of order k, and a k-tuple π_1, …, π_k of permutations from A_C. We will show that ρ[π_1, … , π_k] belongs to A_C.Fix a permutation τ∈ C and two embeddings f_1 and f_2 of the singleton permutation into ρ[π_1, … , π_k] and τ. We aim to find a permutation σ∈ C and two embeddings g_1 and g_2 of ρ[π_1, … , π_k] and τ into σ such that g_1 ∘ f_1 = g_2 ∘ f_2. We again view ρ[π_1, … , π_k] as a concatenation of k blocks, the i-th block being order-isomorphic to π_i. Suppose that the image of f_1 is in the j-th block. Let us decompose f_1 into an embedding h_1 of the singleton permutation into ρ whose image is the j-th element of ρ, and an embedding h_2 of the singleton permutation into π_j. Since π_j belongs to A_C, there is a permutation σ' with embeddings g_1' and g_2' of π_j and τ such that g_1' ∘ h_2 = g_2' ∘ f_2.Define σ = ρ[π_1, …, π_j-1 , σ', π_j+1, …, π_k] and let us define mapping g_1 in the following way. Every block of ρ[π_1, … , π_k] except for the j-th one gets mapped to the corresponding block of σ, and the j-th block is mapped using the embedding g_1' to the j-th block of σ. Then define mapping g_2 simply by mapping τ to the j-th block of σ using g_2'. It is easy to see that both g_1 and g_2 are in fact embeddings of ρ[π_1, … , π_k] and τ into σ. Furthermore, the images of g_1 ∘ f_1 and g_2 ∘ f_2 both lie in the j-th block of σ. Their equality then follows from the construction since g_1' ∘ h_2 = g_2' ∘ f_2.It remains to show that if C is inflation-closed then A_C = C. But if C is inflation-closed, then it is closed under ·[C], so A_C is also closed under ·[C]. And since A_C trivially contains the singleton permutation, for every π∈ C we have that π = 1[π] also belongs to A_C. As noted by Jelínek and Valtr <cit.>, it follows from the results of Nešetřil <cit.> that if a permutation class C is unsplittable then C is also 1-amalgamable. Using the same argument, we get the following stronger proposition relating the classes U_C and A_C. Let C be a permutation class, then U_C ⊆ A_C. Let π be an unavoidable permutation in C and let τ be a permutation from C.By the definition of U_C, there is a permutation σ∈ C such that any red-blue coloring of σ has a red copy of τ or a blue copy of π.We claim that σ contains every 1-amalgamation of π and τ.Suppose for a contradiction that there are two embeddings f_1 and f_2 of the singleton permutation 1 into π and τ such that there are no embeddings g_1 and g_2 of π and τ into σ that would satisfy g_1 ∘ f_1 = g_2 ∘ f_2.Let f_1(1) = a and f_2(1) = b.We aim to color the elements of σ to avoid both a red copy of τ and a blue copy of π.We color an element σ_i red if and only if there is an embedding of π which maps π_a to σ_i.Trivially, we cannot obtain a blue copy of π, since we must have colored the image of π_a red.On the other hand, suppose we obtained a red copy of τ.Then the image of τ_b was painted red which means that there is an embedding of π which maps π_a to the same element.We assumed that such a pair of embeddings does not exist, therefore we defined a coloring of σ that contains neither a red copy of τ nor a blue copy of π.§ LEFT-TO-RIGHT MINIMA We say that the element π_i covers the element π_j if i < j and simultaneously π_i < π_j. The i-th element of a permutation π is then a left-to-right minimum, or shortly LR-minimum, if it is not covered by any other element.Similarly we could define LR-maxima, RL-minima and RL-maxima.However we can easily translate between right-to-left and left-to-right orientation by looking at the reverses of the permutations, and similarly between maxima and minima by looking at the complements of the permutations. Therefore we restrict ourselves to dealing only with LR-minima from now on. Suppose that π∈ S_n is a permutation with k LR-minima and let σ_1, …, σ_k be a k-tuple of non-empty permutations. The LR-inflation of π by the sequence σ_1, …, σ_k is the permutation resulting from the inflation of the LR-minima of π by σ_1, …, σ_k. We denote this by π⟨σ_1, …, σ_k⟩.We say that a permutation class C is closed under LR-inflations if for every π∈ C with k LR-minima, and for every k-tuple σ_1,…,σ_k of permutations from C, the LR-inflation π⟨σ_1,…,σ_k⟩ belongs to C. The closure of C under LR-inflations, denoted C, is the smallest class which contains C and is closed under LR-inflations. Recall that one can characterize inflation-closed classes by a basis that consists of simple permutations. We can derive a similar characterization in the case of classes closed under LR-inflations. We say that a permutation is LR-simple if it cannot be obtained by LR-inflations except for the trivial ones. Using the same arguments, it is easy to see that a permutation class is closed under LR-inflations if and only if every permutation in its basis is LR-simple. §.§ LR-splittability We aim to define a stronger version of splittability that would help us connect the properties of permutation classes and their LR-closures. A natural way to do that is to consider an operation similar to the regular merge, with LR-minima being shared between both parts. We say that a permutation π is a LR-merge of permutations τ and σ, if its non LR-minimal elements can be partitioned into two disjoint sequences, such that one of them is, together with the sequence of LR-minima of π, an occurrence of τ, and the other is, together with the sequence of LR-minima of π, an occurrence of σ. For two permutation classes A and B, we write AB for the class of all LR-merges of a permutation from A with a compatible permutation from B. Trivially, AB is again a permutation class. Note that we can also look at LR-merges as a special red-blue colorings of permutations in which the LR-minima are both blue and red at the same time. Naturally we can use this definition of LR-merge to define LR-splittability in the same way that the concept of regular merge gives rise to the definition of splittability. We say that a multiset of permutation classes { P_1, …, P_m} forms a LR-splitting of a permutation class C if C ⊆ P_1 ⋯ P_m. We call P_i the parts of the LR-splitting. The LR-splitting is nontrivial if none of its parts is a superset of C, and the LR-splitting is irredundant if no proper submultiset of { P_1, …, P_m} forms an LR-splitting of C. A permutation class C is then LR-splittable if C admits a nontrivial LR-splitting. Clearly, every LR-splittable class is splittable. Moreover, some properties of LR-splittability are analogous to the properties of splittability, as shown by the following lemma. We omit the proof as it uses the very same (and easy) arguments as the proof of Lemma <ref>.For a class C of permutations, the following properties are equivalent: * C is LR-splittable.* C has a nontrivial LR-splitting into two parts.* C has an LR-splitting into two parts, in which each part is a proper subclass of C.* C has a nontrivial LR-splitting into two parts, in which each part is a principal class.Now we can state some of the results connecting splittability and LR-splittability of permutation classes and their LR-closures. Let C be a permutation class that is closed under LR inflations. Then C is splittable if and only if C is LR-splittable. Trivially, LR-splittability implies splittability since we can take the corresponding red-blue coloring and simply assign an arbitrary color to each of the LR-minima. Now suppose that C admits splitting { D, E } for some proper subclasses D and E. We aim to prove that also C ⊆ DE. Let us first show that C contains a permutation τ that belongs neither to D nor to E. From the definition of splittability, there are permutations τ_D ∈ C ∖ D and τ_E ∈ C ∖ E. Define τ as the LR-inflation of τ_D with τ_E, which clearly lies outside both subclasses D and E.Let us suppose that there is some π∈ C not belonging to DE , i.e., there is no red-blue coloring of π which proves it is an LR-merge of a permutation α∈ D and a permutation β∈ E. Let π' be the permutation created by inflating each LR-minimum of π with τ. Since π' belongs to C, it has a regular red-blue coloring with the permutation corresponding to the red elements π'_R ∈ D and the permutation corresponding to the blue elements π'_B ∈ E. However there must be both colors in each block created by inflating a LR-minimum of π with τ, and therefore there is a valid red-blue coloring of π that assigns both colors to the LR-minima. Finally, we want to show that, under modest assumptions, the LR-splittability of a permutation class implies the LR-splittability (and thus the splittability) of its LR-closure.If C, D and E are permutation classes satisfying C ⊆ DE, then C⊆DE. Consequently, if neither D nor E contain the whole class C, then its closure C is LR-splittable into parts D and E. We will inductively construct a valid red-blue coloring which proves that C⊆DE. First, any permutation in C that cannot be obtained from shorter permutations using LR-inflations must belong to C and we simply use the red-blue coloring that witnesses the inclusion C⊆ D E.Now take π∈C that can be obtained by LR-inflation from shorter permutations as π = α⟨β_1, …, β_k⟩. We can already color the permutation α and all the permutations β_i and we construct a coloring of π in the following way: color the inflated blocks β_i according to the coloring of β_i and the remaining uninflated elements of α get the color according to the coloring of α. It remains to show that the permutation π_R corresponding to the red elements of π belongs to D and the permutationπ_B corresponding to the blue elements of π belongs to E. Since the LR-minima of α are both red and blue, the permutation π_R is an LR-inflation of the red elements of α by the red elements of the permutations β_i. All these permutations belong to D and thus their LR-inflation also belongs to D. Using the very same argument we can show that π_B belongs to E.It remains to show that the splitting of C into D and E is nontrivial. However that follows from the assumption that neither D nor E contain the whole class C.§.§ LR-amalgamability Similarly to the situation with LR-splittability we want to describe a property of permutation classes which would imply 1-amalgamability of their respective LR-closures. We say that a permutation class C is LR-amalgamable if for any two permutations τ_1, τ_2 ∈ C and any two mappings f_1 and f_2, where f_i is an embedding of the singleton permutation into τ_i and its image is not an LR-minimum of τ_i, there is a permutation σ∈ C and two mappings g_1 and g_2 such that g_i is an embedding of τ_i into σ, g_1 ∘ f_1 = g_2 ∘ f_2 and moreover g_i preserves the property of being a LR-minimum. Observe that LR-amalgamability does not imply 1-amalgamability since it does not guarantee 1-amalgamation over LR-minima and conversely, 1-amalgamability does not imply LR-amalgamability because it may not preserve the property of being an LR-minimum. However, we can at least prove that LR-amalgamability implies 1-amalgamability for classes that are closed under LR-inflations. Recall that we actually derived equivalence between LR-splittability and splittability in Proposition <ref>. Let C be a permutation class that is closed under LR inflations. If C is LR-amalgamable then C is also 1-amalgamable. Let π_1 and π_2 be arbitrary permutations from C and f_1, f_2 embeddings of the singleton permutation into π_1 and π_2 respectively. If neither of the images of f_1 and f_2 is an LR-minimum of the respective permutation we obtain their 1-amalgamation directly since C is LR-amalgamable.Now we can assume without loss of generality that the single element in the image of f_1 is a LR-minimum of π_1. We can create the resulting 1-amalgamation by simply inflating this LR-minimum by the permutation π_2. It is then easy to derive the mappings g_1 and g_2 that show it is the desired 1-amalgamation. We conclude this section by relating LR-amalgamability of a permutation class and 1-amalgamability of its LR-closure.If a permutation class C is LR-amalgamable then its LR-closure C is LR-amalgamable and thus also 1-amalgamable. Let π_1, π_2 ∈C be permutations and f_1, f_2 embeddings of the singleton permutation, f_i into π_i such that the image of f_i avoids the LR-minima of π_i. We aim to prove by induction on the length of π_1 and π_2 that there is a corresponding LR-amalgamation of π_1 and π_2. Consider two cases. If neither of the two permutations π_1 and π_2 can be obtained as an LR-inflation of a shorter permutation then they bothbelong to C. And since C itself is LR-amalgamable they have a desired LR-amalgamation that belongs to C.Without loss of generality we can now assume that π_1 can be obtained by LR-inflations as π_1 = α⟨β_1, …, β_k ⟩ where the permutations α, β_1, …, β_k are all strictly shorter than π_1. Again we consider two separate cases. First, assume that the image of the embedding f_1 lies inside the block corresponding to the j-th inflated LR-minimum of α, which is order-isomorphic to β_j. From induction we get a LR-amalgamation σ of β_j and π_2 for the embeddings f_1' and f_2, where f_1' is the embedding f_1 restricted to the inflated block of β_j. Observe that the permutation α⟨β_1, …, β_j-1, σ, β_j+1, …, β_k ⟩ is precisely the LR-amalgamation of π_1 and π_2 we were looking for.Finally we have to deal with the situation when the image of the embedding f_1 lies outside of the blocks corresponding to the inflated LR-minima of π_1. We can obtain from induction a LR-amalgamation σ of α and π_2 for the embeddings f_1” and f_2, where f_1” is the embedding f_1 restricted to the permutation α. Let g_1 be the corresponding embedding of α into σ that preserves the LR-minima. We construct the desired LR-amalgamation of π_1 and π_2 in the following way: take σ and for every LR-minimum of α inflate its image under g_1 with the corresponding permutation β_i. The resulting permutation is clearly a 1-amalgamation of π_1 and π_2, and it also preserves the LR-minima.Lemma <ref> implies that C is also 1-amalgamable. § MAIN RESULT Now we are ready to prove that 1-amalgamability and unsplittability are not equivalent by exhibiting as a counterexample the LR-closure of 123. First, let us show that this class actually has a nice basis consisting of only two patterns.The class 1423, 1342 is the closure of 123 under LR-inflation.First, let us show that any permutation from the LR-closure of 123 avoids both 1423, 1342. Because both of these patterns contain 123, they would have to be created by the LR-inflations. However, that is not possible since there is no nontrivial interval in either 1423 or 1342 which contains the minimum element.Now, let π be a permutation from 1423, 1342.We will show by induction that this permutation can be obtained by a repeated LR-inflation of permutations from 123.If π does not contain 123 the statement is trivially true.Otherwise, consider the set of the right-to-left maxima of π.We want to show that the remaining elements of π can be split into a descending sequence of intervals.If this holds then we can get π as an LR-inflation of a 123-avoiding permutation by permutations order-isomorphic to the intervals.And by induction these shorter permutations can be obtained as repeated LR-inflations of 123-avoiding permutations.Let us show that there is no occurrence of the pattern 132 that maps only the letter 2 on an RL-maximum. For a contradiction suppose we have such an occurrence and a corresponding embedding f of 132 into π.Then there must be an element covered by π_f(3) since it is not an RL-maximum, i.e., an element π_k such that k > f(3) and π_k > π_f(3).However, π restricted to these four indices would form the pattern 1342. Using the same argument, we can also show that there is no occurrence of the pattern 132 which maps only the letter 3 on an RL-maximum as we would get an occurrence of the pattern 1423 together with the RL-maximum covered by the image of 2.And finally, we conclude by showing that the elements of π that are not RL-maxima can indeed be split into a descending sequence of intervals.Let I = { i_1, …, i_m } be the index set of the RL-maxima of π and furthermore define i_0 = 0 and π_0 = n+1.Let us represent the remaining elements of π as a set A of n - m points on a plane A = {(i, π_i) |}. We define a partition of A into sets A_j,k for any 1 ≤ j < k ≤ m A_j,k = { (x,y) |(x, y) ∈Ai_j-1 < x < i_jπ_i_k < y < π_i_k-1}. For any j, k and l, every element of A_j,k is larger than all the elements of A_j+1,l in the second coordinate since otherwise we would get a 132 occurrence with the letter 3 mapped to π_i_j.Similarly for any j, k and l, every element of A_j,k is to the left of all the elements of A_l,k+1 as otherwise we would get a 132 occurrence with the letter 2 mapped to π_i_k.This transitively implies that all non-empty sets A_j,k correspond to a sequence of descending intervals. In order to show that 1423, 1342 is splittable, we shall first prove the LR-splittability of 123 and then apply the results we have obtained in Subsection <ref>.The class 123 is LR-splittable, and more precisely, it satisfies123⊆463152463152.Let π be a permutation from 123. Clearly π is a merge of two descending sequences, its LR-minima and the remaining elements. The idea is to decompose the non-minimal elements into runs such that for every run there is a specific LR-minimum covering each element of the run but covering none from the following run.This can be done easily by the following greedy algorithm. In one step of the algorithm, let π_i be the first non-minimal element which was not used yet and let j be the maximum integer such that π_j is an LR-minimum covering π_i. The next run then consists of all non-minimal elements starting from π_i that are covered by π_j.We color each run blue or red such that adjacent runs have different colors.We obtained a red-blue coloring of the non-minimal elements and it only remains to check whether the monochromatic permutations form a proper subclass of 123. Observe that the first elements of two adjacent runs cannot be covered by a single LR-minimum, which implies that two elements from different non-adjacent runs cannot be covered by a single LR-minimum. By this observation, in the monochromatic permutations π_B and π_R any two elements covered by the same LR-minimum must belong to the same run.We claim that a monochromatic copy of the pattern 463152 ∈123 can never be created this way. Assume for contradiction that there is a permutation π∈123 on which the algorithm creates a monochromatic copy of 463152 and let f be the corresponding embedding of 463152 into π. Observe that every LR-minimum of 463152 is covering some other element and therefore f must preserve the property of being an LR-minimum, otherwise we would get an occurrence of the pattern 123. Following our earlier observations, the elements π_f(6), π_f(5) and π_f(2) must fall into the same run since π_f(5) shares LR-minima with both of the other two elements. And because elements of the same run are covered by a single LR-minimum, there is an LR-minimum π_i covering π_f(6) and π_f(2). However, π_i must then also cover π_f(3) which contradicts the fact that π_f(3) itself is an LR-minimum of π. The class 1423, 1342 is splittable. In the previous Lemma <ref> we showed that 123 is LR-splittable, more precisely that 123⊆463152463152 . Since the permutation 463152 is LR-simple, we get the splittability of 123 from Proposition <ref>. Finally, owing to Proposition <ref>, we know that 123 and 1423, 1342 are in fact identical. Our final task is to show that 1423, 1342 is 1-amalgamable by proving the LR-amalgamability of 123. In order to do that we will use the following result which is due to Waton <cit.>. Note that Waton in fact proved the equivalent claim for parallel lines of positive slope and the permutation class 321.The class of permutations that can be drawn on any two parallel lines of negative slope is 123. The class 123 is LR-amalgamable. Fix arbitrary two parallel lines of negative slope in the plane. Let π_1 and π_2 be permutations avoiding 123 and f_1 and f_2 be mappings where f_i is an embedding of the singleton permutation into π_i and its image is not an LR-minimum of π_i. According to Proposition <ref> both π_1 and π_2 can be drawn from our fixed parallel lines. Fix sets of points A_1 and A_2 which lie on these lines whose corresponding respective permutations are π_1 and π_2. Moreover, we can choose the sets such that the elements in the images of f_1 and f_2 share the same coordinates. Otherwise we could translate one of the sets in the direction of the lines to align these two points. Finally, if a point x ∈ A_1 and a point y ∈ A_2 share one identical coordinate we can move x a little bit in the direction of the lines without changing the permutation corresponding to the set A_1.We may easily see that the permutation corresponding to the union A_1 ∪ A_2 with the natural mappings of π_1 and π_2 is the desired LR-amalgamation of π_1 and π_2.Applying Proposition <ref>, we get the desired result that the class Av(1423, 1342) is indeed 1-amalgamable.The class 1423, 1342 is 1-amalgamable. § FURTHER DIRECTIONS Using our results about LR-inflations, we proved that a single class 1423, 1342 is both 1-amalgamable and splittable. Naturally, the same holds for its three symmetrical classes, i.e. 3241, 2431, 4132, 4213 and 2314, 3124, since both splittability and 1-amalgamability is preserved when looking at the reverses or complements of the permutations. However, the question remains whether these results can be used to find more classes that are both 1-amalgamable and splittable or even infinitely many such classes. It would be particularly interesting to find other such classes with small basis.Our method of obtaining a splittable 1-amalgamable class was based on the notion of LR-inflations, and the related concepts of LR-amalgamations and LR-splittings. These notions can be generalized to a more abstract setting as follows: suppose that we partition every permutation π into `inflatable' and `non-inflatable' elements, in such a way that for any embedding of a permutation σ into π, the non-inflatable elements of σ are mapped to non-inflatable elements of π. We might then consider admissible inflations of π (in which only the inflatable elements can be inflated), admissible splittings of π (which are based on two-colorings in which each inflatable element receives both colors), as well as admissible amalgamations (where we amalgamate by identifying non-inflatable elements, and the amalgamation must preserve the inflatable elements of the two amalgamated permutations). In this paper, we only considered the special case when the inflatable elements are the LR-minima; however, the main properties of LR-inflations, LR-splittings and LR-amalgamations extend directly to the more abstract setting. | http://arxiv.org/abs/1704.08732v3 | {
"authors": [
"Vít Jelínek",
"Michal Opler"
],
"categories": [
"math.CO",
"05A05"
],
"primary_category": "math.CO",
"published": "20170427201327",
"title": "Splittability and 1-amalgamability of permutation classes"
} |
Three-dimensional structure of the magnetic field in the disk of the Milky Way A. Ordog1 J.C. Brown1R. Kothes2T.L. Landecker2 ===============================================================================================§ INTRODUCTIONIn this paper, we propose efficient methods for computing the followingintegrals appearing in electronic structure calculations<cit.>:I(μ,τ) ≡-1πlim_η↓ +0Im∫_-∞^∞W(x; μ, τ) G(x+ iη) dx,where G(z) is the Green's function which is defined asG(z)=b^* (zI-H)^-1b,where b is a vector and H is the so-called Hamiltonian matrix (a Hermitian matrix), and W(x; μ, τ) is the Fermi-Dirac function:W(x; μ, τ)=1/1+exp(x-μτ),where μ is a real number, and τ is a small positive number. The Green's function G(z) is expanded as follows:G(z)=∑_j=1^N c_j/z-λ_j,where N is the order of H,c_j (j=1, 2, ⋯, N) are non-negative real numbers,and λ_j(j=1, 2, ⋯, N) arethe eigenvalues (real numbers)of H, which correspond to the energy levels in the material.We assume that λ_j's are labeled in increasing order: λ_1<λ_2< ⋯<λ_N. We also assume that a lower estimate for the smallest eigenvalue, λ_1,is known, although λ_j (j=1, 2, ⋯, N) are unknown. In <cit.>, the trapezoidal rule is applied to evaluate the integralin I(μ,τ),by taking η as a very small number, and by setting the interval ofintegration adequately wide. It is evident that many sampling points are necessary, since G(z) hasmany poles on the real axis. However, due to a limited time of computation, the number of the samplingpoints is not as many as supposed to be. Therefore, the accuracy of the computed results is not enough.In this paper, we propose two methods for computing I(μ, τ)efficiently,both of which consist of two parts: the first part is to represent the integrals as contour integrals and the second one is to evaluate the contour integrals by the Clenshaw-Curtis quadrature.The paper is organized as follows. In Sec. 2, we introduce one of the proposed methods, which we call Method 1,and give a numerical example. Subsequently, in Sec. 3, we present another method, which we call Method 2,together with a numerical example.In Sec. 4, we develop a method for computing I(μ,τ)for many distinct values of μ, which situation sometimes arises. Finally, in Sec. 5, we make concluding remarks.§METHOD 1 It is easily verified that I(μ,τ) is equal to ∑_jc_jW(λ_j; μ, τ). Thus, by setting the contour C as a simply closed curve which encloses all the poles of G(z) but does not of W(z) (Fig. <ref>), the contour integral representation of I is obtained as follows:I(μ,τ)=1/2π i∫_C W(z; μ, τ)G(z) dz .It is expected that we will perform the numerical integration efficiently with this contour integral, because we can set the contour far from the poles of G(z) on the real axis.Taking account of easiness of numerical integration, we now set the contour ofthe integral as L_1+L_2+L_3+C_4 illustrated in Fig. <ref>, where *ℓ is a real number that is smaller than the smallest pole of G(z), i.e.,λ_1(note that the possibility of setting up this number is guaranteed by the assumption that a lower estimate for λ_1 is known),and also such that W(ℓ; μ, τ) ≈ 1, i.e., |W(ℓ; μ, τ) - 1| is small enough (in the numerical examples below,we set |W(ℓ; μ, τ) - 1| ≤ 10^-40); * u is a real number such that W(u; μ, τ) ≈ 0 (in the numerical examples below,we set |W(u; μ, τ)| ≤ 10^-40); * L_1=[u+(πτ/2) i,ℓ+(πτ/2) i];* L_2=[ℓ+(πτ/2) i,ℓ-(πτ/2) i];* L_3=[ℓ-(πτ/2) i, u-(πτ/2) i];* C_4 is a curve connecting the points u-(πτ/2) i and u+(πτ/2) i and such that L_1+L_2+L_3+C_4 encloses all the poles of G(z) . Then, denoting W(z; μ, τ)G(z) by F(z; μ, τ), we have 12π i∫_L_1F(z; μ, τ) dz+ 12π i∫_L_3F(z; μ, τ) dz =-1π ∫_ℓ^uImF(x+πτ/2i; μ, τ) dx, 1/2π i∫_L_2 F(z; μ, τ)dz≈ -1/π∫_0^πτ/2 Re G(ℓ+yi)dy, 1/2π i∫_C_4F(z; μ, τ) dz ≈ 0 .Thus,we obtain I(μ, τ)≈ I_ h+I_ vwhere I_ h =-1π∫_l^u Im F(x+πτ/2i; μ, τ)dx, I_ v =-1π∫_0^πτ/2 ReG(ℓ+yi)dy .For calculation of I_ h and I_ v we adopt the Clenshaw-Curtis quadrature <cit.>, because the integrands are analytic over the intervals of integration. Numerical example 1 We consider the case where the Green's function is given byG(z)=∑_j=1^46861/z-λ_j,which appears in an electronic structure calculation of a nanoscale amourphous-like conjugated polymer<cit.>． The values of λ_1, λ_2, ⋯ , λ_4686 (λ_1≃-1.16, λ_4686≃5.58) are given on the website <cit.> as the data set ^^ ^^ APF4686”.(The actual computation of G(z) is done by using the expression (<ref>), that is, by solving the large system of linear equations (zI-H)x=b, which leads to relatively large numerical errors. Since we here concentrate on examining numerical errors caused by the numerical integration, we give G(z) as the rational expression as above,the computation of which produces small numerical errors.) We first set μ as μ=(λ_2343+λ_2344)/2=-0.3917431575916144,where λ_2343= -0.4258775547956950, λ_2344= -0.3576087603875338,and τ=0.01. In this case, the exact value of I(μ, τ) is 2342.992785654893⋯. We evaluate the integrals I_ h and I_ vwith the Clenshaw-Curtis quadrature, setting ℓ=-1.5,u=0.6.For I_ v, whose integrandhas no poles near the interval of integration, a very rapid convergence of the Clenshaw-Curtis quadrature is observed:the relative error 10^-15 is attained with 7 sampling points. For I_ h, whose integrandhas many poles nearthe interval of integration,the convergence behavior is shownin Fig. <ref>. Exponential convergence is observed,as expected from the convergence theory ofthe Clenshaw-Curtis quadrature.Next, we set μ the same as above and τ=0.001. The exact value of I(μ, τ) is 2343.000000000000⋯. We evaluate I_ h and I_ v with the Clenshaw-Curtis quadrature, setting ℓ=-1.5,u=-0.3. For I_ v, the relative error 10^-15 is achievedwith 6 sampling points. For I_ h, Fig. <ref> shows the convergence behavior.Exponential convergence is observed, which is slower than that of the case of τ=0.01.§METHOD 2Fig. <ref> shows that Method 1 requires a large number of sampling points, that is, function evaluations for computing the integral I_h, which caused by the proximity of the paths L_1, L_3 to the poles of W(z)G(z). To solve this problem, we take the contour L'_1+L'_2+L'_3+C'_4 so that it is far from the poles of W(z)G(z), as shown in Fig.<ref>.In this setting, we should consider the residues of W(z; μ, τ) at z=μ±πτ i, thus the contour integral becomes12 π i ∫_L'_1+L'_2+L'_3+C'_4 F(z; μ, τ)dz=I(μ, τ)+Res(F, μ+πτ i)+Res(F, μ-πτ i) =I(μ, τ)-2 τReG(μ+πτ i).Calculating the left-hand side similarly as in Method 1, we obtainI(μ, τ) ≈ I_ h'+I_ v'+ 2 τRe G(μ+πτ i), where I_ h' =-1π∫_l^uIm F(x+2πτ i; μ, τ)dx ,I_ v' =-1π∫_0^2πτ ReG(l+yi)dy.For evaluation of I_ h' and I_ v', we use the Clenshaw-Curtis quadrature. Numerical example 2 We set G(z), μ, and τ as the same as in Numerical example 1. First, in the case of τ=0.01, we evaluatethe integrals I_ h' and I_ v' with the Clenshaw-Curtis quadrature, setting ℓ=-1.5,u=0.6.For I_ v', the relative error 10^-15 is attained with 12sampling points. For I_ h', Fig. <ref> shows the convergence behavior ofthe Clenshaw-Curtis quadrature. Next, in the case of τ=0.001, we compute the integrals I_ h' andI_ v' with the Clenshaw-Curtis quadrature, setting ℓ=-1.5,u=-0.3. For I_ v' the relative error 10^-15 is achievedwith 6 sampling points. For I_ h', Fig. <ref> shows the convergence behavior ofthe Clenshaw-Curtis quadrature. We can see that the performance is much improved.Remark 1Setting such contour contributes not only to fast computations of integrations, but also to the actual computation of G(z). In fact, the actual computation of G(z) requires to solvethe equation (zI-H)x=b with Krylov subspace methods such as the COCG method. And it is known that the fartherthe distances between z and the eigenvalues of H, i.e., λ_j's,the faster the convergence of Krylov subspace methods in general.§METHOD FOR COMPUTING I(Μ,Τ) WITH VARIOUS VALUES OF ΜIt is often the case that we need to compute I(μ,τ)for many distinct values of μ. Computing separately for each value of μ with Method 1 or 2, costs a massive amount of calculation in total. Instead wepropose an efficient methodbased on Method 1.It is supposed here that the range of required μ, say [μ_ min, μ_ max], is known.In Method 1, it is evident that the computation of I_ h(μ, τ) for many values of μ causes the massive amount of computation.Hence we reduce the amount of the computation of I_ h(μ, τ) for many values of μ.The key is to use common ℓ and u (See (<ref>)) in the computation of I_ h(μ,τ) for many values of μ.In fact, we can use ℓ_ min and u_ max as the common ℓand u respectively, where ℓ_ min is a real numberthat is less than or equal to the value of ℓ determined in the case of μ=μ_ min, and u_ max is a real number that is greater than or equal to the valueof u determined in the case of μ=μ_ max. Then,I_ h =-1π∫_ℓ_ min^u_ max Im W(x+πτ/2i; μ, τ) G(x+πτ/2i)dx .(I_ v =-1π∫_0^πτ/2 ReG(ℓ_ min+yi)dy) It follows that the computation of G, which costs a large amount ofcalculation, is independent of μ.Thus, once we compute I_h for an appropriately large μ and store the computed values of G(x+(πτ/2)i) for reuse, we can immediately obtain the result of I_ h for another value of μ, by multiplying the stored values of G(x+(πτ/2)i) by W(x+(πτ/2)i; μ, τ), the cost of computation of which is low. This device enables us to compute I(μ,τ) for many of distinct μ efficiently. Numerical example 3We consider the case where G(z) is the same as in Numerical example 1 and τ = 0.01. We compute I(μ,τ) for μ =(λ_10+λ_11)/2=-1.147817562299727,⋮ μ =(λ_4650+λ_4651)/2=3.379941668485607.We apply the Clenshaw-Curtis quadrature to evaluate I_ h and I_ v with ℓ_ min=-1.5,u_ max=6.0.Note that I_ v is the same as in Numerical example 1, for whichthe Clenshaw-Curtis quadrature attains the relative error 10^-15 with 6 sampling points.For μ=(λ_4650+λ_4651)/2=3.3799⋯,we compute I_ h and store the computed values of G(z).Then we compute I_ h for another value of μ, by using the stored value of G(z).Fig. <ref>shows the convergence behaviors ofthe Clenshaw-Curtis quadrature for I_ h withμ=(λ_10+λ_11)/2=-1.1478⋯, μ=(λ_2343+λ_2344)/2=-0.39174⋯ andμ=(λ_4650+λ_4651)/2=3.3799⋯. Remark 2As for Method 2, we can use a similar device for computing I_ h' with various values of μ.In fact, once we compute I_ h' for a suitable μ and store the values of G(x+2πτ i), then we can obtain the results of I_ h' for distinct values of μ, by multiplying the stored values of G(x+2πτ i) byW(x+2πτ i; μ, τ), which does not need much computation.However, it should be noted that an additionalcomputation of G(μ+πτ i) is needed for the calculation of I. The fact that Method 2 is faster than Method 1, tells us thatMethod 2 with this device can be effective when the total cost of computationof G(μ+πτ i) is not large.§ CONCLUDING REMARKSIn this paper, we proposed methods for computing integrals appearingin electronic structure calculations and showed that the proposed methods are efficient through numerical experiments. We would like to note that the proposed methods are also efficient for the case where the Green's function is given by G(z) =∑_j=1^4686c_j/z-λ_jwhere {c_j} are uniform random numbers on [0,1], although the results are not contained here, due to a limited number of pages.We applied the proposed methods to the Green's function G(z)given as rational expression, but we should treat the Green's function G(z) with (<ref>) using Hamiltonian matrix, which is left for the future work. 99 TAKAYAMA-2006 R. Takayama, T. Hoshi, T. Sogabe, S.-L. Zhang, and T. Fujiwara,Linear algebraic calculation of the Green's function for large-scale electronic structure theory, Phys. Rev. B, 73 (2006), 165108, 1-9. Hoshi2011 H. Teng, T. Fujiwara, T. Hoshi, T. Sogabe, S.-L. Zhang, S. Yamamoto, Efficient and accurate linear algebraic methods for large-scale electronic structure calculations with nonorthogonal atomic orbitals, Physical Review B, 83 (2011), 165103, 1-12. Trefethen2013 Lloyd N. Trefethen, Approximation Theory and Apporoximation Practice, SIAM, Oxford, 2013. Hoshi2012 T. Hoshi, S. Yamamoto, T. Fujiwara, T. Sogabe, S.-L. Zhang, An order-N electronic structure theory with generalized eigenvalue equationsand its application to a ten-million-atom system,J. Phys.: Condens. Matter 24 (2012), 165502, 1-5. Elses http://www.elses.jp/matrix/ | http://arxiv.org/abs/1704.08644v1 | {
"authors": [
"Hisashi Kohashi",
"Kosuke Sugita",
"Masaaki Sugihara",
"Takeo Hoshi"
],
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"primary_category": "cond-mat.mtrl-sci",
"published": "20170427163036",
"title": "Efficient methods for computing integrals in electronic structure calculations"
} |
Spreading law on a completely wettable spherical substrate: The energy balance approach Masao Iwamatsu December 30, 2023 ========================================================================================= Existing question answering methods infer answers either from a knowledge base or from raw text.While knowledge base (KB) methods are good at answering compositional questions, their performance is often affected by the incompleteness of the KB. Au contraire,web text contains millions of facts that are absent in the KB, however in an unstructured form. Universal schema can support reasoning on the union of both structured KBs and unstructured text by aligning them in a common embedded space. In this paper we extend universal schema to natural language question answering, employing memory networks to attend to the large body of facts in the combination of text and KB. Our models can be trained in an end-to-end fashion on question-answer pairs.Evaluation results on fill-in-the-blank question answering dataset show that exploiting universal schema for question answering is better than using either a KB or text alone. This model also outperforms the current state-of-the-art by 8.5 F_1 points.[Code and data available in <https://rajarshd.github.io/TextKBQA>]§ INTRODUCTION Question Answering (QA) has been a long-standing goal of natural language processing. Two main paradigms evolved in solving this problem: 1) answeringquestions on a knowledge base; and 2) answering questions using text.Knowledge bases (KB) contains facts expressed in a fixed schema, facilitating compositional reasoning.These attracted research ever since the early days of computer science, e.g., BASEBALL <cit.>. This problem has matured into learning semantic parsers from parallel question and logical form pairs <cit.>, to recent scaling of methods to work on very large KBs like Freebase using question and answer pairs <cit.>.However, a major drawback of this paradigm is that KBs are highly incomplete <cit.>. It is also an open question whether KB relational structure is expressive enough to represent world knowledge <cit.>The paradigm of exploiting text for questions started in the early 1990s <cit.>.With the advent of web, access to text resources became abundant and cheap.Initiatives like TREC QA competitions helped popularizing this paradigm <cit.>.With the recent advances in deep learning and availability of large public datasets, there has been an explosion of research in a very short time <cit.>. Still, text representation is unstructured and does not allow the compositional reasoning which structured KB supports. An important but under-explored QA paradigm is where KB and text are exploited together <cit.>. Such combination is attractive because text contains millions of facts not present in KB, and a KB's generative capacity represents infinite number of facts that are never seen in text. However QA inference on this combination is challenging due to the structural non-uniformity of KB and text.Distant supervision methods <cit.> address this problem partially by means of aligning text patterns with KB.But the rich and ambiguous nature of language allows a fact to be expressed in many different forms which these models fail to capture.Universal schema <cit.> avoids the alignment problem by jointly embedding KB facts and text into a uniform structured representation, allowing interleaved propagation of information. <Ref> shows a universal schema matrix which has pairs of entities as rows, and Freebase and textual relations in columns. Although universal schema has been extensively used for relation extraction, this paper shows its applicability to QA. Consider the question USA has elected _blank_, our first african-american president with its answer Barack Obama. While Freebase has a predicate for representing presidents of USA, it does not have one for `african-american' presidents. Whereas in text, we find many sentences describing the presidency of Barack Obama and his ethnicity at the same time. Exploiting both KB and text makes it relatively easy to answer this question than relying on only one of these sources.Memory networks (MemNN; ) are a class of neural models which have an external memory component for encoding short and long term context.In this work, we define the memory components as observed cells of the universal schema matrix, and train an end-to-end QA model on question-answer pairs. The contributions of the paper are as follows (a) We show that universal schema representation is a better knowledge source for QA than either KB or text alone, (b) On the SPADES dataset <cit.>, containing real world fill-in-the-blank questions, we outperform state-of-the-art semantic parsing baseline, with 8.5 F_1 points. (c) Our analysis shows how individual data sources help fill the weakness of the other, thereby improving overall performance.§ BACKGROUND Problem DefinitionGiven a question q with words w_1, w_2, …, w_n, where these words contain one _blank_ and at least one entity, our goal is to fill in this _blank_ with an answer entity q_a using a knowledge base 𝒦 and text 𝒯. Few example question answer pairs are shown in Table <ref>.Universal Schema Traditionally universal schema is used for relation extraction in the context of knowledge base population. Rows in the schema are formed by entity pairs (e.g. USA, NYC), and columns represent the relation between them. A relation can either be a KB relation, or it could be a pattern of text that exist between these two entities in a large corpus. The embeddings of entities and relation types are learned by low-rank matrix factorization techniques. USchema:13 treat textual patterns as static symbols, whereas recent work by pat:15 replaces them with distributed representation of sentences obtained by a RNN. Using distributed representation allows reasoning on sentences that are similar in meaning but different on the surface form.We too use this variant to encode our textual relations. Memory Networks MemNNs are neural attention models with external and differentiable memory. MemNNs decouple the memory component from the network thereby allowing it store external information. Previously, these have been successfully applied to question answering on KB where the memory is filled with distributed representation of KB triples <cit.>, or for reading comprehension <cit.>, where the memory consists of distributed representation of sentences in the comprehension. Recently, key-value MemNN are introduced <cit.> where each memory slot consists of a key and value. The attention weight is computed only by comparing the question with the key memory, whereas the value is used to compute the contextual representation to predict the answer. We use this variant of MemNN for our model.KVMemNN, in their experiments, store either KB triples or sentences as memories but they do not explicitly model multiple memories containing distinct data sources like we do.§ MODELOur model is a MemNN with universal schema as its memory. <Ref> shows the model architecture.Memory: Our memory ℳ comprise of both KB and textual triples from universal schema. Each memory cell is in the form of key-value pair. Let (s,r,o) ∈𝒦 represent a KB triple. We represent this fact with distributed key 𝐤∈ℝ^2dformed by concatenating the embeddings 𝐬∈ℝ^d and 𝐫∈ℝ^d of subject entity s and relation r respectively. The embedding 𝐨∈ℝ^d of object entity o is treated as its value 𝐯.Let (s, [w_1,…,arg_1,…, arg_2,w_n], o) ∈𝒯 represent a textual fact, where arg_1 and arg_2 correspond to the positions of the entities `s' and `o'. We represent the key as the sequence formed by replacing arg_1 with `s' and arg_2 with a special `_blank_' token, i.e., k = [w_1, …,s, …, _blank_, w_n] and value as just the entity `o'. We convert k to a distributed representation using a bidirectional LSTM <cit.>, where 𝐤∈ℝ^2d is formed by concatenating the last states of forward and backward LSTM, i.e., 𝐤 = [LSTM(k);LSTM(k)]. The value 𝐯 is the embedding of the object entity o. Projecting both KB and textual facts to ℝ^2d offers a unified view of the knowledge to reason upon.In <Ref>, each cell in the matrix represents a memory containing the distributed representation of its key and value. Question Encoder: A bidirectional LSTM is also used to encode the input question q to a distributed representation𝐪∈ℝ^2d similar to the key encoding step above. Attention over cells:We compute attention weight of a memory cell by taking the dot product of its key 𝐤 with a contextual vector 𝐜 which encodes most important context in the current iteration. In the first iteration, the contextual vector is the question itself. We only consider the memory cells that contain at least one entity in the question. For example, for the input question in <Ref>, we only consider memory cells containing USA. Using the attention weights and values of memory cells, we compute the context vector 𝐜_𝐭 for the next iteration t as follows:𝐜_𝐭 = 𝐖_𝐭( 𝐜_𝐭-1 + 𝐖_𝐩∑_(k,v) ∈ℳ (𝐜_𝐭-1·𝐤) 𝐯)where 𝐜_0 is initialized with question embedding𝐪, 𝐖_𝐩 is a projection matrix, and𝐖_𝐭 represents the weight matrix which considers thecontext in previous hop and the values in the current iteration based on their importance (attention weight). This multi-iterative context selection allows multi-hop reasoning without explicitly requiring a symbolic query representation. Answer Entity Selection: The final contextual vector c_t is used to select the answer entity q_a (among all 1.8M entities in the dataset) which has the highest inner product with it.§ EXPERIMENTS§.§ Evaluation Dataset We use Freebase <cit.> as our KB, and ClueWeb <cit.> as our text source to build universal schema. For evaluation, literature offers two options: 1) datasets for text-based question answering tasks such as answer sentence selection and reading comprehension; and 2) datasets for KB question answering.Although the text-based question answering datasets are large in size, e.g., SQuAD <cit.> has over 100k questions, answers to these are often not entities but rather sentences which are not the focus of our work. Moreover these texts may not contain Freebase entities at all, making these skewed heavily towards text.Coming to the alternative option, WebQuestions <cit.> is widely used for QA on Freebase. This dataset is curated such that all questions can be answered on Freebase alone.But since our goal is to explore the impact of universal schema, testing on a dataset completely answerable on a KB is not ideal. WikiMovies dataset <cit.> also has similar properties.gardner_openvocabulary_2017 created a dataset with motivations similar to ours, however this is not publicly released during the submission time.Instead, we use Spades <cit.> as our evaluation data which contains fill-in-the-blank cloze-styled questions created from ClueWeb. This dataset is ideal to test our hypothesis for following reasons: 1) it is large with 93K sentences and 1.8M entities; and 2) since these are collected from Web, most sentences are natural. A limitation of this dataset is that it contains only the sentences that have entities connected by at least one relation in Freebase, making it skewed towards Freebase as we will see ( <ref>). We use the standard train, dev and test splits for our experiments. For text part of universal schema, we use the sentences present in the training set. §.§ ModelsWe evaluate the following models to measure the impact of different knowledge sources for QA.OnlyKB: In this model, MemNN memory contains only the facts from KB. For each KB triple (e_1,r,e_2), we have two memory slots, one for (e_1,r,e_2) and the other for its inverse (e_2,r^i,e_1). OnlyTEXT:contains sentences with blanks.We replace the blank tokens with the answer entities to create textual facts from the training set. Using every pair of entities, we create a memory cell similar to as in universal schema. Ensemble This is an ensemble of the above two models. We use a linear model that combines the scores from, and use an ensemble to combine the evidences from individual models. UniSchema This is our main model with universal schema as its memory, i.e., it contains memory slots corresponding to both KB and textual facts. §.§ Implementation DetailsThe dimensions of word, entity and relationembeddings, and LSTM states were set to d=50.The word and entity embeddings were initialized with word2vec <cit.> trained on 7.5 million ClueWeb sentences containing entities in Freebase subset of . The network weights were initialized using Xavier initialization <cit.>. We considered up to a maximum of 5k KB facts and 2.5k textual facts for a question.We used Adam <cit.> with the default hyperparameters (learning rate=1e-3, β_1=0.9, β_2=0.999, ϵ=1e-8) for optimization. To overcome exploding gradients, we restricted the magnitude of the ℓ_2 norm of the gradient to 5. The batch size during training was set to 32.To train the UniSchema model, we initialized the parameters from a trained OnlyKB model. We found that this is crucial in making the UniSchema to work.Another caveat is the need to employ a trick similar to batch normalization <cit.>.For each minibatch, we normalize the mean and variance of the textual facts and then scale and shift to match the mean and variance of the KB memory facts. Empirically, this stabilized the training and gave a boost in the final performance. §.§ Results and Discussions <Ref> shows the main results on . UniSchema outperforms all our models validating our hypothesis that exploiting universal schema for QA is better than using either KB or text alone. Despite creation process being friendly to Freebase, exploiting text still provides a significant improvement. <Ref> shows some of the questions which UniSchema answered but OnlyKB failed. These can be broadly classified into (a) relations that are not expressed in Freebase (e.g., african-american presidents in sentence 1); (b) intentional facts since curated databases only represent concrete facts rather than intentions (e.g., threating to leave in sentence 2); (c) comparative predicates like first, second, largest, smallest (e.g., sentences 3 and 4); and (d) providing additional type constraints (e.g., in sentence 5, Freebase does not have a special relation for father. It can be expressed using the relation parent along with the type constraint that the answer is of gender male).We have also anlalyzed the nature of UniSchema attention. In 58.7% of the cases the attention tends to prefer KB facts over text. This is as expected since KBs facts are concrete and accurate than text.In 34.8% of cases, the memory prefers to attend text even if the fact is already present in the KB. For the rest (6.5%), the memory distributes attention weight evenly, indicating for some questions, part of the evidence comes from text and part of it from KB. <Ref> gives a more detailed quantitative analysis of the three models in comparison with each other.To see how reliable is UniSchema, we gradually increased the coverage of KB by allowing only a fixed number of randomly chosen KB facts for each entity. As Figure <ref> shows, when the KB coverage is less than 16 facts per entity, UniSchema outperforms OnlyKB by a wide-margin indicating UniSchema is robust even in resource-scarce scenario, whereas OnlyKB is very sensitive to the coverage. UniSchema also outperforms Ensemble showing joint modeling is superior to ensemble on the individual models. We also achieve the state-of-the-art with 8.5 F_1 points difference.use graph matching techniques to convert natural language to Freebase queries whereas even without an explicit query representation, we outperform them. § RELATED WORK A majority of the QA literature that focused on exploiting KB and text either improves the inference on the KB using text based features <cit.> or improves the inference on text using KB <cit.>.Limited work exists on exploiting text and KB jointly for question answering. gardner_openvocabulary_2017 is the closest to ours who generate a open-vocabulary logical form and rank candidate answers by how likely they occur with this logical form both in Freebase and text. Our models are trained on a weaker supervision signal without requiring the annotation of the logical forms.A few QA methods infer on curated databases combined with OpenIE triples <cit.>. Our work differs from them in two ways: 1) we do not need an explicit database query to retrieve the answers <cit.>; and 2) our text-based facts retain complete sentential context unlike the OpenIE triples <cit.>.§ CONCLUSIONSIn this work, we showed universal schema is a promising knowledge source for QA than using KB or text alone. Our results conclude though KB is preferred over text when the KB contains the fact of interest, a large portion of queries still attend to text indicating the amalgam of both text and KB is superior than KB alone.§ ACKNOWLEDGMENTSWe sincerely thank Luke Vilnis for helpful insights. This work was supported in part by the Center for Intelligent Information Retrieval and in part by DARPA under agreement number FA8750-13-2-0020. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the sponsor.acl_natbib | http://arxiv.org/abs/1704.08384v1 | {
"authors": [
"Rajarshi Das",
"Manzil Zaheer",
"Siva Reddy",
"Andrew McCallum"
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"published": "20170427000302",
"title": "Question Answering on Knowledge Bases and Text using Universal Schema and Memory Networks"
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1]M. M. Romanova 1]A. A. Blinova 2]G. V. Ustyugova 3]A. V. Koldoba 1,4]R. V. E. Lovelace[1]Department of Astronomy, Cornell University, Ithaca, NY 14853, email:[email protected], tel: +1(607)255-6915, fax: +1(607)255-3433 [2]Keldysh Institute of Applied Mathematics RAS, Miusskaya sq., 4, Moscow, 125047, Russia, email: [email protected] [3]Moscow Institute of Physics and Technology, Dolgoprudnyy, Moscow region, 141700, Russia, email: [email protected] [4]Also Department of Applied and Eng. Physics We investigate the properties of magnetized stars in the propeller regime using axisymmetric numerical simulations. We were able to model the propeller regime for stars with realistically large magnetospheres (5-7 stellar radii) and relatively thin accretion disks, H/r≈ 0.15, so that our results could be applied to different types of magnetized stars, including Classical T Tauri stars (CTTSs), cataclysmic variables (CVs), and accreting millisecond pulsars (MSPs).A widerange of propeller strengths has been studied, from very strong propellers (where the magnetosphere rotates much more rapidly than the inner disk and most of the inner disc matter is redirected into the wind) to very weak propellers (where the magnetosphere rotates only slightly faster than the inner disc, and only a small part of the inner disc matter is redirected into the wind). In both the strong and weak propellers, matter is accumulated at the inner disc for the majority of the time, while episodes of accretion onto the star and ejection into the wind are relatively brief. The efficiency of the propeller, which characterizes the part of inner disk matter flowing into the wind, strongly depends on the fastness parameter ω_s, which is the ratio of the stellar angular velocity to the inner disc Keplerian velocity: propeller efficiency increases with ω_s. The properties of the winds are different in strong and weak propellers. In the strong propellers, matter is accelerated rapidly above the escape velocity and flows at a relatively small opening angle of 40-45 degrees. This matter leaves the system, forming the large-scale outflows. In the weak propellers (during episodes of ejection into the wind), matter may flow faster or slower than the escape velocity and at a large opening angle of 60-70 degrees. Most of this matter is expected to either fall back to the disk or form a magnetic turbulent corona above the disk. A star-disk system loses energy and angular momentum.A part of the rotational energy of the star is ejected to the magnetically-dominated (Poynting flux) jet, which is only present in the strong propellers. The other part of the energy flows from the inner disk into a propeller-driven wind. A star spins down partly due to the flow of angular momentum from the star to the corona (in weak propellers) or to the Poynting flux jet (in strong propellers) along the open field lines, and partly due to the flow of angular momentum to the inner disk along the closed field lines. accretion; accretion disks; MHD; stars: neutron; stars: magnetic; magnetohydrodynamics § INTRODUCTIONDifferent magnetized stars are expected to be in the propeller regime if the magnetosphere rotates more rapidly than the inner disk (e.g., ). This regimeis expected, e.g., when the accretion rate decreases and the magnetosphere expands. If the inner disk matter penetrates through the magnetosphere, then itacquires angular momentum and can be ejected from the disk-magnetosphere boundary in the form of a wind. Signs of the propeller regime have been observed in Classical T Tauri stars (CTTSs) (e.g., ), in cataclysmic variable AE Aqr (e.g., ), and in a few accreting millisecond pulsars (MSPs) at the ends of their outbursts, when the accretion rate decreases and the disk moves away from the star (e.g., ). Recently, transitional millisecond pulsars were discovered, where a millisecond pulsar transits between the state of an accreting MP, where the accretion disk moves close to the star, andthat of a radiopulsar, where the accretion disk moves to larger distances from the star (e.g., ) [Transitional MSPs were predicted long ago (e.g., ), but were not discovered until recently.].In these types of stars, the propeller regime is inevitable. In fact, different observational properties of transitional MSPs may possibly be connected with the propeller state, such as the highly variable X-ray radiation (e.g., ), γ-ray flares(e.g., ) and radiation in the radio band with a flat spectrum, which indicates the presence ofoutflows or jets (e.g., ). Many observational properties of propeller candidate stars were not well-understood, such as the accretion-induced pulsations observed at very low accretion rates in some transitional MSPs and the CV AE Aqr. According to theoretical estimates, at low accretion rates the inner disk should be far away from the star and accretion should be blocked by the centrifugal barrier of the propelling star (e.g., ). However, observations show that a small amount of matter accretes in spite of the centrifugal barrier. This and other issues require further understanding, so the propeller regime should be studied in greater detail.The propeller regime has been studied in a number of theoretical works and numerical simulations. <cit.> and <cit.> investigated the strong propeller regime analytically. They suggested that the propelling star ejects all of the accreting matter into the wind, and no matter accretes onto the star.In otheranalytical works and 1D numerical simulations it was suggested that the magnetosphere rotates only slightly faster than the inner disk, that is, the propeller is relatively weak, and there are no outflows (e.g., ). In their models, the excess angular momentumis transferred back to the disk, forming a dead disk, and matter of the inner disk accretes onto the star quasi-periodically due to the cyclic process of matter accumulation and accretion.The propeller regime has been studied in a number of axisymmetric (2.5D) simulations, where a magnetized, rapidly rotating star interacts with an accretion disk (e.g., ). Simulations have shown that * In the propeller regime, both accretion and outflows are present. Matter accretes onto the star in cycles.For the major part of the cycle, matter accumulates in the inner disk and slowly moves inward. Then, it partially accretes onto the star and is partially ejected into the wind. Subsequently, the magnetosphere expands. Therefore, both accretion and outflows occur in brief episodes (spikes); * Accretion onto the star is typically accompanied byoutflow of matter from the inner disk. However, the outflowsmay also be present at other times, such as when accretion is blocked by the centrifugal barrier;* In strong propellers, a two-component outflow has been observed: a relatively slow and dense, conically shaped inner disk wind, which carries away most of the inner disk matter, and a low-density, high-velocity collimated jet, which carries away significant energy and angular momentum; * A starspins down due to the outward flow of angular momentum along the open and closed field lines. In earlier models, accretion from laminar α-disks <cit.> had been considered, where the accretion rate in the disk was regulatedby the α-parameter of viscosity, α_v, while the rate of the field line diffusion through the disk was regulatedby a similar parameter, α_ diff<cit.>.More recently, simulations were performedfor turbulent disks <cit.>, where the turbulence is driven by the magneto-rotational instability (MRI, e.g., ). Also, in contrast with the earlier works, simulations were performed in both the top and bottom hemispheres (no equatorial symmetry). These simulations show similarresults to those obtained with the α-disks. However, the accretion funnels are not symmetric about the equatorial plane, and the outflows aretypically one-sided. In these earlier studies, only stars with relatively small magnetospheres were modeled (r_m≲ 3 R_⋆, where R_⋆ is the stellar radius)[Note that modeling the propeller regime is numerically challenging and time-consuming, because the magnetic and velocity gradients can be large compared with the cases of slowly-rotating stars. It is somewhat easier to model stars with smaller magnetospheres.]. However,most of the propeller candidate stars have larger magnetospheres, so the earlier models could only be applied to a limited range of stars. This is why we adjusted the model in such ways as to allow us to model the stars with larger magnetospheres, r_m≈ (5-7)R_⋆.In addition, the earlier numerical simulations were mainly focused on very strong propellers, where the magnetosphere rotates much more rapidly than the inner disk,(e.g., ).However, propellers of lower strengths have not been systematically studied. Some of the major questions are: (1) What are the properties of outflows in propellers of different strengths? In particular, (2) Which parts of the inner disk matter flow into the winds? (3) What is the velocity of matter in the wind? (4) What is the opening angle of the wind? (5) How much energy flows into the inner disk wind and the Poynting flux jet? (6) What is the rate of stellar spin-down? (7) How do these properties depend on the strength of the propeller ?To answer these questions, we performed a number of axisymmetric simulations of propellers of different strengths, ranging from very weak propellers to very strong propellers, and studied the properties of matter, energy and angular momentum flow. As a base case, we used a model with a turbulent disk similar to that used by <cit.>. However, compared with <cit.>, we (1) took the disk to be a few times thinner, with an aspect ratio of H/r≈ 0.15, which is closer to realistic (thin) disks; (2) considered magnetospheres of larger sizes, r_m≈ (5-7) R_⋆ (compared with r_m≈ 3 R_⋆ in <cit.>), so that the model could be applied to propelling stars with larger magnetospheres; (3) suggested that the 3D instabilities are efficient at the disk-magnetosphere boundary and added a diffusivity layer at r≤ 7R_⋆, where the diffusivity is high. The diffusivity is very low (numerical) in the rest of the disk. [Note that in <cit.> the diffusivity has been very low (numerical) in the entire simulation region, excluding a few test cases where the diffusive layer was added, as in our current simulations (see Appendix in ).] (4) investigated the properties of propellers of different strengths. Our simulations show that the properties of strong versus weak propellers are qualitatively different, and are expected to provide different observational properties.The main goal of this new research was to develop a series of models with parameter values similar to those expected in propeller candidate stars, such as transitional millisecond pulsars, intermediate polars, and Classical T Tauri stars. The results of the simulations are presented in dimensionless form and can be applied to all types of stars. We also provide convenient formulae for the conversion of dimensionless values to dimensional values in application to thesestars. We plan to apply the results of our models to particular propeller candidate stars in future papers. The plan of the paper is the following. In Sec. <ref> we discuss the theoretical background of the problem. We describe our numerical model in Sec. <ref> and show the main results of simulations and analysis in Sec. <ref>-<ref>. In Sec. <ref> we provide examples of applications and convenient formulae for different types of stars. We conclude in Sec. <ref>. <ref> and <ref> provide the details of the numerical model and the variation of different variables with timefor a number of representative models.§ THEORETICAL BACKGROUNDFor investigation of propellers of different strengths, it is important to find the main parameters which determine the strengths of propellers and which determine the main properties of propellers. In case of slowly-rotating (non-propelling) magnetized stars, such a parameter is the fastness parameter ω_s (e.g., ). In the study of the propeller regime, we also use the fastness parameter as the main parameter of the problem. §.§ Fastness parameter ω_sThe fastness parameter is determined as the ratio between the angular velocity of the star Ω_⋆ and the angular velocity of the inner disk at the disk-magnetosphere boundary r=r_m (e.g., ):ω_s= Ω_⋆/Ω_K(r_m) ,where,Ω_K(r_m) is the Keplerian angular velocity of the inner disk at r=r_m. An importance of the fastness parameter can also be shown through the simplified analysis of forces. In case of a thin (cold) accretion disk, the matter pressure force is small, and the main forces acting on the matter of the inner disk are the gravitational, centrifugal and magnetic forces. In case of strong propellers (ω_s>>1) the centrifugal force is often the main force driving matter to the outflows (e.g., ), so that the total force acting to the unit mass of the disk is dominated by the effective gravity:g_ eff=g+g_c , where g=-GM_⋆/r^2 and g_c=Ω_⋆^2 r are the gravitational and centrifugal acceleration, respectively. Taking into account that at the inner disk, r=r_m,g(r_m)=-GM_⋆/r_m^2=-Ω_K(r_m)^2 r_m, we obtaing_ eff=-Ω_K^2(r_m) r_m + Ω_⋆^2 r_m = Ω_K^2(r_m) r_m (ω_s^2-1) . One can see that in stars of the same mass M_⋆ and the same magnetospheric radius r_m, the main force acting onto the inner disk matter depends only on the fastness parameter: g_ eff∼ (ω_s-1). In cases of relatively strong propellers, ω_s>>1, the power-law dependence g_ eff∼ω_s^2 is expected. We should note that the magnetic force also contributes to driving and acceleration/collimation of matter in the wind, so that the above power-law dependence on ω_s can be different in real situation. Above analysis shows that the strength of propeller and processes at the disk-magnetosphere boundary should depend on the fastness parameter ω_s. That is why we chose this parameter as the main parameter of the model and investigate different properties of propellers as a function of ω_s. §.§ Convenient form for fastness parameter ω_sIn the case of a Keplerian disk, Ω_K(r_m)=(GM_⋆/r_m^3)^1/2, and the fastness parameter can be presented in the form:ω_s = (r_m/r_ cor)^3/2 ,where r_ cor is the corotation radius at which the angular velocity of the star matches the Keplerian angular velocity in the inner disk, Ω_⋆=Ω_K:r_ cor=(GM_⋆/Ω_⋆^2)^1/3 .The magnetospheric radius r_m is the radius at which the magnetic stress in the magnetosphere is equal to the matter stress in the disk:B_p^2 + B_ϕ^2/8π=ρ (v_p^2 + v_ϕ^2) + p .Here, ρ is density, p is thermal pressure, v_p, v_ϕ and B_p, B_ϕ are the poloidal and azimuthal components of velocity and the magnetic field, respectively.In cases of slowly-rotating stars (not propellers), the magnetospheric radius has been derived theoretically from the balance of the largest components of the stresses : B_ dip^2/8π=ρ v_ϕ^2, where B_ dip is the magnetic field of the star which is suggested to be a dipole field, and v_ϕ is the Keplerian angular velocity in the inner disk:r_m = k [μ_⋆^4/(Ṁ^2 GM_⋆)]^1/7, k∼ 1 ,where μ_⋆=B_⋆ R_⋆^3 is the magnetic moment of the star with a surface field of B_⋆, Ṁ is the accretion rate in the disk, and M_⋆and R_⋆ are the mass and radius of the star, respectively (e.g., ) [Comparisons of the magnetospheric radii obtained in the axisymmetric simulations with Eq. <ref> provided the values k≈0.5 <cit.> and k≈0.6 <cit.>. 3D simulations of multiple cases have shown a slightly different power (1/10 instead of 1/7) in Eq. <ref> due to the compression of the magnetosphere <cit.>.]. However, in the propeller regime, the magnetosphere departs from the dipole shape and the poloidal velocity v_p may become comparable to or larger than the azimuthal velocity v_ϕ. In addition, the process of disk-magnetosphere interaction is non-stationary, so all variables vary in time. This is why we calculate the magnetospheric radius r_m using the general equationfor balance of stresses (Eq. <ref>), where both poloidal and azimuthal components of velocity and magnetic field are taken into account. § THE NUMERICAL MODELWe performed axisymmetric simulations of disk accretion onto a rotating magnetized star in the propeller regime. The model is similar to that used in the simulations of <cit.>, but with a few differences. Below, we briefly discuss the main features of the numerical model and also the differences between our model and that of <cit.>. More technical details of the model are described in <ref>.We consider accretion onto a magnetized star from a turbulent accretion disk, where the turbulence is driven by the magneto-rotational instability (MRI, e.g., ), which is initiated by a weak poloidal magnetic field placed inside the disk (see Fig. <ref>). The accretion disk is cold and dense, while the corona is hot and rarefied. The disk is 3,000 times cooler and denserthan the corona. The disk is geometrically thin, with an aspect ratio of h/r≈ 0.15, where h is the semi-thickness of the disk. This disk is about 2.7 times thinner than that used in <cit.>. To achieve an accretion rate in the new thin disk comparable with that in the <cit.> thicker disk, we increased the reference density in the disk by a factor of three. [Note, that the first set of simulations was performed for the thicker disk and only later was recalculated for the thinner disk. Comparisons did not show any significant differences between the results. However, the current paper is based on the simulations of accretion from the thinner disk, because a thinner disk is closer to realistic disks expected in different accreting magnetized stars.] A star with an aligned dipole magnetic field is placed at the center of the coordinate system. The disk is placed at a distance of 10 stellar radii from the center of the star, whichrotates slowly, with the Keplerian angular velocity corresponding to corotation radius r_ cor=10 R_⋆. Then, we gradually spin up the star (over the period of 100 rotations of the inner disk at r=R_⋆) to a higher angular velocity, Ω_⋆, corresponding to the propeller regime. The corresponding corotation radius, r_ cor, is a parameter of the model. See <ref> for details of initial and boundary conditions.The disk-magnetosphere interaction in the propeller regime requires some kind of diffusivity, so that matter of the inner disk can penetrate through the rapidly-rotating layers of the stellar magnetosphere.This type of penetration may be connected with the magnetic Rayleigh-Taylor (magnetic interchange) instability (e.g., ).The magnetic interchange instability has been observed in 2D simulations of propellers, performed in polar coordinates <cit.>, as well as in global three-dimensional (3D) simulations of accretion onto slowly-rotating stars (e.g., ) and in the local 3D simulations (e.g., ). However, in our current 2.5D axisymmetric simulations, this instability is suppressed by the axisymmetry of the problem. This is why we added a diffusivity term into the code and suggested the presence of large diffusivity at the disk-magnetosphere boundary. We determined the coefficient of diffusivity in analogy with the coefficient of viscosity in the α-disk model: η_m=α_ diff c_s^2/v_K, where c_s is the local sound speed, and α_ diff is the α-coefficient of diffusivity. We chose the largest value, permitted by the α-disk theory, α_ diff=1 <cit.> (see details in <ref>). This is different from the <cit.> simulations, where no diffusivity term was added in most of the simulation runs, and where a small numerical diffusivity was responsible for the disk-magnetosphere interaction. Test simulation runs have shown that, when such a diffusive layer is added, more matter is ejected into the outflows(see Appendix in ). In the current paper, we suggest high diffusivity in all simulation runs. The equations were solved in dimensionless units, so that the model could be applied to different types of stars, from young stars to neutron stars (see <ref> and Tab. <ref> for details of the dimensionalization procedure). The results of simulations are shown in dimensionless units, excluding those in Sec. <ref>.One of the important dimensionless parameters is μ, which determines the typical size of the dimensionless magnetosphere, r_m/R_⋆. Simulations of <cit.> were performed at a relatively small value of μ (μ=10), which provided typical values of the magnetospheric radius, r_m≈ 3 R_⋆.[Note that in application of the model to realistic stars, <cit.> suggested that the inner boundary R_0=2R_⋆, that is a star is located inside the inner boundary. These provided the twice as larger efficient magnetosphere of the star, r_m/R_⋆. In current paper, stars with larger magnetospheres are calculated, and we take R_0=R_⋆ during dimensionaliztion procedure.]. In the current paper, we consider larger values of μ: μ=30, 60 and 100, which provide larger sizes of the magnetosphere, r_m≈ (4-7) R_⋆. Larger magnetospheric sizes are needed to model the propeller regime in different types of stars, some of which may have relatively large magnetospheres. A fine grid resolution is used, with grid compression in the regions of the disk and the magnetosphere. A Godunov-type numerical code in cylindrical coordinates has been developed by <cit.>, which incorporates an HLLD numerical solver of <cit.>.See <ref> for details of our numerical model.§ VARIABILITY AND TIME-AVERAGED VALUESWe investigated the properties of propellers of different strengths,from very weak propellers (in which the magnetosphere rotates only slightly faster than the inner disk) to very strong propellers (where the magnetosphere rotates much more rapidly than the inner disk). To achieve different strengths of propeller, we varied the corotation radius in the range of r_ cor=1.3-6. We also varied themagnetospheric parameter μ (which determines the dimensionless size of the magnetosphere, r_m/R_⋆), and performed calculations for three values, μ=30, 60, and 100. Table <ref> shows parameters for a number of representative models and also results of simulations.The disk-magnetosphere interaction in the propeller regime is a non-stationary process, where the inner disk radius r_m oscillates, and the matter fluxes to the star, Ṁ_s, and to the wind, Ṁ_w, are also strongly variable.The variability is connected with the cycle of matter accumulation, accretion/ejection, and magnetosphere expansion.§.§ A cycle of accumulation-accretion/ejection-expansion To demonstrate the non-stationary nature of the propeller regime and our procedure for time-averaging, we show the results obtained for one of our models of a strong propeller, μ60c1.5. The top panels of Fig. <ref> show a close-in view of the inner part of the simulation region during one episode of the accumulation-accretion/ejection-expansion cycle. (1) At t=723, matter is accumulated at the inner disk, which gradually moves inward (towards the star). The stellar field lines threading the disk inflate and expand. Inflation is stronger in the part of the magnetosphere that is below the equator. (2) At t=745, part of the matter starts accreting onto the star above the magnetosphere, while another part starts flowing away from the star below the magnetosphere.It is very typical for matter to accrete on one side of the magnetosphere while forming outflows on the other side of the magnetosphere. The magnetosphere is slightly compressed on the accreting side, while the field lines strongly inflate on the outflowing side. One can see that a significant part of the magnetic flux inflates in the direction away from the star below the equatorial plane.(3) After an accretion/ejection event, the magnetosphere expands (see 3rd panel at t=763). (4) Subsequently, the inner disk gradually moves inward (see 4th panel at t=780) and the process repeats. At stage (2), accretion is possible because only a part of the magnetosphere (where the field lines are closed) rotates more rapidly than the inner disk and represents a centrifugal barrier [ We should note that some matter is also ejected into the outflows without significant accretion onto the star (see, e.g., a burst to an outflow at t≈ 830 in Fig. <ref>). In this type of outflow, matter of the inner disk penetrates through the magnetosphere, acquires sufficient angular momentum and flows away from the star.]§.§ Magnetospheric radius r_mWe calculated the position of the inner disk (magnetospheric) radius r_m using Eq. <ref>. To calculate the magnetospheric radius r_m(t) at some moment t in time, we take the values of density, pressure andcomponents of velocity and magnetic field in the equatorial plane from the simulations, and find the radius r_m(t), at which the balance of stresses (Eq. <ref>) is satisfied. The magnetosphere is often asymmetric about the equatorial plane. To take this issue into account, we initially calculated the magnetospheric radius at the surfaces z=± R_⋆, which are above and below the equatorial plane, and then took the half-averaged value r_m=[r_m(z=R_⋆) + r_m(z=-R_⋆)]/2 as the main magnetospheric radius in the model. We observed from the simulations that at this radius, r_m, the density in the disk drops from the high values in the disk down to very low values in the magnetosphere, while the angular velocity changes from the Keplerian angular velocity in the disk to the angular velocity of the magnetosphere. Simulations show that the magnetospheric radius varies in time.The middle panel of Fig. <ref> shows that it varies between r_m≈ 4.9 and r_m≈ 9. To characterize the magnetospheric radius in each model, we introduce the time-averaged magnetospheric radius:⟨ r_m(t)⟩ = ∫_t_i^t dt' r_m(t')/∫_t_i^t dt' .We show this time-averaged radius as a dashed line in Fig. <ref>. This radius slightly varies in time. For consistency, we start averaging at moment t=200 in each simulation run (so as to exclude the effects of the initial conditions) and take this radius at moment t=1,000 for each model. In the model shown in Fig. <ref>, we obtain ⟨ r_m⟩≈ 5.8. The right panels of Figs. <ref>, <ref> and <ref> show variation of the inner disk radius in our representative models. The figures show that the amplitudes of disk oscillations are larger in models with larger ω_s values (strongest propellers) and also increase with the magnetospheric parameter μ. In the weaker propellers, the amplitude of the oscillations is much smaller, and the radius varies only slightly. Tab. <ref> shows the time-averaged magnetospheric radii ⟨ r_m⟩for our representative models. Fig. <ref> shows the density distribution in three models with the same corotation radius, r_ cor=1.5, but different values of the magnetospheric parameter μ: μ=30, 60, 100, at the times when the magnetospheric radius is approximately equal to the time-averaged radius. These radii are approximately twice as large as the radii in the models of <cit.>,§.§ Time-averaged fastness parameterWe use the time-averaged radius ⟨ r_m⟩ to calculate the time-averaged fastness parameter:⟨ω_s⟩ = (⟨ r_m ⟩/r_ cor)^3/2 .Subsequently, in this paper, we use this parameter, as the main parameter of the model. For convenience, we remove the brackets ⟨ ⟩ and simply use the variable ω_s. Tab. <ref> shows the values of the time-averaged fastness parameter for our representative models. One can see that the fastness parameter ranges from very low values, ω_s=1.2, to very high values, ω_s=10.2.Fig. <ref> shows the dependence of the time-averaged magnetospheric radius, ⟨ r_m⟩, calculated for all models, on the time-averaged fastness parameter ω_s. One can see that in a set of models with the same parameter μ, the magnetospheric radius ⟨ r_m⟩ slightly increases with ω_s. The dependencies are the following: (1) at μ=30, ⟨ r_m⟩≈ 4.5 ω_s^0.076; (2) at μ=60, ⟨ r_m⟩≈ 5.7ω_s^0.04; (3) at μ=100, ⟨ r_m⟩≈ 5.85ω_s^0.073. In each set, the radii are larger at larger values of μ. We took the dependence on ω_s corresponding to μ=30 and μ=100, and found an approximate general relationship for all models: ⟨ r_m⟩≈ 5.7 μ_60^0.21ω_s^0.07 ,where μ_60=μ/60. A comparison of the radii obtained with this formula with the values of ⟨ r_m⟩ observed in the simulations shows that the typical deviation of the observed radiifrom those obtained with the formula is ∼ 5-10%.§.§ Why does matter accrete in the propeller regime? In the sample model shown in Fig. <ref>, the magnetospheric radius is always larger than the corotation radius, r_m > r_ cor, and the time-averaged magnetospheric radius, ⟨ r_m⟩≈ 5.8, is also larger than the corotation radius, r_ cor=1.5. In spite of this, matter accretes onto the star. In all other models, matter also accretes onto the star (see Tab. <ref> for ⟨Ṁ_s⟩). This is different from the generally-accepted definition that, in the propeller regime, accretion is possible if r_m<r_ cor and is completely forbidden otherwise.Below, we describe the main reasons why accretion becomes possible even in the cases of very strong propellers: * In theoretical studies and one-dimensional models, it is suggested that the centrifugal barrier is an infinite vertical wall (e.g., ). However, two-dimensional simulations show that only the closed part of the magnetosphere rotates more rapidly than the inner disk and represents the centrifugal barrier. That is why, at favorable conditions, matter can flow above or below the magnetosphere and accrete onto the star at condition r_m>r_ cor. * The process is non-stationary. Most of the time, accretion is blocked by the centrifugal barrier and matter does not accrete onto the star. However, when the disk comes closer to the star, conditions become favorable for accretion and matter accretes onto the star in a brief episode. This explains why the time-averaged matter flux to the star can be so low (much lower than the matter flux to the wind).These two-dimensional, non-stationary propeller models help explain whypropellers can accrete apart of the disk matter, in spite of the fact that their magnetospheres rotatemore rapidly than their inner disks. §.§ Matter fluxes We also calculated the matter fluxes onto the star and to the wind:Ṁ_s(t)=∫_S_ starρ v_p dS , Ṁ_w(t)=∫_S_ windρ v_p dS .The matter flux to the star has been calculated through the stellar surface, S_ star=S(r=R_⋆,z=± R_⋆) which is a cylinder with radius r=R_⋆ and height z=± R_⋆ centered on the star, while the matter flux to the wind has been calculated throughcylindrical surface S_ wind=S(r=10,z=±10)with radius r=10 and height z=± 10 [The surface S(r=10,z=±10) is located relatively close to the star, at a distancethat is only slightly larger than the time-averaged values of the magnetospheric radii in our models, ⟨ r_m ⟩≈ 4.6-7.1 (see Tab. <ref>). It helps us select the propeller-driven wind from the disk-magnetosphere boundary and deselect the slow winds from the other parts of the disk. ]. To exclude the slow motions in the turbulent disk, we placed the condition that the poloidal velocity in the wind should be larger than some minimum value v_ min=k v_ esc, where v_ esc=(2GM_⋆/r)^1/2 is the local escape velocity, and k≤ 1. Our simulations show that only instrong propellers is matter ejected from the disk-magnetosphere boundary with a velocity comparable to the local escape velocity. In most cases, the initial outflow velocityis low. It can be as low as 0.1 v_ esc. In spite of that, matter flows away from the simulation region, driven mainly by the magnetic force of the inflating field lines. That is why we chose the condition v_ min = 0.1 v_ esc for the calculation of the outflows. Using this condition, we take into account both the fast and slow winds from the disk-magnetosphere boundary.The bottom panel of Fig. <ref> shows the flux ontothe star (in red) and the flux to the wind (in blue). One can see thatthe fluxesare “spiky", because most of the time accretion onto the star is stopped by the centrifugal barrier of the propelling star.The left panels of Figs. <ref>, <ref> and <ref> show variation of the matter fluxes in our representative models. To characterize fluxes in each model, we introduced the time-averaged matter fluxes:⟨_s(t)⟩ = ∫_t_i^t dt' _s(t')/∫_t_i^t dt' , ⟨_w(t)⟩ = ∫_t_i^t dt' _w(t')/∫_t_i^t dt' .The dashed lines in the bottom left panel of Fig. <ref> show the time-averaged matter fluxes onto the star ⟨Ṁ_s(t)⟩ and to the wind ⟨Ṁ_w(t)⟩.We observed that the matter fluxes areaffected by the initial conditions during the first∼200 rotations, which is why we calculated the time-averagedvalues of the fluxesstarting at t=200. We observed that the time-averaged fluxes vary only slightly in time.We chose a late moment in time, t=1,000, which is near the end of most simulation runs [Much longer simulation runs were performed in a few test cases. However, they did not show new information compared with shorter runs. That is why most of simulations were stopped shortly after time t=1,000 (to save computing time). ], and took the flux values at this moment to be the typical averaged fluxes for any given model. The values of these averaged fluxes are ⟨_s⟩ = 0.31 and ⟨_w⟩ = 0.82 in the sample model shown in Fig. <ref>.Left panels of Figures <ref>, <ref> and <ref> of <ref>) show examples of the fluxes in our representative models. The dashed lines show their time-averaged values.Tab. <ref> shows the time-averaged matter fluxes for a number of calculated models.§.§ Propeller EfficiencyPropellers of different strengthseject different amounts of matter into the wind. To characterize the relative matter flux ejected into the wind, we introduce propeller efficiency:f_ eff =⟨_w⟩/⟨_s⟩ + ⟨_w⟩,where ⟨_w⟩ and ⟨_s⟩ are the time-averaged matter fluxes to the wind and to the star, respectively.For each model, we calculated the time-averaged matter fluxes and the value of propeller efficiency f_ eff using Eq. <ref>.We also calculated the time-averaged value of the fastness parameter for each model using eq. <ref>. Fig. <ref> shows the plot of efficiency f_ eff versus the averaged fastness, ω_s, for all models, where each point corresponds to a single model. The set of models includes a wide variety of propeller strengths, from very weak propellers (bottom left corners of the plots)to very strong propellers (top right corners of the plots), and different values of magnetospheric parameter μ, which correspond to different magnetospheric sizes, r_m/R_⋆, from relatively small magnetospheres (μ=30, marked as squares) to large magnetospheres (μ=100, marked as circles). The triangles show models with intermediate magnetospheric sizes (μ=60). We calculated the propeller efficiency taking into account only the faster component of the outflowing matter (to exclude the slow motions in the inner disk), with poloidal velocities v_p>v_ min, wherev_ min=0.1v_ esc, v_ min=0.3v_ esc, and v_ min=1.0v_ esc (see top, middle and bottom panels of Fig. <ref>). The top panel of Fig. <ref> shows that, at condition v_p>v_ min=0.1 v_ esc, efficiency is high in both the strong propellers, f_ eff≈ 0.85 (see top right corner of the plot), and the weak propellers, f_ eff≈ 0.6 (bottom left corner of the plot). This means that, in propellers of different strengths, most of the inner disk matter flows into the wind. This wind can beslow in the cases of weak propellers and much faster in the strong propellers (see Sec. <ref>). The middle panel of Fig. <ref> shows that, if we only include the relatively fast outflows, withv_p>v_ min=0.3 v_ esc, then efficiency becomes lower, f_ eff≈ 0.15-0.2, in the weak propellers. The bottom panel of Fig. <ref> shows that, if we only consider the outflows with super-escape velocities (v_p>v_ min=1.0 v_ esc), then, in the weak propellers, efficiency becomes very low, f_ eff≈ 10^-3, but increases sharply with ω_s. One can see that, in all three cases, the dependency f_ eff on ω_s can be approximated as a power law:f_ eff≈ Kω_s^α. Tab. <ref> shows these dependencies for different v_ min/v_ esc values and different values of μ. One can see that, at v_ min=0.1 v_ esc and v_ min=0.3 v_ esc, the efficiency is slightly lower for larger magnetospheres (μ=100) compared with the smaller magnetospheres (μ=30 and μ=60).The above analysis shows that, in propellers of different strengths, a significant amount of the inner disk matter is lifted above the disk plane and flows into the wind. However, the fate of this wind is different in the cases of strong and weak propellers. Below, we analyze the properties of the wind.§ PROPERTIES OF PROPELLER WINDS§.§ Matter flow in strong and weak propellersTo demonstrate typical matter flow in strong and weak propellers, we took two models (μ60c1.5 and μ60c3.7) with the same magnetic moment, μ=60, but different corotation radii r_ cor=1.5 and r_ cor=3.7. Fig. <ref> shows several snapshots of matter flow in the strong propeller regime, taken during or after an episode of matter outflow. The color background shows matter flux density ρ| v_p| and the lines are sample poloidal field lines. One can see that most of the external field lines are open and matter flows into conical-shaped wind at an angle of Θ_ wind≈ 40^∘-45^∘. The dashed red line shows an approximate direction of the outflow.The bottom left panel shows the matter fluxes to the star, Ṁ_s, and to the wind, Ṁ_w, and their time-averaged values (dashed lines), which were used to calculate the efficiency of the propeller, f_ eff≈ 0.82. The bottom right panel shows variation of the inner disk radius r_m and its time-averaged value ⟨ r_m⟩≈ 5.8, which was used to calculate the fastness parameter: ω_s=(⟨ r_m⟩/r_ cor)^3/2≈ 7.6.Fig. <ref> shows matter flow to the wind in a relatively weak propeller (model μ60c3.7) during several outbursts to the wind. One can see that the magnetic field lines connecting the inner disk with the star inflate and matter is ejected into the wind at a larger opening angle, Θ_ wind≈ 60^∘, compared with the case of the stronger propeller. The bottom left panel shows that the matter fluxes to the star and to the wind look somewhat similar to those of the strong propeller shown in Fig. <ref>. The efficiency of the weaker propeller, f_ eff≈ 0.66, is only slightly lower than that of the stronger propeller. This similarity is due to the fact that, in both models, the outflows include any matter that flows through surface S(r=10,z=±10) with velocity v>v_ min=0.1 v_ esc. However, in the weak propeller, the velocity of matter flow into the wind is much lowerthan in the strong propeller (see Sec. <ref>). The bottom right panel of Fig. <ref> shows that the magnetospheric radius r_m oscillates and the time-averaged value ⟨ r_m ⟩≈ 5.4, which is only slightly smaller than that in the above exampleof the strong propeller [This approximate equality of the radii ⟨ r_m ⟩ is due to the fact that the magnetospheric radius is determined by the balance between the magnetic and matter pressures, where ρ v_ϕ^2 term dominates over ρ v_p^2 term at the disk-magnetosphere boundary.]. Note, however, that the fastness parameter, ω_s≈ 1.8, is much smaller than that of the strong propeller.The fastness parameter is one of the main factors contributing to the differences in the properties of the windsin strong versus weak propellers. §.§ Velocities in the windWe investigated the velocities of matter in the wind component of propellers of different strengths. Most of the matter flows from the disk-magnetosphere boundary into the conical wind. To find the typical velocity of matter flow in each model, we chose a surface S(r=20,z=±20),which is a cylinder with dimensions r=20 and z=± 20 (see Fig. <ref>) [Note, that this cylinder is larger than that used for the calculation of matter fluxes, because matter in the wind often has low velocities at the disk-magnetosphere boundary, but is accelerated at larger distances from the star due to the magnetic force], and searched for the maximum poloidal velocity v_ max at this surface. The maximum velocity was calculated for the parts of the wind where the density is not very low (ρ>0.001), so as to deselect the regions of very low density and high velocity flow in the axial regions. We also deselected the matter which moves at low velocities in the disk by the conditionv > v_ min=0.1 v_ esc. We calculated the ratio v_ max/v_ esc, which shows whether the maximum poloidal velocity in the wind is larger or smaller than the local escape velocity v_ esc. Fig. <ref> (top panels) shows variation of the ratio v_ max/v_ esc with time in a strong (left panel) and weak (right panel) propeller. One can see that in the case of a strong propeller, this ratio varies in the range of v_ max/v_ esc≈ 2-5 during the bursts. In the case of a relatively weak propeller, the maximum velocity during the bursts is either slightly larger or slightly smaller than the escape velocity, so that v_ max/v_ esc≈ 1. Some of this matterescapes the star's gravity, while some of it returns back to the star or falls onto the disk at some distance from the star. In even weaker propellers, the maximum velocity in the wind is lower than the escape velocity, so that matter will fall back onto the star. Alternatively, it may contribute to the slowly-expanding turbulent magnetic corona. The left panels of Figs. <ref>, <ref>, and <ref> from <ref> show the variation of v_ max/v_ esc with time for some of the representative models.To characterize the velocity of the outflows in each model, we calculated the time-averaged maximum velocity⟨ v_ max/v_ esc⟩ = ∫_t_i^t dt' v_ max(t')/v_ esc/∫_t_i^t dt' . These averaged velocities are approximately 2-3 times lower than the maximum velocities during the bursts to the wind (compare the dashed lines in top panels of Fig. <ref> with the velocity maxima). For each model, we take the averaged velocity at time t=1,000 and use it for comparisons with other models. Fig. <ref> (left panel) showsthat the averaged velocity increases with fastness exponentially (see Tab. <ref> for dependencies). Note that the dependencies are approximately the same for magnetospheres of different sizes (different values of μ).§.§ Opening Angle of the windWe also calculated the opening angle of the wind, Θ_ wind, which we determined as the angle between the line connecting the inner disk to the point of maximum wind velocity (v=v_ max, located at the surface S(r=20,z=±20)) and the vertical line crossing the inner disk, r=r_m (see right panel of Fig. <ref>). Simulations show that, in both the strong and weak propellers, the opening angle strongly oscillates (see Fig. <ref>, bottom panels). This angle is smaller in the cases of stronger propellers. Figures <ref>, <ref>, and <ref> from <ref> show the variation of Θ_ wind with time in our representative models.We calculated the time-averaged value of the opening angle for each model:⟨Θ_ wind⟩ = ∫_t_i^t dt' Θ_ wind/∫_t_i^t dt' .The dashed horizontal lines in Fig. <ref> (bottom panels) show the time-averaged opening angles in our sample cases of strong and weak propellers. One can see that the time-averaged angle ⟨Θ_ wind⟩ is approximately 45^∘ and 60^∘ in our examples of strong and weak propellers, respectively.Fig. <ref> (right panel) shows the dependence of the time-averaged opening angles, ⟨Θ_ wind⟩, taken for all models (at t=1,000), on the fastness parameter, ω_s. One can see that ⟨Θ_ wind⟩ decreases withω_s. The dependencies can be approximated by a power law (see Tab. <ref>). One can see that the dependencies are similar for μ=30 and μ=60. However, in the models with the largest magnetospheres, μ=100, the slope is not as steep as in the other two cases.The opening angle is large, ⟨Θ_ wind⟩≈ 60^∘-65^∘, in the weak propellers. Velocities of outflows into the wind are also lower in the weak propellers, and, during the outbursts, can be comparable to or lower than the escape velocity. This wind matter may fall back to the disk at some distance from the star. In the weak propeller regime, a significant amount of matter may be recycledthrough the process of ejection from the inner disk boundary, the fall of this matter onto the disk at larger distances from the star, and subsequent inward accretion in the disk.§ ANGULAR MOMENTUM AND ENERGY In the propeller regime, a star-disk system loses angular momentum and energy. §.§ Angular momentum flow and the spin-down rateIn the propeller regime, a star loses its angular momentum and spins down (e.g., ). Angular momentum flows from the surface of the star along the field lines connecting the star with the disk and the corona. In addition, angular momentum flows from the inner parts of the accretion disk along the open field lines of the dipole, which have foot-points at the disk. The angular momentum flux is calculated by integrating the angular momentum flux densities through some surface S:= _m + _f = ∫_S d𝐒· (_ Lm+_ Lf) ,where _ Lm and _ Lf are the angular momentum flux densities carried by matter and magnetic field, respectively, and given by_ Lm = r ρ v_ϕ v_p , _ Lf= - r B_ϕ𝐁_p/4 π .Here, the normal vector to the surface d𝐒 points inward towards the star. To estimate the rate of stellar spin-down, L̇_ sd, we calculated the angular momentum flux through the surface of the star, (r=R_⋆,z=± R_⋆) . We observed that the angular momentum flux is carried from the stellar surface by the magnetic field [We forbid the outflow of matter from the stellar surface and therefore the flux carried by matter is zero.]. The red lines in Fig. <ref> show variation of this flux in the cases of strong and weak propellers. We also calculated the fluxes carried by matter, L̇_m, and by the field, L̇_f, through surface S(r=10, z=±10) [Here, we place the surface S close to the inner disk, so that to take into account angular momentum which flows back to the disk along the closed field lines.]. Fig. <ref> shows these fluxes in blue and green,respectively. All fluxes strongly vary with time. Comparisons with the matter fluxesshow that angular momentum fluxes are the largest during episodes of matter outflow.To compare the fluxes calculated for different models, we calculated the time-averaged values using a formula similar to Eq. <ref>. We calculated separately the fluxes of angular momentum carried from the surface of the star,⟨L̇_ sd⟩, and the fluxes carriedthrough surface S(r=10,z=±10) by the magnetic field, ⟨L̇_f⟩, and by matter, ⟨L̇_m⟩. Fig. <ref> shows these fluxes (taken at t=1,000) as a function of the fastness parameter, ω_s. The dependencies can be approximately described by power laws (see solid lines in Fig. <ref> and dependencies in Tab. <ref>). One can see that the angular momentum fluxes are larger at larger values of μ. The left panel of Fig. <ref> and Tab. <ref> show that, in the models with the same values of μ, the angular momentum carried from the star, ⟨L̇_ sd⟩, is approximately twice as large as the angular momentum carried by the field,⟨L̇_f⟩, through surface S(r=10,z=±10). This means that only a part of the angular momentum flows from the star to the inflated field lines. The other part (approximately half of ⟨L̇_ sd⟩) flows into the disk along the closed field lines [This result is in agreement with that obtained by <cit.>.].In the weak propellers, magnetic angular momentum flux ⟨L̇_f⟩ is associated with the inflation of the field lines and the outward propagation of magnetic flux. However, in the strong propellers, the magnetic flux acquires the form of a magnetic (Poynting flux) jet, where magnetic pressure accelerates the low-density plasma to high velocities inside the simulation region. This jet is magnetically-driven and also magnetically-collimated. The matter component of the flux through surface S(r=10,z=± 10), ⟨L̇_m⟩, is associated with the centrifugally-driven conical component of the wind coming from the inner disk.The right panel of Fig. <ref> shows that, at small values of ω_s, the angular momentum flux associated with matter,⟨L̇_m⟩, is larger than that associated with the field, ⟨L̇_f⟩. However, at large values of ω_s they become comparable. In summary, the magnetic field carries angular momentum away from the star, while both matter and magnetic field carry angular momentum away from the star-disk system. §.§ Energy fluxesPropeller-driven winds and jets also carry energy out of the system. We calculated the energy fluxes carried by matter and magnetic field through surface S(r=10,z=±10): Ė = Ė_m + Ė_f = ∫_S d𝐒· (_ Em+_ Ef) ,where _ Em and _ Ef are the energy flux densities carried by matter and magnetic field, given by_ Em=(ρ v^2/2 + γ p/γ-1 )v_p , _ Ef=1/4 π( 𝐁^2v_p - (𝐁· v) 𝐁_p ) .Here, the normal vector to the surface d𝐒 points inward towards the star. Fig. <ref> shows an example of the temporal variations of energy fluxes in strong and weak propellers. One can see that the energy fluxes strongly vary in time, that is, energy is ejected into the outflows in the form of bursts. Fig. <ref> shows the time-averaged energy fluxes ⟨Ė_m⟩ and ⟨Ė_f⟩ (taken at t=1,000) for all models. One can see that the fluxes increase with ω_s and are larger at larger values of μ. Table <ref> shows the power law dependencies for fluxesat different values of μ.In both the strong and weak propellers, some energy is carried by matter from the inner disk into a conically-shaped wind. Additionally, in both cases, inflation of the field lines leads to the flow of magnetic energy out of the star. However, inthe strong propellers, magnetic energy also flows into a non-stationary, magnetically-driven and magnetically-collimated low-density jet. Fig. <ref> shows the distribution of energy flux density in a strong propeller (model μ60c1.5). The left panel shows that the energy carried by matter flows into the conically-shaped wind. The right panel shows the energy flux density associated with the magnetic field. One can see that this energy flux is large and is more collimated than the matter energy flux. This is the magnetically-dominated (Poynting flux) jet, where matter is accelerated and collimated by the magnetic field.§ SUMMARY OF WIND PROPERTIES IN STRONG AND WEAK PROPELLERS To summarize the specifics of matter flow in propellers of different strengths, we show two sketches that demonstrate the properties of strong and weak propellers during accretion/outburst events (see left and right panels of Fig. <ref>). * In the strong propeller regime (left panel), matter flows from the inner disk into the disk wind along the open field lines connecting the disk with the corona. This wind has high super-escape velocities and relatively small opening angles, ⟨Θ_ wind⟩≈ 40^∘-45^∘. Such a windmay flow to large distances from the star, forming large-scale wind structures. Alternatively, it may be collimated by the external medium, forming a jet. There is also a low-density, high-velocity, magnetically-dominated and magnetically-driven Poynting flux jet,whose matter flows along the stellarfield lines.This jetcarries a significant amount ofenergy and angular momentum away from the star. In a typical case of non-stationary ejections, shock waves are expected to form alongthe flow, where particlesmay be accelerated to high energies. In summary, a strong, two-component outflow is expected in strong propellers. * In the weak propeller regime (right panel), matter flows from the inner disk into the low-velocity wind, where the maximum velocity is comparable with or lower than the local escape velocity. Matter flows intothe conical wind at large opening angles, ⟨Θ_ wind⟩≈ 60^∘-70^∘. This wind partly forms the large-scale outflow structures, and partly falls back onto the disk at some distance from the star. The fallen matter then accretes back towards the disk-magnetosphere boundary andis again ejected into the slow, conically-shaped wind.Such recycling of inner disk matter is expected in many weak propellers. In addition, inflation and reconnection of the field lines lead to the formation of magnetic islands, which are ejected at low, sub-escape velocities,forming the slow, magnetically-dominated wind. Some of this matter may accrete back to the star, driven by gravitational force.We should note that, in both the strong and weak propellers, matter accretes (and is ejected) during brief episodes, and the inner disk strongly oscillates. Therefore, in both cases, strong variability in the light curves is expected.§ TIME INTERVALS BETWEEN ACCRETION/EJECTION EVENTS Here, we estimate the characteristic time intervals between accretion/ejection events. Simulations show that accretion/ejection events are typically associated with a cycle in which (1) Matter accumulates at the inner disk and slowly moves inward, (2) Matter of the inner disk penetrates through the external regions of the magnetosphere and the field lines begin to inflate,(3) Matter partly accretes onto the star and partly flows into the wind, (4) The magnetosphere expands and the cycle repeats. An outflow becomes possible when the field lines inflate and open.Here, we consider two possible scenarios: (1)The diffusivity at the disk-magnetosphere boundary is relatively high, and matter penetrates through the disk-magnetosphere boundary rapidly (we observe this in most of our simulation runs); (2)The diffusivity is low, so that the inner disk matter gradually penetrates through the magnetosphere.§.§ High Diffusivity ScenarioLet's suggest that, after an event of accretion/ejection, the magnetosphere is “empty" and the inner disk is located at some radius r_m. Then, matter of the inner disk gradually penetrates through the external layers of the magnetosphere due to some 3D instabilities. The depth of the penetration is unknown. However, we can suggest that this matter penetrates into the magnetosphere up to some depth Δ r and then stops at some distance from the star due to an even stronger centrifugal barrier of the propelling star [This scenario is observed in most of our simulation runs.] In parallel, new matter comes in from the disk to its inner parts with an accretion rate Ṁ. It carries the angular momentum fluxL̇_ m=r_m^2 (Ω_⋆ - Ω_d) Ṁ .The dipole magnetic field of the star inflates due to the difference between the angular velocity of the star, Ω_*, and the disk, Ω_d. Angular momentum flows from a unit length of the disk at radius r to the inflating field lines. Its value (per unit length) is:L_f = B_m^2 r_m^2/(Ω_* - Ω_d) ,where B_m is the magnetic field at r=r_m.Inflation occurs when the angular momentum of matter in the disk is larger than the angular momentum required for inflation:r_m^2 (Ω_⋆ - Ω_d) ṀΔ t > B_m^2 r_m^2/(Ω_* - Ω_d)Δ r .Therefore, the next episode of inflation will occur after an interval of time Δ t, ifΔ t > B_m^2 Δ r/Ṁ (Ω_⋆ - Ω_d)^2 .Taking into account the fact that B_m=B_⋆ (R_⋆/r_m)^3 and definition of fastness, ω_s=Ω_s/Ω_d, we obtain:Δ t > μ_⋆^2 Δ r/Ṁ r_m^3 G M_⋆ (ω_s - 1)^2 . To find thecharacteristic time interval between inflation events in our simulations, we re-write this condition in dimensionless form using the dimensionalization procedure from<ref>: Δ r = R_0 Δr, Δ t = P_0 Δ t (where P_0=2π t_0 is the period of rotation at r=R_0), etc., and obtain Eq. <ref> in dimensionless form:Δ t > μ^2 Δ r/2πṀ r_m^3 (ω_s - 1)^2 .Here, we take into account the fact that B_⋆/B_0=μ (R_⋆/R_0)^3=μ and remove all tildes above the dimensionless variables. §.§ Low Diffusivity ScenarioAt a low diffusivity rate, matter of the inner disk slowly penetrates through the external layers of the magnetosphere (in the direction of the star),and the depth of penetration is proportional to the time interval Δ t:Δ r = √(η_m Δ t), where η_m is the diffusivity coefficient. During this time interval Δ t, matter accretes towards the inner disk and is accumulated in the amount of ṀΔ t. This matter carries the angular momentum flux describedby Eq. <ref>, and the inflating field lines (which thread the ring with width Δ r) carry away the angular momentum flux described by Eq. <ref>. Inflation becomes possible when the angular momentum flux carried by matter becomes larger than the angular momentum flux carried by the field:r_m^2 (Ω_⋆ - Ω_d) ṀΔ t > B_m^2 r_m^2/(Ω_⋆ - Ω_d)√(η_m Δ t) .Therefore, the next episode of inflationoccurs after an interval of time Δ t, ifΔ t > η_m μ^4 /2πṀ^2 r_m^6 (ω_s - 1)^4 .Here, we have already converted the time interval to dimensionless units and removed the tilde's.One can see that the time interval is proportional to the diffusivity coefficient, η_m, and all the variables that Δ t is dependent on have coefficient powers that are twice as high as those in the high-diffusivity scenario (see Eq. <ref>).§.§ Comparison with simulations In our model, the diffusivity is high, so we take Eq. <ref> (for the high diffusivity scenario) and compare the time intervals obtained with this formula with the time intervals between episodes of accretion obtained in our simulations. Eq. <ref> shows that in stars with a larger magnetospheric parameter μ the time interval Δ t should be larger. To check the dependence on μ, we compare the time intervals between the bursts in models with different values of parameter μ (see left panels of Figures <ref>, <ref> and <ref> from <ref>). One can see that, in the models with comparable values of ω_s, the time intervals between the main bursts are larger at μ=100 than at μ=60 and μ=30. This trend is even more clear if we compare the variation of the inner disk radius (compare the right panels of the same figures). Eq. <ref> also shows that the time interval Δ t should increase when the difference (ω_s-1) becomes small. Therefore, the time intervals between bursts of accretion should be larger in the weaker propellers.The left panels of Figures <ref>, <ref> and <ref> show that, for each value of μ, the time interval Δ t between accretion/ejection events increaseswhen the fastness parameter ω_s decreases.Therefore,Eq. <ref> describes the dependenceon these two parameterscorrectly.Eq. <ref> also shows that the time intervalΔ t increases when the accretion rate Ṁ decreases. In our simulations, we did not varythe initial accretion rate in the disk. However, we observed that in some long simulation runs the accretion rate decreasesand the time interval increases, which is consistent with the theoretical dependence. § APPLICATIONS TO DIFFERENT TYPES OF STARSIn this section, we provide convenient estimates and formulae for the application of our model to different types of stars.§.§ Application to accreting millisecond pulsars For accreting millisecond pulsars, we take the mass and radius of the star to be M_⋆=1.4 M_⊙ and R_⋆ = 10km, respectively, and the magnetic field to be B_⋆=10^8G. Using equations <ref>, <ref> and <ref>, we obtain the reference values for matter, angular momentum and energy fluxes in the following form:_0= ρ_0 v_0 R_0^2 =3.22× 10^-12μ_60^-2B_8^2 R_6^5/2 M_1.4^-1/2M_⊙/ yr , _0= _0 v_0 R_0 =2.80× 10^30μ_60^-2B_8^2 R_6^3ergs , _0= _0 v_0^2 =3.83× 10^34μ_60^-2B_8^2 R_6^3/2M_1.4^1/2 ergs/ s ,where M_1.4=M_⋆/1.4 M_⊙, R_6=R_⋆/10^6 cm, B_8=B_⋆/10^8G, and μ_60=μ/60. For example, to obtain the dimensional matter fluxes to the star, Ṁ_s, and to the wind, Ṁ_w, one should take the dimensionless values⟨Ṁ_s⟩ and ⟨Ṁ_w⟩ from Tab. <ref> and multiply them by Ṁ_0:Ṁ_s = Ṁ_0 ⟨Ṁ_s⟩ , Ṁ_w = Ṁ_0 ⟨Ṁ_w⟩ .Analogously, we can find the energy fluxes to the wind/jet associated with matter and magnetic field:Ė_m = Ė_0 ⟨Ė_m⟩ , Ė_f = Ė_0 ⟨Ė_f⟩ .A star in the propeller regime spins down. The spin-down energy flux (spin-down luminosity) is:Ė_ sd=L̇_ sdΩ_⋆ =L̇_0 ⟨L̇_ sd⟩Ω_⋆=L̇_0⟨L̇_ sd⟩ 2π/P_⋆ = =1.76× 10^34μ_60^-2B_8^2 R_6^3 ⟨L̇_ sd⟩ P_-3^-1 ergs/s ,where P_-3 is the period of a neutron star in milliseconds. The spin-down time scale can be estimated ast_ sd = L_⋆/_ sd= I Ω_⋆/L̇_0 ⟨_ sd⟩ ,where L_⋆=IΩ_⋆ is the angular momentum of the star, I=k M_⋆ R_⋆^2≈ 1.12× 10^45 k_0.4 M_1.4 R_6^2 g cm^2 is the star's moment of inertia, k_0.4=k/0.4. Substituting in _0, Ω_⋆=Ω_0 Ω̃_⋆, and taking Ω_0 from Tab. <ref>, we obtain:t_ sd≈ 1.73× 10^11 k_0.4 M_1.4^3/2R_6^-5/2B_8^-2μ_60^2 Ω̃_⋆/⟨_ sd⟩ yr .We can re-write the last term of Eq. <ref>, Ω̃_⋆/⟨_ sd⟩, in the following way.From Tab. <ref> we note that the spin-down flux ⟨_ sd⟩ is proportional to the fastness parameter, ω_s, and there is also an approximately linear dependence on μ: ⟨_ sd⟩≈ 0.83 μ_60ω_s. Onthe other hand, using the definition of the fastness parameter, ω_s=Ω_⋆/Ω_K (r_m)=Ω̃_⋆/Ω̃_K (r_m)≈Ω̃_⋆/⟨ r_m⟩^-3/2, we can re-write Ω̃_⋆ as Ω̃_⋆=⟨ r_m ⟩ ^-3/2ω_s and obtain the following relationship:Ω̃_⋆/⟨L̇_ sd⟩≈ 0.10 (⟨ r_m⟩/5)^-3/2μ_60^-1 ,and the time scale in the form oft_ sd≈ 1.74× 10^10 k_0.4 M_1.4^3/2R_6^-5/2B_8^-2μ_60(⟨ r_m ⟩/5)^-3/2 yr .The spin down time-scale does not depend on the angular velocity of the star, Ω_⋆, because the faster rotators have larger spin-down rates but alsoa larger amount of initial angular momentum.Eq. <ref> shows that in stars with the same mass, radius and magnetic field, the spin-down time scaleis roughly the same in models with the same magnetospheric radius ⟨ r_m⟩.Using the values of ⟨ r_m ⟩ from Table <ref>, we obtain the time scales in the range of t_ sd=(1.0-2.0)× 10^10 yrs. In application to millisecond pulsars, the rate of spin-down is often measured as the rate of variation of frequency ν with time, ν̇= dν/dt (in Hz/s). Taking into account the fact that the angular momentum of the star L_⋆=I_⋆Ω_⋆ and the angular momentum flux from the star L̇_ sd=I_⋆Ω̇_⋆, we obtain:dν/dt = (1/2π)Ω̇_⋆ = Ω_⋆/(2π) (L̇_ sd/L_⋆) = ν_⋆/t_ sd = =1.82× 10^-15ν_3 R_6^5/2B_8^2 (⟨ r_m ⟩/5)^3/2/k_0.4 M_1.4^3/2μ_60 Hz/s ,where ν_3=ν/1000 Hz.The time interval between accretion/ejection events varies in the range of Δt̃=30-200 in dimensionless units. To convert to dimensional units of time, we multiply this value by the reference period of rotation, P_0=0.46s, and obtain Δ t≈ (14-92) ms. §.§ Application to cataclysmic variables For cataclysmic variables, we take the mass and radius of the star to be M_⋆=M_⊙ and R_⋆ = 5000km, respectively, and the magnetic field to be B_⋆=10^6G. We obtain the reference values for matter, angular momentum and energy fluxes in the following form:_0=2.11× 10^-9μ_60^-2B_6^2 R_5000^5/2 M_⊙^-1/2M_⊙/ yr , _0=3.46× 10^34μ_60^-2B_6^2 R_5000^3ergs , _0=3.57× 10^34μ_60^-2B_6^2 R_5000^3/2M_1^1/2 ergs/ s ,where whereR_5000=R_⋆/5,000km, and B_6=B_⋆/10^6G.We can obtain the spin-down luminosity of the star in a convenient form:Ė_ sd=L̇_ sdΩ_⋆ =L̇_0⟨L̇_ sd⟩ 2π/P_⋆ = = 2.2× 10^35μ_60^-2B_6^2 R_5000^3 ⟨L̇_ sd⟩ P^-1 ergs/s ,where P is the period of the star in seconds.Using Eq. <ref> and the value for themoment of inertia for white dwarf,I=k M_⋆ R_⋆^2=2.0× 10^50 k_0.4 M_⊙ R_5000^2g cm^2, we obtain the spin-down time scale in a form similar to that obtained formillisecond pulsars:t_ sd≈ 1.89× 10^7 k_0.4 M_⊙^3/2R_5000^-5/2B_6^-2μ_60 (⟨ r_m ⟩/5)^-3/2 yr .Using the values of ⟨ r_m ⟩ from Table <ref>, we obtain the time scales in the range of t_ sd=(1.47-1.68)× 10^7 yrs. The time interval between accretion/ejection events varies in the range of Δt̃=30-200 in dimensionless units. To convert to dimensional units of time, we multiply this value by the reference period of rotation, P_0=6.08s, and obtain Δ t≈ (180-1220) s. §.§ Application to CTTSs In application to Classical T Tauri stars,we take the mass and radius of the star to be M_⋆=0.8 M_⊙ and R_⋆ = 2R_⊙, respectively, and the magnetic field to be B_⋆=10^3G.We obtain the reference values for matter, angular momentum and energy fluxes in the following form:_0=2.1× 10^-9μ_60^-2B_3^2 R_2R_⊙^5/2 M_0.8^-1/2 M_⊙/ yr , _0=3.5× 10^34μ_60^-2B_3^2 R_2R_⊙^3 ergs , _0=3.6× 10^34μ_60^-2B_3^2 R_2R_⊙^3/2M_0.8^1/2 ergs/ s ,whereM_0.8=M_⋆/0.8 M_⊙, R_2R_⊙=R/2R_⊙, and B_3=B_⋆/10^3G. Using Eq. <ref> and the value for the moment of inertia of CTTSs,I=k M_⋆ R_⋆^2≈ 1.25× 10^55 k_0.4 M_0.8 R_2R_⊙^2g cm^2, we obtain the spin-down time scale in a form similar to that obtained formillisecond pulsars and CVs:t_ sd≈ 1.03× 10^7k_0.4M_0.8^1.5R_2^-5/2B_3^-2μ_60(⟨ r_m⟩/5)^-3/2 yr .Using the values of ⟨ r_m ⟩ from Table <ref>, we obtain the time scales in the range of t_ sd=(5.3× 10^6 - 1.3× 10^7) yrs.These time scales are in agreement with the observations of CTTSs, which show that CTTSs are already slow rotators after 1-10 million years. The time interval between accretion/ejection events varies in the range of Δt̃=30-200 in dimensionless units. To convert to dimensional units of time, we multiply these values by the reference period of rotation, P_0=0.37 days, and obtain Δ t≈ (11-74) days. A recent analysis of the light-curves of accreting young stars in the ρ Oph and Upper Sco regions of star formation (obtained with the K-2 Kepler mission) has shown that bursts of accretionoccur every 3-80 days <cit.>. Stars with infrequent bursts may be in the propeller regime.§ CONCLUSIONS AND DISCUSSIONSWe performed axisymmetric simulations of accretion onto rotating magnetized stars in the propeller regime, ranging from very weak to very strong propellers. We used the fastness parameter ω_s to characterize the strength of the propellers. We observed that many properties of the propellers depend on the fastness parameter. §.§ Main conclusions The main conclusions are the following:1. Both accretion and outflows are observed in propellers of different strengths. The relative amount of matter ejected into the outflows (propeller efficiency, f_ eff, see Eq. <ref>) increases with ω_s as a power law.2. The accretion/ejection cycle is observed at different propeller strengths. In this cycle: (a) Matter of the inner disk slowly moves inward and penetrates through the field lines of the external magnetosphere, (b) The magnetic field lines inflate and open. Matter partly accretes onto the star and is partly ejected into the outflows along the inflated field lines. (c) The magnetosphere expands and the cycle repeats. Most of the time matter accumulates in the inner disk, while the accretion/ejection events occur during brief intervals of time.3. The inner disk oscillates.The time-averaged inner disk (magnetospheric) radius ⟨ r_m ⟩ is larger than the corotation radius r_ cor. In spite of this, matter accretes onto the star. Accretion is possible due to the fact that (a) only the closed part of the magnetosphere represents the centrifugal barrier, (b) the process is non-stationary: accretion occursin brief episodes as the inner disk moves closer to the star.4. The velocity of matter ejected into the wind is different in propellers of different strengths: (a) In strong propellers, the maximum velocity of ejecting matter is a few times larger than the local escape velocity; (b) In weak propellers, the maximum velocity is slightly larger or smaller than the escape velocity; (c) In very weak propellers, matter is ejected at sub-escape velocities, forming a turbulent corona above the disk. The time-averaged velocity of matter ejected into the wind increases with the fastness parameter (ω_s) exponentially.5. The time-averaged opening angle of the wind ⟨Θ_ wind⟩ is also different in propellers of different strengths: (a) In strong propellers, this angle is relatively small, ⟨Θ_ wind⟩≈ 40^∘-45^∘. (b) In weak propellers, it is larger, ⟨Θ_ wind⟩≈ 60^o. The opening angle decreases with ω_s as a power law.6. A star in the propeller regime spins down due to the outward angular momentum flow along the field lines. Approximately half of the angular momentum flows to the disk along the closed field lines. The other half flows along the open field lines connecting the star with the corona.7. A star-disk system loses mass, angular momentum and energy. Most of the matter flows from the inner disk into a conically-shaped wind, which carries the energy and angular momentum associated with that matter. In addition, the inflating field lines carry angular momentum and energy associated with the magnetic field. In thestrong propellers, the field lines originating at the star wind up rapidly and form a magnetically-dominated and magnetically-driven (Poynting flux) jet, which accelerates a small amount of matter to high velocities. This jettakes a significant amount of angular momentum out of the star. In addition, it carries angular momentum and energy out of the system. Ejections to the conical wind and Poynting flux jet are strongly non-stationary, so the formation of shocks is expected at some distances from the star.§.§ Application to propeller candidate starsOur research shows that our models of propellers can explain the different observational properties of propeller candidate stars:* Strong variability in the light-curves, which can be associated with(1) variable accretion rate onto the star, (2) variable ejection rate to the wind, (3) oscillations of the inner disk. * Accretion of matter onto the stellar surface in the low-luminosity (low accretion rate) regime, when the magnetospheric radius r_m is larger than the corotation radius r_ cor. Our models show that a small amount of matter accretes onto a star even in the strongest propeller regime. * Outflows from propeller candidates stars. These outflows can be associated with conical winds andmore collimated magnetic jets. * Flares of high-energy radiation (e.g., the gamma-ray flares observed in some transitional MSPs) can be associated with acceleration of particles in shocks, which form during non-stationary ejections to jets and winds in the strong propeller regime.In future studies, we plan to model the propeller candidate stars individually (using the known stellar parameters) and to compare our models with observations in detail. §.§ Comparisons with other models Our model is somewhat similar to the model of <cit.>, who suggested that the field lines connecting the star and the disk should inflate and reconnect quasi-periodically. This modelwas developed for accreting (non-propelling) stars. However, differential rotation between the foot-points of the field lines and their inflation is expected in both regimes (see also ) [Inflation of the field lines has been observed, e.g., in simulations by <cit.>. The signs of such inflation were observed in CTTS AA Tau <cit.>.]. Axisymmetric MHD numerical simulations byconfirmed this type of instability.In their simulations, they observed several cycles in which matter accumulated, the field lines inflated and subsequentlyreconnected, matter accreted onto the star and then was ejected into the winds, and the magnetosphere expanded. A similar cycle has been observed in axisymmetric simulations of the propeller regime <cit.>. However, both types of simulations (for slowly and rapidly-rotating stars) have only been performed in the top part of the simulation region (above the equatorial plane).In these models, reconnection of the field lines has been necessary for the subsequent accretion of matter onto the star. More recent simulations by <cit.> have shown that modeling the entire simulation region (above and below the equator) leads to a newphenomenon: the magnetic flux inflates in one direction (above or below the disk), but matter accretes onto the star on the opposite sideof the equator [This phenomenon has been initially observed in simulations by , where accretion onto stars with complex fields has been modeled.]. This phenomenon leads to the fact that reconnection is not required for accretion: matter of the inner disk accretes above the magnetosphere (on the opposite side of inflation relative to the equator), where the magnetic flux does not block its path (see, e.g., Fig. <ref>). In our current studies of the propeller regime, we observed a similar phenomenon in the models with larger magnetospheres and thinner disks. We should note that, in the models of slowly-rotating stars (e.g., ), accretion is blocked by the magnetic flux of the inflated field lines, while in the models of propellers the centrifugal barrier of the rapidly-rotating star is a more important factor in blocking accretion. Our model also has some similarities with the “dead disk" model, proposed by <cit.> and further developed by <cit.>: in these models, matter of the inner disk is blocked by the centrifugal barrierfor some interval of time, and the periods of matter accumulation alternate with episodes of matter accretion onto the star. However, compared with their models, our model is two-dimensional andtakes into account (1) inflation of the field lines, (2) formation of outflows and jets, which can be driven by both centrifugal and magnetic forces, and (3) the possibility of accretion above or below the centrifugal barrier (which has the shape of a closed magnetosphere). Also, in <cit.>, it is suggested that the magnetospheric radius should be near the corotation radius. In our models, the position of the magnetospheric radius does not depend much on the rotation of the star, but is instead determined by the balance of magnetic and matter stresses, while the position of the corotation radius is determined by the period of the star. We modeled propellers with different ratios of these two radii, which are in the range of ⟨ r_m⟩/r_ cor=1.1-4.7 (see Tab. <ref>). Moreover, in each model, the magnetospheric radiustypically varies strongly. In spite of these differences, cyclicaccretion is also observed in our models. However, in our models, we observe several time-scalesassociated with more complex processes of disk-magnetosphere interaction. §.§ Restrictions of the model and future work Current simulations are axisymmetric. This restricts us from modeling instabilities at the disk-magnetosphere boundary, which determine the rate of matter penetration through the external magnetosphere. 3D instabilities are shown to be effective in cases of slowly-rotating magnetized stars (e.g., ).In this paper, we suggested that similar instabilities may also operate and provide an effective diffusivity at the disk-magnetosphere boundary. We used the α-diffusivity approach and took the maximum possible value of α_ diff=1 (acting only inside the spherical radius R=7, which typically includes the disk-magnetosphere boundary). We observed that this diffusivity provides rapid penetration of matter through the external layers of the magnetosphere. However, the effective diffusivity may depend, for example, on the fastness parameter ω_s, and can be high at some values of ω_s and low at other values (as in the cases of slowly-rotating stars, see ). Fortunately, the results of the propeller model do not depend too much on the value of diffusivity. Our earlier studies of propellers, performed at different values of the diffusivity parameter α_ diff (see Appendix B in ), have shown that the process of disk-magnetosphere interaction is similar in the cases of high and low diffusivity.However, at very low diffusivity, α_ diff=0.01, matter is accumulated at the disk-magnetosphere boundary for longer time before it accretes onto the star. In this case, accretion is more “spiky" (see left panel of Fig. B2 of ). In the oppositescenario, when the diffusivity is high, α_ diff=1, matter of the inner disk penetrates more rapidly through the external layers of the magnetosphere, acquires angular momentum and is ejected into the winds (see right panel of Fig. B2 of ). In this case, the accretion rate is smaller. As a result, efficiency is higherat higher diffusivity values. However, the difference in not very large: f_ eff=0.70 in the low-diffusivity case versus f_ eff=0.86 in the high-diffusivity case. Overall, the results of <cit.> obtained at a very low diffusivity [Most of results in <cit.> simulations were obtained using ideal MHD code with no diffusivity term added, and where only a small numerical diffusivity determined penetration of the disk through the magnetosphere. The numerical diffusivity in the code corresponded to α_ diff≈ 0.01-0.003.] do not differ qualitatively from the results obtained in the current paper. The issue of diffusivity should be further studied in 3D simulations. On the other hand, in three dimensions, the magnetic axis of the dipole can be tilted about the rotational axis of the disk. 3D MHD simulations of accreting stars have shown that the magnetospheric radius r_m is approximately the same in stars with different tilts of the magnetic axis <cit.>, and therefore the centrifugal barriershould be located at the same distance as in the 2D simulations. However, the centrifugal barrier will have a slightly different shape, which may be against accretion. On the other hand, the tilted dipole is more favorable for accretion. Therefore, the efficiency of the propeller may be somewhat different compared with the axisymmetric case. 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R., & Horne, K. 1997, MNRAS, 286, 436 [Zanni and Ferreira2013]ZanniFerreira2013Zanni, C., & Ferreira, J. 2013, A&A, 550, 99 § DESCRIPTION OF NUMERICAL MODEL§.§ Initial and boundary conditions *Initial Conditions: In this work, the initial conditions for the hydrodynamic variables are similarto thoseused in our previous works (e.g., ), wherethe initial density and entropy distributionswere calculatedby balancing the gravitational, centrifugal and pressure forces. The disk isinitially cold and dense, with temperature T_d and density ρ_d. The coronais hot and rarified, with temperature T_c = 3× 10^3 T_d and density ρ_c = 3.3× 10^-4ρ_d.In the beginning ofthe simulations,the inner edge of the disk is placedat r_d = 10, and the star rotates with Ω_i = 0.032 (corresponding to r_ cor = 10), so that the magnetosphere and the inner disk initially corotate. This condition helps to ensure that the magnetosphere and the disk are initially in near-equilibrium at the disk-magnetosphere boundary.The star is gradually spun up from Ω_i to the final state with angular velocity Ω_⋆, corresponding to r_ cor (given in Table <ref>). The initial pressure distribution in the simulation is determined from the Bernoulli equation:F(p) + Φ + Φ_c = B_0 =constant, where Φ = -GM_⋆/(r^2+z^2)^1/2 is the gravitational potential, Φ_c=-kGM_⋆/r is the centrifugal potential, k is a Keplerian parameter[We take k slightly greater than unity to balance the disk pressure gradient (k=1+0.003).] andF(p) =R T_d ln(p/p_b), p > p_br > r_d, R T_c ln(p/p_b), p ≤ p_br ≤ r_d,where p_b is the pressure at the boundary that separates the disk from the corona. We assume the system to be initially barotropic, and determine the density from the pressure:ρ(p) =p/ R T_d, p > p_br > r_d, p/ R T_d, p ≤ p_br ≤ r_d.To initialize the MRI, 5% velocity perturbations are added to v_ϕ inside the disk. *Initial magnetic field configuration: Initially, the disk is threaded by the dipole magnetic field of the star. We also add a small “tapered” poloidal field inside the disk (see left panel in Fig. <ref>), which is given byΨ=B_0 r^2/2cos(πz/2h), h=√((GM_*/Φ_c(r)-E)^2 - r^2),where h is the half-thickness of the disk and E is a constant of integration in the initial equilibrium equation (see ). This tapered field helps initialize the MRI in the disk and has the same polarity as the stellar field at the disk-magnetosphere boundary. *Boundary Conditions: Stellar surface: all the variables on the surface of the star have “free” boundary conditions, such that ∂ (...)/∂ n=0 along the entire surface. Wedo not allow for the outflow of matter from the star (i.e. we prohibit stellar winds), and adjust the matter velocity vectors to be parallel to the magnetic field vectors. This models the frozen-in condition on the star.Top and bottom boundaries: all variables have free boundary conditions along the top and bottom boundaries. In addition, we implement outflow boundary conditions on velocity to prohibit matter from flowing back into the simulation region once it leaves.Outer side boundary: the side boundary is divided into a “disk region” (|z| < z_ disk) and a “coronal region" (|z| > z_ disk), withz_ disk = h(R_ out) = √((GM_*/Φ_c(R_ out)-E)^2 - R_ out^2),where R_ out is the external simulation radius. The matter along the disk boundary (|z| < z_ disk) is allowed to flow inward with a small radial velocityv_r=-δ3/2p/ρ v_K(R_ out), δ=0.02,anda poloidal magnetic field corresponding to the calculated magnetic field at r=R_ out. Theremaining variables have free boundary conditions. The coronal boundary (|z| > z_ disk) has the same boundary conditions as the top and bottom boundaries.§.§ Grid and code description *Grid description: The axisymmetric grid is in cylindrical (r, z) coordinates with mesh compression towardsthe equatorial plane and the z-axis, so that there is a larger number of cells in the disk plane and near the star. In the models presented here, we use a non-uniform grid with 190 × 306 grid cells corresponding to a grid that is 43 by 82 stellar radii in sizeAt r=20, the number of grid cells that cover the disk in the vertical direction is about 60. *Code description: We use a Godunov-type numerical method with a five-wave Riemann solver similar to the HLLD solver developed by <cit.>. The MHD variables are calculated in four states bounded by five MHD discontinuities: the contact discontinuity, two Alfvén waves and two fast magnetosonic waves. Unlike <cit.>, our method solves the equation for entropy instead of the full energy equation. This approximation is valid in cases (such as ours) where strong shocks are not present.We ensure that the magnetic fields are divergence-free by introducing the ϕ-component of the magnetic field potential, which is calculated using the constrained transport scheme proposed by <cit.>. The magnetic field is split into the stellar dipole and the calculated components, B=B_ dip + B';we omit the terms of the order B_ dip^2 which do not contribute to the Maxwellian stress tensor <cit.>. No viscosity terms have been included in the MHD equations, and hence we only investigate accretion driven by the resolved MRI-turbulence. Our code has been extensively tested and has been previously utilized to study different MHD problems (seefor tests and some astrophysical examples).Table <ref> shows sample reference values for three different types of accreting stars: to apply the simulation results to a particular class of star, multiply the dimensionless value by the reference value. The dependence on μ is also shown.§.§ Reference unitsThe simulations are performed in dimensionless units and are applicable to stars over a wide range of scales. There are four free parameters: we choose the values of the stellar mass M_*, radius R_*, magnetic field B_* and dimensionless magnetospheric parameterμ̃ and derive reference values from these parameters. The magnetic moment μ_⋆ = μ̃μ_0ẑ is used to initialize the stellar dipole field𝐁_ dip = 3(μ·𝐑)𝐑 - μR^2/R^5,where 𝐑 is the radius in spherical coordinates. In this work, we take μ̃ = 30, 60 and 100.The reference units are as follows: length R_0=R_*, magnetic moment μ_0 = B_0 R_0^3, magnetic field B_0 = B_*/μ̃× (R_*/R_0)^3 (the equatorial field dipole strength at r = R_0) , velocity v_0=√(GM_*/R_0) (the Keplerian orbital velocity at r = R_0), time t_0=2π R_0/v_0 (the Keplerian orbital period at r = R_0), angular velocity Ω_0 = v_0/R_0, pressure p_0 = B_0^2, density ρ_0=p_0/v_0^2, temperature T_0 = p_0/ρ_0 × m_H/k_B where m_H is the mass of hydrogen and k_B is the Boltzmann constant, force per unit mass f_0=v_0^2/R_0. Accretion rate _0=ρ_0 v_0 R_0^2, angular momentum flux _0 = _0 v_0 R_0 and energy flux _0 = _0 v_0^2. We should stress out that in our dimensionalization procedure, the reference magnetic field and many other referencevariables depend on the parameter μ̃. Matter flux and other reference fluxes also depend on this parameter. For practical purposes, we provide a useful form for reference fluxes:_0=ρ_0 v_0 R_0^2 = (B_⋆/μ̃)^2 (R_0^2/v_0)(R_⋆/R_0)^6 , _0=_0 v_0 R_0 = (B_⋆/μ̃)^2 R_0^3 (R_⋆/R_0)^6 , _0=_0 v_0^2 = (B_⋆/μ̃)^2 (R_0^2 v_0)(R_⋆/R_0)^6 .All fluxes depend on parameter μ̃ as ∼μ̃^-2. For example, in case of matter flux, this means that at larger values of μ̃, the matter flux is smaller, and(at fixed B_⋆) the magnetospheric radius is expected to be larger, because the general dependence r_m∼ (μ_⋆^2/Ṁ)^1/7 is approximately satisfied. That is why in our model we use parameter μ̃ to regulate the dimensionless size r_m/R_⋆ of the magnetosphere: the magnetosphere is largest in case of μ̃=100, and smallest in case of μ̃=30. In Tab. <ref>, we use the normalized value μ_60=μ/60.§ VARIATION OF THE INNER DISK RADIUS AND MATTER FLUXES IN REPRESENTATIVE RUNSLeft-hand panels of Figures <ref>, <ref>, and <ref> show temporal variation of thematter fluxes to the star Ṁ_⋆ and to the wind Ṁ_wind in cases of magnetospheres with different sizes (different parameter μ) and different corotation radii (different r_ cor). One can see that in all cases accretion occurs in relatively brief bursts. Matter flux to the wind also occurs in bursts.The dashed lines show the time-averaged values ⟨Ṁ_⋆⟩ and ⟨Ṁ_wind⟩. Right-hand panels show temporal variation of theinner disk radius, r_m. The dashed lines show the time-averaged values ⟨ r_m ⟩.Left-hand panels of Figs. <ref>, <ref> and <ref> show temporal variation of the normalized maximum velocityv_ max/v_ esc in the matter-dominated component of the wind. Dashed lines show the time-averaged values, ⟨ v_ max⟩/v_ esc. Figures show that the maximum velocity rapidly decreases.hen r_ cor increases. Right-hand panels of Figs. <ref>, <ref> and <ref>show variation of the opening angle of the wind, Θ_ wind, with time. One can see that the opening angle systematically increases, when r_ cor increases. Dashed lines show variation of the time-averaged values, ⟨Θ_ wind⟩. Figures show that the opening angle systematically increases when r_ cor increases. | http://arxiv.org/abs/1704.08336v1 | {
"authors": [
"Marina M. Romanova",
"Alisa A. Blinova",
"Galina V. Ustyugova",
"Alexander V. Koldoba",
"Richard V. E. Lovelace"
],
"categories": [
"astro-ph.SR",
"astro-ph.HE"
],
"primary_category": "astro-ph.SR",
"published": "20170426200618",
"title": "Properties of Strong and Weak Propellers from MHD Simulations"
} |
[pages=1-last]quadreg_transport_arxiv.pdf | http://arxiv.org/abs/1704.08200v4 | {
"authors": [
"Montacer Essid",
"Justin Solomon"
],
"categories": [
"math.OC",
"cs.NA",
"math.NA"
],
"primary_category": "math.OC",
"published": "20170426163742",
"title": "Quadratically-Regularized Optimal Transport on Graphs"
} |
A Generalization of Convolutional Neural Networks to Graph-Structured Data Yotam Hechtlinger, Purvasha Chakravarti & Jining Qin Department of Statistics Carnegie Mellon University{yhechtli,pchakrav,jiningq}@stat.cmu.eduDecember 30, 2023 ================================================================================================================================================================================ This paper introduces a generalization of Convolutional Neural Networks (CNNs) from low-dimensional grid data, such as images, to graph-structured data. We propose a novel spatial convolution utilizing a random walk to uncover the relations within the input, analogous to the way the standard convolution uses the spatial neighborhood of a pixel on the grid. The convolution has an intuitive interpretation, is efficient and scalable and can also be used on data with varying graph structure. Furthermore, this generalization can be applied to many standard regression or classification problems, by learning the the underlying graph. We empirically demonstrate the performance of the proposed CNN on MNIST, and challenge the state-of-the-art on Merck molecular activity data set. § INTRODUCTION Convolutional Neural Networks (CNNs) are a leading tool used to address a large set of machine learning problems(<cit.>, <cit.>). They have successfully provided significant improvements in numerous fields, such as image processing, speech recognition, computer vision and pattern recognition, language processing and even the game of Go boards ( <cit.>,<cit.>, <cit.>, <cit.>, <cit.> respectively). The major success of CNNs is justly credited to the convolution. But any successful application of the CNNs implicitly capitalizes on the underlying attributes of the input. Specifically, a standard convolution layer can only be applied on grid-structured input, since it learns localized rectangular filters by repeatedly convolving them over multiple patches of the input. Furthermore, for the convolution to be effective, the input needs to be locally connective, which means the signal should be highly correlated in local regions and mostly uncorrelated in global regions. It also requires the input to be stationary in order to make the convolution filters shift-invariant so that they can select local features independent of the spatial location. Therefore, CNNs are inherently restricted to a (rich) subset of datasets. Nevertheless, the impressive improvements made by applying CNNs encourage us to generalize CNNs to non-grid structured data that have local connectivity and stationarity properties. The main contribution of this work is a generalization of CNNs to general graph-structured data, directed or undirected, offering a supervised algorithm that incorporates the structural information present in a graph. Moreover our algorithm can be applied to a wide range of regression and classification problems, by first estimating the graph structure of the data and then applying the proposed CNN on it. Active research in learning graph structure from data makes this feasible, as demonstrated by the experiments in the paper.The fundamental hurdle in generalizing CNNs to graph-structured data is to find a corresponding generalized convolution operator. Recall that the standard convolution operator picks the neighboring pixels of a given pixel and computes the inner product of the weights and these neighbors. We propose a spatial convolution that performs a random walk on the graph in order to select the top p closest neighbors for every node, as Figure <ref> shows. Then for each of the nodes, the convolution is computed as the inner product of the weights and the selected p closest neighbors, which are ordered according to their relative position from the node. This allows us to use the same set of weights (shared weights) for the convolution at every node and reflects the dependency between each node and its closest neighbors. When an image is considered as an undirected graph with edges between neighboring pixels, this convolution operation is the same as the standard convolution. The proposed convolution possesses many desired advantages: * It is natural and intuitive. The proposed CNN, similar to the standard CNN, convolves every node with its closest spatial neighbors, providing an intuitive generalization. For example, if we learn the graph structure using the correlation matrix, then selecting a node's p nearest neighbors is similar to selecting its p most correlated variables, and the weights correspond to the neighbors' relative position to the node (i.e. i^th weight globally corresponds to the i^th most correlated variable for every node).* It is transferable. Since the criterion by which the p relevant variables are selected is their relative position to the node, the convolution is invariant to the spatial location of the node on the graph. This enables the application of the same filter globally across the data on all nodes on varying graph structures. It can even be transfered to different data domains, overcoming a known limitation of many other generalizations of CNNs on graphs. * It is scalable. Each forward call of the graph convolution requires O(N· p) flops, where N is the number of nodes in the graph or variables. This is also the amount of memory required for the convolution to run. Since p≪ N, it provides a scalable and fast operation that can efficiently be implemented on a GPU.* It is effective. Experimental results on the Merck molecular activity challenge and the MNIST data sets demonstrates that by learning the graph structure for standard regression or classification problems, a simple application of the graph convolutional neural network gives results that are comparable to state-of-the-art models.To the best of our knowledge, the proposed graph CNN is the first generalization of convolutions on graphs that demonstrates all of these properties.§ LITERATURE REVIEW Graph theory and differential geometry have been heavily studied in the last few decades, both from mathematical and statistical or computational perspectives, with a large body of algorithms being developed for a variety of problems. This has laid the foundations required for the recent surge of research on generalizing deep learning methods to new geometrical structures. <cit.> provide an extensive review of the newly emerging field. Currently, there are two main approaches generalizing CNNs to graph structured data, spectral and spatial approaches (<cit.>). The spectral approach generalizes the convolution operator using the eigenvectors derived from the spectral decomposition of the graph Laplacian. The motivation is to create a convolution operator that commutes with the graph Laplacian similar to the way the regular convolution operator commutes with the Laplacian operator. This approach is studied by <cit.> and <cit.>, which used the eigenvectors of the graph Laplacian to do the convolution, weighting out the distance induced by the similarity matrix. The major drawback of the spectral approach is that it is graph dependent, as it learns filters that are a function of the particular graph Laplacian. This constrains the operation to a fixed graph structure and restricts the transfer of knowledge between different different domains. <cit.> introduce ChebNet, which is a spectral approach with spatial properties. It uses the k^th order Chebyshev polynomials of the Laplacian, to learn filters that act on k-hop neighborhoods of the graph, giving them spatial interpretation. Their approach was later simplified and extended to semi-supervised settings by <cit.>. Although in spirit the spatial property is similar to the one suggested in this paper, since it builds upon the Laplacian, the method is also restricted to a fixed graph structure. The spatial approach generalizes the convolution using the graph's spatial structure, capturing the essence of the convolution as an inner product of the parameters with spatially close neighbors. The main challenge with the spatial approach is that it is difficult to find a shift-invariance convolution for non-grid data. Spatial convolutions are usually position dependent and lack meaningful global interpretation. The convolution proposed in this paper is spatial, and utilizes the relative distance between nodes to overcome this difficulty.Diffusion Convolutional Neural Network (DCNN) proposed by <cit.> is a similar convolution that follows the spatial approach. This convolution also performs a random walk on the graph in order to select spatially close neighbors for the convolution while maintaining the shared weights. DCNN's convolution associates the i^th parameter (w_i) with the i^th power of the transition matrix (P^i), which is the transition matrix after i steps in a random walk. Therefore, the inner product is considered between the parameters and a weighted average of all the nodes that can be visited in i steps. In practice, for dense graphs the number of nodes visited in i steps can be quite large, which might over-smooth the signal in dense graphs.Furthermore, <cit.> note that implementation of DCNN requires the power series of the full transition matrix, requiring O(N^2) complexity, which limits the scalability of the method. Another example of a spatial generalization is provided by <cit.>, which uses multi-scale clustering to define the network architecture, with the convolutions being defined per cluster without the weight sharing property. <cit.> on the other hand, propose a neural network to extract features or molecular fingerprintsfrom molecules that can be of arbitrary size and shape by designing layers which are local filters applied to all the nodes and their neighbors. In addition to the research generalizing convolution on graph, there is active research on the application of different types of Neural Networks on graph structured data. The earliest work in the field is the Graph Neural Network by Scarselli and others, starting with <cit.> and fully presented in <cit.>. The model connect each node in the graph with its first order neighbors and edges and design the architecture in a recursive way inspired by recursive neural networks. Recently it has been extended by <cit.> to output sequences, and there are many other models inspired from the original work on Graph Neural Networks. For example, <cit.> introduce "interaction networks" studying spatial binary relations to learn objects and relations and physics.The problem of selecting nodes from a graph for a convolution is analogous to the problem of selecting local receptive fields in a general neural network. The work of <cit.>suggests selecting the local receptive fields in a feed-forward neural network using the closest neighbors induced by the similarity matrix, with the weights not being shared among the different hidden units.In contrast to previous research, we suggest a novel scalable convolution operator that captures the local connectivity within the graph and demonstrates the weight sharing property, which helps in transferring it to different domains. We achieve this by considering the closest neighbors, found by using a random walk on the graph, in a way that intuitively extends the spatial nature of the standard convolution. § GRAPH CONVOLUTIONAL NEURAL NETWORKThe key step which differentiates CNNs on images from regular neural networks is the selection of neighbors on the grid in a p × p window combined with the shared weight assumption. In order to select the local neighbors of a given node, we use the graph transition matrix and calculate the expected number of visits of a random walk starting from the given node. The convolution for this node, is then applied on the top p nodes with highest expected number of visits from it. In this section, we discuss the application of the convolution in a single layer on a single graph. It is immediate to extend the definition to more complex structures, as will be explicitly explained in section<ref>. We introduce some notation in order to proceed into further discussion. Notation: Let 𝒢=(𝒱,ℰ) be a graph over a set of N features, 𝒱=(X_1,…,X_N), and a set of edges ℰ.Let P denote the transition matrix of a random walk on the graph, such that P_ij is the probability to move from node X_i to X_j. Let the similarity matrix and the correlation matrix of the graph be given by S and R respectively. Define D as a diagonal matrix where D_ii = ∑_j S_ij. §.§ Transition matrix and expected number of visits§.§.§ Transition matrix existenceThis work assumes the existence of the graph transition matrix P. If graph structure of the data is already known, i.e. if the similarity matrix S is already known, then the transition matrix can be obtained, as explained in <cit.>, byP = D^-1S.If the graph structure is unknown, it can be learned using several unsupervised or supervised graph learning algorithms. Learning the data graph structure is an active research topic and is not in the scope of this paper. The interested reader can start with <cit.>and <cit.> discussing similarity matrix estimation.We use the absolute value of the correlation matrix as the similarity matrix, following <cit.> who showed that correlation between the features is usually enough to capture the geometrical structure of images. That is, we assumeS_ij = |R_ij| ∀ i, j. §.§.§ Expected number of visitsOnce we derive the transition matrix P, we define Q^(k):=∑_i=0^kP^k, where [P^k]_ij is the probability of transitioning from X_i to X_j in k steps.That is,Q^(0)=I, Q^(1)=I+P,⋯ , Q^(k)=∑_i=0^kP^k.Note that Q^(k)_ij is also the expected number of visits to node X_j starting from X_i in k steps. The i^th row, Q_i·^(k) provides a measure of similarity between node X_i and its neighbors by considering a random walk on the graph. As k increases we incorporate neighbors further away from the node, while the summation gives appropriate weights to the node and its closest neighbors. Figure <ref> provides a visualization of the matrix Q over the 2-D grid. §.§ Convolutions on graphs As discussed earlier, each row of Q^(k) can be used to obtain the closest neighbors of a node. Hence, it seems natural to define the convolution over the graph node X_iusing the i^th row of Q^(k). In order to do so, we denote π_i^(k) as the permutation order of the i^th row of Q^(k) in descending order. That is, for every i = 1, 2, ..., Nand every k,π_i^(k) : { 1,2, ..., N}⟶{ 1,2, ..., N},such that Q_i π_i^(k)(1) > Q_i π_i^(k)(2) > ... > Q_i π_i^(k)(N).The notion of ordered position between the nodes is a global feature of all graphs and nodes. Therefore, we can take advantage of it to satisfy the desired shared weights assumption, enabling meaningful and transferable filters. We define Conv_1 as the size p convolution over the graph G with nodes x = (x_1, …, x_N)^T ∈ R^N and weights w∈ R^p, for the p nearest neighbors of each node, as the inner product:Conv_1( x) = [[ x_π_1^(k)(1)⋯ x_π_1^(k)(p); x_π_2^(k)(1)⋯ x_π_2^(k)(p);⋮⋱⋮; x_π_N^(k)(1)⋯ x_π_N^(k)(p) ]]·[[ w_1; w_2; ⋮; w_p ]] Therefore the weights are decided according to the distance induced by the transition matrix. That is, w_1 will be convolved with the variable which has the largest value in each row of the matrix Q^(k). For example, when Q^(1)=I+P, w_1 will always correspond to the node itself and w_2 will correspond to the node's closest neighbor. For higher values of k, the order will be determined by the graph's unique structure.It should be noted that Conv_1 doesn't take into account the actual distance between the nodes, and might be susceptible (for example) to the effects of negative correlation between the features. For that reason, we have also experimented with Conv_2, defined as: Conv_2( x) =[[ y_1,π_1^(k)(1)⋯ y_1,π_1^(k)(p); y_2,π_2^(k)(1)⋯ y_2,π_2^(k)(p);⋮⋱⋮; y_N,π_N^(k)(1)⋯ y_N,π_N^(k)(p) ]]·[[ w_1; w_2; ⋮; w_p ]],where x = [[ x_1; x_2; ⋮; x_N ]] andy_i j =sign(R_ij)Q^k_i j x_j.In practice the performance of Conv_1 was on par with Conv_2, and the major differences between them were smoothed out during the training process. As Conv_1 is more intuitive, we decided to focus on using Conv_1. §.§ Selection of the power of TEXTThe selection of the value of k is data dependent, but there are two main components affecting its value. Firstly, it is necessary for k to be large enough to detect the top p neighbors of every node. If the transition matrix P is sparse, it might require higher values of k. Secondly, from properties of stochastic processes, we know that if we denote π as the Markov chain stationary distribution, thenlim_k→∞Q_ij^(k)/k=π_j∀ i,j.This implies that for large values of k, local information will be smoothed out and the convolution will repeatedly be applied on the features with maximum connections. For this reason, we suggest keeping k relatively low (but high enough to capture sufficient features).§.§ Implementation§.§.§ The convolutionAn important feature of the suggested convolution is the complexity of the operation. For a graph with N nodes, a single p level convolution only requires O(N · p) flops and memory, where p≪ N. Furthermore, similar to standard convolution implementation <cit.>, it is possible to represent the graph convolution as a tensor dot product, transferring most of the computational burden to the GPU using highly optimized matrix multiplication libraries.For every graph convolution layer, we have as an input a 3D tensor of M observations with N features at depth d. We first extend the input with an additional dimension that includes the top p neighbors of each feature selected by Q^(k), transforming the input dimension from 3D to 4D tensor as(M, N, d)→(M, N, p, d).Now if we apply a graph convolution layer with d_new filters, the convolution weights will be a 3D tensor of size (p,d,d_new). Therefore application of a graph convolution which is a tensor dot product between the input and the weights along the (p, d) axes results in an output of size: ((M, N),(p, d)) ∙((p, d),(d_new)) =(M, N, d_new). We have implemented the algorithm using Keras <cit.> and Theano <cit.> libraries in Python, inheriting all the tools provided by the libraries to train neural networks, such as dropout regularization, advanced optimizers and efficient initialization methods. The source code is publicly available on Github [<https://github.com/hechtlinger/graph_cnn>].§.§.§ The selection of neighborsThe major computational effort in this algorithm is the computation of Q, which is performed once per graph structure as a pre-processing step. As it is usually a one-time computation, it is not a significant constraint.However, for very large graphs, if done naively, this might be challenging. An alternative can be achieved by recalling that Q is only needed in order to calculate the expected number of visits from a given node after k steps in a random walk. In most applications, when the graph is very large, it is also usually very sparse. This facilitates an efficient implementation of Breadth First Search algorithm (BFS). Hence, the selection of the p neighbors can be parallelized and would only require O(N · p) memory for every unique graph structure, making the method scalable for very large graphs, when the number of different graphs is manageable. Any problem that has many different large graphs is inherently computationally hard. The graph CNN reduces the memory required after the preprocessing from O(N^2) to O(N · p) per graph. This is because the only information required from the graph is the p nearest neighbors of every node.§ EXPERIMENTS In order to test the feasibility of the proposed CNN on graphs, we conducted experiments on well known data sets functioning as benchmarks: Merck molecular activity challenge and MNIST. These data sets are popular and well-studied challenges in computational biology and computer vision, respectively.In our implementations, in order to enable better comparisons between the models and reduce the chance of over-fitting during the model selection process, we consider shallow and simple architectures instead of deep and complex ones. The hyper-parameters were chosen arbitrarily when possible rather than being tuned and optimized. Nevertheless, we still report state-of-the-art or competitive results on the data sets.In this section, we denote a graph convolution layer with k feature maps by C_k and a fully connected layer with k hidden units by FC_k. §.§ Merck molecular activity challenge The Merck molecular activity is a Kaggle [Challenge website is <https://www.kaggle.com/c/MerckActivity>]challenge which is based on 15 molecular activity data sets. The target is predicting activity levels for different molecules based on the structure between the different atoms in the molecule. This helps in identifying molecules in medicines which hit the intended target and do not cause side effects.Following <cit.>, we apply our algorithm on the DPP4 dataset. DPP4 contains 6148 training and 2045 test molecules. Some of the features of the molecules are very sparse and are only active in a few molecules. For these features, the correlation estimation is not very accurate. Therefore, we use features that are active in at least 20 molecules (observations), resulting in 2153 features. As can be seen in Figure <ref>, there is significant correlation structure between different features. This implies strong connectivity among the features which is important for the application of the proposed method. The training in the experiments was performed using Adam optimization procedure <cit.> where the gradients are derived by the back-propagation algorithm, using the root mean-squared error loss (RMSE). We used learning rate α=0.001, fixed the number of epochs to 40 and implemented dropout regularization on every layer during the optimization procedure. The absolute values of the correlation matrix were used to learn the graph structure. We found that a small number of nearest neighbors (p) between 5 to 10 works the best, and used p=5 in all models. Following the standard set by the Kaggle challenge, results are reported in terms of the squared correlation (R^2), that is, R^2 = Corr(Y, Ŷ)^2,where Y is the actual activity level and Ŷ is the predicted one. The convergence plot given in Figure <ref> demonstrates convergence of the selected architectures. The contribution of the suggested convolution is explained in view of the alternatives: * Fully connected Neural Network: Models first applying convolution followed by a fully connected hidden layer, converge better than more complex fully connected models. Furthermore, convergence in the former methods are more stable in comparison to the fully connected methods, due to the parameter reduction.* Linear Regression: Optimizing over the set of convolutions is often considered as automation of the feature extraction process. From that perspective, a simple application of one layer of convolution, followed by linear regression, significantly outperforms the results of a standalone linear regression.Table <ref> provides more thorough R^2 results for the different architectures explored, and compares it to two of the winners of the Kaggle challenge, namely the Deep Neural Network and the random forest in <cit.>. We perform better than both the winners of the Kaggle contest. The models in <cit.> and <cit.> use a spectral approach and currently are the state-of-the-art. In comparison to them, we perform better than the Spectral Networks CNN on unsupervised graph structure, which is equivalent to what was done by using the correlation matrix as similarity matrix. The one using Spectral Networks on supervised graph structure holds the state-of-the-art by learning the graph structure. This is a direction we have not yet explored, as graph learning is beyond the scope of this paper, although it will be straightforward to apply the proposed graph CNN in a similar way to any learned graph.§.§ MNIST data The MNIST data often functions as a benchmark data set to test new machine learning methods. We experimented with two different graph structures for the images. In the first experiment, we considered the images as observations from an undirected graph on the 2-D grid, where each pixel is connected to its 8 adjoining neighbor pixels. This experiment was done, to demonstrate how the graph convolution compares to standard CNN on data with grid structure. We used the convolutions over the grid structure as presented in Figure <ref> usingQ^(3) with p=25 as the number of nearest neighbors. Due to the symmetry of the graph, in most regions of the image, multiple pixels are equidistant from the pixel being convolved. In order to solve this, if the ties were broken in a consistent manner, the convolution would be reduced to the regular convolution on a 5 × 5 window. The only exceptions to this would be the pixels close to the boundary. To make the example more compelling, we broke ties arbitrarily, making the training process harder compared to regular CNN. Imitating LeNet <cit.>, we considered an architecture withC_40, Pooling_(2×2), C_80, Pooling_(2×2) , FC_100 followed by a linear classifier that resulted in a 0.87% error rate. This is comparable to a regular CNN with the same architecture that achieves an error rate of about 0.75%-0.8%. We outperform a fully connected neural network which achieves an error rate of around 1.4%, which is expected due to the differences in the complexities of the models. In the second experiment, we used the correlation matrix to estimate the graph structure directly from the pixels. Since some of the MNIST pixels are constant (e.g the corners are always black), we restricted the data only to the active 717 pixels that are not constant. We used Q^(1) with p=6 as the number of neighbors. This was done in order to ensure that the spatial structure of the image no longer effected the results. With only 6 neighbors, and a partial subset of the pixels under consideration, the relative location of the top correlated pixels necessarily varies from pixel to pixel. As a result, regular CNNs are no longer applicable on the data whereas the convolution proposed in this paper is. We compared the performance of our CNN to fully connected Neural Networks.During the training process, we used adropout rate of 0.2 on all layers to prevent over-fitting. In all the architectures the final layer is a standard softmax logistic regression classifier.Table <ref> presents the experimental results. The Graph CNN performs on par with the fully connected neural networks, with fewer parameters. A single layer of graph convolution followed by logistic regression greatly improves the performance of logistic regression, demonstrating the potential of the graph convolution for feature extraction purposes. As with regular convolutions, C_20 - FC_512 required over 7 million parameters as each convolution uses small amount of parameters to generate different maps of the input. This suggests that the graph convolution can be made even more effective with the development of an efficient spatial pooling method on graphs, which is a known but unsolved problem.§ CONCLUSIONS We propose a generalization of convolutional neural networks from grid-structured data to graph-structured data, a problem that is being actively researched by our community. Our novel contribution is a convolution over a graph that can handle different graph structures as its input. The proposed convolution contains many sought-after attributes; it has a natural and intuitive interpretation, it can be transferred within different domains of knowledge, it is computationally efficient and it is effective. Furthermore, the convolution can be applied on standard regression or classification problems by learning the graph structure in the data, using the correlation matrix or other methods. Compared to a fully connected layer, the suggested convolution has significantly fewer parameters while providing stable convergence and comparable performance. Our experimental results on the Merck Molecular Activity data set and MNIST data demonstrate the potential of this approach. Convolutional Neural Networks have already revolutionized the fields of computer vision, speech recognition and language processing. We think an important step forward is to extend it to other problems which have an inherent graph structure.§.§.§ Acknowledgments We would like to thank Alessandro Rinaldo, Ruslan Salakhutdinov and Matthew Gormley for suggestions, insights and remarks that have greatly improved the quality of this paper.icml2017 | http://arxiv.org/abs/1704.08165v1 | {
"authors": [
"Yotam Hechtlinger",
"Purvasha Chakravarti",
"Jining Qin"
],
"categories": [
"stat.ML",
"cs.AI",
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"published": "20170426153750",
"title": "A Generalization of Convolutional Neural Networks to Graph-Structured Data"
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AIP/123-QED [email protected]@gmail.comFaculty of Physics and Chemistry, Alzahra University, P. O. Box 19938-93973, Tehran, Iran In this paper, by comparing the time scales associated with the velocityrelaxation and correlation time of the random force due to dust charge fluctuations, memory effects in the velocity relaxation of an isolated dust particle exposed to the random force due to dust charge fluctuations are considered, and the velocity relaxation process of the dust particle is considered as a non-Markovian stochastic process. Considering memory effects in the velocity relaxation process of the dust particle yields a retarded friction force, which is introduced by a memory kernel in the fractional Langevin equation. The fluctuation-dissipation theorem for the dust grain is derived from this equation. The mean-square displacement and the velocity autocorrelation function of the dust particle are obtained, and their asymptotic behavior, the dust particle temperature due to chargefluctuations, and the diffusion coefficient are studied in the long-time limit. As an interesting feature, it is found that by considering memory effects in the velocity relaxation process of the dust particle, fluctuating force on the dust particle can cause an anomalous diffusion in a dusty plasma. In this case, the mean-square displacement of the dust grain increases slower than linearly with time, and the velocity autocorrelation function decays as a power-law instead of the exponential decay. Finally, in the Markov limit, these results are in good agreement with those obtained from previous works for Markov (memoryless) process of the velocity relaxation. Memory effects in the velocity relaxation process of the dust particlein dusty plasma H. Hakimi Pajouh December 30, 2023 =======================================================================================§ INTRODUCTIONDust particles in a dusty plasma acquire a net electric charge by collecting electrons and ions from the background plasma. The dust grain charge fluctuates in time because of the discrete nature of charge carriers <cit.>. Electrons and ions arrive at the dust surface at random times. For this reason, the charge fluctuates. These fluctuations always exist even in a steady-state uniform plasma <cit.>.Dust charge fluctuations have been investigated by many researchers <cit.>. In a dusty plasma, there are many phenomena that dust charge fluctuations can be considered as a reason for them such as heating of dust particles system <cit.>, instability of lattice oscillations in a low-pressure gas discharge <cit.>, and the formation of the shock waves in dusty plasmas <cit.>. Also, the motion of dust particles under the influence of the random force due to dust charge fluctuations has been investigated in some studies <cit.>. In these studies, the motion of dust particles has been modeled by Brownian motion based on the Fokker-Planck or Langevin equations, assuming that dust charge fluctuations are fast. It means that the relaxation timescale for charge fluctuations is much shorter than the relaxation time for the dust velocity; hence, they have considered the stochastic motion of dust particles as a Markov process with no memory. It means that the stochastic motion of the dust after the time t is entirely independent of its history before the time t, i.e., the dust particle has no memory of the past.The values of the relaxation timescale for dust charge fluctuations and the dust velocity entirely depend on the dusty plasma parameters. For example, Hoang et al. <cit.> showed that the relaxation time of dust charge fluctuations τ_c is comparable to the relaxation time of the dust velocity τ_r, i.e., τ_c≈τ_r for very small dust particles with radius a⩽ 5×10^-8 cm in the interstellar medium. In addition to space dusty plasmas, in laboratory dusty plasmas depending on the plasma parameters, such as pressure or density of the neutral gas, τ_c is comparable to τ_r. As a result, in such situations, the main assumptions of fast charge fluctuations and the Markov process for the velocity relaxation of the dust particle become inappropriate. Thus, memory effects are important in the velocity relaxation of dust particles as a non-Markov process, and they cannot generally be neglected.In this paper, we study memory effects in the velocity relaxation of the dust particle exposed to the random force due to charge fluctuations. We present an analytic model based on a fractional Langevin equation. We will show that in the presence of memory effects in the velocity relaxation, dust charge fluctuations can cause an anomalous diffusion of the dust particle in a dusty plasma. The anomalous diffusion of dust particles has been experimentally observed in laboratory dusty plasmas <cit.>, and our research provides a possible reason for this behavior based on the memory effects. It is important to note here that the diffusion of a dust particle means a process of random displacements of a dust particle in a specified time interval.The paper is organized as follows. In section <ref>, we introduce major timescales characterizing dynamics of the dust particle. In section <ref>, a model based on the fractional Langevin equation is presented and solved using the Laplace transform technique. In section <ref>, we calculate the mean-square displacement and the velocity autocorrelation function of the dust grain. Section <ref> is devoted to the analysis of the asymptotic behavior of the results. Section <ref>contains summary and conclusions.§ TIMESCALESLet us consider an isolated spherical dust particle in the sheath, and study the relaxation timescales of dust charge fluctuations and the dust velocity. The particle is charged by collecting electrons and ions from the plasma. The particle charge fluctuates about the steady-state value because of the discrete nature of the electron and ion currents <cit.>.It was shown that the autocorrelation function of dust charge fluctuations has the following form <cit.>⟨δ Z(t)δ Z(t')⟩=⟨δ Z^2⟩ exp(-β| t-t'|),where δ Z(t)=Z(t)–Z_0, Z(t) is the instantaneous charge, Z_0 is the steady-state (equilibrium) charge, ⟨δ Z^2⟩ is the square of the amplitude of random charge fluctuations, and charging frequency β is defined as the relaxation frequency for small deviations of the charge from the equilibrium value Z_0. For the isolated dust particle under the condition a≪λ_D≪λ_mfp, where a is the dust radius, λ_D is the screening length due to electrons and ions, and λ_mfp is the mean free path for electron-neutral or ion-neutral collisions, the charging frequency can be calculated by usingthe orbital- motion-limited (OML) theory <cit.>β=-d(I_i-I_e)/dZ|_Z=Z_0=1+z/√(2π)a/λ_Diω_piwhere I_i=√(8π)a^2n_iv_Ti(1-Ze^2/aT_i) is the ion flux, I_e=√(8π)a^2n_ev_Teexp(Ze^2/aT_e) is the electron flux to the particle surface, v_Ti(e)=(T_i(e)/m_i(e))^1/2 is the ion (electron) thermal velocity, T_i(e), m_i(e), and n_i(e) are ion (electron) temperature, mass, and number density, respectively, z=| Z | e^2/aT_e is the absolute magnitude of the particle charge in the units aT_e/e^2, λ_Di=√(T_i/4π e^2n_i) is the ionic Debye radius, and ω_pi=v_Ti/λ_Di is the ion plasma frequency.Let us now consider an isolated dust particle with fluctuating charge in the sheath. To separate the dust particle transport due to dust charge fluctuations from other processes, which in turn can influence the dust motion in the plasma sheath, we only consider thegravitational, electric field, and neutral drag forces on the dust. We hence neglect the ion drag, electron drag, and thermophoretic forces, collisions between dust particles, and other processes, which can be included in more realistic models <cit.>. The motion of this dust particle is treated as a stochastic process because of the stochastic nature of the force due to dust charge fluctuations, and it can generally be modeled by a normal Langevin equation of the form <cit.> v̇(t)+γ v(t)=f(t)+ξ(t),where v(t) is the velocity of the dust particle, -γ v(t) is the neutral drag force per unit mass,γ is the damping rate due to neutral gas friction, ξ(t) is the stochastic Langevin force per unit mass due to collisions with the neutral gas molecules, and f(t) is the random force per unit mass representing the effect of dust charge fluctuations. Here, the forces acting on a dust are the electric force due to the sheath electric field, the gravitational force, and the stochastic Langevin force, i.e., F(t)=F_Z(t)+F_g+ξ(t). The electric force is given by F_Z(t)=eEZ(t), where Z(t)=Z_0+δ Z(t) is the instantaneous dust charge, and E is the electric field. Note that, for simplicity, we have neglected the fluctuations of the electric field. The force can be written as F(t)=F_0+f(t)+ξ(t), where F_0=eEZ_0+F_g. Thus, the random force due to dust charge fluctuations has the following formf(t)=eEδ Z(t).In a steady state eEZ_0+F_g=0, so that F(t)=f(t)+ξ(t). By using Eqs. (<ref>) and (<ref>), one can see that the random force due to dust charge fluctuations per unit mass has following properties⟨ f(t)⟩=0,⟨ f(t)f(t')⟩=e^2E^2/m_d^2⟨δ Z^2⟩ exp(-β|t-t'|)where m_d is the dust particle mass. The stochastic Langevin force per unit mass has following properties⟨ξ(t)⟩=0,⟨ξ(t)ξ(t')⟩=S_nδ(t-t'),where S_n is the intensity of the stochastic Langevin force. By solving the Eq. (<ref>), the dust velocity is obtained in the following formv(t)=v(0)e^-γ t+∫_0^t (f(t')+ξ(t'))e^-γ(t-t') dt';then, mean-square velocity is obtained in the form of⟨ v^2(t)⟩=v(0)e^-2γ t+∫_0^t∫_0^tdt'dt” e^-γ(t-t') e^-γ(t-t”)× ⟨(f(t')+ξ(t'))(f(t”)+ξ(t”))⟩.Then, by using the fact that the stochastic Langevin force and the random force due to dust charge fluctuations come from different sources; therefore, they are uncorrelated and independent, so that ⟨ f(t')ξ(t”)⟩=⟨ξ(t') f(t”)⟩=0,then by substituting in Eq. (<ref>), and using Eqs. (<ref>) and (<ref>), we obtain⟨ v^2(t)⟩=v(0)e^-2γ t+∫_0^t∫_0^tdt'dt” e^-γ(t-t') e^-γ(t-t”)× (S_nδ(t'-t”)+e^2E^2/m_d^2⟨δ Z^2⟩ e^-β(t'-t”)).By taking the integral and applying the limit t→∞, we obtain the long-time behavior of the mean-square velocity in the following form⟨ v^2(t)⟩_t→∞=S_n/2γ+e^2E^2/2m_d^2βγ⟨δ Z^2⟩.The mean-square velocity in long-time limit is representative of the dust temperatureT_d=m_d⟨ v^2(t)⟩_t→∞therefore, from Eq. (<ref>), one can see that the dust temperature has two part. One part is due to collisions with the neutral gas molecules, T_n, and another part is due to dust charge fluctuations, T_f, so thatT_d=T_n+T_f,where T_n=m_dS_n/2γ, T_f=e^2E^2/2m_dβγ⟨δ Z^2⟩.Note that since we want to study the role of dust charge fluctuations in the dust particle transport, we neglect the stochastic Langevin force, and we assume that the random force due to dust charge fluctuations is important. Therefore, in the absence of stochastic Langevin force (ξ(t)=0), T_d=T_f, and the Langevin equation Eq. (<ref>) reduces tov̇(t)+γ v(t)=f(t), Now, we study the major timescales characterizing dynamics of the charged grain. First, we introduce a dimensionless parameter ϵ, which characterizes memory effects in the velocity relaxation for stochastic processesϵ=τ_r/τ_c, τ_r=1/C_v(0)∫_0^∞ C_v(t)dt,τ_c=1/C_f(0)∫_0^∞ C_f(t)dtwhere τ_r is the relaxation time of the velocity, τ_c is the correlation (relaxation) time of the random force, C_v(t)=⟨ v(0)v(t)⟩ is the velocity autocorrelation function (VACF), and C_f(t)=⟨ f(0)f(t) ⟩ is the random force autocorrelation function. In general, stochastic processes are classified into two types of Markov and non-Markov processesbased on the timescales <cit.>: * A stochastic process is said to have memory effects in the velocity relaxation, if its relaxation time of the velocity is comparable to the relaxation time of the random force, i.e., τ_r≈τ_c. Then, according to Eq. (<ref>), we find that ϵ≈ 1. In this situation, the stochastic process is called a non-Markov process with memory effects in the velocity relaxation. The memory in the velocity means that the velocity of the particle at the current time depends on its velocity at all past times.* However, the situation τ_r≫τ_c corresponds to a memoryless behavior, meaning that the velocity at the current time is entirely independent of the velocity at all past times. In this case, the stochastic process is called a Markov process, and according to Eq. (<ref>), we find that ϵ→∞. Let us calculate the relaxation timescales for the dust velocity and the random force. We obtain the dust velocity from Eq. (<ref>) in the following formv(t)=v(0)e^-γ t+∫_0^t f(t')e^-γ(t-t') dt'.Multiplying Eq. (<ref>) by v(0), and performing an appropriate ensemble average ⟨...⟩, and by using ⟨ v(0)f(t)⟩=0, we obtainC_v(t)=⟨ v(0)^2⟩ e^-γ t;then, by substituting Eq. (<ref>) into (<ref>), we findτ_r=1/⟨ v(0)^2⟩∫_0^∞⟨ v(0)^2⟩ e^-γ tdt=1/γ.The random force autocorrelation function is obtained by substituting t'=0 into Eq. (<ref>)C_f(t)=e^2E^2/m_d^2⟨δ Z^2⟩ exp(-β t),then, by substituting C_f(t) into Eq. (<ref>), one findsτ_c=∫_0^∞ e^-β tdt=1/β.When the damping rate of the neutral gas is much smaller than the dust charging frequency, i.e., γ≪β, the stochastic process for the velocity v(t) of the dust particle can be considered as a Markov process, and memory effects in the velovity relaxation can reasonably be neglected. In this case, the Langevin equation (Eq. (<ref>)) is appropriate for the description of the dust motion under the influence of the random force due to charge fluctuations. In general, the values of the damping rate and the charging frequency entirely depend on the plasma parameters. For example, we consider three different types of plasmas with neon, argon, and kryptonneutral gases, and estimate the values of γ and β using typical experimental values for various parameters. We assumethe neutral gases at room temperature and in the range of pressures 0.5–1.0 Torr. We also assume T_e=4 eV, T_e/T_i=40, n_i=10^8 cm^-3, and the silica dust particle with radius a=0.5 μm and the mass density ρ=2 g/cm^3. For neon, argon, and krypton neutral gases with T_e/T_i=40, the absolute magnitudes of the dust charge are z∼2.6, 2.8, and 3.2, respectively <cit.>. With the given parameters, we calculate the charging frequency from Eq. (<ref>). The values of the relaxation time of the random force due to dust charge fluctuations τ_c=β^-1 (in seconds) are listed in the last column of Table <ref>.To calculate the timescale τ_r from Eq. (<ref>), we first need to find the damping rate. When the neutral gas mean free path is long compared to the dust grain radius, it is appropriate to use the Epstein drag force to calculate γ <cit.>. The mean free path values of neon, argon, and krypton atoms at the maximal pressure used (1.0 Torr) are approximately equal to 92, 60, and 40 μm, respectively. These are about 184, 120, and 80 times larger than the dust grain radius (0.5 μm). Thus, the damping rate is given by the Epstein formulaγ=δ√(8/π)(T_g/m_g)^-1/2P/ρ a,where m_g, T_g, P are the mass, temperature, and pressure of the neutral gas, respectively. The parameter δ is 1 for specular reflection or 1.39 for diffuse reflection of the neutral gas atoms from the dust grain <cit.>. We use δ=1.39 and the given parameters to calculate the damping rate from Eq. (<ref>). The relaxation time values of the dust velocityτ_r=γ^-1 (in seconds) are tabulated in Table <ref> for each gas at various pressures. As shown in Table <ref>,with increasing the pressure (or equivalently increasing the density) of the neutral gas, the relaxation time of the dust velocity becomes comparable to the relaxation time of the random force. Thus, in this situation, the assumption of the fast rate for charge fluctuations becomes inappropriate, and memory effects in the velocity of the dust particle cannot be neglected. As a result, the normal Langevin equation becomes inappropriate for the description of the dust motion because this equation is built on the Markovian assumption with no memory. Whenτ_r becomes comparable to τ_c, the memory effects in the relaxation of the dust velocity become important, because at the times of the same order ofτ_c, the random forces due to dust charge fluctuations are correlated. As a result, the velocity of the dust particle at the current time depends on its velocity at past times, and it means that the process of the dust velocity relaxation is retarded, and these retarded effects (or equivalently memory effects) are characterized with memory kernel in the friction force within Langevin equation. Now, this equation with retarded friction is called the fractional Langevin equation. Hence, the fractional Langevin equation is built on the non-Markovian assumption, while the Langevin equation without the existence a retarded friction is built on the Markov assumption. Note that, as shown in Table <ref>, we study the regime τ_r≈τ_c not τ_r≪τ_c, because in the regimeτ_r≪τ_c, the damping rate of the neutral gas is very high and the kinetic energy transferred to the dust (due to charge fluctuations) is totally dissipated by friction with the neutral gas, as mentioned in Ref. [10]. In the next section, we introduce a model based on fractional Langevin equation for the evolution process of the dust particle as a non-Markov process with memory effects in the velocity. § BASIC EQUATION AND FLUCTUATION-DISSIPATION THEOREMNow, we consider memory effects in the velocity relaxation of an isolated dustparticle exposed to the random force due to dust charge fluctuations. We model the motion of the dust grain based on the fractional Langevin equation (FLE) because this equation includes a retarded friction force with a memory kernel function, which is non-local in time, and shows memory effects in the velocity relaxation of the dust particle. The FLE is as follows <cit.>v̇(t)+γ̅/Γ(1-α)∫_0^t (| t-t'|/τ_c)^-αv(t')dt'=f(t),where 0<α<1, and Γ(1-α) is the gamma function. γ̅ is the scaling factor with physical dimension (time)^-2, and must be introduced to ensure the correct dimension of the equation. We define γ̅=(τ_rτ_c)^-1, so that for α=1, and according to the Dirac generalized function, δ(t-t')=| t-t'|^-1/Γ(0) <cit.>, Eq. (<ref>) reduces to the normal Langevin equation (13).Equation (<ref>) can be rewritten in the following formv̇(t)+∫_0^tγ(t-t')v(t')dt'=f(t),whereγ(t-t')=(τ_rτ_c)^-1/Γ(1-α)(| t-t'|/τ_c)^-α is often called the memory kernel function. It is interesting to know that the name fractional in the fractional Langevin equation originates from the fractional derivative, which is defined in the Caputo sense as follows <cit.>d^αf(t)/dt^α=_0𝒟_t^α-1(df(t)/dt),where _0𝒟_t^α-1 is the Riemann-Liouville fractional integral <cit.>_0𝒟_t^α-1f(t)=1/Γ(1-α)∫_0^t(t-t')^-αf(t')dt';consequently, the fractional Langevin equation readsv̇(t)+γ̅/τ_c^-αd^αx(t)/dt^α=f(t);therefore, the name fractional Langevin equation is confirmed. Now, we need to derive a suitable relation between the memory kernel and the autocorrelation function of the random force. To this end, we first define the Fourier transforms for the velocity and random force as followsv(ω)=∫_-∞^∞v(t) e^iω tdt,f(ω)=∫_-∞^∞f(t) e^iω tdt.Using Eq. (<ref>) and taking the Fourier transform of Eq. (<ref>) yieldsv(ω)=f(ω)/-iω+γ(ω)where γ(ω), defined by γ(ω)=∫_0^∞γ(t) e^iω tdt, denotes the Fourier-Laplace transform of the memory kernel γ(t). The velocity autocorrelation function is defined by the formula:C_v(τ)=⟨ v(t)v(t+τ)⟩=lim_θ→∞1/θ∫_-θ/2^θ/2v(t)v(t+τ)dt,where θ is the time interval for the integration. Generally, the power spectrum S(ω) and the autocorrelation function of a stochastic process C(τ) are related by the Wiener-Khintchine theorem <cit.> S(ω)=∫_-∞^∞C(τ) e^iωτdτ, C(τ)=1/2π∫_-∞^∞S(ω) e^-iωτdω. By substituting C_v(τ) from Eq. (<ref>) into (<ref>) and using Eq. (<ref>), we obtain the velocity power spectrum asS_v(ω)=lim_θ→∞1/θ| v(ω)|^2;then, by using Eq. (<ref>), we obtain the relation between the power spectrums of the velocity and random forceS_v(ω)=S_f(ω)/|γ(ω)-iω|^2,where S_f(ω)=lim_θ→∞1/θ| f(ω)|^2 is the power spectrum of the random force. Note that Eq. (<ref>) was obtained by using the fractional Langevin equation for the non-Markov velocity process. In the same way, we obtain the relation between the power spectrums of the velocity and random force for the Markov velocity process by using the normal Langevin equation in the following formS_v(ω)=S_f(ω)/|γ-iω|^2.When the power spectrum of the random force S_f(ω) is given, the above equation yields the velocity power spectrum S_v(ω) from which the VACF is obtained by Eq. (<ref>). If VACF should include the velocity in the thermal equilibrium (i.e., at sufficiently long times), S_f(ω) is required to satisfy a special condition. Now, we obtain this condition. In the long-time limit (| t-t'|≫τ_c), the autocorrelation function of the random force obtained from Eq. (<ref>) reduces to the following formC_f(τ)=⟨ f(t)f(t+τ)⟩=e^2E^2/m_d^2β⟨δ Z^2⟩δ(τ);then, by using Eq. (<ref>), the force power spectrum readsS_f=e^2E^2/m_d^2β⟨δ Z^2⟩.The mean-square velocity ⟨ v^2⟩ can be evaluated by substituting τ=0 into Eq. (<ref>). Thus, by using Eqs. (<ref>), (<ref>), and (<ref>), one finds⟨ v^2⟩=S_f/2γ=e^2E^2⟨δ Z^2⟩/2m_d^2βγ.As we mentioned before, the mean-square velocity in the long-time limit (the equilibrium velocity) is representative of the dust temperatureT_d=m_d⟨ v^2⟩.Thus, by using Eqs. (<ref>) and (<ref>), the special condition for the power spectrum of the random force S_f(ω) is obtained byS_f=2T_dγ/m_d.It is important to note that this condition, i.e., Eq. (<ref>), was obtained by using Eq. (<ref>) for the Markov velocity process. For the non-Markov velocity process given by the FLE , the relation between the power spectrums of the velocity and random force is given by Eq. (<ref>) instead of (<ref>); hence, the condition for the power spectrum S_f(ω), in this case, is a generalization of Eq. (<ref>) as followsS_f(ω)=2T_d/m_dRe(γ(ω));then, by using Eqs. (<ref>), (<ref>), and (<ref>), we obtain⟨ f(ω)f^*(ω')⟩=2π S_f(ω)δ(ω-ω') ⟨ f(ω)f^*(ω')⟩=4π T_d/m_dRe(γ(ω)δ(ω-ω')),where f^*(ω') is the complex conjugate of the function f(ω'). Therefore, we obtain the relation between the autocorrelation function of the random force and the memory kernel by using the inverse Fourier transform ofEq. (<ref>)⟨ f(t)f(t')⟩=T_d/m_dγ(t-t')=S_f/2γγ(t-t').We call this the fluctuation-dissipation theorem for the dust particle because dust charge fluctuations (S_f∝⟨δ Z^2⟩, according to Eq. (<ref>)) are necessarily accompanied by the friction (γ in the denominator). It is important to note that in FLE, we have normalized the time | t-t'| to the correlation time τ_c. The reason for this can be found from the fluctuation-dissipation theorem. According to Eq. (<ref>), for | t-t'|≫τ_c, the memory kernel relaxes to zero. Consequently, by using the fluctuation-dissipation theorem, ⟨ f(t)f(t')⟩ relaxes to zero. Thus, the characteristic time of the memory relaxation is the correlation time of the random force due to charge fluctuations.Now, we solve Eq. (<ref>) for the dust velocity with the initial conditions x_0=x(0) and v_0=v(0), by using the Laplace transform technique. We obtainv(t)=⟨ v(t)⟩+∫_0^t g(t-t')f(t')dt',where ⟨ v(t)⟩=v_0g(t) is the mean dust velocity, and the function g(t) is the inverse Laplace transform ofĝ(s)=1/s+γ̂(s)=1/s+γ(τ_c s)^α -1,where γ̂(s)=γ(τ_c s)^α -1 is the Laplace transform of the memory kernel γ(t), and the Laplace transform of the function f(t) is defined by f̂(s)=∫_0^∞ f(t)e^-stdt. To find the function g(t), we first need to introduce the Mittag-Leffler (ML) function E_α(at^α) as follows <cit.>E_α(at^α)=∑_n=0^∞(at^α)^n/Γ(α n+1), α>0.The ML function is a generalization of the exponential function. For α=1, we have the exponential function, E_1(at)=e^at. The Laplace transform of the ML function is given by <cit.>∫_0^∞ E_α(at^α)e^-stdt=s^-1/1-as^-α.Thus, by using Eqs. (<ref>) and (<ref>), one findsg(t)=E_2-α(-γτ_c(t/τ_c)^2-α). The dust particle velocity is the derivative of the dust position. Therefore, the position of the dust particle is obtained by x(t)=⟨ x(t)⟩+∫_0^t G(t-t')f(t')dt',where ⟨ x(t)⟩=x_0+v_0G(t) is the mean dust particle position, and G(t)=∫_0^t g(t')dt'. To calculate the function G(t) in terms of the ML function, we use the following identity <cit.>d/dtt^β-1E_α,β(at^α)=t^β-2E_α,β-1(at^α),where E_α,β(at^α)=∑_n=0^∞(at^α)^n/Γ(α n+β), α,β>0is the generalization of the ML function that for β=1 reduces to Eq. (<ref>), i.e., E_α,1(at^α)=E_α(at^α) <cit.>. Then, by using Eqs. (<ref>), (<ref>), and also using g(t)=dG(t)/dt, one readsG(t)=tE_2-α,2(-γτ_c(t/τ_c)^2-α).It is worth mentioning that the functions G(t) and g(t) help us find the mean-square displacement (MSD) and the velocity autocorrelation function of the dust particle. In the next section, we evaluate the MSD and VACF for the dust particle. § MEAN-SQUARE DISPLACEMENT AND VELOCITY AUTOCORRELATION FUNCTIONBelow, we calculate two diagnostics to characterize the random motion of the dust particle due to random charge fluctuations. The first diagnostic is MSD, and it is the most common quantitative tool used to investigate random processes. The MSD of a dust particle is defined by the relationMSD=⟨ (x(t)-x_0)^2 ⟩=(⟨ x(t) ⟩-x_0)^2+σ_x^2,where σ_x^2=⟨ x^2(t) ⟩- ⟨ x(t) ⟩^2 is the variance of the displacement, and ⟨...⟩ denotes an average over an ensemble of random trajectories. By using Eqs. (<ref>), (<ref>), and (<ref>), we haveσ_x^2=e^2E^2⟨δ Z^2⟩/2m_d^2βγ ∫_0^t dt'∫_0^tdt”G(t-t')× G(t-t”)γ(t'-t”).Using the double Laplace transform technique <cit.>, and after some calculations, we obtain the following expression for the varianceσ_x^2=e^2E^2⟨δ Z^2⟩/2m_d^2βγ(2I(t)-G^2(t)),where I(t)=∫_0^tG(t')dt'. The function I(t) is obtained from Eqs. (<ref>) and (<ref>) as followsI(t)=t^2E_2-α,3(-γτ_c(t/τ_c)^2-α).By substituting Eq. (<ref>) into (<ref>), and using x_0=0 and ⟨ x(t) ⟩=v_0G(t), we haveMSD=G^2(t)(v_0^2-e^2E^2⟨δ Z^2⟩/2m_d^2βγ) +e^2E^2⟨δ Z^2⟩/m_d^2βγI(t);then, from Eqs. (<ref>) and (<ref>), we getMSD=e^2E^2⟨δ Z^2⟩/m_d^2βγt^2 E_2-α,3(-γτ_c(t/τ_c)^2-α) +t^2E^2_2-α,2(-γτ_c(t/τ_c)^2-α)(v_0^2-e^2E^2⟨δ Z^2⟩/2m_d^2βγ).The second diagnostic, VACF, is the average of the initial velocity of a dust particle multiplied by its velocity at a later time. To calculate the VACF of the dust grain, we first calculate the double Laplace transform ⟨v̂(s)v̂(s')⟩ of the function ⟨v̂(t)v̂(t')⟩, and we obtain⟨v̂(s)v̂(s')⟩=(v_0^2-e^2E^2⟨δ Z^2⟩/2m_d^2βγ)ĝ(s)ĝ(s') + e^2E^2⟨δ Z^2⟩/2m_d^2βγĝ(s)+ĝ(s')/s+s',where 𝑣̂(s), ĝ(s), and Ĝ(s) are Laplace transforms of 𝑣(t),𝑔(t), and G(t), respectively. Taking the inverse double Laplace transform of Eq. (<ref>) yields⟨ v(t)v(t')⟩=(v_0^2-e^2E^2⟨δ Z^2⟩/2m_d^2βγ)g(t)g(t') + e^2E^2⟨δ Z^2⟩/2m_d^2βγg(t-t');therefore, by substituting t'=0 into Eq. (<ref>), the VACF can be written asC_v(t)=v_0^2g(t)=v_0^2E_2-α(-γτ_c(t/τ_c)^2-α).By substituting α=1 into Eq. (<ref>), we obtainC_v(t)=v_0^2 e^-γ t,which corresponds to the function C_v(t) in Eq. (<ref>) obtained by the Langevin equation (13). As we expected, the velocity autocorrelation function of dust particle decays exponentially with time for the memoryless case.§ ASYMPTOTIC BEHAVIORSIn order to gain physical insight, we investigate the long-time behavior of the MSD and VACF. We first study the asymptotic behavior of the function g(t) for t≫τ_c.The asymptotic behavior of the Mittag-Leffler functions is given by <cit.>E_α(y)∼-y^-1/Γ(1-α), y→∞, E_α,β(y)∼-y^-1/Γ(β-α), y→∞.By using Eqs. (<ref>) and (<ref>), we haveg(t)∼1/γτ_cΓ(α-1)(t/τ_c)^α-2.Also, by using Eqs. (<ref>), (<ref>), and (<ref>), we readG(t)∼1/γΓ(α)(t/τ_c)^α-1,and I(t)∼τ_c/γΓ(1+α)(t/τ_c)^α. To calculate the dust temperature due to charge fluctuations, by substituting t'=t into Eq. (<ref>); then, by using Eq. (<ref>), we obtain⟨ v^2(t)⟩∼(v_0^2-e^2E^2⟨δ Z^2⟩/2m_d^2βγ)(t/τ_c)^2α-4/(γτ_cΓ(α-1))^2 + e^2E^2⟨δ Z^2⟩/2m_d^2βγ.We observe that a slow power-law decay like t^2α-4 for the mean-square velocity. In the long-time limit, by applying t≫τ_c to Eq. (<ref>), we obtain the temperature of the dust particle as followsT_d=m_d⟨ v^2⟩=e^2E^2⟨δ Z^2⟩/2m_dβγ,which coincides with the temperature obtained from the Markov velocity process <cit.>. To find the long-time behavior of the MSD and VACF of the dust particle, without loss of generality, we can choose the initial conditions in the following formx_0=0,v_0^2=e^2E^2⟨δ Z^2⟩/2m_d^2βγ,where v_0^2 is the mean-square velocity in the thermal equilibrium given by Eq. (<ref>). Thus, by using Eqs. (<ref>), (<ref>), and (<ref>), MSD can be written asMSD=⟨ x^2(t) ⟩∼e^2E^2⟨δ Z^2⟩/m_d^2β^2γ^2Γ(1+α)(t/τ_c)^α.The MSD obtained from Eq. (<ref>) is a nonlinear function of time because the exponent α takes the values 0<α<1. It implies that fluctuating force by considering memory effects in the velocity relaxation of the dust particle leads to the anomalous diffusion of the dust particle. We emphasize here that the diffusion of a dust particle means a process of random displacements of a dust particle in a specified time interval. In Figure <ref>, we display the curves ⟨ x^2(t) ⟩ for the values α=0.25, 0.5, and 0.75. For the given parameters α, the MSD increases asymptotically slower than linearly with time (the signature of anomalous diffusion).Diffusion coefficient for the anomalous diffusion processes is given by the generalized Green-Kubo relation in the following form <cit.>D_α=1/Γ(1+α)∫_0^∞_0𝒟_t^α-1C_v(t)dt,where D_α is often called the generalized diffusion coefficient, and _0𝒟_t^α-1 is the Riemann-Liouville fractional integral [see Eq. (<ref>)] . For α=1, this relation reduces to the standard Green-Kubo relation, which holds for the normal diffusion <cit.>D=∫_0^∞ C_v(t)dt.Generalized diffusion coefficient can easily be obtained by the Laplace transform of Eq. (<ref>). The Laplace transform of the Riemann-Liouville fractional integral is given by <cit.>∫_0^∞_0𝒟_t^α-1 C_v(t)e^-stdt=s^α-1Ĉ_v(s);then, by using lim_s → 0∫_0^∞_0𝒟_t^α-1 C_v(t)e^-stdt=∫_0^∞_0𝒟_t^α-1 C_v(t)dt,and using Eq. (<ref>), we findD_α=lim_s → 01/Γ(1+α)s^α-1Ĉ_v(s).The function Ĉ_v(s) is obtained by using Eqs. (<ref>), (<ref>), and (<ref>) as followsĈ_v(s)=v_0^2ĝ(s)=e^2E^2⟨δ Z^2⟩/2m_d^2βγ1/s+γ(τ_c s)^α-1;therefore, by substituting Eq. (<ref>) into (<ref>), we find the generalized diffusion coefficient of the dust particleD_α=e^2E^2⟨δ Z^2⟩/2Γ(1+α)m_d^2β^2γ^2τ_c^α,which has a physical dimension (length)^2/(time)^α.Moreover, using Eqs. (<ref>), (<ref>), and (<ref>), the long-time behavior of the velocity autocorrelation function of the dust particle can be written asC_v(t)∼e^2E^2⟨δ Z^2⟩/2m_d^2γ^2Γ(α-1)(t/τ_c)^α-2.Thus, the function C_v(t) for long times exhibits a power-law decay instead of the exponential decay [see Eq. (<ref>)]. The power-law decay can obviously be seen in Figure <ref> that we show the curves C_v(t) for the values α=0.25, 0.5, and 0.75. For the given values of the parameter α , the function Γ(α-1) is negative, and the curves C_v(t) have a negative tail at all times, C_v(t)<0. As we mentioned,the function C_v(t) is the average of the velocity of a dust particle at the time t multiplied by its velocity at a later time. When C_v(t) is negative, this means that there are anti-correlations between the dust particle velocities at the time t and the later time t+t', i.e., the diffusing dust particle tends to change the direction of its motion and goes back, which indicates an anti-persistent motion for the dust particle. In other words, the motion of the dust particle in a direction at the time t will be followed by its motion in the opposite direction at the later time, and the dust particle prefers to continually change its direction instead of continuing in the same direction; as a result, the diffusion of the dust grain is slower than normal case in normal diffusion.As we expected, in the very long times limit, the velocity autocorrelation function of the dust particle decays to zero. It can also be seen by applying the limit t→∞ to Eq. (<ref>), and this result is consistent with the result C_v(t)→0 obtained from Markov dynamics (by applying the limit t→∞ to C_v(t)=v_0^2e^-γ t from Eq. (<ref>), C_v(t) goes to zero). The reason for this consistency between the results (i.e., vanishing C_v(t) in the very long times limit) obtained from non-Markov and Markov dynamics can be understood as follows. As we showed, at the times of the order of τ_r≈τ_c, memory effects in the velocity relaxation process of the dust particle become important, and their effects yield to power-law behavior for MSD and VACF. However, in very long times, i.e., when the observation time t is much longer than all characteristic time scales including the correlation time of the random force due to dust charge fluctuations τ_c and the relaxation time of the dust velocity τ_r, there are no longer the memory effects in the velocity of the dust particle, and the dynamics of the dust particle in the very long times limit is a Markov dynamics with no memory effects in the velocity. Therefore, as in Markov dynamics in the limit t→∞, the function C_v(t) goes to zero, the function C_v(t) obtained from Eq. (<ref>) in the limit t→∞ approaches to zero.§ SUMMARY AND CONCLUSIONSWe have studied memory effects in the velocity relaxation of an isolated dust particle.First, we have compared the relaxation timescales of the fluctuating force and the dust velocity, and have shown that they can be of the same order of the magnitude, depending on the plasma parameters.Thus, the fast charge fluctuations assumption does not always hold, and memory effects in the velocity relaxation should generally be considered.We have developed a model based on the fractional Langevin equation for the evolution of the dust particle. Memory effects in the velocity relaxation of the dust particle have been introduced by using the memory kernel in this equation.We have derived a suitable fluctuation-dissipation theorem for the dust grain, which relates the autocorrelation function of the random force to the memory kernel. Then, the fractional Langevin equation has been solved by the Laplace transform technique, and the mean-square displacement and the velocity autocorrelation function of the dust grain have been obtained in terms of the generalized Mittag-Leffler functions.We have investigated the asymptotic behaviors of the MSD, VACF, generalized diffusion coefficient, and the dust temperature due to charge fluctuations in the long-time limit. We have found that in the presence of memory effects in the relaxation of the dust velocity, dust charge fluctuations can cause the anomalous diffusion of the dust particle, which has been experimentally observed in laboratory dusty plasmas <cit.>. In this case, the mean-square displacement of the dust grain has a nonlinear dependence on time, and the velocity autocorrelation function decays as a power-law instead of the exponential decay.* | http://arxiv.org/abs/1704.07980v1 | {
"authors": [
"Zahra Ghannad",
"Hossein Hakimi Pajouh"
],
"categories": [
"physics.plasm-ph"
],
"primary_category": "physics.plasm-ph",
"published": "20170426060142",
"title": "Memory effects in the velocity relaxation process of the dust particle in dusty plasma"
} |
[ [ December 30, 2023 ===================== SHONOSUKE SUGASAWARisk Analysis Research Center, The Institute of Statistical MathematicsAbstract. Parametric empirical Bayes (EB) estimators have been widely used in variety of fields including small area estimation, disease mapping. Since EB estimator is constructed by plugging in the estimator of parameters in prior distributions, it might perform poorly if the estimator of parameters is unstable. This can happen when the number of samples are small or moderate. This paper suggests bootstrapping averaging approach, known as “bagging" in machine learning literatures,to improve the performances of EB estimators. We consider two typical hierarchical models, two-stage normal hierarchical model and Poisson-gamma model, and compare the proposed method with the classical parametric EB method through simulation and empirical studies. Key words: Bagging; Hierarchical model; Mean squared error; Poisson-gamma model§ INTRODUCTIONThe parametric empirical Bayes estimators (Morris, 1983) are known to be a useful method producing reliable estimates of multidimensional parameters. This technique is widely used in variety of fields such as small area estimation (Rao and Molina, 2015) and disease mapping (Lawson, 2013). Let þ_1,…,þ_m be the multiple parameters of interest, and y_1,…,y_m be the independent observations generated from the distribution f_i(y_i|þ_i),i=1,…,m. To carry out an empirical Bayes estimation, it is assumed that the parameters þ_1,…,þ_m independently follows the distribution g(þ_i; ), whereis a vector of unknown parameters. Therefore, we obtain a two stage model:y_i|þ_i ∼ f_i(y_i|þ_i),þ_i∼ g(þ_i;), i=1,…,m,which are independent for i=1,…,m. Under the setting, the posterior distribution of þ_i is given by π(θ_i |y_i;)=f_i(y_i | θ_i)g(θ_i;)/∫ f_i(y_i | θ_i)g(θ_i;)dþ_i, i=1,…,m.The Bayes estimator _i of þ_i under squared error loss is the conditional expectation (posterior mean) of þ_i given y_i, that is_i≡[þ_i|y_i;]=∫þ_if_i(y_i | θ_i)g(θ_i;)dþ_i/∫ f_i(y_i | θ_i)g(θ_i;)dþ_i, i=1,…,m.However, the Bayes estimator _i depends on unknown model parameters , which can be estimated from the marginal distribution of all the data y={y_1,…,y_m}, given byL()=∏_i=1^m∫ f_i(y_i | θ_i)g(θ_i;)dþ_i.Using the marginal distribution of y, one can immediately define the maximum likelihood (ML) estimator as the maximizer of L(). Based on the estimator , we obtain the empirical Bayes (EB) estimator of þ_i as _i=[þ_i|y_i;].The variability of the EB estimator _i can be measured by the integrated mean squared error (MSE) [(_i-þ_i)^2], where the expectation is taken with respect to þ_i's and y_i's following the model (<ref>).Since _i is the conditional expectation as given in (<ref>), the MSE can be decomposed as [(_i-þ_i)^2]=R_1+R_2 with R_1=[(_i-þ_i)^2] and R_2=[(_i-_i)^2]. The first term R_1 is not affected by the estimation ofwhereas the second term R_2 reflects the variability of the ML estimator , so that the second term can be negligibly small when m is large. However, in many applications, m might be small or moderate, in which the contribution of the second term to the MSE cannot be ignored. Hence, the EB estimator might perform poorly depending on the ML estimator . To overcome this problem, we propose to use the bootstrap averaging technique, known as “bagging" (Breiman, 1996) in machine learning literatures. This method produces many estimators based on bootstrap samples, and average them to produce a stable estimator. We adapt the bagging method to the EB estimation to improve the performances of EB estimators under small or moderate m.This paper is organized as follows: In Section <ref>, we consider mean squared errors of EB estimators and propose a bootstrap averaging empirical Bayes (BEB) estimator for decreasing the mean squared error.In Section <ref> and Section <ref>, we apply the BEB estimators in well-known two-stage normal hierarchical model and Poisson-gamma model, respectively, and compare the performances between BEB and EB estimators through simulation and empirical studies. In Section <ref>, we provide conclusions and discussions.§ BOOTSTRAP AVERAGING EMPIRICAL BAYES ESTIMATORSAs noted in the previous section, the performances of the EB estimators depend on the variability of the estimator , which cannot be ignored when m is not large. To reduce the variability of the empirical Bayes estimator _i, we propose to average many empirical Bayes estimators with bootstrap estimates ofrather than computing one empirical Bayes estimator from the observation Y={y_1,…,y_m}. Specifically, letting Y_(b)={y_1^(b),…,y_m^(b)} be a bootstrap samples of the original observation Y, we define _(b) be an estimator ofbased on the bootstrap sample Y_(b). Then the bagging empirical Bayes (BEB) estimator is given by_i^=1/B∑_b=1^B_i(y_i,_(b)). Similarly to Breiman (1996), we note that 1/B∑_b=1^B{_i(y_i,_(b))-þ_i}^2=1/B∑_b=1^B_i(y_i,_(b))^2-2_i^þ_i+þ_i^2≥{1/B∑_b=1^B_i(y_i,_(b))}^2-2_i^þ_i+þ_i^2 =(_i^-þ_i)^2.By taking expectation with respect to the model (<ref>), we have1/B∑_b=1^B[{_i(y_i,_(b))-þ_i}^2] ≥[(_i^-þ_i)^2],which means that the integrated MSE of BEB estimator (<ref>) is smaller than bootstrap average of the integrated MSE of the EB estimator. Hence, the BEB estimator is expected to perform better than the EB estimator. The amount of improvement depends on 1/B∑_b=1^B_i(y_i,_(b))^2-{1/B∑_b=1^B_i(y_i,_(b))}^2 =1/B∑_b=1^B{_i(y_i,_(b))-_i^}^2,which is the bootstrap variance of the EB estimator and it vanishes as m→∞ but it would not be negligible when m is not large. Therefore, when m is small or moderate, the BEB estimator would improve the performance of the EB estimator. In the subsequent section, we investigate the performances of the EBE estimator compared with the EB estimator in the widely-used hierarchical models.§ TWO-STAGE NORMAL HIERARCHICAL MODEL §.§ Model descriptionWe first consider the two-stage normal hierarchal model to demonstrate the proposed bagging procedure. The two-stage normal hierarchical model is described as y_i|þ_i∼ N(þ_i, D_i), þ_i∼ N(_i^t,A), i=1,…,m,where D_i is known sampling variance, _i andare a vector of covariates and regression coefficients, respectively, A is an unknown variance.Let =(^t,A)^t be the vector of unknown parameters. The model (<ref>) is known as the Fay-Herriot model (Fay and Herriot, 1979) in the context of small area estimation.Under the model (<ref>), the Bayes estimator of þ_i is_i(y_i;)=_i^t+D_i/A+D_i(y_i-_i^t).Concerning the estimation of unknown parameter , we here consider the maximum likelihood estimator for simplicity. Since y_i∼ N(_i^t,A+D_i) under the model (<ref>), the maximum likelihood estimatoris defined as the maximizer of the function:Q()=∑_i=1^mlog(A+D_i)+∑_i=1^m(y_i-_i^t)^2/A+D_i.While several other estimating methods are available, we here only consider the maximum likelihood estimator for presentational simplicity. Using the maximum likelihood estimator , we obtain the EB estimator of þ_i as _i(y_i;). §.§ Simulation studyWe here evaluate the performances of the BEB estimator together with the EB estimator under the normal hierarchical model (<ref>) without covariates, namely _i^t=μ. We considered m=10, 15,…,40. For each m, we set D_i as equally spaced points between 0.5 and 1.5. Concerning the true parameter values, we used μ=0 and four cases for A, namely A=0.1, 0.3, 0.5 and 0.7. The simulated data was generated from the model (<ref>) in each iteration, and computed the EB and BEB estimates of þ_i. Based on R=5000 simulation runs we calculated the simulated mean squared errors (MSE) defined as MSE=1/mR∑_i=1^m∑_r=1^R(_i^(r)-þ_i^(r))^2, where _i^(r) is the EBE or EB estimates and þ_i^(r) is the true value of þ_i in the rth iteration. In Figure <ref>, we present the simulated MSE of the EB estimator as well as the three BEB estimator using 25, 50 and 100 bootstrap samples under various settings of A and m. It is observed that the BEB estimator performs better than the EB estimator on the whole. In particular, the improvement is greater when A is small compared with D_i, which is often arisen in practice. Moreover, as the number of m gets larger, the MSE differences get smaller since the variability of estimatingvanishes when m is sufficiently large. We also found that the ML estimator of A often produces 0 estimates when m is small, in which the EB estimator is known to perform poorly. However, the BEB estimator can avoid the problem since the BEB estimator is aggregated by B bootstrap estimators and at least one bootstrap estimates should be non-zero. In fact, by investigating the case where the ML estimator produces 0 estimates of A, the some bootstrap estimates of A were away from 0. This would be one of the reason why the BEB estimator performs better than the EB estimator in this setting.§.§ Example: corn dataWe next illustrate the performances of the BEB estimator by using the corn and soybean productions in 12 Iowa counties, which has been used as an example in the context of small area estimation. Especially, we use the area-level data set given in table 6 in Dass et al. (2012) and we here focus only on corn productions for simplicity. The data set consists of m=8 areas with sample sizes in each area ranging from 3 to 5, and survey data of corn production y_i, sampling variance D_i and the satellite data of corn x_i as the covariate observed in each area. We considered the following hierarchical model:y_i|þ_i∼ N(þ_i,D_i),þ_i∼ N(β_0+β_1x_i,A), i=1,…,m,where β_0,β_1 and A are unknown parameters. For the data set, we computed the BEB as well as EB estimators. We used 1000 bootstrap samples for computing the BEB estimator. In Figure <ref>, we present the histogram of of the bootstrap estimates used in the BEB estimates and the maximum likelihood (ML) estimates used in the EB estimators. We can observe that the bootstrap estimates vary depending on the bootstrap samples. Moreover, in Table <ref>, we show the BEB and EB estimates of þ_i, which shows that the BEB estimator produces different estimates from the EB estimator since the number of areas m is only 8.§ POISSON-GAMMA MODEL §.§ SetupThe Poisson-gamma model (Clayton and Kalder, 1987) is described asz_i|þ_i∼Po(n_iþ_i),þ_i∼Γ(ν m_i,ν), i=1,…,m,where m_i=exp(_i^t), _i andare a vector of covariates and regression coefficients, respectively, ν is an unknown scale parameter.This model is used as the standard method of disease mapping. Let =(^t,ν)^t be the vector of unknown parameters. The model (<ref>) is known as the Poisson-Gamma model considered in Clayton and Kaldor (1987) and used in disease mapping. Under the model (<ref>), the Bayes estimator of þ_i is given by_i(y_i;)=z_i+ν m_i/n_i+ν.Since the Bayes estimator depends on unknown , we need to replaceby its estimator. Noting that the gamma prior of þ_i is a conjugate prior for the mean parameter in the Poisson distribution, the marginal distribution of y_i is the negative binomial distribution with the probability function:f_m(y_i;)=Γ(z_i+ν m_i)/Γ(z_i+1)Γ(ν m_i)(n_i/n_i+ν)^z_i(ν/n_i+ν)^ν m_i.Then the maximum likelihood estimator ofis defined as =argmax_∑_i=1^m log f_m(y_i;), which enables us to obtain the empirical Bayes estimator _i(y_i;). §.§ Simulation studyWe next evaluated the performances of the BEB estimator under the Poisson-gamma model without covariates, described as z_i|þ_i∼Po(n_iþ_i),þ_i∼(νμ,ν), i=1,…,m,where we set μ=1 and ν=40, 60, 80 and 100. Note that ν is a scale parameter and (þ_i)=μ/ν, so that random effect variance (þ_i) is a decreasing function of ν. Regarding the number of areas, we considered m=10,15,…,40. For each m, we set n_i as rounded integers of equally spaced numbers between 10 and 50.Similarly to Section <ref>, using (<ref>) with R=5000 simulation runs, we calculated the MSE of the BEB estimator as well as the EB estimator of þ_i.The results are presented in Figure <ref>, which show that the BEB estimator tends to perform better than the EB estimator. In particular, the amount of improvement is greater when m is not large as we expected. Moreover, we can also observe that the MSE difference tends larger as ν gets larger, which corresponds to the case where the random effect variance gets smaller. This is consistent to the results in the normal model given in Section <ref>.§.§ Example: Scottish lip cancerWe applied the BEB and EB method to the famous Scottish lip cancer data during the 6 years from 1975 to 1980 in each of the m=56 counties of Scotland. For each county, the observed and expected number of cases are available, which are respectively denoted by z_i and n_i. Moreover, the proportion of the population employed in agriculture, fishing, or forestry is available for each county, thereby we used it as a covariate AFF_i, following Wakefield (2007).For each area, i=1,…,m, we consider the Poisson-gamma model:z_i|þ_i∼Po(n_iþ_i),þ_i∼Γ(νexp(β_0+β_1AFF_i),ν),where þ_i is the true risk of lip cancer in the ith area, and β_0,β_1 and A are unknown parameters. For the data set, we computed the BEB as well as EB estimates of þ_i, where we used 1000 bootstrap samples for computing the BEB estimator. In Figure <ref>, we present the quantiles of the bootstrap estimates used in the BEB estimates and the maximum likelihood (ML) estimates used in the EB estimators. We can observe that the bootstrap estimates vary depending on the bootstrap samples while the variability seems small compared with Figure <ref>. This might comes from that the number of areas in this case is much larger than the corn data in Section <ref>. Finally, in Figure <ref>, we show the scatter plot of percent relative difference between the BEB and EB estimates, that is, 100(^-_i)/_i, against the number of expected number of cases n_i. Figure <ref> shows that the differences get larger as n_i gets small since the direct estimator y_i=z_i/n_i of þ_i is shrunk toward the regression mean exp(β_0+β_1AFF_i) in areas with small n_i. § CONCLUSION AND DISCUSSIONWe have proposed the use of bootstrap averaging, known as “bagging" in the context of machine learning, for improving the performances of empirical Bayes (EB) estimators. We focused on two models extensively used in practice, two-stage normal hierarchical model and Poisson-gamma model. In both models, the simulation studies revealed that the bootstrap averaging EB (BEB) estimator performs better than the EB estimator.In this paper, we considered the typical area-level models as an application of the BEB estimator. However, the BEB method would be extended to the more general case, for example, generalized linear mixed models. The detailed comparison in such models will be left to a future study.100 Breiman, L. (1996).Bagging predictors. Machine Learning, 24, 123-140.Clayton, D. and Kaldor, J. (1987).Empirical Bayes estimates of age-standardized relative risks for use in disease mapping.Biometrics, 43, 671-681. Dass, S. C., Maiti, T., Ren, H. and Sinha, S. (2012).Confidence interval estimation of small area parameters shrinking both means and variances.Survey Methodology, 38, 173-187.Fay, R. and Herriot, R. (1979). Estimators of income for small area places: An application of James-Stein procedures to census. Journal of the American Statistical Association, 74, 341-353.Lawson, A. B. (2013). Bayesian disease mapping: hierarchical modeling in spatial epidemiology, 2nd Edition. Chapman and Hall/CRC press. Morris, C. N. (1983). Parametric empirical Bayes inference: theory and applications. Journal of the American Statistical Association, 78, 47-65. Rao, J.N.K. and Molina, I. (2015) Small Area Estimation, 2nd Edition. Wiley.Wakefield, J. (2007). Disease mapping and spatial regression with count data. Biostatistics, 8, 158-183. | http://arxiv.org/abs/1704.08440v1 | {
"authors": [
"Shonosuke Sugasawa"
],
"categories": [
"stat.ME"
],
"primary_category": "stat.ME",
"published": "20170427054948",
"title": "On Bootstrap Averaging Empirical Bayes Estimators"
} |
Spreading law on a completely wettable spherical substrate: The energy balance approach Masao Iwamatsu December 30, 2023 ========================================================================================= Diffusion approximations have been a popular tool for performance analysis in queueing theory, with the main reason being tractability and computational efficiency. This dissertation is concerned with establishing theoretical guarantees on the performance of steady-state diffusion approximations of queueing systems. We develop a modular framework based on Stein's method that allows us to establish error bounds, or convergence rates, for the approximations. We apply this framework three queueing systems: the Erlang-C, Erlang-A, and M/Ph/n+M systems. The former two systems are simpler and allow us to showcase the full potential of the framework. Namely, we prove that both Wasserstein and Kolmogorov distances between the stationary distribution of a normalized customer count process, and that of an appropriately defined diffusion process decrease at a rate of 1/√(R), where R is the offered load. Futhermore, these error bounds are universal, valid in any load condition from lightly loaded to heavily loaded. For the Erlang-C model, we also show that a diffusion approximation with state-dependent diffusion coefficient can achieve a rate of convergence of 1/R, which is an order of magnitude faster when compared to approximations with constant diffusion coefficients. Anton received his Bachelors degree in Statistics and Mathematics from the University of Toronto in 2012.I am grateful to my parents for raising me to set high standards in life, and to my grandfather for being uncompromising in his attitude towards education. I am eternally grateful to Jim Dai for taking me on as an apprentice, and showing me how to stand on my own two feet. An even deeper sentiment goes out to my friends, who always provided me with an escape to normality whenever it was needed.Half of this research was fueled by my grandmother's delicious cooking.arabic cornell 0.5CHAPTER: INTRODUCTION Diffusion approximations have been a popular tool for performance analysis in queueing theory, with the main reason being tractability and computational efficiency. As an example, in<cit.>, an algorithm was developed to compute the stationary distribution of the diffusion approximation of the M/H_2/500+M system, which is a many-server queue with 500 servers, Poisson arrivals, hyper-exponential service times and customer abandonment.The approximation is remarkably accurate; see, for example, Figure 1 there. It was demonstrated there that computational efficiency, in terms of both time and memory, can be achieved by diffusion approximations. For example, in the M/H_2/500+M system, it took around 1 hour and peak memory usage of 5 GB to compute the stationary distribution of the customer count.On the same computer,it took less than 1 minute to compute the stationary distribution of the corresponding diffusion approximation, and peak memory usage was less than 200 MB. This dissertation is concerned with establishing error bounds on steady-state diffusion approximations of queueing systems.The main technical driver of our results is a mathematical framework known as Stein's method. Stein's method is a powerful method used for studying approximations of probability distributions, and is best known for its ability to establish convergence rates. It has been widely used in probability, statistics, and their wide range of applications such as bioinformatics; see, for example, the survey papers <cit.>, the recent book <cit.> and the references within. Applications of Stein's method always involve some unknown distribution to be approximated, and an approximating distribution. For instance, the first appearance of the methodin <cit.> involved the sum of identically distributed dependent random variables as the unknown, and the normal as approximating distribution. Other approximating distributions include the Poisson <cit.>, binomial <cit.>, and multinomial <cit.> distributions, just to name a few. To begin our discussion, we provide an example to illustrate the type of result that will be frequently encountered in this document.§ A TYPICAL RESULT Consider the Erlang-A and Erlang-C queuing systems. Both systems have n homogeneous servers that serve customers in a first-come-first-serve manner. Customers arrive according to a Poisson process with rate λ, and customer service times are assumed to be i.i.d. having exponential distribution with mean 1/μ. In the Erlang-A system, each customer has a patience time and when his waiting time in queue exceeds his patience time, he abandons the queue without service; the patience times are assumed to be i.i.d. having exponential distribution with mean 1/α. We consider the birth-death processX={X(t), t≥ 0},where X(t) is the number of customers in the system at time t.In the Erlang-A system, α is assumed to be positive and therefore the mean patience time is finite. This guarantees that the CTMC X is positive recurrent. In the Erlang-C system, α=0, and in order for the CTMC to be positive recurrent we need to assume that the offered load to the system, defined as R = λ / μ, satisfies R< n.For both Erlang-A and Erlang-C systems, we use X(∞) to denote the random variable having the stationary distribution of X. Consider the case when α = 0 and (<ref>) is satisfied. Set X̃(∞) = (X(∞) - R) /√(R), and let Y(∞) denote a continuous random variable onhaving densityκexp(1/μ∫_0^xb(y)dy),x ∈,where κ>0 is a normalizing constant that makes the density integrate to one,b(x) = ((x+ζ)^–ζ^-)μ for x ∈,and ζ =(R -n)/√(R).Although our choice of notation does not make this explicit, we highlight that the random variable Y(∞) depends on λ, μ, and n, meaning that we are actually dealing with a family of random variables {Y^(λ, μ, n)(∞)}_(λ, μ, n). The following theorem illustrates the type of result that can be obtained by Stein's method.Consider the Erlang-C system. For all n ≥ 1, λ > 0, and μ > 0 satisfying 1 ≤ R < n,d_W(X̃(∞), Y(∞)) :=sup_h(x) ∈| h(X̃(∞)) -h(Y(∞))|≤190/√(R),where ={h: →, h(x)-h(y)≤x-y, x,y ∈}. Several points are worth mentioning. First, we note that Theorem <ref> is not a limit theorem. Steady-state approximations are usually justified by some kind of limit theorem. That is, one considers a sequence of queueing systems and proves that the corresponding sequence of steady-state distributions converges to some limiting distribution as traffic intensity approaches one, or as the number of servers goes to infinity. In contrast, our theorem holds for any finite parameter choices of λ, n, and μ satisfying (<ref>) and R ≥ 1. Second, the error bound in (<ref>) is universal, as it does not assume any relationship between λ,n, and μ, other than the stability condition (<ref>) and the condition that R ≥ 1.Universal approximations were previously studied in <cit.>. One consequence of universality is that the error bound holdswhenparameters λ,n, and μfall inone of thefollowing asymptotic regimes:n = ⌈ R+ β R⌉,n = ⌈ R + β√(R)⌉, orn = ⌈ R + β⌉, where β > 0 is fixed, whileR →∞. The first two parameter regimes above describe the quality-driven (QD), and quality-and-efficiency-driven (QED) regimes, respectively. The last regime is the nondegenerate-slowdown (NDS) regime, which was studied in <cit.>. Third, as part of the universality of Theorem <ref>, we see that | X(∞) - ( R + √(R) Y(∞) ) | ≤ 190.For a fixed n, let ρ = R/n ↑ 1. One expects that X(∞) be on the order of 1/(1-ρ). Conventional heavy-traffic limit theorems often guarantee that the left hand side of (<ref>) is at most o(1/√(1-ρ)), whereas our error is bounded by a constant regardless of the load condition. This suggests that the diffusion approximation for the Erlang-C system is accurate not only as R →∞, but also in the heavy-traffic setting when R → n. Table <ref> contains some numerical results where we calculate the error on the left side of (<ref>). The constant 190 in (<ref>) is unlikely to be a sharp upper bound. In this thesis, we do not focus on optimizing such upper bounds, as Stein's method is not known for producing sharp constants.Theorem <ref> provides rates of convergence under the Wasserstein metric <cit.>. The Wasserstein metric d_W(·, ·) is one of the most commonly studied metrics when Stein's method is concerned. This is because the the spaceis relatively simple to work with, but is also rich enough so that convergence under the Wasserstein metric implies the convergence in distribution <cit.>. § OUTLINE OF THE STEIN FRAMEWORK Having seen the example in the previous section, let us briefly outline the key components of Stein's method. These are the Poisson equation, generator comparison, gradient bounds, moment bounds, and state-space collapse (SSC).The generator comparison idea is also known as the generator approach, and is attributed to Barbour <cit.> and Götze <cit.>. Chapter <ref> is devoted to a detailed walkthrough of the first four of these components, while the SSC component is not required until Chapter <ref>.Consider two sequences of stochastic processes {X^(ℓ)}_ℓ =1^∞ and {Y^(ℓ)}_ℓ=1^∞ indexed by ℓ, where X^(ℓ) = {X^(ℓ)(t)∈^d, t ≥ 0} is a continuous-time Markov chain (CTMC) and Y^(ℓ) = {Y^(ℓ)(t) ∈^d, t ≥ 0} is a diffusion process. Suppose X^(ℓ)(∞) and Y^(ℓ)(∞) are two random vectors having the stationary distributions of X^(ℓ) and Y^(ℓ), respectively. Let G_X^(ℓ) and G_Y^(ℓ) be the generators of X^(ℓ) and Y^(ℓ), respectively;for a diffusion process, G_Y^(ℓ) is a second order elliptic differential operator. For a function h: ^d → in a "nice" (but large enough) class, we wish to boundh(X^(ℓ)(∞)) - h(Y^(ℓ)(∞)). The first component is to set up the Poisson equation G_Y^(ℓ) f_h(x) = h(Y^(ℓ)(∞)) -h(x),x ∈^d.We then take the expectation of both sides above to see thath(Y^(ℓ)(∞)) -h(X^(ℓ)(∞))=G_Y^(ℓ) f_h(X^(ℓ)(∞)). When d = 1, the Poisson equation (<ref>) is an ordinary differential equation (ODE), and when d > 1, it is a partial differential equation (PDE). To execute Stein's method, we require bounds on the derivatives of f_h(x) (usually up to the third derivative). We refer to these as gradient bounds.The next step is to rely on the following relationship between the generator and stationary distribution of a CTMC. One can check that a random vector X^(ℓ)(∞) ∈^d has the stationary distribution of the CTMC X^(ℓ) if and only ifG_X^(ℓ) f (X^(ℓ)(∞)) = 0for all functions f:^d→ that have compact support.For a given h(x), the corresponding Poisson equation solution f_h(x) does not have compact support, but it is typically not hard to prove that (<ref>) continues to hold for f_h(x).Thus, it follows from (<ref>) and (<ref>) thath(Y^(ℓ)(∞))-h(X^(ℓ)(∞)) =[G_Y^(ℓ) f_h(X^(ℓ)(∞)) - G_X^(ℓ) f_h (X^(ℓ)(∞))]. The focus now falls on bounding the right side of(<ref>). To do so, we studyG_X^(ℓ)f_h(x) -G_Y^(ℓ)f_h(x)for each x in the state space of X^(ℓ). By performing Taylor expansion on G_X^(ℓ)f_h(x), we find that the difference in (<ref>) involves the product of partial derivatives of f_h(x) and terms related to the transition structure of X^(ℓ). The former are why we need gradient bounds, and the latter can typically be bounded by a polynomial of x.Therefore, we also need bounds on various moments of X^(ℓ)(∞),which we refer to as moment bounds. The main challenge is that both gradient and moment bounds must be uniform in ℓ. Once we have both gradient and moment bounds, the right hand side of (<ref>) can be bounded. We point out that this procedure can also be carried out when X^(ℓ) itself is not a CTMC, but a function of some higher dimensional CTMC U^(ℓ) = {U^(ℓ)(t) ∈𝒰, t ≥ 0}, where the dimension of the state space 𝒰 is strictly greater than d. When this is the case, the CTMC U^(ℓ) must exhibit some form of SSC. This is the case in Chapter <ref>, where we study the M/Ph/n+M system. This difference in dimensions is partly responsible forthe computational speedup in diffusion approximations; most complex stochastic processing systems exhibit some form of SSC <cit.>.Let G_U be the generator of U^(ℓ) and U^(ℓ)(∞) have its stationary distribution. Now, BAR (<ref>) becomesG_U^(ℓ) F(U^(ℓ)(∞))=0 for each `nice' F:𝒰→. Furthermore, (<ref>) becomesh(X^(ℓ)(∞)) -h(Y^(ℓ)(∞)) =[G_Y^(ℓ) f_h(X^(ℓ)(∞)) - G_U^(ℓ) F_h(U^(ℓ)(∞))],where F_h: 𝒰→ is the lifting of f_h:^d→ defined by letting x ∈^d be the projection of u ∈𝒰 and then setting F_h(u) = f_h(x).As before, we can perform Taylor expansion on G_U^(ℓ)F_h(u) to simplify the difference G_U^(ℓ)F_h(u)-G_Y^(ℓ)f_h(x).To use this difference to bound the right side of (<ref>), we need a steady-state SSC result for U^(ℓ)(∞), which tells us how to approximate U^(ℓ)(∞) from X^(ℓ)(∞) and guarantees that this approximation error is small. Typically, such an SSC result relies heavily on the structure of U^(ℓ). The SSC component will not appear until Chapter <ref>.§ RELATED LITERATUREThis dissertation lies at the intersection of two mathematical communities: the queueing theory, andStein method communities. It is therefore appropriate to separate the literature review into two parts. We begin with the literature from queueing theory.Diffusion approximations are a popular tool in queueing theory, and are usually “justified” by heavy traffic limit theorems. For example, a typical limit theorem would say that an appropriately scaled and centered version of the process X in (<ref>) converges to some limiting diffusion process as the system utilization ρ tends to one.Proving such limit theorems has been an active area of research in the last 50 years; see, for example, <cit.> for single-class queueing networks, <cit.> for multiclass queueing networks, <cit.> for bandwidth sharing networks, <cit.> for many-server queues. The convergence used in these limit theorems is the convergence in distribution on the path space ([0, ∞), ^d), endowed with Skorohod J_1-topology <cit.>. The J_1-topology on ([0,∞), ^d) essentially means convergence in ([0, T], ^d) for each T>0. In particular, it says nothing about the convergence at “∞”. Therefore, these limit theorems do not justify steady-state convergence.The jump from convergence on ([0, T], ^d) to convergence of stationary distributions was first established in the seminal paper <cit.>, where the authors prove an interchange of limits for generalized Jackson networks of single-server queues.The results in <cit.> were improved and extended by various authors for networks of single-servers <cit.>, for bandwidth sharing networks <cit.>, and for many-server systems <cit.>.These “interchange of limits” theorems are qualitative and thus do not provide rates of convergence as in Theorem <ref>.The first paper to have established convergence rates for steady-state diffusion approximations was <cit.>, which studied the Erlang-A system (many-server queue with customer abandonment) using an excursion based approach. Their approximation error bounds are universal. Although the authors in <cit.> did not study the Erlang-C system, their approach appears to be extendable to it as well. However, their method is not readily generalizable to the multi-dimensional setting. Following <cit.>, Gurvich <cit.> develops an approach to prove statements similar to Theorem <ref> for various queueing systems. Along the way, he independently rediscovers many of the ideas central to Stein's method in the setting of steady-state diffusion approximations. In particular, Gurvich's results also rely on the Poisson equation, generator comparison, gradient and moment bounds components discussed in Section <ref>. Gurvich packages the necessary conditions to establish convergence rates into a single condition which requires the existence of a uniform Lyapunov function for the diffusion processes. In particular, this Lyapunov function provides the necessary moment and gradient bounds to establish convergence rates. However, his results are no longer immediately applicable when the SSC component is required, i.e. when dim(𝒰) > d. In contrast, Stein's method is a modular framework. It allows one to separate a problem into its components, e.g. gradient bounds, moment bounds etc., and treat the difficulties of each component in isolation.We now discuss the relevant literature in the Stein method community. The first uses of Stein's method for stationary distributions of Markov processes traces back to <cit.>, where it is pointed out that Stein's method can be applied anytime the approximating distribution is the stationary distribution of a Markov proccess.That paper considers the multivariate Poisson, which is the stationary distribution of a certain multi-dimensional birth-death process. One of the major contributions of <cit.> was to show how viewing the Poisson distribution as the stationary distribution of a Markov chain could be exploited to establish gradient bounds using coupling arguments; cf. the discussion around (<ref>) of this document. A similar idea was subsequently used for the multivariate normal distribution through its connection to the multi-dimensional Ornstein–Uhlenbeck process in <cit.>.Of the papers that use the connection between Stein's method and Markov processes, <cit.> and the more recent <cit.> are the most relevant to this work. The former studies one-dimensional birth-death processes, with the focus being that many common distributions such as the Poisson, Binomial, Hypergeometric, Negative Binomial, etc., can be viewed as stationary distributions of a birth-death process. Although the Erlang-A and Erlang-C models are also birth-death processes, the focus in Chapter <ref> is on how well these models can be approximated by diffusions, e.g. qualitative features of the approximation like the universality in Theorem <ref>. Diffusion approximations go beyond approximations of birth-death processes, with the real interest lying in cases when a higher-dimensional Markov chain collapses to a one-dimensional diffusion, e.g. <cit.>, or when the diffusion approximation is multi-dimensional <cit.>.In <cit.>, the authors apply Stein's method to one-dimensional diffusions. The motivation is again that many common distributions like the gamma, uniform, beta, etc.,happen to be stationary distributions of diffusions. Their chief result is to establish gradient bounds for a very large class of diffusion processes, requiring only the mild condition that the drift of the diffusion be a decreasing function. However, their result cannot be applied here, because it is impossible to say how their gradient bounds depend on the parameters of the diffusion. Detailed knowledge of this dependence is crucial, because we are dealing with a family of approximatingdistributions;cf. (<ref>) and the comments below (<ref>).Outside the diffusion approximation domain, Ying has recently successfully applied Stein's framework to establish error bounds for steady-state mean-field approximations <cit.>.There is one additional recent line of work <cit.> that deserves mention, where the theme is corrected diffusion approximations using asymptotic series expansions. In particular, <cit.> considers the Erlang-C system and<cit.> considers the Erlang-A system. In these papers, the authors derive series expansions for various steady-state quantities of interest like the probability of waiting (X(∞) ≥ n). These types of series expansions are very powerful because they allow one to approximate steady-state quantities of interest within arbitrary precision. However, while accurate, these expansions vary for different performance metrics (e.g. waiting probability, expected queue length), and require non-trivial effort to be derived. They also depend on the choice of parameter regime, e.g. Halfin-Whitt. In contrast, the results provided by the Stein approach can be viewed as more robust because they capture multiple performance metrics andmultiple parameter regimes at the same time.§ OUTLINE OF DISSERTATIONThe rest of this document is structured as follows. Chapter <ref> serves as an introduction to Stein's method, where we outline the main steps of the procedure and carry them out on the Erlang-A and Erlang-C models. In Chapter <ref> we work in the setting of the Erlang-C model. We prove that we can achieve a faster convergence rate by using a diffusion approximation with a state dependent diffusion coefficient. Finally, in Chapter <ref>, we apply Stein's method to the M/Ph/n+M queueing system, which is a significantly more complicated model than both the Erlang-A and Erlang-C systems. Each of the chapters requires its own moment and gradient bounds. We aggregate all moment bounds in Appendix <ref>, and all gradient bounds in Appendix <ref>. § NOTATION All random variables and stochastic processes are defined on a common probability space (Ω, ℱ, ℙ) unless otherwise specified. For a sequence of random variables {X^n}_n=1^∞, we write X^n ⇒ X to denote convergence in distribution (also known as weak convergence) of X^n to some random variable X. If a > b, we adopt the convention that ∑_i=a^b (·) = 0. For an integer d ≥ 1, ^d denotes the d-dimensional Euclidean space and _+^d denotes the space of d-dimensional vectors whose elements are non-negative integers. For a,b ∈, we define a ∨ b = max{a,b} and a ∧ b = min{a,b}. For x ∈, we define x^+ = x ∨ 0 and x^- = (-x)∨ 0.For x ∈^d, we use x_i to denote its ith entry and x to denote its Euclidean norm. For x,y ∈^d, we write x ≤ y when x_i ≤ y_i for all i and when x ≤ y we define the vector interval [x,y] = {z: x ≤ z ≤ y}. All vectors are assumed to be column vectors. We let x^T and A^T denote the transpose of a vector x and matrix A, respectively. For a matrix A, we use A_ij to denote the entry in the ith row and jth column. We reserve I for the identity matrix, e for the vector of all ones and e^(i) for the vector that has a one in the ith element and zeroes elsewhere; the dimensions of these vectors willbe clear from the context. §.§ Probability MetricsFor two random variables U and V, define their Wasserstein distance, or Wasserstein metric, to be d_W(U, V) = sup_h(x) ∈[h(U)] -[h(V)],where ={h: →, h(x)-h(y)≤x-y}.It is known, see for example <cit.>, convergence under theWasserstein metric implies convergence in distribution. We can replacein (<ref>) by H_K={1_(-∞, a](x): a∈},and define d_K(U, V) = sup_h(x) ∈ H_K[h(U)] -[h(V)].This quantity is known as the Kolmogorov distance, or Kolmogorov metric.CHAPTER: INTRODUCTION TO STEIN'S METHOD VIA THE ERLANG-A AND ERLANG-C MODELSThe goal of this Chapter is to introduce the reader to the main ideas behind Stein's method, and specifically in the context of steady-state diffusion approximations. We use the Erlang-A and Erlang-C systems as working examples to illustrate the technical aspects of the method. We begin this chapter with Section <ref>, where we recall some details about the Erlang-A and Erlang-C models. In Section <ref>, we list the main results of this chapter. In Section <ref>, we outline the key steps of the Stein framework: the Poisson equation, generator comparison, gradient bounds and moment bounds. In Section <ref> we prove Theorem <ref>, which is a result about the Wasserstein distance. In Section <ref>, we discuss the Kolmogorov distance and the additional difficulties typically associated with it. Finally, we briefly discuss the approximation of higher moments in Section <ref>.This chapter is based on <cit.>. The author would like to acknowledge Jiekun Feng, who contributed significantly to the contents of this chapter, and in particular to the results about the Erlang-A model.§ CHAPTER INTRODUCTION Section <ref> already describes much of the focus of this chapter. We quickly recall some of the details about the Erlang-C and Erlang-A systems introduced there. Both systems have n homogeneous servers that serve customers in a first-come-first-serve manner. Customers arrive according to a Poisson process with rate λ, and customer service times are assumed to be i.i.d. having exponential distribution with mean 1/μ. In the Erlang-A system, each customer has a patience time and when his waiting time in queue exceeds his patience time, he abandons the queue without service; the patience times are assumed to be i.i.d. having exponential distribution with mean 1/α. Recall that X={X(t), t≥ 0} is the customer count process. This process is positive recurrent when α > 0, or if α = 0 and the offered load R = λ/μ satisfies R < n. We use X(∞) to denote the random variable having the stationary distribution of X, and set X̃(∞) = (X(∞) - R) /√(R). Theorem <ref> states that d_W(X̃(∞), Y(∞)) =sup_h(x) ∈| h(X̃(∞)) -h(Y(∞))|≤190/√(R),where Y(∞) is the random variable defined in (<ref>).In addition to the discussion on universality below Theorem <ref> in Section <ref>, there are two additional aspects that we will focus on in this chapter. From (<ref>), we know that the first moment of X̃(∞) can be approximated universally by the first moment of Y(∞). It is natural to ask what can be said about the approximation of higher moments. We performed some numerical experiments in which we approximate the second and tenth moments of X̃(∞) in a system with n = 500. The results are displayed in Table <ref>. One can see that the approximation errors grow as the offered load R gets closer to n. We will see in Section <ref> that this happens because the (m-1)th moment appears in the approximation error of the mth moment. A similar phenomenon was first observed for the M/GI/1+GI model in Theorem 1 of <cit.>. The rate of convergence in (<ref>) is for theWasserstein metric <cit.>, which is usually the simplest metric to work with.Another metric commonly studied in problems involving Stein's method is the Kolmogorov metric, which measures the distance between cumulative distribution functions of two random variables. The Kolmogorov distance between X̃(∞) and Y(∞) issup_h(x) ∈_K| h(X̃(∞)) -h(Y(∞))|,whereH_K={1_(-∞, a](x): a∈}.Theorems <ref> and <ref> of Section <ref> involve the Kolmogorov metric. A general trend in Stein's method is that establishing convergence rates for the Kolmogorov metric often requires much more effort than establishing rates for the Wasserstein metric, and our problem is no exception. The extra difficulty always comes from the fact that the test functions belonging to the class _K are discontinuous, whereas the ones inare Lipschitz-continuous. In Section <ref>, we describe how to overcome this difficulty in our model setting. We now move on to state the main results of this chapter. § MAIN RESULTS Recall the offered load R = λ/μ. For notational convenience we define δ>0 asδ = 1/√(R) = √(μ/λ).Let x(∞) be the unique solution to the flow balance equationλ = (x(∞)∧ n )μ + (x(∞)-n)^+ α.Here, x(∞) is interpreted as the equilibrium number of customers in the corresponding fluid model, and is the point at which the arrival rate equals the departure rate. The latter is the sum of the service completion rate and the customer abandonment rate with x(∞) customers in the system. One can check that the flow balance equation has a unique solution x(∞) given by x(∞) =n + λ-nμ/αifR ≥ n,R ifR < n.By noting that the number of busy servers x(∞) ∧ n equals n minus the number of idle servers (x(∞)-n)^-, the equation in (<ref>) becomesλ - n μ = (x(∞)-n)^+α - (x(∞)-n)^-μ.We note that x(∞) is well-defined even when α = 0, because in that case we always assume that R < n. We consider the CTMCX̃ = {X̃(t) := δ(X(t) - x(∞)), t ≥ 0},and let the random variable X̃(∞) have its stationary distribution. Define ζ =δ(x(∞) -n),andb(x) = ((x+ζ)^–ζ^-)μ -((x+ζ)^+ -ζ^+)α forx∈,with convention that α is set to be zero in the Erlang-C system. For intuition about the quantity ζ, we note that in the Erlang-C system satisfying R < n, n = R -ζ√(R).Thus,-ζ=ζ>0 is precisely the“safety coefficient” in the square-root safety-staffing principle <cit.>. We point out that the event {X̃(t) = -ζ} corresponds to the event {X(t) = n}. Throughout this chapter, let Y(∞) denote a continuous random variable onhaving densityν(x)= κexp(1/μ∫_0^xb(y)dy),where κ>0 is a normalizing constant that makes the density integrate to one. Note that these definitions are consistent with (<ref>) and (<ref>).Consider the Erlang-A system (α > 0). There exists an increasing function C_W : _+ →_+ such that for all n ≥ 1, λ > 0, μ>0, and α>0 satisfying R ≥ 1,d_W(X̃(∞), Y(∞)) ≤ C_W(α/μ)δ.The proof of Theorems <ref> and <ref> uses the same ideas. Therefore, for the sake of brevity, we only give an outline for the proof of Theorem <ref>in Section <ref>, without filling in all the details. It is for this reason that we do not write out the explicit form of C_W(α/μ), although it can be obtained from the proof. The same is true for Theorem <ref> below. Given two random variables U and V, <cit.> implies that when V has a density that is bounded by C>0, d_K(U, V) ≤√(2C d_W(U, V)).At best, (<ref>) and Theorems <ref> and <ref> imply a convergence rate of √(δ) for d_K(X̃(∞), Y(∞)). However, this bound is typically too crude, and the following two theorems show that convergence happens at rate δ. Theorem <ref> is proved in Section <ref>. The proof of Theorem <ref> is outlined in Section <ref>. Consider the Erlang-C system (α = 0). For all n ≥ 1, λ > 0, and μ>0 satisfying 1 ≤ R < n,d_K(X̃(∞), Y(∞)) ≤ 156δ.Consider the Erlang-A system (α > 0). There exists an increasing function C_K : _+ →_+ such that for all n ≥ 1, λ > 0, μ>0, and α>0 satisfying R ≥ 1,d_K(X̃(∞), Y(∞)) ≤ C_K(α/μ)δ. Theorems <ref> and <ref> are new, but versions of Theorems <ref> and <ref> were first proved in the pioneering paper <cit.> using an excursion based approach. However, our notion of universality in those theorems is stronger than the one in <cit.>, because most of their results require μ and α to be fixed. The only exception is in Appendix C of that paper, where the authors consider the NDS regime with μ= μ(λ) = β√(λ) and λ = nμ + β_1 μ for some β> 0 and β_1 ∈.We emphasize that both constants C_W and C_K are increasing in α/μ. That is, for an Erlang-A system with a higher abandonment rate with respect to its service rate, our error bound becomes larger. The reader may wonder why these constants depend on α/μ, while the constant in the Erlang-C theorems does not depend on anything. Despite our best efforts, we were unable to get rid of the dependency on α/μ. The reason is that the Erlang-C model depends on only three parameters (λ, μ, n), while the Erlang-A model also depends on α.As a result, both the gradient bounds and moment bounds have an extra factor α/μin the Erlang-A model. For example, compare Lemma <ref> in Section <ref> with Lemma <ref> in Section <ref>. § OUTLINE OF THE STEIN FRAMEWORK In this section we introduce the main tools needed to prove Theorems <ref>–<ref>. However, the framework presented here is generic, and is not limited to the Erlang-A or Erlang-C systems. It can be appliedwhenever one compares a Markov chain to a diffusion process; the content here will be referred to liberally in all chapters of this dissertation. The following is an informal outline of the rest of this section. We know that X̃(∞) follows the stationary distribution of the CTMC X̃, and that this CTMC has a generator G_X̃. To Y(∞), we will associate a diffusion process with generator G_Y. We will start by fixing a test function h :→ and deriving the identity |h(X̃(∞)) -h(Y(∞)) | = |G_X̃ f_h(X̃(∞)) - G_Y f_h(X̃(∞)) |,where f_h(x) is a solution to the Poisson equationG_Y f_h(x) =h(Y(∞)) - h(x),x ∈.We then focus on bounding the right hand side of (<ref>), which is easier to handle than the left hand side. This is done by performing a Taylor expansion of G_X̃ f_h(x) in Section <ref>. To bound the error term from the Taylor expansion, we require bounds on various moments of | X̃(∞) |, as well as the derivatives of f_h(x). We refer to the former as moment bounds, and the latter as gradient bounds. These are presented in Sections <ref> and <ref>, respectively.§.§ The Poisson Equation of a Diffusion ProcessA one-dimensional diffusion process can be described by its generator G f(x) =b̅(x)f'(x) +1/2a̅(x) f”(x)for x ∈,f ∈ C^2().The functions b̅(x) and a̅(x) are known as the drift, and diffusion coefficient, respectively. It is typically required that a̅(x) > 0 for all x ∈, and that both b̅(x) and a̅(x) satisfy some regularity condition, e.g. Lipschitz continuity. The random variable Y(∞) in Theorems <ref>–<ref> is well-defined and its density is given in (<ref>). It turns out that Y(∞) has the stationary distribution of a diffusion process Y={Y(t), t≥ 0}. The process Y is the one-dimensional piecewise Ornstein–Uhlenbeck (OU) process, whose generator is given byG_Y f(x) =b(x)f'(x) +μ f”(x)for x ∈,f ∈ C^2(),where b(x) is defined in (<ref>). Clearly,b(0)=0, andb(x) is Lipschitz continuous. Indeed,b(x) -b(y) ≤(α∨μ) x-yforx, y∈.The generator in (<ref>) has a constant diffusion coefficient a̅(x) = 2μ. Since the diffusion process Y depends on parameters λ, n, μ, and α in an arbitrary way, there is no appropriate way to talk about the limit of Y(∞) in terms of these parameters. Therefore, we call Y a diffusion model, as opposed to a diffusion limit. Having a diffusion model whose input parameters are directly taken from the corresponding Markov chain model is critical to achieve universal accuracy. In other words, this diffusion model is accurate in any parameter regime, from underloaded, to critically loaded, and to overloaded. Diffusion models, not limits,of queueing networks with a given set of parameters have been advanced in <cit.>.The main tool we use is known as the Poisson equation. It allows us to say that Y(∞) is a good estimate for X̃(∞) if the generator of Y behaves similarly to the generator of X̃, where X̃ is defined in (<ref>). Let H be a class of functions h : →, to be specified shortly.For each function h(x) ∈H, consider the Poisson equation G_Y f_h(x) =b(x) f_h'(x) + μ f_h”(x) =h(Y(∞)) - h(x), x∈.The solution to the Poisson equation is described by the following generic lemma.Let a̅:→_+ and b̅: → be continuous functions, and assume that a̅(x) > 0 for all x ∈. Assume also that∫_-∞^∞2/a̅(x)exp(∫_0^x2 b̅(u)/a̅(u) du)dx < ∞,and let V be a continuous random variable with density 2/a̅(x)exp(∫_0^x2 b̅(u)/a̅(u) du)/∫_-∞^∞2/a̅(x)exp(∫_0^x2 b̅(u)/a̅(u) du)dx,x ∈.Fix h:→ satisfying h(V) < ∞, and consider the Poisson equation 1/2a̅(x) f_h”(x) + b̅(x) f_h'(x) =h(V) - h(x),x ∈.There exists a solution f_h(x) to this equation satisfying f_h'(x) =e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) ( h(V) - h(y)) e^∫_0^y2b̅(u)/a̅(u)du dy=-e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) ( h(V) - h(y)) e^∫_0^y2b̅(u)/a̅(u)du dy , f_h”(x) = - 2b̅(x)/a̅(x) f_h'(x) +2/a̅(x)( h(V) - h(x) ).The integrals in (<ref>) and (<ref>) are finite because h(V) < ∞. One can verify directly that both forms of f_h'(x) in (<ref>) and (<ref>) satisfy (<ref>). The form of f_h”(x) follows from rearranging (<ref>).Provided h(x), a̅(x), and b̅(x)/a̅(x) are sufficiently differentiable, f_h(x) can have more than two derivatives. For example, f_h”'(x) =-(2b̅(x)/a̅(x))' f_h'(x) - 2b̅(x)/a̅(x) f_h”(x) - 2/a̅(x)h'(x) - 2a̅'(x)/a̅^2(x)(h(V) - h(x)).In this chapter, we take = when we deal with the Wasserstein metric (Theorems <ref> and <ref>), and we choose = _K (defined in (<ref>)) when we deal with the Kolmogorov metric (Theorems <ref> and <ref>). We claim that h(Y(∞)) < ∞. Indeed, when = _K, this clearly holds.When ℋ =, without loss of generality we take h(0)= 0 in (<ref>), and use the Lipschitz property of h(x) to see thath(Y(∞))≤Y(∞) < ∞,where the finiteness of Y(∞) will be proved in (<ref>). From (<ref>), one has|h(X̃(∞)) -h(Y(∞)) | =|G_Y f_h(X̃(∞)) |.In (<ref>), X̃(∞) has the stationary distribution of the CTMC X̃, not necessarily defined on the same probability space of Y(∞).Actually, X̃(∞) in (<ref>) can be replaced by any other random variable, although one does not expect the error on the right side to be small if this random variable has no relationship with the diffusion process Y. §.§ Comparing Generators To prove Theorems <ref>–<ref>, we need to bound the right side of (<ref>).The CTMC X̃ defined in (<ref>) also has a generator.We bound the right side of (<ref>) by showing that the diffusion generator in (<ref>) is similar to the CTMC generator. For any k ∈_+, we define x = x_k = δ(k - x(∞)). Then for any function f:→, the generator of X̃ is given byG_X̃ f(x) = λ (f(x + δ) - f(x)) + d(k) (f(x-δ) - f(x)),whered(k) = μ (k ∧ n) + α (k-n)^+,is the departure rate corresponding to the system having k customers.One may check that b(x) = δ( λ - d(k)).The relationship between G_X̃ and the stationary distribution of X̃ is illustrated by the following lemma. Let f(x): → be a function such that f(x)≤ C(1+x)^3 for some C > 0 (i.e. f(x) is dominated by a cubic function), and assume that the CTMC X̃ is positive recurrent. Then[ G_X̃ f(X̃(∞)) ] = 0.We will see in Lemma <ref> later this section, in Lemmas <ref> and <ref> of Section <ref>, and inLemma <ref> of Section <ref> that there is a family of solutions to the Poisson equation (<ref>) whose first derivatives grow at most linearly in both the Wasserstein and Kolmgorov settings, meaning that these solutions satisfy the conditions of Lemma <ref>.The proof of Lemma <ref> is provided in Section <ref>. Suppose for now that for any h(x) ∈H, the solution to the Poisson equation f_h(x) satisfies the conditions of Lemma <ref>. We can apply Lemma <ref> to (<ref>) to see that|h(X̃(∞)) -h(Y(∞)) | =|G_Y f_h(X̃(∞)) | =|G_X̃ f_h(X̃(∞)) -G_Y f_h(X̃(∞)) |≤| G_X̃ f_h(X̃(∞)) -G_Y f_h(X̃(∞)) |.While the two random variables on the left side of (<ref>) are usually defined on different probability spaces, the two random variables on the right side of (<ref>) are both functions of X̃(∞). Thus, we haveachieved a coupling through Lemma <ref>. Setting up the Poisson equation is a generic first step one performs any time one wishes to apply Stein's method to a problem. The next step is to bound the equivalent of our |G_Y f_h(X̃(∞)) |. This is usually done by using a coupling argument. However, this coupling is always problem specific, and is one of the greatest sources of difficulty one encounters when applying Stein's method. In our case, this generator coupling is natural because we deal with Markov processes X̃ and Y.Since the generator completely characterizes the behavior of a Markov process, it is natural to expect that convergence of generators implies convergence of Markov processes. Indeed, the question of weak convergence was studied in detail, for instance in <cit.>, using the martingale problem of Stroock and Varadhan <cit.>. However, (<ref>) lets us go beyond weak convergence, both because different choices of h(x) lead to different metrics of convergence, and also because the question of convergence rates can be answered. One interpretation of the Stein approach is to view f_h(x) as a Lyapunov function that gives us information about h(x). Instead of searching very hard for this Lyapunov function, the Poisson equation (<ref>) removes the guesswork. However, this comes at the cost of f_h(x) being defined implicitly as the solution to a differential equation. §.§ Taylor ExpansionTo bound the right side of (<ref>),we study the difference G_X̃ f_h(x) - G_Y f_h(x). For that we perform a Taylor expansion on G_X̃f_h(x). To illustrate this, suppose that f_h”(x) exists for all x ∈, and is absolutely continuous. Then for any k ∈_+, and x = x_k = δ(k - x(∞)), we recall that b(x) = δ(λ - d(k)) in (<ref>) to see thatG_X̃ f_h(x) =λ (f_h(x + δ) - f_h(x)) + d(k) (f_h(x-δ) - f_h(x)) =f_h'(x) δ (λ - d(k)) + 1/2δ^2 f_h”(x)(λ + d(k)) + 1/2λδ^2 (f_h”(ξ) - f_h”(x)) + 1/2 d(k) δ^2 (f_h”(η) - f_h”(x)) =f_h'(x)b(x) +1/2δ^2 (2λ - 1/δ b(x)) f_h”(x) + 1/2μ (f_h”(ξ) - f_h”(x)) + 1/2 ( λ -1/δb(x))δ^2 (f_h”(η) - f_h”(x))=G_Y f_h(x) - 1/2δ f_h”(x)b(x) + 1/2μ (f_h”(ξ) - f_h”(x)) + 1/2 ( μ -δ b(x)) (f_h”(η) - f_h”(x)),where ξ∈ [x, x+δ] and η∈ [x-δ,x]. We invoke the absolute continuity of f_h”(x) to get|h(X̃(∞)) -h(Y(∞)) |≤1/2δ[ |f_h”(X̃(∞))b(X̃(∞)) | ] + μ/2[ ∫_X̃(∞)^X̃(∞) + δf_h”'(y)dy] + μ/2[ ∫_X̃(∞)-δ^X̃(∞)f_h”'(y)dy] + 1/2δ[| b(X̃(∞))| ∫_X̃(∞)-δ^X̃(∞)f_h”'(y)dy].As one can see, to show that the right hand side of (<ref>) vanishes as δ→ 0, we must be able to bound the derivatives of f_h(x); we refer to these as gradient bounds. Furthermore, we will also need bounds on moments of | X̃(∞) |; we refer to these as moment bounds. Both moment and gradient bounds will vary between the Erlang-A or Erlang-C setting, and the gradient bounds will be different for the Wasserstein, and Kolmogorov settings. Moment bounds will be discussed shortly, and gradient boundsin the Wasserstein setting will be presented in Section <ref>.We discuss the Kolmogorov setting separately in Section <ref>. In that case we face an added difficulty because f_h”(x) has a discontinuity, and we cannot use (<ref>) directly.§.§ Moment Bounds The following lemma presents the necessary moment bounds to bound (<ref>) in the Erlang-C model, and is proved in Appendix <ref>. These moment bounds are used for both the Wasserstein and Kolmogorov metrics. Consider the Erlang-C model (α = 0). For all n ≥ 1, λ > 0, and μ>0 satisfying 0 < R < n,[(X̃(∞))^2 1(X̃(∞) ≤ -ζ)] ≤4/3 + 2δ^2/3,[ |X̃(∞) 1(X̃(∞) ≤ -ζ)| ] ≤√(4/3 + 2δ^2/3), [ |X̃(∞) 1(X̃(∞) ≤ -ζ)| ] ≤ 2ζ[|X̃(∞)1(X̃(∞) ≥ -ζ)| ] ≤1/ζ + δ^2/4ζ + δ/2, (X̃(∞) ≤ -ζ) ≤ (2+δ)ζ.We see that (<ref>) immediately implies that when δ≤ 1, ζ( X̃(∞) ≥ -ζ) ≤ζ∧[| X̃(∞) 1(X̃(∞) ≥ -ζ)| ] ≤ζ∧(1/ζ + δ^2/4ζ + δ/2)≤7/4,where to get the last inequality we considered separately the cases where ζ≤ 1 and ζ≥ 1. This bound will be used in the proofs of Theorems <ref> and <ref>. One may wonder why the bounds are separated using the indicators 1{X̃(∞) ≤ -ζ} and 1{X̃(∞) ≥ -ζ}. This is related to the drift b(x) appearing in (<ref>), and the fact that b(x) takes different forms on the regionsx≤ -ζ and x≥ -ζ. Furthermore, it may be unclear at this point why both (<ref>) and (<ref>) are needed, as the left hand side in both bounds is identical. The reason is that (<ref>) is an O(1) bound (we think of δ≤ 1), whereas (<ref>) is an O(ζ) bound. The latter is only useful when ζ is small, but this is nevertheless an essential bound to achieve universal results. As we will see later, it negates 1/ζ terms that appear in (<ref>) from f_h”(x) and f_h”'(x).For the Erlang-A model, we also require moment bounds similar to those stated in Lemma <ref>. Both the proof, and subsequent usage, of the Erlang-A moment bounds are similar to the proof and subsequent usage of the Erlang-C moment bounds. We therefore delay their precise statement until Lemma <ref> in Section <ref> to avoid distracting the reader with a bulky lemma. §.§ Wasserstein Gradient Bounds Given a function h(x), there are multiple solutions to the Poisson equation (<ref>). Going forward, when we refer to a solution f_h(x), we mean the solution in Lemma <ref> with b̅(x) = b(x) and a̅(x) = 2μ. The following lemma presents Wasserstein gradient bounds for the Erlang-C model. It is proved in Section <ref>. Consider the Erlang-C model (α = 0), and fix h(x) ∈. Then f_h(x) is twice continuously differentiable, with an absolutely continuous second derivative. Furthermore, for all n ≥ 1, λ > 0, and μ>0 satisfying 0 < R < n, f_h'(x)≤ 1/μ(7.5 + 5/ζ),x ≤ -ζ, 1/μ1/ζ(x + 1 + 2/ζ),x ≥ -ζ.f_h”(x)≤34/μ( 1 + 1/ζ),x ≤ -ζ, 1/μζ,x ≥ -ζ,and for those x ∈ where f_h”'(x) exists, f_h”'(x)≤1/μ(17 + 10/ζ) ,x ≤ -ζ,2/μ,x ≥ -ζ. This lemma validates the Taylor expansion used to obtain (<ref>)becausef_h”(x) is absolutely continuous. Furthermore, f_h(x)satisfies the conditions of Lemma <ref>, because f_h'(x) grows at most linearly.Gradient bounds, also known as Stein factors, are central to any application of Stein's method. The problem of gradient bounds for diffusion approximations can be divided into two cases: the one-dimensional case, and the multi-dimensional case. In the former, the Poisson equation is an ordinary differential equation (ODE) corresponding to a one-dimensional diffusion process. In the latter, the Poisson equation is a partial differential equation (PDE) corresponding to a multi-dimensional diffusion process.The one-dimensional case is simpler, because the explicit form of f_h(x) is given to us by Lemma <ref>. To bound f_h'(x) and f_h”(x) we can analyze (<ref>)–(<ref>) directly, as we do in the proof of Lemma <ref>. In Appendix <ref>, we see that this direct analysis can be used as a go-to method for one-dimensional diffusions. However, it fails in the multi-dimensional case, because closed form solutions for PDE's are not typically known. In this case, it helps toexploit the fact that f_h(x) satisfiesf_h(x) = ∫_0^∞([h(Y(t)) | Y(0) = x] -h(Y(∞))) dt,where Y = {Y(t), t≥ 0} is a diffusion process with generator G_Y <cit.>. To bound derivatives of f_h(x) based on (<ref>), one may use coupling arguments to bound finite differences of the form 1/s(f_h(x+ s) - f_h(x)). For examples of coupling arguments, see <cit.>. A related paper to these types of gradient bounds is <cit.>, where the author used a variant of (<ref>) for the fluid model of a flexible-server queueing system as a Lyapunov function.As an alternative to coupling, one may combine (<ref>) with a-priori Schauder estimates from PDE theory, as was done in <cit.>. Just like we did with the moment bounds, we delay the Erlang-A gradient bounds to Lemma <ref> in Section <ref>. We are now ready to prove Theorem <ref>. § PROOF OF THEOREM <REF> (ERLANG-C WASSERSTEIN) In this section we prove Theorem <ref>. Fixing h(x) ∈, we see from Lemma <ref> that f_h”(x) is absolutely continuous, implying that (<ref>) holds. We recall it here as|h(X̃(∞)) -h(Y(∞)) |≤1/2δ[ |f_h”(X̃(∞))b(X̃(∞)) | ] + μ/2[ ∫_X̃(∞)^X̃(∞) + δf_h”'(y)dy] + μ/2[ ∫_X̃(∞)-δ^X̃(∞)f_h”'(y)dy] + 1/2δ[| b(X̃(∞))| ∫_X̃(∞)-δ^X̃(∞)f_h”'(y)dy],where δ = 1/√(R) = √(μ/λ). The proof of Theorem <ref> simply involves applying the moment bounds and gradient bounds to show that the error bound in (<ref>) is small.Throughout the proof we assume that R ≥ 1, or equivalently, δ≤ 1. We bound each of the terms on the right side of (<ref>) individually. We recall here that the support of X̃(∞) is a δ-spaced grid, and in particular this grid contains the point -ζ. In the bounds that follow, we will often consider separately the cases where X̃(∞) ≤ -ζ - δ, and X̃(∞) ≥ -ζ. We recall that b(x) = μ((x+ζ)^–ζ^-) = -μ x,x ≤ -ζ, μζ,x ≥ -ζ,and apply the moment bounds (<ref>), (<ref>), and the gradient bound (<ref>), to see that[ |f_h”(X̃(∞))b(X̃(∞)) | ] ≤34(1 + 1/ζ)[|X̃(∞) | 1(X̃(∞) ≤ -ζ - δ) ] + (X̃(∞) ≥ -ζ)≤34(1 + 1/ζ)(2ζ∧√(4/3 + 2δ^2/3)) + 1≤34(√(4/3 + 2δ^2/3) + 2) + 1≤34(√(2) + 2) + 1 ≤ 118.Next, we use (<ref>) and the gradient bound in (<ref>) to getμ/2[∫_X̃(∞)^X̃(∞)+δf_h”'(y) dy ]≤δ/2((17 + 10/ζ)(X̃(∞) ≤ -ζ - δ) + 2(X̃(∞) ≥ -ζ))≤δ/2(17 + 10/ζ(3 ζ) ) ≤ 24δ.By a similar argument, we can show thatμ/2[∫_X̃(∞)-δ^X̃(∞)f_h”'(y) dy ] ≤ 24δ,with the only difference in the argument being that we consider the cases when X̃(∞) ≤ -ζ and X̃(∞) ≥ -ζ + δ, instead of X̃(∞) ≤ -ζ -δ and X̃(∞) ≥ -ζ. Lastly, we use the form of b(x), the moment bounds (<ref>), (<ref>), and (<ref>), and the gradient bound (<ref>) to get δ/2[ | b(X̃(∞)) | ∫_X̃(∞)-δ^X̃(∞)f_h”'(y) dy ]≤δ^2/2( (17 + 10/ζ)[ | X̃(∞)| 1(X̃(∞) ≤ -ζ) ] + 2ζ(X̃(∞) ≥ -ζ + δ)) ≤δ^2/2( (17 + 10/ζ)(2ζ∧√(4/3 + 2δ^2/3)) + 14/4 ) ≤δ^2/2(17 √(2) + 20 + 14/4 ) ≤ 24δ^2.Hence, from (<ref>) we conclude that for all R ≥ 1, and h(x) ∈,|h(X̃(∞)) -h(Y(∞)) | ≤δ (118 + 24 + 24 + 24 δ) ≤ 190δ,which proves Theorem <ref>.§ PROOF OUTLINE FOR THEOREM <REF> (ERLANG-A WASSERSTEIN)We begin by stating some necessary moment and gradient bounds, and thenoutline the proof of Theorems <ref>. §.§ Erlang-A Moment and Gradient Bounds The following lemma states the necessary moment bounds for the Erlang-A model. The underloaded and overloaded cases have to be handled separately. Since the drift b(x) is different between the Erlang-A and Erlang-C models, the quantities bounded in the following lemma will resemble those in Lemma <ref>, but will not be identical. Its proof is outlined in Appendix <ref>.Consider the Erlang-A model (α>0). Fix n ≥ 1, λ > 0,μ > 0, and α > 0. If 0 < R ≤ n (an underloaded system), then[ (X̃(∞))^2 1(X̃(∞)≤ -ζ)] ≤1/3(α/μδ^2 + δ^2 + 4 ), [ |X̃(∞)1(X̃(∞)≤ -ζ)|] ≤√(1/3(α/μδ^2 + δ^2 + 4 )), [ |X̃(∞)1(X̃(∞)≤ -ζ)|] ≤ 2ζ + α/μ√(1/3(μ/αδ^2 + μ/α4 + δ^2)),[ |X̃(∞)1(X̃(∞)≥ -ζ)|] ≤(1 + δ^2/4 +δ/2√(1/3(α/μδ^2 + δ^2 + 4 ))) (μ/μ∧α∧1/ζ), [ (X̃(∞)+ζ)^2 1(X̃(∞)≥ -ζ)]≤1/3(μ/αδ^2 + μ/α4 + δ^2), [ (X̃(∞)+ζ)1(X̃(∞)≥ -ζ)]≤√(1/3(μ/αδ^2 + μ/α4 + δ^2)), [ (X̃(∞)+ζ)1(X̃(∞)≥ -ζ)]≤1/ζ( δ^2/4α/μ + δ^2/4 + 1 ),(X̃(∞)≤ -ζ) ≤ (2+δ)(ζ + α/μ√(1/3(μ/αδ^2 + μ/α4 + δ^2))).and if n ≤ R (an overloaded system), then[ |X̃(∞)1(X̃(∞)≤ -ζ)|]≤√(1/α∧μ( αδ^2/4 + μ) ), [ |X̃(∞)1(X̃(∞)≤ -ζ)|]≤1/ζ(δ^2/4+μ/α), [(X̃(∞))^2 1(X̃(∞) ≥ -ζ)] ≤1/3(δ^2 + 4μ/α), [ |X̃(∞)1(X̃(∞)≥ -ζ)|] ≤√(1/3(δ^2 + 4μ/α)), [| (X̃(∞)+ζ)1(X̃(∞)≤ -ζ)|]≤1/ζ(δ^2/4+1), [ (X̃(∞)+ζ)^21(X̃(∞)≤ -ζ)]≤δ^2/4α/μ+1, [| (X̃(∞)+ζ)1(X̃(∞)≤ -ζ)|]≤√(δ^2/4α/μ+1), [| (X̃(∞)+ζ)1(X̃(∞)≤ -ζ)|]≤α/μ√(1/3(δ^2 + 4μ/α)), (X̃(∞) ≤ -ζ)≤ (3+δ)16/√(2)(δ^2/4+1)((1/ζ∨α/μ)∧√(α/μ)).The following Wasserstein gradient bounds are proved in Appendix <ref>. Consider the Erlang-A model (α > 0), and fix h(x) ∈. Then f_h(x) given in Lemma <ref> is twice continuously differentiable, with an absolutely continuous second derivative. Furthermore, there exists a constant C > 0 independent of λ, n, μ, and α such that for all n ≥ 1, λ > 0, μ>0, and α > 0 satisfying 0 < R ≤ n (an underloaded system), f_h'(x)≤ C (√(μ/α)∧1/ζ+1)1/μ, x≤ -ζ,C (μ/α+√(μ/α)∧1/ζ+1)1/μ, x≥ -ζ,f_h”(x)≤C (√(μ/α)∧1/ζ+1)1/μ,x≤ 0, C [(α/μ+√(α/μ)+1)(√(μ/α)∧1/ζ)+1]1/μ, x∈ [0,-ζ],C(α/μ+√(α/μ)+1)(√(μ/α)∧1/ζ)1/μ, x≥ -ζ,and for those x ∈ where f_h”'(x) exists,f_h”'(x)≤ C(√(μ/α)∧1/ζ+1)1/μ,x≤ 0,C(√(μ/α)∧1/ζ+α/μ+√(α/μ)+1)1/μ,x∈ [0,-ζ], C (α/μ+√(α/μ)+1)1/μ, x≥ -ζ,and for all n ≥ 1, λ > 0, μ>0, and α > 0 satisfying n ≤ R (an overloaded system),f_h'(x)≤C (1/μ+ 1/√(α)1/√(μ) + ζ/μ∧1/α),x≤-ζ,C (1/μ+1/√(α)1/√(μ)+1/α),x≥ -ζ,f_h”(x)≤C(1/μ+ 1/√(α)1/√(μ) + ζ/μ∧1/α), x≤-ζ, C(α/μ+√(α/μ)+1)1/μ|x| + C(1/μ+1/√(α)1/√(μ)), x≥ -ζ,and for those x ∈ where f_h”'(x) exists,f_h”'(x)≤C/μ(1+√(μ/α)+ζ∧μ/α), x≤ -ζ, f_h”'(x)≤C/μ(α/μ+√(α/μ)+1) (1+α/μx^2) +C/μ(α/μ+√(α/μ))x, x≥ -ζ, f_h”'(x)≤ C/μ(α/μ+√(α/μ)+1) + C/μ(α/μ+√(α/μ)+1)^2 x , x≥ -ζ. §.§ Proof Outline Proving Theorem <ref> consists of bounding the four error terms in (<ref>). Since the procedure is very similar to the proof of Theorem <ref>, we will only outline which gradient and moment bounds need to be used to bound each error term.We start with the underloaded case, when R ≤ n. To bound the first term in (<ref>), we use moment bounds (<ref>), (<ref>), and (<ref>), together with the gradient bounds in (<ref>). For the second and third terms, we use moment bound (<ref>) and the gradient bounds in (<ref>). For the fourth term, we use moment bounds (<ref>)–(<ref>), and the gradient bounds in (<ref>).We now prove the overloaded case, when R ≥ n. To bound the first term in (<ref>), we use moment bounds (<ref>)–(<ref>), together with the gradient bounds in (<ref>). For the second and third terms, we use moment bounds (<ref>),(<ref>), and (<ref>), together with the gradient bounds in (<ref>) and (<ref>). For the fourth term, we use moment bounds (<ref>)–(<ref>), and gradient bounds in (<ref>) and (<ref>). § THE KOLMOGOROV METRIC In this section we prove Theorem <ref>, which is stated in the Kolmogorov setting. The biggest difference between theWasserstein and Kolmogorov settings is that in the latter, the test functions h(x) used in the Poisson equation (<ref>) are discontinuous. For this reason, new gradient bounds need to be derived separately for the Kolmogorov setting; we present these new gradient bounds in Section <ref>. Furthermore, the solution to the Poisson equation no longer has a continuous second derivative, meaning that the Taylor expansion we used to derive the upper bound in (<ref>) is invalid. We discuss an alternative to (<ref>) in Section <ref>. This alternative bound contains a new error term that cannot be handled by the gradient bounds, nor the moment bounds. This term appears because the solution to the Poisson equation has a discontinuous second derivative, and to bound it we present Lemma <ref>. We then prove Theorem <ref>inSection <ref>, and outline the proof for Theorem <ref> in Section <ref>. §.§ Kolmogorov Gradient BoundsRecall that in the Kolmogorov setting, we take the class of test functions for the Poisson equation (<ref>) to be _K defined in (<ref>). For the statement of the following two lemmas, we fix a ∈ and set h(x) = 1_(-∞, a](x). Recall that the Poisson equation has multiple solutions, but going forward we always work with the one from Lemma <ref>. Furthermore, we use f_a(x) instead of f_h(x) to denote the solution to the Poisson equation. Consider the Erlang-C model (α = 0). Then f_a(x) is continuously differentiable, with an absolutely continuous derivative. Furthermore, for all n ≥ 1, λ > 0, and μ > 0 satisfying 0 < R< n, f_a'(x)≤ 4/μ ,x ≤ -ζ,1/μζ,x ≥ -ζ,and for all x ∈,f_a”(x)≤ 2/μ,where f_a”(x) is understood to be the left derivative at the point x = a.Consider the Erlang-A model (α > 0). Then f_a(x) is continuously differentiable, with an absolutely continuous derivative.Fix n ≥ 1, λ > 0, μ > 0, and α > 0. If 0 < R≤ n (an underloaded system), thenf_a'(x)≤1/μ√(2π)e^1/2 ,x ≤ -ζ,1/μ(√(π/2μ/α)∧1/ζ),x ≥ -ζ,and if n ≤ R (an overloaded system), thenf_a'(x)≤1/μ√(π/2) ,x ≤ -ζ,1/μ√(π/2)(1 + √(μ/α)),x ≥ -ζ.Moreover,for allλ>0, n ≥ 1, μ > 0, and α > 0, and all x ∈,f_a”(x)≤ 3/μ,where f_a”(x) is understood to be the left derivative at the point x = a.Lemmas <ref> and <ref> are proved in Appendix <ref>. Unlike the Wasserstein setting, these lemmas do not guarantee that f_a”(x) is absolutely continuous. Indeed, for any a ∈,substituting h(x) = 1_(-∞, a](x) into (<ref>) gives usμ f_a”(x) =(Y(∞) ≤ a) - 1_(-∞, a](x) - b(x) f_a'(x).Since b(x) f_a'(x) is a continuous function, the above equation implies that f_a”(x) is discontinuous at the point x = a. Thus, we can no longer use the error bound in (<ref>), and require a different expansion of G_X̃ f_a(x). §.§ Alternative Taylor Expansion To get an error bound similar to (<ref>), we first define ϵ_1(x) =∫_x^x+δ (x+δ -y)(f_a”(y)-f_a”(x-))dy,ϵ_2(x) =∫_x-δ^x (y-(x-δ))(f_a”(y)-f_a”(x-))dy.Now observe thatf_a(x+δ) -f_a(x) = f_a'(x) δ+ ∫_x^x+δ (x+δ -y)f_a”(y)dy = f_a'(x) δ+1/2δ^2 f_a”(x-)+∫_x ^x+δ (x+δ -y)(f_a”(y)-f_a”(x-))dy = f_a'(x) δ+ 1/2δ^2 f_a”(x-) +ϵ_1(x),and(f_a(x-δ) -f_a(x))=- f_a'(x) δ+∫_x-δ^x (y-(x-δ))f_a”(y)dy=-f_a'(x) δ+1/2δ^2 f_a”(x-) +∫_x-δ^x (y-(x-δ))(f_a”(y)-f_a”(x-))dy=-f_a'(x) δ+1/2δ^2 f_a”(x-) + ϵ_2(x).For k ∈_+ andx=x_k=δ(k-x(∞)), we recall the forms of G_Y f_a(x) and G_X̃ f_a(x) from (<ref>) and (<ref>) to see thatG_X̃f_a(x) =λδ f_a'(x) + λ1/2δ^2 f_a”(x-) + λϵ_1(x) - d(k) δ f_a'(x) + d(k) 1/2δ^2 f_a”(x-) + d(k) ϵ_2(x)=b(x) f_a'(x) + λ1/2δ^2 f_a”(x-) + λϵ_1(x)+ (λ -1/δb(x)) 1/2δ^2 f_a”(x-) + (λ-1/δb(x) ) ϵ_2(x)=G_Y f(x) - b(x)1/2δ f_a”(x-) + λ(ϵ_1(x)+ϵ_2(x)) - 1/δb(x)ϵ_2(x),where in the second equality we used the fact that b(x) = δ( λ - d(k)), and in the last equality we use that δ^2 λ = μ. Combining this with (<ref>), we have an error bound similar to (<ref>):| (X̃(∞) ≤ a) - (Y(∞) ≤ a) |≤1/2δ[ |f_a”(X̃(∞)-)b(X̃(∞)) | ] + λ[ | ϵ_1(X̃(∞)) | ]+ λ[ |ϵ_2(X̃(∞))| ] + 1/δ[ | b(X̃(∞))ϵ_2(X̃(∞))| ],where ϵ_1(x) and ϵ_2(x) are as in (<ref>) and (<ref>), respectively. To bound the error terms in (<ref>) that are associated with ϵ_1(x) and ϵ_2(x), we need to analyze the difference f_a”(y) - f_a”(x-) for x-y≤δ. Since f_a(x) is a solution to the Poisson equation (<ref>), we see that for any x, y ∈ with y ≠ a, f_a”(y)-f_a”(x-) = 1/μ[ 1_(-∞, a](x) - 1_(-∞, a](y) + b(x)f_a'(x) - b(y)f_a'(y) ].Therefore, for any y ∈ [x, x + δ]with y ≠ a,f_a”(y)-f_a”(x-) ≤1/μ[ 1_(a-δ, a](x) + b(x)f_a'(x) -f_a'(y) + b(x)-b(y))f_a'(y)] ≤1/μ[ 1_(a-δ, a](x) + δb(x)f” + b(x)-b(y))f_a'(y)],and likewise, for any y ∈ [x-δ, x] with y ≠ a, f_a”(y)-f_a”(x-) ≤1/μ[ 1_(a, a + δ](x) + b(x)f_a'(x) -f_a'(y) + b(x)-b(y))f_a'(y)] ≤1/μ[ 1_(a, a + δ](x) + δb(x)f” + b(x)-b(y))f_a'(y)].The inequalities above contain the indicators 1_(a-δ, a](x) and 1_(a, a + δ](x). When we consider the upper bound in (<ref>), these indicators will manifest themselves as probabilities (a - δ < X̃(∞) ≤ a) and (a< X̃(∞) ≤ a+δ). To this end we present the following lemma, which will be used in the proof of Theorem <ref>.Consider the Erlang-C model (α = 0). Let W be an arbitrary random variable with cumulative distribution function F_W:→ [0,1]. Let ω(F_W) be the modulus of continuity of F_W, defined as ω(F_W) = sup_x, y ∈x ≠ yF_W(x)-F_W(y)/x-y.Recall that d_K(X̃(∞), W) is the Kolmogorov distance between X(∞) and W. Then for any a ∈, n ≥ 1, and 0 < R< n,( a - δ < X̃(∞) ≤ a + δ) ≤ω(F_W)2δ+ d_K(X̃(∞), W) + 9δ^2 + 8δ^4. This lemma is proved in Section <ref>. We will apply Lemma <ref> with W = Y(∞) in the proof of Theorem <ref> that follows. The following lemma guarantees that the modulus of continuity of the cumulative distribution function of Y(∞) is bounded by a constant independent of λ, n, and μ. Its proof is provided in Section <ref>. Consider the Erlang-C model (α = 0), and let ν(x) be the density of Y(∞), defined in (<ref>). Then for for all n ≥ 1, λ > 0, and μ>0 satisfying 0 < R<n,ν(x)≤√(2/π) ,x ∈. Lemmas <ref> and <ref> are stated for the Erlang-C model, but one can easily repeat the arguments in the proofs of those lemmas to prove analogues for the Erlang-A model. Therefore, we state the following lemmas without proof.Consider the Erlang-A model (α > 0). Let W be an arbitrary random variable with cumulative distribution function F_W:→ [0,1]. Let ω(F_W) be the modulus of continuity of F_W. Then for any a ∈, α > 0, n ≥ 1, and R > 0,( a - δ < X̃(∞) ≤ a + δ)≤ω(F_W)2δ+ d_K(X̃(∞), W) + 9 ( α/μ∨ 1 )δ^2 + 8( α/μ∨ 1 )^2δ^4. Consider the Erlang-A model (α > 0), and let ν(x) be the density of Y(∞). Fix n ≥ 1, λ > 0, μ>0, and α > 0. If 0 < R ≤ n, thenν(x)≤√(2/π) ,x ∈,andif n≤ R, thenν(x)≤√(2/π)√(α/μ),x ∈. §.§ Proof of Theorem <ref> (Erlang-C Kolmogorov) Throughout the proof we assume that R ≥ 1, or equivalently, δ≤ 1. For h(x) = 1_(-∞, a](x), we let f_a(x) be a solution the Poisson equation (<ref>) with parameter a_2 = 0. In this proof we will show that for all a ∈, (X̃(∞) ≤ a) - (Y(∞) ≤ a)≤1/2(a - δ < X̃(∞) ≤ a + δ) + 59δ,The upper bound in (<ref>) is similar to (<ref>), however (<ref>) has the extra term1/2(a - δ < X̃(∞) ≤ a + δ).The reason this term appears in the Kolmogorov setting but not in the Wasserstein setting is because f_a”(x) is discontinuous in the Kolmogorov case, as opposed to the Wasserstein case where f_h”(x) is continuous. Applying Lemmas <ref> and <ref> to the right hand side of (<ref>), and taking the supremum over all a ∈ on both sides, we see that d_K(X̃(∞), Y(∞)) ≤1/2d_K(X̃(∞), Y(∞)) + 2√(2/π)δ+ 9δ^2 + 8δ^4 +59δ,or d_K(X̃(∞), Y(∞)) ≤ 156δ.We want to add that Lemma <ref> makes heavy use of the birth-death structure of the Erlang-C model, and that it is not obvious how to handle (<ref>) more generally. To prove Theorem <ref> it remains to verify (<ref>), which we now do. The argument we will use is similar to the argument used to prove (<ref>) in Theorem <ref>. We will bound each of the terms in (<ref>), which we recall here as | (X̃(∞) ≤ a) - (Y(∞) ≤ a) |≤1/2δ[ |f_a”(X̃(∞)-)b(X̃(∞)) | ] + λ[ | ϵ_1(X̃(∞))| ]+ λ[ | ϵ_2(X̃(∞))| ] + 1/δ[ | b(X̃(∞))ϵ_2(X̃(∞))| ].We also recall the form of b(x) from (<ref>). We use the moment bounds (<ref>) and (<ref>), and the gradient bound (<ref>) to see that[ |f_a”(X̃(∞)-)b(X̃(∞)) |] ≤2/μ[|b(X̃(∞)) | ] =2[|X̃(∞) 1(X̃(∞) ≤-ζ - δ)| ]+ 2 ζ (X̃(∞) ≥ -ζ)≤2√(4/3 + 2δ^2/3) + 2(ζ∧[ |X̃(∞)| 1(X̃(∞) ≥ -ζ) ])≤2√(2) + 14/4≤ 7.Next, we use (<ref>), (<ref>), and the gradient bound (<ref>) to getλ[ |ϵ_1(X̃(∞))| ] =λ[∫_X̃(∞)^X̃(∞)+δ (X̃(∞)+δ -y) |f_a”(y)-f_a”(X̃(∞)-)|dy]≤λ/μ[ 1_(a-δ, a](X̃(∞))∫_X̃(∞)^X̃(∞)+δ (X̃(∞)+δ -y)dy] + λ/μδ^3 [ | b(X̃(∞)) | ]f_a” + λ/μδ[∫_X̃(∞)^X̃(∞)+δ|b(X̃(∞))-b(y))| |f_a'(y)|dy ]≤1/2( a - δ < X̃(∞) ≤ a) + 7 δ +4δ=1/2( a - δ < X̃(∞) ≤ a) + 11δ,where in the last inequality we used the fact that for y ∈ [X̃(∞), X̃(∞) + δ], b(X̃(∞)) - b(y) = μδ 1(X̃(∞) ≤ -ζ - δ).By a similar argument, one can check thatλ[ |ϵ_2(X̃(∞))| ]≤1/2( a< X̃(∞) ≤ a+δ) +11δ,with the only difference in the argument being that we consider the cases when X̃(∞) ≤ -ζ and X̃(∞) ≥ -ζ + δ, instead of X̃(∞) ≤ -ζ -δ and X̃(∞) ≥ -ζ.Lastly, we use the first inequality in (<ref>) to see that1/δ[ | b(X̃(∞))ϵ_2(X̃(∞))| ] ≤1/μ[ | b(X̃(∞))| ∫_X̃(∞)-δ^X̃(∞)[ 1_(a,a+δ](X̃(∞)) +|b(X̃(∞))|(|f_a'(X̃(∞))| + |f_a'(y)| )+ |b(X̃(∞))-b(y))||f_a'(y)| ]dy ] ≤δ1/μ[ |b(X̃(∞))| ] + δ1/μ[ |b^2(X̃(∞))f_a'(X̃(∞))| ] +1/μ[ | b^2(X̃(∞))| ∫_X̃(∞)-δ^X̃(∞)f_a'(y)dy] +4δ^2 [| X̃(∞)1(X̃(∞) ≤ -ζ)| ] ≤7/2δ+ δ1/μ[ |b^2(X̃(∞))f_a'(X̃(∞))| ] +1/μ[ | b^2(X̃(∞))| ∫_X̃(∞)-δ^X̃(∞)f_a'(y)dy] +4 √(2)δ^2,where in the last inequality we used (<ref>) and the moment bound (<ref>). Now by (<ref>) and (<ref>),δ1/μ[ | b^2(X̃(∞))f_a'(X̃(∞))| ]≤4δ[ X̃^2(∞)1( X̃(∞) ≤ -ζ ) ] + δζ(X̃(∞) ≥ -ζ + δ) ≤8 δ + δ7/4≤ 10δ,and similarly,1/μ[ | b^2(X̃(∞))| ∫_X̃(∞)-δ^X̃(∞)f_a'(y)dy]≤10 δ.Therefore,1/δ[ | b(X̃(∞))ϵ_2(X̃(∞))| ] ≤7/2δ+ 20δ + 4√(2)δ^2 ≤ 30 δ.This verifies (<ref>) and concludes the proof of Theorem <ref>. §.§ Outline for Theorem <ref> (Erlang-A Kolmogorov) The proof of Theorem <ref> is nearly identical to the proof of Theorem <ref>. Therefore, we only outline the key steps and differences. The goal is to obtain a version of (<ref>), from which the theorem follows by applying Lemmas <ref> and <ref>. To get a version of (<ref>), we bound each of the terms in (<ref>), just like we did in the proof of Theorem <ref>. The proof varies between the underloaded and overloaded cases.We begin with the underloaded case (1 ≤ R ≤ n). To bound the first term in (<ref>), we use moment bounds (<ref>), (<ref>), and (<ref>), together with gradient bound (<ref>). For the second and third terms in (<ref>) we use the gradient bound in (<ref>). For the fourth error term, we use gradient bound (<ref>), and moment bounds (<ref>), (<ref>), and[ ( b(X̃(∞)))^2 1( X̃(∞) ≥ -ζ) ] =α^2[ ( X̃(∞)+ ζ)^2 1( X̃(∞) ≥ -ζ) ]+ μ^2 ζ^2(X̃(∞) ≥ -ζ)+ 2αμζ[ (X̃(∞) + ζ)1( X̃(∞) ≥ -ζ)]≤α^2 1/3(μ/αδ^2+ μ/α4 + δ^2)+ μ^2 ζ^2(X̃(∞) ≥ -ζ) + 2αμ( δ^2/4α/μ+ δ^2/4 + 1 ),where the last inequality follows from moment bounds (<ref>) and (<ref>).In the overloaded case (n ≤ R), to bound the first term in (<ref>) we use moment bounds (<ref>), (<ref>), and (<ref>) with gradient bound (<ref>). To bound the second and third terms in (<ref>) we use gradient bound (<ref>). To bound the fourth term in (<ref>), we use gradient bound (<ref>), with moment bounds (<ref>) and[ ( b(X̃(∞)))^2 1( X̃(∞) ≤ -ζ) ] =μ^2[ ( X̃(∞)+ ζ)^2 1( X̃(∞) ≤ -ζ) ]+ α^2 ζ^2(X̃(∞) ≤ -ζ)+ 2αμζ[| (X̃(∞) + ζ)1( X̃(∞) ≤ -ζ)|]≤μ^2 (δ^2/4α/μ+1) + α^2 (δ^2/4+μ/α) + 2αμ(δ^2/4+1),where the last inequality follows from moment bounds (<ref>), (<ref>), and (<ref>).§ EXTENSION: ERLANG-C HIGHER MOMENTS In this section we consider the approximation of higher moments for the Erlang-C model. We begin with the following result. Consider the Erlang-C system (α = 0), and fix an integer m > 0. There exists a constant C = C(m), such that for all n ≥ 1, λ > 0, and μ>0 satisfying 1 ≤ R < n,| (X̃(∞))^m -(Y(∞))^m|≤ (1+1/ζ^m-1)C(m)δ, where ζ is defined in (<ref>).The proof of this theorem follows the standard Stein framework in Section <ref>, but we do not provide it in this document. The most interesting aspect of (<ref>) is the appearance of 1/ζ^m-1 in the bound on the right hand side, which of course only matters when ζ is small. To check whether the bound is sharp, we performed some numerical experiments illustrated in Table <ref>. The results suggest that the approximation error does indeed grow like 1/ζ^m-1.A better way to understand the growth parameter 1/ζ^m-1 is through its relationship with (X̃(∞))^m-1. We claim that(X̃(∞))^m-1≈ 1/ζ^m-1for small values of ζ. The following lemma, which is proved in Section <ref>,is needed. For any integer m ≥ 1, and all n ≥ 1, λ > 0, and μ>0 satisfying R < n,lim_ζ↑ 0ζ^m (Y(∞))^m = m!. Multiplying both sides of (<ref>) by ζ^m and applying Lemma <ref>, we see that for all n ≥ 1, λ > 0, and μ>0 satisfying 1 ≤ R < n,lim_ζ↑ 0ζ^m (X̃(∞))^m = m!.In other words,we can rewrite (<ref>) as | (X̃(∞))^m -(Y(∞))^m| ≤(1+1/ζ^m-1| (X̃(∞))^m-1|| (X̃(∞))^m-1|)C(m)δ ≤(1+| (X̃(∞))^m-1|)C̃(m)δ,where C̃(m) is a redefined version of C(m). That the approximation error in Table <ref> increases is then attributed to the fact that X̃(∞) increases as ζ↑ 0. As we mentioned before, the appearance of the (m-1)th moment in the approximation error of the mth moment was also observed recently in <cit.> for the virtual waiting time in the M/GI/1+GI model, potentially suggesting a general trend. § CHAPTER APPENDIX §.§ Miscellaneous LemmasIn this section we prove Lemmas <ref>, <ref>,<ref>, and <ref>. §.§.§ Proof of Lemma <ref>Letf(x): → satisfy f(x)≤ C(1+x)^3. A sufficient condition to ensure that[ G_X̃ f(X̃(∞)) ] = 0is given by <cit.> (alternatively, see <cit.>). Namely, we require that [| G_X̃ (X̃(∞), X̃(∞)) f(X̃(∞))| ] < ∞,where G_X̃ (x,x) is the diagonal entry of the generator matrix G_X̃ corresponding to state x. In the Erlang-C model, the transition rates of X̃ are bounded by λ + nμ. Since f(x)≤ C(1+x)^3, it suffices to show that (X̃(∞))^3 < ∞, or that (X(∞))^3 < ∞, where X(∞) has the stationary distribution of the CTMC X. Consider the function V(k) = k^4, where k ∈_+. Let G_X be the generator of X, which is a simple birth death process with constant birth rate λ and departure rate μ (k ∧ n) in state k ∈_+. Then for k ≥ n, G_XV(k) =λ ( (k + 1)^4 - k^4) + nμ ((k-1)^4 - k^4)=λ (4k^3 + 6k^2 + 4k + 1) + nμ(-4k^3 + 6k^2 - 4k + 1) =-4k^3 ( nμ - λ) +6k^2 ( nμ + λ) -4k( n μ - λ ) + (λ + nμ).It is not hard to see that there exists some k_0 ∈_+, and a constant c > 0 (that depends on λ, n, and μ), such that for all k ≥ k_0,-4k^3 ( nμ - λ) +6k^2 ( nμ + λ) -4k( n μ - λ ) ≤ -ck^3. We combine (<ref>)–(<ref>) to conclude that there exists some constant d > 0 (that depends on λ, n, and μ) satisfying G_X V(k)≤ -c k^3 + d 1(k < (k_0 ∨ n)),and invoking <cit.>, we see that (X(∞))^3 < ∞. The case of the Erlang-A model is not very different. When α > 0, the transition rates of the CTMC depend linearly on its state. Hence, to satisfy (<ref>) we need to show that (X(∞))^4 < ∞. This is readily proven by repeating the procedure above with the Lyapunov function V(k) = k^5, and we omit the details.§.§.§ Proof of Lemma <ref>We let F_W(w) and F_X̃(x) be the distribution functions of W and X̃(∞), respectively. For any a ∈, letã = δ(a - x(∞)). We want to show that( ã - δ <X̃(∞) ≤ã + δ) =F_X̃(ã + δ) - F_X̃(ã - δ)≤2δω(F_W)+ d_K(X̃(∞), W) + 9δ^2 + 8δ^4.Let {π_k}_k=0^∞ be the distribution of X(∞), andk^* = inf{k ≥ 0 : π_k ≥ν_j,for all j ≠ k}.Then for any ã∈, F_X̃(ã + δ) - F_X̃(ã - δ) ≤ 2π_k^*,because X̃(∞) takes at most two values in the interval (ã - δ, ã + δ]. Observe that by the flow balance equations, we know that for any k ∈_+, π_k=d(k+1)/λπ_k+1,where d(k) is defined in (<ref>). Since k^* is the maximizer of {π_k}, we know that d(k^*) ≤λ≤ d(k^*+1) ≤λ + μ,where in the last inequality we have used the fact that the increase in departure rate between state k^* and k^*+1 is at most μ. Likewise, d(k^*+i) ≤λ + i μ for i = 2,3. Hence,π_k^*=d(k^*+1)/λπ_k^*+1≤(1 + μ/λ)π_k^*+1≤π_k^*+1 + δ^2, π_k^*=d(k^*+1)/λd(k^*+2)/λπ_k^*+2 ≤(1 + δ^2)(1 + 2δ^2)π_k^*+2≤π_k^*+2 + 3δ^2+ 2δ^4, π_k^*+1 =d(k^*+2)/λd(k^*+3)/λπ_k^*+3 ≤(1 + 2δ^2)(1 + 3δ^2)π_k^*+3≤π_k^*+3 + 5δ^2 + 6δ^4,which implies that for any ã∈, F_X̃(ã + δ) - F_X̃(ã - δ) ≤ 2π_k^*≤π_k^* + π_k^* + 1 + δ^2 =F_X̃(k̃^* + δ) - F_X̃(k̃^* - δ) + δ^2. There are now 4 cases to consider, with the first three being simple to handle. Recall that ω(F_W) is the modulus of continuity of F_W(w).*If F_W(k̃^* - δ) ≤ F_X̃(k̃^* - δ) and F_W(k̃^* + δ) ≥ F_X̃(k̃^* + δ), then F_X̃(k̃^* + δ) - F_X̃(k̃^* - δ) ≤ F_W(k̃^* + δ) - F_W(k̃^* - δ) ≤ 2δω(F_W). *If F_W(k̃^* - δ) ≤ F_X̃(k̃^* - δ) but F_W(k̃^* + δ) < F_X̃(k̃^* + δ), then F_X̃(k̃^* + δ) - F_X̃(k̃^* - δ)≤ F_X̃(k̃^* + δ) - F_W(k̃^* + δ) +F_W(k̃^* + δ) - F_W(k̃^* - δ)≤ 2δω(F_W)+ d_K(X̃(∞), W). *Similarly, if F_W(k̃^* - δ) > F_X̃(k̃^* - δ) and F_W(k̃^* + δ) ≥ F_X̃(k̃^* + δ), then F_X̃(k̃^* + δ) - F_X̃(k̃^* - δ)≤ F_W(k̃^* + δ) - F_W(k̃^* - δ) + F_W(k̃^* - δ) - F_X̃(k̃^* - δ) ≤ 2δω(F_W)+ d_K(X̃(∞), W). * Suppose F_W(k̃^* - δ) > F_X̃(k̃^* - δ) and F_W(k̃^* + δ) < F_X̃(k̃^* + δ), then we need to use a different approach. We know thatF_X̃(k̃^* + δ) - F_X̃(k̃^* - δ) =π_k^* + π_k^*+1 ≤π_k^*+2 + π_k^* + 3 + 8δ^2 + 8δ^4 =F_X̃(k̃^* + 3δ) - F_X̃(k̃^* + δ)+ 8δ^2 + 8δ^4.SinceF_W(k̃^* + δ) ≤ F_X̃(k̃^* + δ), we are either incase <ref>or <ref> for the differenceF_X̃(k̃^* + 3δ) - F_X̃(k̃^* + δ), and hence we haveF_X̃(k̃^* + 3δ) - F_X̃(k̃^* + δ) ≤ 2δω(F_W)+ d_K(X̃(∞), W). This proves (<ref>), concluding the proof of this lemma.§.§.§ Proof of Lemma <ref> In the Erlang-C model,ν(x) = a_- e^-1/2x^2,x ≤ - ζ,a_+ e^-ζ x,x ≥ -ζ.To bound this density, we need to bound a_- and a_+. We know that ν(x) must integrate to one, which implies that a_- ∫_-∞^-ζ e^-1/2y^2 dy + a_+ ∫_-ζ^∞ e^-ζ y dy = 1Furthermore, since ν(x) is continuous at x = -ζ, a_- e^-1/2ζ^2 = a_+ e^-ζ^2.Combining these two facts, we see that a_- = 1/∫_-∞^-ζ e^-1/2y^2 dy + e^1/2ζ^2∫_-ζ^∞ e^-ζ y dy≤1/∫_-∞^0 e^-1/2y^2 dy = √(2/π),and a_+ = 1/e^-1/2ζ^2∫_-∞^-ζ e^-1/2y^2 dy +∫_-ζ^∞ e^-ζ y dy≤1/e^-1/2ζ^2∫_-∞^0 e^-1/2y^2 dy = e^1/2ζ^2√(2/π).Therefore, for x ≤ -ζ, ν(x)≤ a_- ≤√(2/π),and for x ≥ -ζ, we recall that ζ < 0 to see thatν(x)≤ a_+ e^-ζ x≤√(2/π)e^1/2ζ^2e^-ζ x≤√(2/π).§.§.§ Proof of Lemma <ref> The density of Y(∞) is given in (<ref>), and so(Y(∞))^m = a_- ∫_-∞^-ζ y^me^-1/2y^2 dy + a_+ ∫_-ζ^∞ y^me^-ζ y dy,where a_- and a_+ are as in (<ref>) and (<ref>). In particular, a_- = 1/∫_-∞^-ζ e^-1/2y^2 dy + e^1/2ζ^2∫_-ζ^∞ e^-ζ y dy= 1/∫_-∞^-ζ e^-1/2y^2 dy + 1/ζ e^-1/2ζ^2,which implies that lim_ζ↑ 0ζ^m a_- ∫_-∞^-ζ y^me^-1/2y^2 dy = 0.Furthermore, a_+ = 1/e^-1/2ζ^2∫_-∞^-ζ e^-1/2y^2 dy +∫_-ζ^∞ e^-ζ y dy = 1/e^-1/2ζ^2∫_-∞^-ζ e^-1/2y^2 dy + 1/ζ e^-ζ^2,and using integration by parts,∫_-ζ^∞ y^me^-ζ y dy =e^-ζ^2∑_j=0^mm!/(m-j)!1/ζ^j+1ζ^m-j=e^-ζ^2∑_j=0^m-1m!/(m-j)!1/ζ^j+1ζ^m-j +e^-ζ^2m!/ζ^m+1.Hence, lim_ζ↑ 0ζ^m a_+ ∫_-ζ^∞ y^me^-ζ y dy =m!. CHAPTER: STATE DEPENDENT DIFFUSION COEFFICIENT: FASTER CONVERGENCE RATESChoosing a diffusion approximationinvolves selecting a drift b̅(x) and a diffusion coefficient a̅(x).When choosing a diffusion approximation of a Markov chain, one would think that best course of action would be to choose b̅(x) and a̅(x) based on the infinitesimal drift and variance of the Markov chain, respectively. While the drift of the diffusion b̅(x) is usually matched exactly to the infinitesimal drift of the Markov chain, the diffusion coefficient a̅(x) is often taken to be a constant, even when the infinitesimal variance of the Markov chain is state dependent; see <cit.> just to name a few. However, not everyone uses a constant a̅(x). State-dependent diffusion coefficients are used for example in strong approximation theorems in <cit.>; see <cit.> for further discussion.In <cit.>,the authors compare two diffusion approximations, onewith constant and one with state-dependent a̅(x). Numerically, they find that the latter does perform a little better, but overall they are unenthusiastic about promoting its use. The main reason being that a state-dependent diffusion coefficient makes the transient behavior of the diffusion process more difficult to compute, and their observed accuracy gains are not sufficient to justify this extra difficulty.The purpose of this chapter is to strongly promote the use of state-dependent diffusion coefficients a̅(x) that more accurately capture the infinitesimal variance of the Markov chain. Working in the setting of the Erlang-C model, we prove in Theorem <ref> that the error from an approximation with a state-dependent diffusion coefficient goes to zero an order of magnitude faster than the error from an approximation with a constant diffusion coefficient. We will also see that a state-dependent diffusion coefficient does not increase the difficulty of computing the stationary distribution of the diffusion. Going forward, the reader is assumed to be familiar with the content of Chapter <ref>. In particular, we assume familiarity with the Stein framework from Section <ref>. We begin the chapter with Section <ref>, where we present Theorem <ref> and some numerical results that go along with it. In Section <ref>, we present the ingredients needed to prove Theorem <ref> and carry out the proof in Section <ref>. Section <ref> is a shortappendix for the chapter. § MAIN RESULT We adopt the notation of Chapter <ref>, which we recall briefly below. The Erlang-C system has n servers, arrival rate λ, and service rate μ. The quantity R = λ/μ is known as the offered load, and we set δ = 1/√(R) for convenience. The customer count process is X = {X(t), t ≥ 0} and the scaled and centered process is X̃ = {δ(X(t) - R), t ≥ 0}. When R < n, these processes are positive recurrent, andX(∞) and X̃(∞) are the random variables having the respective stationary distributions. The process X̃ has generator G_X̃ f(x) = λ (f(x + δ) - f(x)) + d(k) (f(x-δ) - f(x)),where k ∈_+, x = x_k = δ(k - x(∞)), andd(k) = μ (k ∧ n),is the departure rate corresponding to the system having k customers.We also recall ζ = δ(R - n), which was defined in (<ref>). The approximation to X̃(∞) was Y(∞), a continuous random variable with density ν(x) given in (<ref>). The random variable Y(∞) corresponds to a diffusion process with driftb(x) = -μ x,x ≤ -ζ, μζ,x ≥ -ζ,and diffusion coefficient 2μ.In this chapter, we propose a different diffusion approximation. Namely, let Y_S(∞) be the continuous random variable with density ν_S(x)= κ/a(x)exp(∫_0^x 2b(y)/a(y)dy),x ∈,where κ > 0 is a normalization constant, and a(x)=μ ,x ≤ -1/δ,μ (2 + δ x),x ∈ [-1/δ, -ζ],μ (2 + δζ),x ≥ -ζ, . One may check that for k ∈_+ and x = δ(k-R), b(x) = δ( λ - d(k)),anda(x) = δ(λ + d(k) 1(k > 0)).The random variable Y_S(∞) has the stationary distribution of a diffusion process on the real line with drift b(x) and state dependent diffusion coefficient a(x). In contrast, in Chapter <ref> we used a constant diffusion coefficient of 2μ. The following is the main result of this chapter. There exists a constant C > 0 (independent of λ, n, and μ), such that for all n ≥ 1, λ > 0, and μ > 0 satisfying 1 ≤ R < n,d_W_2(X̃(∞), Y_S(∞)) := sup_h(x) ∈ W_2|h(X̃(∞)) -h(Y_S(∞)) | ≤C/R,where W_2 = {h: → |h(x), h'(x) ∈}. Theorem <ref> should be compared with Theorem <ref> of Chapter <ref>. The former has a convergence rate of 1/R versus the 1/√(R) rate of the latter. The class of functions W_2 in (<ref>) is not significantly smaller than , meaning that the two statements are comparable. We will see in Section <ref> that W_2 is a rich enough class of functions to imply convergence in distribution. Theorem <ref> can also be compared to the results in <cit.> and Chapter <ref> (which is based in <cit.>), all of which study convergence rates for steady-state diffusion approximations of various models. A rate of 1/R is an order of magnitude better than the rates in any of the previously mentioned papers, whose rates are equivalent to 1/√(R) in our model.§.§ Numerical Study Before moving on to the proof of Theorem <ref>, we present some numerical results to complement the theorem. The results in this section show that Y_S(∞) consistently outperforms Y(∞). In Table <ref> we see that for large or heavily loaded systems, i.e. when R is either large or close to n, the approximation Y(∞) performs reasonably well, and the accuracy gained from using Y_S(∞) is not as impressive. However, the accuracy gain of Y_S(∞) is much more significant for smaller systems with lighter loads. In Table <ref> we see that the errors of Y(∞) and Y_S(∞) indeed decrease at a rate of 1/√(R) and 1/R, respectively. Furthermore, the table suggests that the approximation error of the second moment also decreases at a rate of 1/R, even though (<ref>) does not guarantee this. Numerically, we observed a rate of 1/R for higher moments as well. This is not surprising, as there is nothing preventing us from repeating the analysis in this chapter for higher moments.Furthermore, although Theorem <ref> is only stated in the context of the W_2 metric, we show that Y_S(∞) is a superior approximation to Y(∞) when it comes to estimating the both the probability mass function (PMF), and cumulative distribution function (CDF). Let {π_k}_k=0^∞ be the distribution of X(∞). For k ∈_+ define π^Y_k = (Y(∞) ∈[δ(k - R)-δ/2, δ (k - R)+δ/2]), π^Y_S_k = (Y_S(∞) ∈[δ(k - R)-δ/2, δ (k - R)+δ/2]).Results for the PMF are displayed in Figure <ref> and Table <ref>, and results for the CDF are in Table <ref>. We observe numerically that the Kolmogorov distance converges to zero at a rate of 1/√(R) as opposed to 1/R. However, Y_S(∞) still performs better.§ PROOF COMPONENTS The proof of Theorem <ref> uses the Stein framework developed in Section <ref>. We assume familiarity with that section, and now state the main ingredients needed to prove Theorem <ref>. As we mentioned,the random variable Y_S(∞) is associated to a diffusion process with generatorG_Y_S f(x) =b(x)f'(x) +1/2a(x) f”(x)for x ∈,f ∈ C^2(),where b(x) and a(x) are defined in (<ref>) and (<ref>), respectively. Fix h(x) ∈ W_2 with h(0) = 0, and consider the Poisson equation G_Y_S f_h(x)=h(Y(∞)) - h(x), x∈.We use the Lipschitz property of h(x) to see thath(Y_S(∞))≤|Y_S(∞)| < ∞,where the finiteness of |Y_S(∞)| will be proved in (<ref>). Just as was done in (<ref>), we can take expected values on both sides of (<ref>) with respect to X̃(∞) and apply Lemma <ref> to get|h(X̃(∞)) -h(Y_S(∞)) | =|G_Y_S f_h(X̃(∞)) | =|G_X̃ f_h(X̃(∞)) -G_Y_S f_h(X̃(∞)) |≤| G_X̃ f_h(X̃(∞)) -G_Y_S f_h(X̃(∞)) |.We will shortly see in Lemma <ref> that there is indeed asolution f_h(x) to the Poisson equation (<ref>) with a bounded second derivative. This means that it can be bounded by a quadratic polynomial, and hence satisfy the conditions of Lemma <ref>. The following section presents the necessary moment and gradient bounds. §.§ Moment Bounds and Gradient Bounds Recall that ζ < 0. We begin with several moment bounds. For all n ≥ 1, λ > 0, and μ > 0 satisfying 1 ≤ R < n,(1+1/ζ)[|X̃(∞) 1(X̃(∞) ≤ -ζ) | ] ≤√(2) + 2,(1+1/ζ)[(X̃(∞))^21(X̃(∞) ≤ -ζ) ] ≤9, ζ(X̃(∞) ≥ -ζ) ≤2, ζ^2(X̃(∞) ≥ -ζ) ≤20,and if 0 < R < n, then(X̃(∞) ≤ -ζ) ≤ (2+δ)ζ.Let {π_k}_k=0^∞ be the distribution of X(∞). For all n ≥ 1, λ > 0, and μ > 0 satisfying 0 < R < n,π_0 ≤4(2+δ)δ^2ζ,when ζ≤ 1,and π_n ≤δζ. Lemmas <ref> and <ref> are proved in Section <ref>. Next we present the gradient bounds, which are proved in Section <ref>. Fix h(x) ∈ W_2 with h(0) = 0 and consider the Poisson equation (<ref>). There exists a solution f_h(x) such that f_h”(x) is absolutely continuous, f_h”'(x) exists and is continuous everywhere except the points x = -1/δ and x=-ζ, and lim_u ↑ x f_h”'(u) and lim_u ↓ x f_h”'(u) both exist at those two points. Moreover, there exists a constant C > 0 independent of λ, n, and μ, such that for all n ≥ 1, λ > 0, and μ > 0 satisfying 1 ≤ R < n,f_h'(x)≤ C/μ(1 +1/ζ),x ≤ -ζ, C/μζ(x + 1 + 1/ζ),x ≥ -ζ,f_h”(x)≤C/μ(1 +1/ζ),x ≤ -ζ, C/μζ,x ≥ -ζ,and f_h”'(x)≤C/μ(1 + 1/ζ),x ≤ -ζ, C/μ,x > -ζ,where f_h”'(x) is interpreted as the left derivative at the points x = -1/δ and x = -ζ.The gradient bounds in Lemma <ref> involve only the first three derivatives of f_h(x), and are not sufficient for us. We require the following bounds on the fourth derivative (when it exists). These are proved in Appendix <ref>. Fix h(x) ∈ W_2 with h(0)=0, and let f_h(x) be the solution to the Poisson equation (<ref>) from Lemma <ref>. Consider only those x ∈ such that x = δ (k - R) for some k ∈_+. Then there exists a constant C>0 independent of λ, n, and μ, such that for all n ≥ 1, λ > 0, and μ > 0 satisfying 1 ≤ R < n,f_h”'(x-) - f_h”'(y)≤Cδ/μ[ 1(x ≤ -ζ)(1 + x)(1 + 1/ζ) + 1(x ≥ -ζ+δ) (1+ ζ)], y ∈ (x-δ, x)and f_h”'(x-) - f_h”'(y) ≤Cδ/μ[ 1(x ≤ -ζ-δ)(1 + x)(1 + 1/ζ) + 1(x ≥ -ζ) (1+ ζ) + 1/δ(1 + 1/ζ)1(x∈{-1/δ, -ζ}) ],y ∈ (x, x + δ).The upper bound in (<ref>) has an extra term compared to the bound in (<ref>). This terms is the result of the discontinuity of f_h”'(x) at x = -1/δ and x = -ζ. §.§ Taylor Expansion In this section we perform a Taylor expansion on G_X̃ f_h(x) to get a handle on the difference G_X̃ f_h(x) - G_Y_S f_h(x). The Taylor expansion here is similar to the one in Section <ref>, except that now we expand to four terms, whereas the expansion in Section <ref> was done only up to three terms. Lemma <ref> guarantees that f_h”(x) is absolutely continuous, and that f_h”'(x) is continuous everywhere except the points x = -1/δ, and x = -ζ. We write f_h”'(x-) to denote lim_u ↑ xf_h”'(u). We first define ϵ̃_1(x) =1/2∫_x^x+δ (x+δ -y)^2(f_h”'(y)-f_h”'(x-))dy,x ∈,ϵ̃_2(x) = -1/2∫_x-δ^x (y-(x-δ))^2(f_h”'(y)-f_h”'(x-))dy,x ∈.Now observe thatf_h(x+δ) -f_h(x) = f_h'(x) δ+1/2δ^2 f_h”(x) +∫_x ^x+δ (x+δ -y)(f_h”(y)-f_h”(x))dy= f_h'(x) δ+1/2δ^2 f_h”(x) + 1/6δ^3 f_h”'(x-) + 1/2∫_x ^x+δ(x+δ -y)^2(f_h”'(y)-f_h”'(x-))dy= f_h'(x) δ+ 1/2δ^2 f_h”(x) + 1/6δ^3 f_h”'(x-)+ϵ̃_1(x),where the first equality is the same as in (<ref>).Similarly, one can check that(f_h(x-δ) -f_h(x)) = -f_h'(x) δ+1/2δ^2 f_h”(x)-1/6δ^3 f_h”'(x-)+ ϵ̃_2(x).Recall from (<ref>)that for any k ∈_+, and x = x_k = δ(k - R),b(x) = δ(λ - d(k)) and a(x) = δ(λ + d(k) 1(k > 0)). Therefore, G_X̃f_h(x) =λδ f_h'(x) + λ1/2δ^2 f_h”(x) + 1/6λδ^3 f_h”'(x-) + λϵ̃_1(x)- d(k) δ f_h'(x) + d(k) 1/2δ^2 f_h”(x) -d(k) 1/6δ^3 f_h”'(x-) + d(k) ϵ̃_2(x) =b(x) f_h'(x) + (λ +d(k)1(x ≥ -1/δ)) 1/2δ^2 f_h”(x)+ 1/6δ^3(λ - d(k))f_h”'(x-) + λϵ̃_1(x)+ (λ-1/δb(x) ) ϵ̃_2(x) =G_Y f_h(x) + 1/6δ^2b(x)f_h”'(x-) + λ(ϵ̃_1(x)+ϵ̃_2(x)) - 1/δb(x)ϵ̃_2(x),and|h(X̃(∞)) -h(Y_S(∞)) | ≤1/6δ^2 [ |f_h”'(X̃(∞)-)b(X̃(∞)) | ] + λ[ | ϵ̃_1(X̃(∞))|] + λ[ | ϵ̃_2(X̃(∞))|] +1/δ[ |b(X̃(∞)) ϵ̃_2(X̃(∞))|].The Taylor expansion in (<ref>) reveals the reason this approximation is better than the one in Chapter <ref>. This approximation is able to capture the entire second order term in the Taylor expansion of G_X̃ f_h(x) (i.e. all the terms that correspond to f_h'(x) and f_h”(x)). In contrast, the constant diffusion coefficient approximation in Chapter <ref> uses a a(0) = 2μ for the diffusion coefficient. Comparing (<ref>) to (<ref>), we see that there is an extra error term of the form 1/2δ^2[ | f_h”(X̃(∞)) ( a(X̃(∞)) - a(0) ) | ] = 1/2δ^2[ | f_h”(X̃(∞)) b(X̃(∞))| ],which turns out to be on the order of δ, not δ^2. We are now ready to prove Theorem <ref>. § PROOF OF THEOREM <REF> (FASTER CONVERGENCE RATES) Fix h(x) ∈W_2 with h(0) = 0, and let f_h(x) be as in Lemma <ref>. We will focus on bounding (<ref>), which we recall here as|h(X̃(∞)) -h(Y_S(∞)) | ≤1/6δ^2 [ |f_h”'(X̃(∞)-)b(X̃(∞)) | ] + λ[ | ϵ̃_1(X̃(∞))|] + λ[ | ϵ̃_2(X̃(∞))|] +1/δ[ |b(X̃(∞)) ϵ̃_2(X̃(∞))|],whereϵ̃_1(x) =1/2∫_x^x+δ (x+δ -y)^2(f_h”'(y)-f_h”'(x-))dy,ϵ̃_2(x) = -1/2∫_x-δ^x (y-(x-δ))^2(f_h”'(y)-f_h”'(x-))dy. Throughout the proof we assume that R ≥ 1, or equivalently, δ≤ 1. We will use C > 0 to denote a generic constant that may change from line to line, but does not depend of λ,n, and μ. Suppose we know that for some positive constants c_1, …, c_4 > 0 independent of λ, n, and μ, |h(X̃(∞)) -h(Y_S(∞)) | ≤δ^2 (c_1 + c_2+c_3+ δ c_4)+ Cδ(π_0+π_n),where {π_k}_k=0^∞ is the distribution of X(∞). Then to prove the theorem we would only need to show that π_0, π_n ≤ C δ.One way to prove this is to appeal to Theorem <ref>, which states that the Kolmogorov distance d_K(X̃(∞), Y(∞)) = sup_a ∈| ( X̃(∞) ≤ a) - ( Y(∞) ≤ a) | ≤ 156 δfor all n ≥ 1 and 1 ≤ R < n, where Y(∞) is the random variable with density ν(x) defined in (<ref>). We would then have that π_n =( -ζ - δ/2 ≤X̃(∞) ≤-ζ + δ/2) =( -ζ - δ/2 ≤ Y(∞) ≤-ζ + δ/2)+ ( -ζ - δ/2 ≤X̃(∞) ≤-ζ + δ/2) - ( -ζ - δ/2 ≤ Y(∞) ≤-ζ + δ/2) ≤δν + 2d_K(X̃(∞), Y(∞)) ≤δ C,where in the last inequality we apply Lemma <ref>, which states that ν(x) is always bounded by √(2/π). The same argument can be used to bound π_0.To conclude the theorem it remains to verify (<ref>), which we do by bounding each of the terms on the right side of (<ref>) individually. We recall here that the support of X̃(∞) is a δ-spaced grid, and in particular this grid contains the points -1/δ and -ζ. In the bounds that follow, we will often consider separately the cases where X̃(∞) ≤ -ζ, and X̃(∞) ≥ -ζ+δ. We recall that b(x) = μ((x+ζ)^- + ζ),and apply the gradient bound (<ref>) together with (<ref>) and (<ref>) of Lemma <ref> to see that [ |f_h”'(X̃(∞)-)b(X̃(∞)) | ] ≤C(1 + 1/ζ)[|X̃(∞) 1(X̃(∞) ≤ -ζ) | ] + Cζ(X̃(∞) ≥ -ζ+δ)≤C(√(2) + 2) + 2C =: c_1.To bound the next term, we use (<ref>) from Lemma <ref> to see thatλ[ | ϵ̃_1(X̃(∞))|] ≤μ/2[∫_X̃(∞)^X̃(∞)+δf_h”'(X̃(∞)-)-f_h”'(y) dy ] ≤Cδ^2 [ 1(X̃(∞) ≤ -ζ -δ)(1 + |X̃(∞))|)(1 + 1/ζ)+ 1(X̃(∞) ≥ -ζ) (1+ ζ)+ 1/δ(1 + 1/ζ)1(X̃(∞)∈{-1/δ, -ζ})] ≤Cδ^2 [[ |X̃(∞)1(X̃(∞) ≤ -ζ -δ) |](1 + 1/ζ) + (X̃(∞) ≤ -ζ -δ)(1 + 1/ζ)+ (X̃(∞) ≥ -ζ) (1+ ζ)+1/δ(1 + 1/ζ)(π_0 + π_n)],where in the last inequality we used the fact that (X̃(∞)=-1/δ) and (X̃(∞)=-ζ) equal π_0 and π_n, respectively. We first use (<ref>), (<ref>), and (<ref>) to see that λ[ | ϵ̃_1(X̃(∞))|] ≤Cδ^2 ((√(2) + 2) + (1 + 3) +(1 + 2))+ Cδ(1 + 1/ζ)(π_0+π_n)≤Cδ^2 + Cδ(π_0+π_n) + Cδ/ζ (π_0+π_n)=Cδ^2 + Cδ(π_0+π_n) + Cδ/ζπ_0 1(ζ≥ 1) + Cδ/ζπ_0 1(ζ≤ 1) + Cδ/ζπ_n≤Cδ^2 + Cδ(π_0+π_n) + Cδ/ζπ_0 1(ζ≤ 1) + Cδ/ζπ_n.Next, we apply the bounds on π_0 and π_n from (<ref>) and (<ref>) to conclude thatλ[ | ϵ̃_1(X̃(∞))|] ≤ c_2δ^2 + Cδ(π_0+π_n).We move on to bound the next term in (<ref>). Using (<ref>) from Lemma <ref>,λ[ | ϵ̃_2(X̃(∞))|] ≤μ/2[∫_X̃(∞)-δ^X̃(∞)f_h”'(X̃(∞)-)-f_h”'(y) dy ] ≤Cδ^2 [ 1(X̃(∞) ≤ -ζ)(1 + |X̃(∞)|)(1+1/ζ) + 1(X̃(∞) ≥ -ζ+δ) (1+ ζ)] ≤Cδ^2 [(X̃(∞) ≤ -ζ)(1+1/ζ)+ [|X̃(∞)1(X̃(∞) ≤ -ζ)| ](1+1/ζ) + (X̃(∞) ≥ -ζ+δ) (1+ ζ)].Now (<ref>), (<ref>), and (<ref>) imply thatλ[ | ϵ̃_2(X̃(∞))|] ≤Cδ^2 (4 + (√(2)+2)+ (1+2)) =: c_3 δ^2.For the last term in (<ref>), we use the form of b(x) together with (<ref>) from Lemma <ref> to see that1/δ[ |b(X̃(∞)) ϵ̃_2(X̃(∞))|] ≤δ/2[|b(X̃(∞))|∫_X̃(∞)-δ^X̃(∞)f_h”'(X̃(∞)-)-f_h”'(y) dy ] ≤Cδ^3 [ [|X̃(∞)(1 + |X̃(∞)|) 1(X̃(∞) ≤ -ζ)|] (1+1/ζ)+ (X̃(∞) ≥ -ζ+δ)ζ (1+ ζ)] ≤Cδ^3 [ [|X̃(∞) 1(X̃(∞) ≤ -ζ)|](1+1/ζ) + [(X̃(∞))^21(X̃(∞) ≤ -ζ)|] (1+1/ζ)+ (X̃(∞) ≥ -ζ+δ)(ζ+ζ^2 )].We apply (<ref>)–(<ref>) from Lemma <ref> to conclude that1/δ[ |b(X̃(∞)) ϵ̃_2(X̃(∞))|] ≤Cδ^3 ( (√(2)+2) + 9 +2 + 20) =: c_4δ^3.Therefore, we have shown that for all R ≥ 1, and h(x) ∈W_2 with h(0)=0, (<ref>) holds, concluding the proof of Theorem <ref>. § CHAPTER APPENDIX§.§ The W_2 Metric Let W_2 be the class of functions defined in (<ref>), i.e. the class of differentiable functions h(x): → such that both h(x) and h'(x) belong to . For two random variables U and V, define their W_2 distance to be d_W_2(U, V) = sup_h∈W_2[h(U)] -[h(V)],and recall the Kolmogorov distance d_K(U, V) defined in (<ref>). In this section we prove the following relationship between the W_2 and Kolmogorov distances. This lemma is a modified version of <cit.>.Let U, V be two random variables, and assume that V has a density bounded by some constant C > 0. If d_W_2(U, V) < 4C, then d_K(U, V) ≤ 5(C/2)^2/3 d_W_2(U, V)^1/3. We wish to combine Lemma <ref> with Theorem <ref>, but to do so we need a bound on the density of Y_S(∞). Let ν_S(x) : → be the density of Y_S(∞), whose form is given in (<ref>). Then for all n ≥ 1, λ > 0, and μ > 0 satisfying 1≥ R < n, ν_S(x) ≤ 4,x ∈. Now combining Theorem <ref> with Lemmas <ref> and <ref> implies that d_K(X̃(∞), Y_S(∞) ) converges to zero at a rate of 1/R^1/3. However, we believe this rate to be sub-optimal, and that d_K(X̃(∞), Y_S(∞) ) actually vanishes at a rate of 1/√(R). This is supported by numerical results in Section <ref>. Fix a ∈ and let h(x) = 1_(-∞, a](x). Now fix ϵ∈ (0, 2) and define the smoothed versionh_ϵ(x) = 1,x ≤ a, -2/ϵ^2(x-a)^2 + 1,x ∈ [a, a + ϵ/2],2/ϵ^2[x - (a + ϵ/2) ]^2 - 2/ϵ(x-a) + 3/2,x ∈ [a + ϵ/2, a + ϵ], 0,x ≥ a + ϵ.Since we chose ϵ < 2, it is not hard to see that h_ϵ'(x)≤4/ϵ^2, h_ϵ”(x)≤4/ϵ^2,x ∈,where h_ϵ”(x) is interpreted as the left derivative of h_ϵ'(x) for x ∈{a, a+ϵ/2, a+ϵ}. Therefore, ϵ^2/4 h_ϵ(x) ∈ W_2. Then h(U) -h(V) = h(U) -h_ϵ(V) + [ h_ϵ(V) - h(V)] ≤ h_ϵ(U) -h_ϵ(V) + C ∫_a^a+ϵ h_ϵ(x) dx = h_ϵ(U) -h_ϵ(V) + Cϵ/2≤4/ϵ^2 d_W_2(U,V) + Cϵ /2,Choose ϵ = (2d_W_2(U,V)/C)^1/3, which lies in (0,2) by our assumption that d_W_2(U,V) < 4C. Thenh(U) -h(V) ≤ 5(C/2)^2/3 d_W_2(U, V)^1/3.Using the function h̃_ϵ(x) := h_ϵ(x+ϵ), a similar argument can be repeated to show that h(V) -h(U) = h(V) - h̃_ϵ(V) + h̃_ϵ(V) -h(U)≤5(C/2)^2/3 d_W_2(U, V)^1/3,concluding the proof.One can check (see also (<ref>) in Section <ref>)that (<ref>) implies ν_S(x) = a_1/μe^-x^2,x ≤ -1/δ, a_2/μ(2 + δ x)e^2/δ^2 [2log(2 + δ x) - δ x],x ∈ [-1/δ, -ζ],a_3/μ(2 + δζ)e^-2ζ x /2+δζ,x ≥ -ζ,where the constants a_1, a_2, a_3 make the ν_S(x) continuous and integrate to one. To prove that ν_S(x) is bounded, we need to bound these three constants. We know that a_1 ∫_-∞^-1/δ1/μe^-y^2 dy + a_2 ∫_-1/δ^-ζ1/μ(2 + δ y)e^2/δ^2 [2log(2 + δ y) - δ y] dy+ a_3 ∫_-ζ^∞1/μ(2 + δζ)e^-2ζ y /2+δζdy = 1.We first bound ν_S(x) when x ≤ -1/δ. Since a_1 and a_2 are chosen to make ν_S(x) continuous at x = -1/δ, we know that a_1 e^-1/δ^2 = a_2 e^2/δ^2, or a_2 = a_1 e^-3/δ^2. Substituting this into (<ref>), we see that a_1 ≤1/e^-3/δ^2∫_-1/δ^-ζ1/μ(2 + δ y)e^2/δ^2 [2log(2 + δ y) - δ y] dy≤2μ/e^-3/δ^2∫_-1/δ^0 e^2/δ^2 [2log(2 + δ y) - δ y] dy.The derivative of e^2/δ^2 [2log(2 + δ y) - δ y] is positive on the interval [-1/δ, 0]. Therefore, on the interval [-1/δ, 0], this function achieves its minimum at y = -1/δ, implying that e^2/δ^2 [2log(2 + δ y) - δ y]≥ e^2/δ^2 for y ∈ [-1/δ, 0], and a_1 ≤2μ/e^-3/δ^2∫_-1/δ^0 e^2/δ^2 [2log(2 + δ y) - δ y] dy≤2μ/e^-3/δ^2∫_-1/δ^0 e^2/δ^2 dy = 2 μδ e^1/δ^2.Hence, for x ≤ -1/δ, ν_S(x) ≤ 2 μδ e^1/δ^21/μe^-x^2≤ 2δ≤ 2,where in the last inequality we used the fact that R ≥ 1, or δ≤ 1. We now bound ν_S(x) when x ∈ [-1/δ, -ζ]. By (<ref>), a_2 ≤1/∫_-1/δ^-ζ1/μ(2 + δ y)e^2/δ^2 [2log(2 + δ y) - δ y] dy≤2μ/∫_-1/δ^0 e^2/δ^2 [2log(2 + δ y) - δ y] dy.Using the Taylor expansion 2log(2 + δ y) = 2log(2) + 2/2δ y - 2/(2 + ξ(δ y))^2(δ y)^2/2,where ξ(δ y) ∈ [δ y, 0], we see thate^2/δ^2 [2log(2 + δ y) - δ y] =e^4/δ^2log(2) e^2/δ^2[ -δ^2 y^2/(2 + ξ(δ y))^2] ≥ e^4/δ^2log(2)e^-2y^2,y ∈ [-1/δ, 0].Therefore, a_2 ≤2μ/∫_-1/δ^0 e^2/δ^2 [2log(2 + δ y) - δ y] dy≤2μ/e^4/δ^2log(2)∫_-1^0 e^-2y^2dy = 2μe^-4/δ^2log(2)/∫_-1^0 e^-2y^2dy,where in the second inequality we used the fact that δ≤ 1. We conclude that for x ∈ [-1/δ, -ζ], ν_S(x) = a_2/μ(2 + δ x)e^2/δ^2 [2log(2 + δ x) - δ x]≤2 e^-4/δ^2log(2)e^2/δ^2 [2log(2 + δ x) - δ x]/∫_-1^0 e^-2y^2dy≤2/∫_-1^0 e^-2y^2dy≤ 4,where in the second last inequality we used the fact that on the interval [-1/δ, -ζ], the function e^2/δ^2 [2log(2 + δ x) - δ x] achieves its maximum at x = 0. This fact can be checked by differentiating the function.Lastly, we bound ν_S(x) when x ≥ -ζ. By (<ref>), a_3 ≤1/∫_-ζ^∞1/μ(2 + δζ)e^-2ζ y /2+δζdy = 2μζe^2ζ^2/2+δζ,which means that for x ≥ -ζ,ν_S(x) = a_3/μ(2 + δζ)e^-2ζ x /2+δζ≤2ζ/2 + δζ≤ζ,which a useful bound only when ζ is small, say ζ≤ 1. Now suppose ζ≥ 1. Since ν_S(x) is continuous at x = -ζ, we havea_2 = a_3 e^-2ζ^2 /2+δζe^-2/δ^2 [2log(2 + δζ) - δζ].We insert this into (<ref>) to see that for x ≥ -ζ, a_3 ≤e^2ζ^2 /2+δζ/∫_-1/δ^-ζ1/μ(2 + δ y)e^2/δ^2 [2log(2 + δ y) - δ y]e^-2/δ^2 [2log(2 + δζ) - δζ] dy ≤e^2ζ^2 /2+δζ/∫_0^-ζ1/μ(2 + δ y) dy≤e^2ζ^2 /2+δζ/∫_0^-ζ1/μ(2 + δζ) dy = μ2 + δζ/ζe^2ζ^2 /2+δζ,where in the second inequality we used that e^2/δ^2 [2log(2 + δ y) - δ y]e^-2/δ^2 [2log(2 + δζ) - δζ]≥ 1 ,y ∈ [0,-ζ],which is true because the derivative of the function e^2/δ^2 [2log(2 + δ y) - δ y] is negative on the interval [0, -ζ]. Therefore, for x ≥ -ζ, ν_S(x) = a_3/μ(2 + δζ)e^-2ζ x /2+δζ≤1/ζ e^2ζ^2 /2+δζ e^-2ζ x /2+δζ≤1/ζ.Together with (<ref>), this implies that ν_S(x) ≤ 1 for x ≥ -ζ. This concludes the proof of this lemma. CHAPTER: MODERATE DEVIATIONS IN THE ERLANG-C MODEL This chapter focuses again on the Erlang-C model. We adopt the notation from previous chapters, and refer the reader to Section <ref> for a quick recap of the model and notation. In Theorem <ref> of Chapter <ref>, we proved a bound on the Kolmogorov distance between the steady-state customer count X̃(∞) and the diffusion approximation Y(∞). Namely, we showed thatd_K(X̃(∞), Y(∞)) = sup_z ∈| (X̃(∞) ≤ z) - (Y(∞) ≤ z) | ≤156/√(R),where R is the offered load to the system. The Kolmogorov distance represents the absolute error between the cumulative distribution functions (CDF). However, when (X̃(∞) ≤ z) is small, the absolute error is a poor indicator of performance, and the relative error becomes more important. It turns out that Stein's method can also be used to prove error bounds on the relative error, and the goal of this chapter is to do this for the Erlang-C model. Our main result is Theorem <ref> contained in Section <ref>, which shows that there exists a constant C>0 independent of λ, n, and μ, such that for z = 1/√(R)(k-R), k > n, k ∈_+,|(X̃(∞) ≥ z)/(Ỹ_S(∞)≥ z) - 1|≤(1-ρ)/ρ+ 2(1-ρ)^2/ρ + 1/R + Ce^ζ^2(1/R+ 1/√(R)1-ρ/ρ + (1-ρ)^2/ρ^2) + Ce^2ζ^2ζ^2(1/R+1/√(R)1-ρ/ρ + (1-ρ)^2/ρ^2) min{(z∨ 1), R(1/ζ+1/√(R))^3},where n is the number of servers in the system,R is the offered load, ρ = R/n is the system utilization, ζ = (R-n)/√(R), andY_S(∞) is the diffusion approximation defined in (<ref>) of Chapter <ref>.In particular, the bound in (<ref>) says that in the quality-and-efficiency-driven (QED) regime where n = ⌈ R + β√(R)⌉ for some β > 0, |(X̃(∞) ≥ z)/(Ỹ_S(∞)≥ z) - 1| ≤C(β)/√(R) + C(β) min{1/R(z∨ 1), 1}. Stein's method has been used to prove bounds on the relative error of the CDF approximation in <cit.>. These results are referred to as moderate deviations results, which date back to Cramér<cit.>, who derived expansions for tail probabilities of sums of independent random variables in terms of the normal distribution. Thefollowing is a typical moderate deviations result<cit.>. If X_1, …, X_m are i.i.d. random variables with X_i = 0, Var(X_i) = 1, and e^t_0 X_i < ∞ for some t_0 > 0, then (X_1 + … + X_m > z)/1 - Φ(z) = 1+ O(1)1+z^3/√(m),0 ≤ z ≤ a_0 m^1/6,where Φ(z) is the CDF of the standard normal, O(1) is bounded a constant, and both the bound on O(1) and a_0 are independent of m. The name “moderate” deviations comes from the restriction z ∈ [0, a_0 m^1/6], which makes the bound valid as long as (X_1 + … + X_m > z) is not too small. This type of range restriction on z is always present in moderate deviations results. In contrast, (<ref>) does not have an upper bound on the value that z can take.The rest of this chapter is structured as follows. We state and prove our main results in Section <ref>, and prove some auxiliary lemmas in Section <ref>. The author would like to thank Xiao Fang, who provided him with a preliminary version of the moderate deviations result for the Erlang-C system.§ MAIN RESULT In this section we state and prove the main result of this chapter. We assume familiarity with the Stein framework introduced in Section <ref>. We also refer the reader to Section <ref> for a quick summary of notation. In addition to the notation used there, we let ρ = λ/nμ be the utilization in the Erlang-C system. Recall that ζ = δ(R-n).There exists a constant C > 0 such that for any k ∈_+, k > n, z = δ (k-R), λ > 0, μ > 0, and n ≥ 1 satisfying ρ≥ 1/2,|(X̃(∞) ≥ z)/(Y_S(∞) ≥ z) - 1|≤(1-ρ)/ρ+ 2(1-ρ)^2/ρ + δ^2 + Ce^ζ^2(δ^2+δ1-ρ/ρ + (1-ρ)^2/ρ^2) + Ce^2ζ^2ζ^2(δ^2+δ1-ρ/ρ + (1-ρ)^2/ρ^2) min{(z∨ 1), 1/δ^2(1/ζ+δ)^3}. To supplement the theorem, we present some numerical results below.Recall that in addition to Y_S(∞), the diffusion approximation with state-dependent diffusion coefficient, we also have Y(∞), the approximation with constant diffusion coefficient; cf. (<ref>). From the results in Chapter <ref>, it is natural to anticipate that Y_S(∞) is a better approximation, and this is correct. Figure <ref> displays the relative error of approximating (X̃(∞) ≥ z) when n = 100 and ρ = 0.9. We see a qualitative difference in the approximation quality of Y(∞) and Y_S(∞). The relative error of the former increases linearly in z, whereas the error of the latter is bounded no matter how large z becomes. These results are consistent for other choices of n and ρ. In contrast to the universal approximation results we saw in the previous chapters, the upper bound in Theorem <ref> only decreases as ρ↑ 1. However, we believe that universality still holds, and that the current statement of Theorem <ref> is simply a shortcoming of the author's proof. To support this, we present Table <ref>, which shows the relative error when n increases while ρ is fixed at 0.6. As we had hoped, the relative error of the approximation decreases as n grows, which suggests that the current statement of Theorem <ref> can be improved upon.§.§ Proof of the Main Result The rest of this section is dedicated to proving Theorem <ref>. To reduce notational clutter, going forward we letW = X̃(∞),andY_S = Y_S(∞).The proof of Theorem <ref> follows the standard Stein framework. We recall the generator G_Y_S defined in (<ref>), as G_Y_S f(x) = b(x) f'(x) + 1/2a(x) f”(x),where a(x) and b(x) are as in (<ref>) and (<ref>), respectively.Fix z ∈ and suppose f_z(w) satisfies the Poisson equationb(w) f_z'(w) + 1/2a(w) f_z”(w) =(Y_S ≥ z) - 1(w ≥ z)Let G_X̃ be the generator of the CTMC associated to W = X̃(∞), whose form can be found in (<ref>). Using the Taylor expansion performed in Section <ref>, one can check that for w = δ(k-R), G_X̃ f_z(w) - G_Ỹ f_z(w) =λ∫_w^w+δ (w+δ -y ) f_z”(y) dy + d(k) ∫_w-δ^w (y-(w-δ))f_z”(y) dy - 1/2a(w) f_z”(w) =∫_0^δ f_z”(w+y) λ(δ -y) dy + ∫_-δ^0 f_z”(w+y)(y+δ) (λ - b(w)/δ) dy- 1/2a(w) f_z”(w),where d(k) = μ(k ∧ n), and f_z”(w) is understood to be the left derivative at the points w = z and w = -δ R. Lemma <ref> tells us that G_W f_z(w) = 0, and we conclude that(Y_S≥ z) - (W ≥ z) = G_Ỹ f_z(W) -G_X̃ f_z(W) =[1/2a(W) f_z”(W) ] - [∫_0^δ f_z”(W+y) λ(δ -y) dy + ∫_-δ^0f_z”(W+y)(λ - b(W)/δ)(y+δ) dy ].The proof of Theorem <ref> revolves around bounding the right hand side above. DefineK_W(y) = (λ - b(W)/δ) (y+δ) ≥ 0,y ∈ [-δ, 0],λ(δ-y) ≥ 0,y ∈ [0,δ].It can be checked that∫_-δ^0 K_W(y) dy =1/2δ^2 λ -1/2δ b(W), ∫_0^δ K_W(y) dy =1/2δ^2 λ, ∫_-δ^δ K_W(y) dy =1/2a(W)Together with (<ref>), the expansion in (<ref>) then implies that(Y_S≥ z) - (W ≥ z)=-[∫_-δ^δ (f_z”(W+y)-f_z”(W)) K_W(y) dy] =[∫_-δ^δ(2b(W+y)/a(W+y)f_z'(W+y) - 2b(W)/a(W)f_z'(W) ) K_W(y) dy]+[∫_-δ^δ( 2/a(W) 1(W≥ z) - 2/a(W+y)1(W+y≥ z) ) K_W(y) dy] +(Y_S ≥ z)[∫_-δ^δ(2/a(W+y) -2/a(W)) K_W(y) dy],where we used f_z”(w) = -2b(w)/a(w)f_z'(w) + 2/a(w)((Y_S ≥ z) - 1(w≥ z) )in the last equation. The following lemma is assumed for now, and will be proved at the end of Section <ref>.There exists a constant C>0, independent of λ, μ or n, such that|[∫_-δ^δ(2b(W+y)/a(W+y)f_z'(W+y) - 2b(W)/a(W)f_z'(W) ) K_W(y) dy]| ≤Ce^ζ^2(δ^2+δ1-ρ/ρ + (1-ρ)^2/ρ^2)+ 2(1-ρ)^2/1+ρ(W≥ z)/(Y_S ≥ z)+ Ce^2ζ^2ζ^2(δ^2+δ1-ρ/ρ + (1-ρ)^2/ρ^2) min{(z∨ 1), 1/δ^2(1/ζ+δ)^3}.We now prove Theorem <ref>.Throughout the proof we will let C>0 be a positive constant that may change from line to line, but will always be independent of λ, n, and μ. We begin by bounding the second and third terms on the right hand side of (<ref>). Since we assumed that z = δ(k-R) and k > n, this implies that z ≥ -ζ + δ. Observe that∫_-δ^δ( 2/a(W) 1(W≥ z) - 2/a(W+y)1(W+y≥ z) ) K_W(y) dy =1(W = z) - 1(W = z)∫_0^δ2/a(W+y)1(W+y≥ z) K_W(y) dy=1(W = z) - 1(W = z) 2/a(-ζ)1/2λδ^2 =1(W = z)1+δζ/2+δζ=1(W = z)1/1+ρ, where in the second equality we used the fact that a(z+y) = a(-ζ) for y ∈ [0,δ], and in the last equality we used the fact that δζ = δ^2(n-R) = 1/ρ - 1. The flow-balance equations of theErlang-C model imply that(W = z) = (1-ρ) (W ≥ z).Therefore,[∫_-δ^δ( 2/a(W) 1(W≥ z) - 2/a(W+y)1(W+y≥ z) ) K_W(y) dy] =(W = z)1/1+ρ = (W ≥ z)1-ρ/1+ρ. To bound the third term in (<ref>), observe that|∫_-δ^δ(2/a(W+y) -2/a(W)) K_W(y) dy| =|2∫_-δ^δ(a(W)-a(W+y)/a(W+y)a(W)) K_W(y) dy |≤2δ^2 μ/μ a(W)∫_-δ^δK_W(y) dy =δ^2,where ininequality we used the fact that K_W(y) ≥ 0, 1/a(w) ≤ 1/μ, anda'(w) ≤μδ for all w ∈. Applying the bounds in (<ref>), (<ref>), and (<ref>) to (<ref>), we arrive at|(W ≥ z)/(Y_S≥ z) - 1| ≤Ce^ζ^2(δ^2+δ1-ρ/ρ + (1-ρ)^2/ρ^2)+ 2(1-ρ)^2/1+ρ(W≥ z)/(Y_S ≥ z)+ Ce^2ζ^2ζ^2(δ^2+δ1-ρ/ρ + (1-ρ)^2/ρ^2) min{(z∨ 1), 1/δ^2(1/ζ+δ)^3}+ δ^2 + (W ≥ z)/(Y_S ≥ z)1-ρ/1+ρ=δ^2 + Ce^ζ^2(δ^2+δ1-ρ/ρ + (1-ρ)^2/ρ^2) + Ce^2ζ^2ζ^2(δ^2+δ1-ρ/ρ + (1-ρ)^2/ρ^2) min{(z∨ 1), 1/δ^2(1/ζ+δ)^3}+ (W ≥ z)/(Y_S ≥ z)1-ρ + 2(1-ρ)^2/1+ρ.It remains to bound (W ≥ z)/(Y_S ≥ z). For convenience, let us define ψ(z) =δ^2 + Ce^ζ^2(δ^2+δ1-ρ/ρ + (1-ρ)^2/ρ^2) + Ce^2ζ^2ζ^2(δ^2+δ1-ρ/ρ + (1-ρ)^2/ρ^2) min{(z∨ 1), 1/δ^2(1/ζ+δ)^3}. Rearranging the inequality above, we see that 1 +ψ(z) ≥(W ≥ z)/(Y_S ≥ z)( 1 -1-ρ + 2(1-ρ)^2/1+ρ) =(W ≥ z)/(Y_S ≥ z)( ρ/1+ρ +ρ - 2(1-ρ)^2/1+ρ)≥(W ≥ z)/(Y_S ≥ z)ρ/1+ρ,where in the last inequality we used the fact that ρ - 2(1-ρ)^2 ≥ 0 for ρ∈ [1/2, 1]. Therefore, (W ≥ z)/(Y_S ≥ z)≤1+ρ/ρ (1+ψ(z)),and we conclude that |(W ≥ z)/(Y_S≥ z) - 1| ≤ψ(z) + 1+ρ/ρ (1+ψ(z))1-ρ + 2(1-ρ)^2/1+ρ=1-ρ + 2(1-ρ)^2/ρ +Cψ(z). § AUXILIARY PROOFSHaving proved Theorem <ref>, we now describe how to prove (<ref>). Attempting to bound the left hand side of (<ref>) in its present form will not yield anything useful. This following lemma manipulates the left hand side into something more manageable using a combination of Taylor's theorem and the Poisson equation. Assume z ≥ -ζ + δ, and let r(w) = 2b(w)/a(w). Then ∫_-δ^δ(2b(W+y)/a(W+y)f_z'(W+y) - 2b(W)/a(W)f_z'(W) ) K_W(y) dy =1/6δ^2 b(W)2b(W)/a(W)f_z”(W) + (2b(W)/a(W))^2 ∫_-δ^δK_W(y)∫_0^y∫_0^s f_z”(W+u) dudsdy +2b(W)/a(W)f_z'(W)∫_-δ^δK_W(y)∫_0^y∫_0^s r'(W+u)duds dy - 1(W=z) 2b(W)/a(W)2/a(W)∫_-δ^0yK_W(y) dy + (Y_S ≥ z)2b(W)/a(W)∫_-δ^δK_W(y)∫_0^y(2/a(W+s) - 2/a(W))dsdy + 1(W = -1/δ)f_z'(W)∫_0^δ K_W(y)∫_0^yr'(W+s)dsdy + 1(W = -ζ) f_z'(W)∫_-δ^0 K_W(y)∫_0^yr'(W+s)dsdy + 1(W ∈ [-1/δ + δ, -ζ - δ]) f_z'(W) ∫_-δ^δ K_W(y)∫_0^y∫_0^sr”(W+u)dudsdy + 1(W ∈ [-1/δ + δ, -ζ - δ]) f_z'(W) r'(W) 1/6δ^2b(W) Examining the right hand side of (<ref>), we see that we will again need moment and gradient bounds to bound its expected value. One of the moment bounds we will need is[(X̃(∞))^2 1(X̃(∞) ≤ -ζ)] ≤4/3 + 2δ^2/3.This was proved in (<ref>) of Chapter <ref>. The following lemma presents the necessary gradient bounds. It is proved in Section <ref>. There exists a constant C > 0 such that for any λ > 0, μ > 0, and n ≥ 1,f_z'(w)≤1/μe^ζ^2(3+ζ),x ≤ -ζ,f_z'(w) =(Y_S ≤ z)/μζ,w ≥ -ζ,1/(Y_S ≥ z)f_z”(w)≤C/μe^ζ^2(1 + ζ + ζ^2),w ≤ -ζ,1/(Y_S ≥ z)f_z”(w)≤C/μe^w2b(-ζ)/a(-ζ)e^ζ^2 (1+ζ + ζ^2),w ∈ [-ζ, z], f_z”(w)= 0 ,w ≥ z. Recall that 2b(-ζ)/a(-ζ) = 2ζ/(2+δζ). The appearance of e^w2b(-ζ)/a(-ζ) in (<ref>) means that we require bounds on the moment generating function of W. The following lemma contains what we need, and is proved in Section <ref>. There exists a constant C > 0 such that for any λ > 0, μ > 0, and n ≥ 1 satisfying ρ≥ 0.1, and any γ > 2+δζ/2ζ, (e^(2ζ/2+δζ - 1/γ)W1 (W ≥ -ζ )) ≤γ C e^2ζ^2/2+δζ, (e^2ζ/2+δζW1 (W ≥ -ζ )) ≤1/δ^2(1/ζ+δ)^3 C e^2ζ^2/2+δζ. We are now ready to prove Lemma <ref>.We prove this lemma by taking expected values on both sides of (<ref>), and bounding the terms on the right hand side one at a time. Namely, we will bound 1/(Y_S ≥ z) times 1/6δ^2 |[b(W)2b(W)/a(W)f_z”(W)] | + |[(2b(W)/a(W))^2 ∫_-δ^δK_W(y)∫_0^y∫_0^s f_z”(W+u) dudsdy ]|+ |[ 2b(W)/a(W)f_z'(W)∫_-δ^δK_W(y)∫_0^y∫_0^s r'(W+u)duds dy]| + |[1(W=z) 2b(W)/a(W)2/a(W)∫_-δ^0yK_W(y) dy ] | + (Y_S ≥ z)| [2b(W)/a(W)∫_-δ^δK_W(y)∫_0^y(2/a(W+s) - 2/a(W))dsdy ]| + | [1(W = -1/δ)f_z'(W)∫_0^δ K_W(y)∫_0^yr'(W+s)dsdy ]| + | [1(W = -ζ) f_z'(W)∫_-δ^0 K_W(y)∫_0^yr'(W+s)dsdy ]|+ | [1(W ∈ [-1/δ + δ, -ζ - δ]) f_z'(W) ∫_-δ^δ K_W(y)∫_0^y∫_0^sr”(W+u)dudsdy ] |+ |[ 1(W ∈ [-1/δ + δ, -ζ - δ]) f_z'(W) r'(W) 1/6δ^2b(W)]|,one line at a time. We begin with the first line in (<ref>):1/(Y_S ≥ z)|[1/6δ^2 b(W)2b(W)/a(W)f_z”(W) ]|≤δ^2C/μe^ζ^2(1 + ζ + ζ^2)[2b^2(W)/a(W) 1(W ≤ -ζ)] + δ^2 C/μe^ζ^2 (1+ζ + ζ^2)2b^2(-ζ)/a(-ζ)[e^W 2ζ/2+δζ 1(W ∈ [-ζ, z])]≤Cδ^2 e^ζ^2(1 + ζ + ζ^2) [W^2 1(W ≤ -ζ)] + δ^2 C/μe^ζ^2 (1+ζ + ζ^2)2b^2(-ζ)/a(-ζ)[e^W 2ζ/2+δζ 1(W ∈ [-ζ, z])]≤Cδ^2 e^ζ^2(1 + ζ + ζ^2)+ δ^2 C/μe^ζ^2 (1+ζ + ζ^2)2b^2(-ζ)/a(-ζ)[e^W 2ζ/2+δζ 1(W ∈ [-ζ, z])] ≤Cδ^2 e^ζ^2(1 + ζ + ζ^2)+ δ^2 C e^ζ^2ζ^2(1+ζ + ζ^2)[e^W 2ζ/2+δζ 1(W ∈ [-ζ, z])],where we used (<ref>) in the third inequality. If z ≤2+δζ/2ζ, then [e^W 2ζ/2+δζ 1(W ∈ [-ζ, z])] ≤ 3. If z > 2+δζ/2ζ, we use (<ref>) with z = γ there to see that[e^W 2ζ/2+δζ 1(W ∈ [-ζ, z])] =[e^W (2ζ/2+δζ- 1/z) e^W/z1(W ∈ [-ζ, z])]≤z C e^2ζ^2/2+δζ≤ z C e^ζ^2.Using (<ref>),[e^W 2ζ/2+δζ 1(W ∈ [-ζ, z])] ≤1/δ^2(1/ζ+δ)^3 C e^2ζ^2/2+δζ≤1/δ^2(1/ζ+δ)^3 C e^ζ^2.Hence, 1/(Y_S ≥ z)|[1/6δ^2 b(W)2b(W)/a(W)f_z”(W) ]|≤Cδ^2 e^ζ^2(1 + ζ + ζ^2) + Cδ^2 e^2ζ^2ζ^2(1 + ζ + ζ^2) min{(z ∨ 1), 1/δ^2(1/ζ+δ)^3}.Moving on to the second line of (<ref>):1/(Y_S ≥ z)|[(2b(W)/a(W))^2 ∫_-δ^δK_W(y)∫_0^y∫_0^s f_z”(W+u) dudsdy ]|≤(2b(-ζ)/a(-ζ))^2 [ ∫_-δ^δ K_W(y)∫_0^y∫_0^s 1( W+u≥ -ζ) f_z”(W+u) du dsdy] +[ (2b(W)/a(W))^2∫_-δ^δ K_W(y)∫_0^y∫_0^s1( W+u≤ -ζ)f_z”(W+u) du dsdy]≤(2b(-ζ)/a(-ζ))^2 [ ∫_-δ^δ K_W(y)∫_0^y∫_0^s1( W+u ∈ [-ζ, z]) ×C/μe^ζ^2(1 + ζ + ζ^2)e^(W+u) 2ζ/2+δζ du dsdy] +[ (2b(W)/a(W))^2 ∫_-δ^δ K_W(y)∫_0^y∫_0^s1( W≤ -ζ )C/μe^ζ^2(1 + ζ + ζ^2) du dsdy],where in the second inequality we used the gradient bounds from Lemma <ref>. To bound the first term, note that (2b(-ζ)/a(-ζ))^2 [ ∫_-δ^δ K_W(y)∫_0^y∫_0^s 1( W+u ∈ [-ζ, z]) ×C/μe^ζ^2(1 + ζ + ζ^2)e^(W+u) 2ζ/2+δζ du dsdy]≤(2b(-ζ)/a(-ζ))^2 [ 1( W ∈[-ζ, z])∫_-δ^δ Cδ^2 K_W(y) 1/μe^ζ^2(1 + ζ + ζ^2)e^(W+δ) 2ζ/2+δζdy] ≤(2b(-ζ)/a(-ζ))^2Cδ^2/μe^ζ^2(1 + ζ + ζ^2)[ 1( W ∈[-ζ, z])e^W 2ζ/2+δζ∫_-δ^δK_W(y)dy]=(2b(-ζ)/a(-ζ))^2Cδ^2/μe^ζ^2(1 + ζ + ζ^2)[ 1( W ∈[-ζ, z])e^W 2ζ/2+δζ1/2 a(W)]=1/2 a(-ζ)(2b(-ζ)/a(-ζ))^2Cδ^2/μe^ζ^2(1 + ζ + ζ^2)[ 1( W ∈[-ζ, z])e^W 2ζ/2+δζ] ≤Cδ^2 e^2ζ^2(1 + ζ + ζ^2) 2b^2(-ζ)/μ a(-ζ)min{z, 1/δ^2(1/ζ+δ)^3} ≤Cδ^2 e^2ζ^2(1 + ζ + ζ^2) ζ^2 min{z, 1/δ^2(1/ζ+δ)^3}.For the second term, [ (2b(W)/a(W))^21( W≤ -ζ )∫_-δ^δ K_W(y)∫_0^y∫_0^sC/μe^ζ^2(1 + ζ + ζ^2) du dsdy]≤Cδ^2 e^ζ^2(1 + ζ + ζ^2) [ (2b(W)/a(W))^21( W≤ -ζ )1/μ∫_-δ^δ K_W(y)dy]=Cδ^2 e^ζ^2(1 + ζ + ζ^2) [ (2b(W)/a(W))^21( W≤ -ζ )a(W)/2μ]=Cδ^2e^ζ^2(1 + ζ + ζ^2)[2b^2(W)/μ a(W) 1(W ≤ -ζ)]≤Cδ^2 e^ζ^2(1 + ζ + ζ^2).Hence, 1/(Y_S ≥ z)|[(2b(W)/a(W))^2 ∫_-δ^δK_W(y)∫_0^y∫_0^s f_z”(W+u) dudsdy ]|≤ Cδ^2 e^ζ^2(1 + ζ + ζ^2) + Cδ^2 e^2ζ^2ζ^2(1 + ζ + ζ^2) min{(z ∨ 1), 1/δ^2(1/ζ+δ)^3}.We now bound the third line in (<ref>):1/(Y_S ≥ z)|[2b(W)/a(W)f_z'(W)∫_-δ^δK_W(y)∫_0^y∫_0^s g'(W+u)duds dy]|≤1/(Y_S ≥ z)|[2b(W)/a(W)f_z'(W)∫_-δ^δK_W(y)∫_0^y∫_0^s 4× 1(W+u ≤ -ζ) duds dy]|≤1/(Y_S ≥ z)|[2b(W)/a(W)f_z'(W) 1(W≤ -ζ)Cδ^2 ∫_-δ^δK_W(y)dy]|≤Cδ^2 [b(W)f_z'(W) 1(W≤ -ζ)]≤Cδ^2 [ μW1/μ e^ζ^2(1+ζ) 1(W≤ -ζ)]≤Cδ^2e^ζ^2(1+ζ),where in the second last inequality we used (<ref>), and in the last inequality we used (<ref>). We now bound the fourth line in (<ref>):1/(Y_S ≥ z)|[1(W=z) 2b(W)/a(W)2/a(W)∫_-δ^0yK_W(y) dy]|≤(W=z)/(Y_S ≥ z)| 2b(z)/a(z)2/a(z)δ∫_-δ^δK_W(y) dy| =(W=z)/(Y_S ≥ z)δ|2b(z)/a(z)| =δ(1-ρ) (W≥ z)/(Y_S ≥ z)2|b(-ζ)|/a(-ζ)=δ(1-ρ)2ζ/2+δζ(W≥ z)/(Y_S ≥ z),where in the second last equality we used (<ref>).We now bound the fifth line in (<ref>):1/(Y_S ≥ z)|[(Y_S ≥ z)2b(W)/a(W)∫_-δ^δK_W(y)∫_0^y(2/a(W+s) - 2/a(W))dsdy]|=[2b(W)/a(W)∫_-δ^δK_W(y)∫_0^y2|a(W)-a(W+s)/a(W+s)a(W)|dsdy] ≤[2b(W)/a(W)∫_-δ^δK_W(y)∫_0^y2μδs/a(W+s)a(W) dsdy] ≤[2b(W)/a(W)∫_-δ^δK_W(y)∫_0^y2μδ^2/μ a(W) dsdy]=δ^3[2b(W)/a(W)2/ a(W)∫_-δ^δK_W(y) dy]=δ^3[2b(W)/a(W)] ≤δ^3C(1+ζ),where in the first inequality we used the fact that a'(w) ≤μδ for all w ∈, and in the last inequality we used (<ref>). We now bound the sixth line in (<ref>):1/(Y_S ≥ z)|[1(W = -1/δ)f_z'(W)∫_0^δ K_W(y)∫_0^yg'(W+s)dsdy]|≤(W = -1/δ)3/μ∫_0^δ K_W(y)∫_0^y|g'(-1/δ+s)|dsdy ≤(W = -1/δ)3/μ∫_0^δ 4δ K_W(y)dy=(W = -1/δ)3/μ 4δλδ^2 /2 ≤Cδ(W = -1/δ)≤Cδ^2,where we obtained the first inequality from (<ref>). The term in the seventh line is bounded similarly:1/(Y_S ≥ z)|[1(W = -ζ) f_z'(W)∫_-δ^0 K_W(y)∫_0^yg'(W+s)dsdy]|≤(W = -ζ)e^ζ^2(3+ζ)/μ∫_-δ^0 4δ K_W(y)dy=(W = -ζ)e^ζ^2(3+ζ)/μ 4δ1/2(δ^2 λ - δ b(-ζ)) ≤Cδ(W = -ζ)e^ζ^2(1+ζ)≤Cδ^2e^ζ^2(1+ζ).We now bound the eighth line in (<ref>):1/(Y_S ≥ z)|[1(W ∈ [-1/δ + δ, -ζ - δ])f_z'(W) ×∫_-δ^δ K_W(y)∫_0^y∫_0^sg”(W+u)dudsdy ]|≤e^ζ^2(3+ζ)/μ[1(W ∈ [-1/δ + δ, -ζ - δ])∫_-δ^δ 8δ^2 K_W(y)dy ]=e^ζ^2(3+ζ)/μ4δ^2 [1(W ∈ [-1/δ + δ, -ζ - δ])a(W) ] ≤e^ζ^2(3+ζ)/μ4δ^2 [1(W ≤ -ζ ) μ (2+δW) ] ≤Cδ^2e^ζ^2(1+ζ).Finally, we bound the ninth line in (<ref>):1/(Y_S ≥ z)|[1(W ∈ [-1/δ + δ, -ζ - δ])f_z'(W)g'(W) 1/6δ^2b(W))]|≤Ce^ζ^2(3+ζ)/μ[1(W ≤ -ζ)δ^2 b(W)] ≤Cδ^2e^ζ^2(1+ζ).Combining these nine bounds together, we arrive at the final bound of Cδ^2e^ζ^2(1+ζ + ζ^2)+δ(1-ρ)2ζ/2+δζ(W≥ z)/(Y_S ≥ z)(W≥ z)/(Y_S ≥ z)+ Cδ^2 e^2ζ^2ζ^2(1 + ζ + ζ^2) min{(z ∨ 1), 1/δ^2(1/ζ+δ)^3}.Combining the above with the fact that δζ = δ^2(n-R) = 1/ρ - 1 concludes the proof. §.§ Proof of Lemma <ref> (Error term)Recall that r(w) = 2b(w)/a(w). Using the forms of a(w) and b(w) in (<ref>) and (<ref>), it is not hard to check that r'(w) =2,w ≤ -1/δ,-4/(2+δ w)^2,w ∈ (-1/δ, -ζ], 0,w > -ζ,where r'(w) is understood to be the left derivative at the points w = -1/δ and w = -ζ. Assume for now that for all y ∈ (-δ, δ), 2b(W+y)/a(W+y)f_z'(W+y) - 2b(W)/a(W)f_z'(W) =y 2b(W)/a(W)f_z”(W) +2b(W)/a(W)∫_0^y∫_0^s(2b(W)/a(W) f_z”(W+u) + r'(W+u)f_z'(W) )duds+ 2b(W)/a(W)∫_0^y(2/a(W+s) 1(W+s ≥ z) - 2/a(W) 1(W ≥ z) )ds + (Y_S ≥ z)2b(W)/a(W)∫_0^y(2/a(W+s) - 2/a(W))ds + f_z'(W) ∫_0^yr'(W+s)ds.We postpone verifying (<ref>) to the end of this proof. Since z ≥ -ζ + δ and a(w) = a(-ζ) for w ≥ -ζ, we see that2b(W)/a(W)∫_0^y(2/a(W+s) 1(W+s ≥ z) - 2/a(W) 1(W ≥ z) )ds =1(W=z) 2b(W)/a(W)2/a(W)∫_0^y(1(s ≥ 0) - 1 )ds=1(W=z) 2b(W)/a(W)2/a(W)(-y1(y ≥ 0)).Combining (<ref>)–(<ref>) with the fact that ∫_-δ^δyK_W(y)dy = 1/6δ^2 b(W), we arrive at∫_-δ^δ(2b(W+y)/a(W+y)f_z'(W+y) - 2b(W)/a(W)f_z'(W) ) K_W(y) dy =1/6δ^2 b(W)2b(W)/a(W)f_z”(W) + (2b(W)/a(W))^2 ∫_-δ^δK_W(y)∫_0^y∫_0^s f_z”(W+u) dudsdy +2b(W)/a(W)f_z'(W)∫_-δ^δK_W(y)∫_0^y∫_0^s r'(W+u)duds dy - 1(W=z) 2b(W)/a(W)2/a(W)∫_-δ^0yK_W(y) dy+ (Y_S ≥ z)2b(W)/a(W)∫_-δ^δK_W(y)∫_0^y(2/a(W+s) - 2/a(W))dsdy + f_z'(W)∫_-δ^δK_W(y) ∫_0^yr'(W+s)dsdyWe are almost done, but the last term on the right hand side above requires some additional manipulations. Since r'(w) = 0 for w ≥ -ζ and K_W(y)=0 for W = -1/δ and y ∈ [-δ, 0],∫_-δ^δ K_W(y)∫_0^yr'(W+s)dsdy=1(W = -1/δ)∫_0^δ K_W(y)∫_0^yr'(W+s)dsdy+ 1(W = -ζ) ∫_-δ^0 K_W(y)∫_0^yr'(W+s)dsdy + 1(W ∈ [-1/δ + δ, -ζ - δ])∫_-δ^δ K_W(y)∫_0^yr'(W+s)dsdy,and for W ∈ [-1/δ + δ, -ζ - δ], ∫_-δ^δ K_W(y)∫_0^yr'(W+s)dsdy =∫_-δ^δ K_W(y)∫_0^y(r'(W+s) - r'(W))dsdy +r'(W) ∫_-δ^δ yK_W(y)dy =∫_-δ^δ K_W(y)∫_0^y∫_0^sr”(W+u)dudsdy + r'(W) 1/6δ^2b(W). To conclude the proof, we verify (<ref>). Sinced/dx(r(x) f_z'(x)) = r(x) f_z”(x) +r'(x) f_z'(x),it follows from the Fundamental Theorem of Calculus that 2b(W+y)/a(W+y)f_z'(W+y) - 2b(W)/a(W)f_z'(W)=∫_0^y(2b(W)/a(W) f_z”(W+s) + r'(W+s)f_z'(W))ds.Now ∫_0^yf_z”(W+s)ds = yf_z”(W) + ∫_0^y(f_z”(W+s) - f_z”(W) )ds =y f_z”(W) +∫_0^y(2b(W+s)/a(W+s)f_z'(W+s) - 2b(W)/a(W)f_z'(W) )ds + ∫_0^y(2/a(W+s) 1(W+s ≥ z) - 2/a(W) 1(W ≥ z) )ds+ (Y_S ≥ z)∫_0^y(2/a(W+s) - 2/a(W))ds,and applying (<ref>) once again, we see that this equals y f_z”(W) +∫_0^y∫_0^s(2b(W)/a(W) f_z”(W+u) + r'(W+u)f_z'(W) )duds + ∫_0^y(2/a(W+s) 1(W+s ≥ z) - 2/a(W) 1(W ≥ z) )ds+ (Y_S ≥ z)∫_0^y(2/a(W+s) - 2/a(W))ds,thus proving (<ref>). §.§ Moment Generating Function BoundThroughout the proof we will let C>0 be a positive constant that may change from line to line, but will always be independent of λ, n, and μ. Recall thatζ = δ(R - n) and that the random variable W lives on the lattice δ(_+ - R). Fix r > 0 and M ∈{δ(k-R) : k ≥ n}. Consider the test function f(w) = e^r ϕ(w), where ϕ(w) = -ζ,w ≤ -ζ, w,w ∈ [-ζ, M], M,w ≥ M.For w = δ(k-R), we haveG_W f(w) =λ(f(w+δ) - f(w))1(w ∈ [-ζ, M-δ]) + μ(k ∧ n) (f(w-δ) - f(w))1(w ∈ [-ζ+δ, M])=λ f(w) (e^δ r-1)1(w ∈ [-ζ, M-δ]) + nμ f(w) (e^-δ r-1)1(w ∈ [-ζ+δ, M]) =λ f(w) (e^δ r-1)1(w ∈ [-ζ, M]) - λ f(M) (e^δ r-1)1(w = M) + nμ f(w) (e^-δ r-1)1(w ∈ [-ζ, M]) - μ n f(-ζ) (e^-δ r-1)1(w=-ζ),Since G_W f(W) = 0, we take the expectation in the equation above to see that- (λ (e^δ r-1) + nμ (e^-δ r-1)) (f(W) 1(W ∈ [-ζ, M])) = - λ f(M) (e^δ r-1)(W = M) +n μ f(-ζ) (1 - e^-δ r)(W=-ζ).First, note that the right hand side is bounded byn μ f(-ζ) (1 - e^-δ r)(W=-ζ) =λ f(-ζ) (1 - e^-δ r)(W=-ζ-δ)≤λ f(-ζ) δ r (W=-ζ-δ) ≤λ f(-ζ) δ r Cδ=rf(-ζ) Cμ ,where the first equality follows from the flow-balance equations of the CTMC corresponding to W, and the last inequality follows from the same logic used to prove (<ref>) of Section <ref>. Now let γ > a(-ζ)/2b(-ζ) and set r = 2b(-ζ)/a(-ζ) - 1/γ. Assume we can prove that-(λ (e^δ r-1) + nμ (e^-δ r-1)) ≥μ(r/γ1+ρ/2ρ + r^4 δ^2/120), ρ≥ 0.1.Then using (<ref>) and (<ref>) we get(f(W) 1(W ∈ [-ζ, M])) ≤γ2ρ/1+ρ f(-ζ) C ≤γ C e^2b(-ζ)/a(-ζ)ζ,r = 2b(-ζ)/a(-ζ) - 1/γ, (f(W) 1(W ∈ [-ζ, M])) ≤1/r^3δ^2 C e^2b(-ζ)/a(-ζ)ζ,r = 2b(-ζ)/a(-ζ),and taking M →∞ then establishes the claim in the lemma. We now verify (<ref>). Using the Taylor expansionse^δ r-1 =rδ + 1/2(rδ)^2 + 1/6(rδ)^3 + 1/24(rδ)^4 +1/120(rδ)^5 e^ξ(δ r)e^-δ r - 1 =-rδ + 1/2(rδ)^2 - 1/6(rδ)^3 + 1/24(rδ)^4 - 1/120(rδ)^5 e^η(-δ r),where ξ(δ r) ∈ [0,δ r] and η (-δ r) ∈ [-δ r, 0] (the fifth order expansion is necessary), we rewrite the left side of (<ref>) as -((λ - nμ ) (r δ +1/6(rδ)^3 ) +( λ + nμ)(1/2(rδ)^2 + 1/24(rδ)^4 ))(f(W) 1(W ∈ [-ζ, M]))- 1/120(rδ)^5 ( λ e^ξ(δ r) - nμ e^η(-δ r))(f(W) 1(W ∈ [-ζ, M])).Recalling that δ(λ - nμ) = μζ, λδ^2 = μ, and nδ ^2 = 1/ρ, the quantity above becomes μ(ζ(r +1/6r^3δ^2 ) -( 1+ 1/ρ)(1/2r^2 + 1/24r^4δ^2 ))(f(W) 1(W ∈ [-ζ, M]))+ μ/120r^5δ^3 ( 1/ρ e^η(-δ r)- e^ξ(δ r))(f(W) 1(W ∈ [-ζ, M]))≥μ(ζ(r +1/6r^3δ^2 ) +( 1+ 1/ρ)(1/2r^2 + 1/24r^4δ^2 )- 1/120r^5δ^3e^δ r)×(f(W) 1(W ∈ [-ζ, M])) =μ(r(ζ - (1+1/ρ) 1/2r) +1/6r^3δ^2(ζ - (1+1/ρ) 1/4r)- 1/120r^5δ^3e^δ r)×(f(W) 1(W ∈ [-ζ, M]))Now if r = 2b(-ζ)/a(-ζ) - 1/γ for some γ > a(-ζ)/2b(-ζ), then ζ - (1+1/ρ) 1/2r =ζ - (1+1/ρ) 1/2(2ζ/2+δζ - 1/γ)=ζ2+δζ - 1-1/ρ/2+δζ + (1+1/ρ) 1/21/γ =ζ2+(1/ρ-1) - 1-1/ρ/2+δζ + (1+1/ρ) 1/21/γ =1/γ1+ρ/2ρ,where in the third equality we used the fact that δζ = δ^2(n-R) = 1/ρ-1. The right hand side of (<ref>) then equals μ(r/γ1+ρ/2ρ+1/6r^3δ^2(1/2ζ + r/z1+ρ/4ρ) - 1/120 r^5δ^3 e^δ r)(f(W) 1(W ∈ [-ζ, M]))≥μ(r/γ1+ρ/2ρ+1/12r^3δ^2ζ- 1/120 r^5δ^3 e^δ r)(f(W) 1(W ∈ [-ζ, M]))≥μ(r/γ1+ρ/2ρ+1/12r^4δ^2- 1/120 r^5δ^3 e^δ r)(f(W) 1(W ∈ [-ζ, M])),where in the last inequality we used the fact that ζ≥2ζ/2 + δζ≥ r. Now rδ≤2δζ/2+δζ = 2(1-ρ)/ρ(2+δζ) = 2(1-ρ)/2ρ + (1-ρ) = 2(1-ρ)/1 + ρ,and so it can be checked that 1/12r^4δ^2- 1/120 r^5δ^3 e^δ r = r^4 δ^2/12(1 - 1/10 rδ e^δ r) ≥r^4 δ^2/12(1 - 1/102(1-ρ)/1 + ρe^2(1-ρ)/1 + ρ) ≥r^4 δ^2/121/10whenever ρ≥ 0.1. CHAPTER: STEADY-STATE DIFFUSION APPROXIMATION OF THE M/PH/N+M MODEL This chapter is based on <cit.>. We ignore any notation defined in previous chapters, and start fresh with notation (although much of the notation will be similar to the previous chapters). In this chapter, we apply the Stein framework introduced in Chapter <ref> to the M/Ph/n+M system, which serves as a building block to model large-scale service systems such as customer contact centers <cit.> and hospital operations <cit.>. In such a system, there are n identical servers, the arrival process is Poisson (the symbol M) with rate λ, the service times are i.i.d. having a phase-type distribution (the symbol Ph) with d phases and mean 1/μ, the patience times of customers are i.i.d. having an exponential distribution (the symbol +M) with mean 1/α<∞. When the waiting time of a customer in queue exceeds her patience time, the customer abandons the system without service; once the service of a customer is started, the customer does not abandon.Let X_i(t) be the number of customers in phase i at time t for i=1, …, d, where d is the number of phases in the service time distribution. Let X(t) be the corresponding vector. Then the system size process X={X(t), t≥ 0} has a unique stationary distribution for any arrival rate λ and any server number n due to customer abandonment; although X is not a Markov chain, it is a function of a Markov chain with a unique stationary distribution,see Section <ref> for details.In Theorem <ref> of this chapter, we prove thatsup_h ∈ℋ[h(X̃^(λ)(∞))] - [h(Y(∞))]≤C/√(λ)for any λ > 0andn ≥ 1satisfyingn μ = λ + β√(λ),where β∈ is some constant and ℋ is some class of functions h:^d→. This is known as the Halfin-Whitt, or quality- and efficiency-driven (QED) regime <cit.>. In (<ref>), X̃^(λ)(∞) is a random vector having the stationary distribution of a properly scaled version of X = X^(λ) that depends on the arrival rate λ, number of servers n, the service time distribution, and the abandonment rate α, and Y(∞) is a random vector having the stationary distribution of a d-dimensional piecewise Ornstein-Uhlenbeck (OU) process Y={Y(t), t≥ 0}. The stationary distribution of X^(λ) exists even when β is negative because α is assumed to be positive. The constant C depends on the service time distribution, abandonment rate α, theconstant β in (<ref>), and the choice of H, but C is independent of the arrival rate λ and the number of servers n.Unlike the results in Chapters <ref> and <ref>, which were universal and did not rely on any particular parameter regime, we do require the QED regime to prove the result in (<ref>). The reason for this is the additional difficulty in establishing gradient and moments bounds due to the multi-dimensional nature of X̃^(λ)(∞) and the approximation Y(∞).Two different classes ℋ will used in our Theorem <ref>. First, we take H to be the class of polynomials up to a certain order. In this case, (<ref>) provides rates of convergence for steady-state moments. Second, H is taken to be W^(d), the class of all 1-Lipschitz functionsW^(d) = {h: ^d→: h(x)-h(y)≤x-y}.In this case, (<ref>) provides rates of convergence for stationary distributions under the Wasserstein metric; convergence under Wasserstein metric implies the convergence in distribution <cit.>.As previously mentioned in Section <ref>, the authors of <cit.> develop an algorithm to compute the distribution of Y(∞).The algorithm is more computationally efficient, in terms of both time and memory, than computing the distribution of X̃^(λ)(∞). For example, in an M/H_2/500+M system studied in <cit.>, where the system has 500 servers and a hyper-exponential service time distribution, it took around 1 hour and peak memory usage of 5 GB to compute the distribution of X^(λ).On the same computer, it took less than 1 minute to compute the distribution of Y(∞),and peak memory usage was less than 200 MB. Theorem <ref> quantifies the steady-state diffusion approximations developed in <cit.>.In <cit.>, the authors prove the convergence of distribution X̃^(λ)(∞) to that of Y(∞) by proving an interchange of limits.The proof technique follows that of the seminal paper <cit.>, where the authors prove an interchange of limits for generalized Jackson networks of single-server queues.The results in <cit.> were improved and extended by various authors for networks of single-servers <cit.>, for bandwidth sharing networks <cit.>, and for many-server systems <cit.>.These “interchange limits theorems” are qualitative and thus do not provide rates of convergence as in (<ref>).Our use of Stein's method in this chapter has two important features that were not present in the previous chapters. Unlike the Erlang-A and Erlang-C models, which are relatively simple one-dimensional birth death processes, the M/Ph/n+M model is a multi-dimensional Markov chain, and the corresponding diffusion approxmiation is also multi-dimensional. This means that our usual approach for deriving gradient bounds does not hold anymore, and we rely on ideas from <cit.> to solve this problem. The second feature of this chapter is state-space collapse (SSC). We will see that the Markov chain representing the M/Ph/n+M system lives in a higher dimensional space than the diffusion approximation. Therefore, certain SSC error bounds need to be established in order for us to carry out Stein's method. In Chapter <ref> we discussed the benefits of using a diffusion approximation with a state-dependent diffusion coefficient. The approximation Y(∞) in(<ref>) is based on a diffusion process with a constant diffusion coefficient. Nothing is proved about the approximation with state-dependent diffusion coefficient, because the multi-dimensional nature of the M/Ph/n+M model makes this task much more difficult. However, this does not prevent us from evaluating the approximation numerically, which we do in Section <ref>. Our observations depend on the type of service-time distribution we use. Namely, we observe a difference between the cases when the first service phase is deterministic or random. In the former case, no SSC is required, and the state-dependent coefficient approximation performs better. Namely, we observe the phenomenon of faster convergence rates of 1/λ,analogous to what was proved in Chapter <ref>. In the latter case, SSC is required, and we do not have faster convergence rates. This is because the SSC error is of order 1/√(λ) and does not vanish with the use of a state-dependent diffusion coefficient. The rest of the chapter is structured as follows. We begin with Section <ref>, where we formally define the M/Ph/n+M system as well as the diffusion process whose steady-state distribution will approximate the system. Section <ref> states our main results. Section <ref> describes the continuous-time Markov chain (CTMC) representation of the M/Ph/n+M system. Section <ref> sets up the Poisson equation, gradient bounds, and Taylor expansion of the CTMC generator. Section <ref> deals with SSC. Moment bounds and the proof of our main result can be found in Section <ref>. Section <ref> contains numerical results evaluating the performance of an approximation with state-dependent diffusion coefficient.§ MODELS In this section, we give additional description of theM/Ph/n+M system and the corresponding diffusion model. §.§ The M/Ph/n+M SystemThe basic description of the M/Ph/n+M queueing system was given in the first paragraph of the introduction. Here, we describe the dynamics of the system.Upon arrival to the system with idle servers, a customer begins service immediately. Otherwise, if all servers are busy, the customer enters an infinite capacity queue to wait for service. When a server completes serving a customer, the server becomes idle if the queue is empty, or takes a customer from the queue under the first-come-first-served service policy if it is nonempty. Recall that the Ph indicates that customer service times are i.i.d. following a phase-type distribution. We shall provide a definition of a phase-type distribution shortly below. The phase-type distribution can approximate any positive-valued distribution <cit.>.§.§.§ Phase-type Service TimeDistributionA phase-type distribution is assumed to have d ≥ 1 phases. Each phase-type distribution is determined by the tuple (p,ν,P), where p ∈^d is a vector of non-negative entries whose sum is equal to one, ν∈^d is a vector of positive entries and P is a d × d sub-stochastic matrix. We assume that P is transient, i.e. (I-P)^-1 exists,and without loss of generality, we also assume that the diagonal entries of P are zero (P_ii=0).A random variable is said to have a phase-type distribution with parameters (p,ν,P) if it is equal to the absorption time of the following CTMC. The state space of the CTMC is {1, ... ,d+1}, with d+1 being the absorbing state. The CTMC starts off in one of the states in {1,...,d} according to distribution p. For i = 1, ... ,d, the time spent in state i is exponentially distributed with mean 1/ν_i. Upon leaving state i, the CTMC transitions to state j= 1, ... ,d with probability P_ij, or gets absorbed into state d+1 with probability 1- ∑_j=1^d P_ij.The CTMC above is a useful way to describe the service times in the M/Ph/n+M system. Upon arrival to the system, a customer is assigned her first service phase according to distribution p. If the customer is forced to wait in queue because all servers are busy, she is still assigned a first service phase, but this phase of service will not start until a server takes on this customer for service. Once a customer with initial phase i enters service, her service time is the time until absorption to state d+1 by the CTMC. We assume without loss of generality that for each service phase i, eitherp_i > 0orP_ji> 0for some j.This simply means that there are no redundant phases. We now define some useful quantities for future use. Define R = (I-P^T)diag(ν) andγ = μ R^-1p,where the matrix diag(ν) is the d× d diagonal matrixwith diagonal entries given by the components of ν. One may verify that ∑_i=1^d γ_i = 1. One can interpret γ_i to be the fraction of phase i service load on the n servers. For concreteness, we provide two examples of phase-type distributions when d=2. The first example is the two-phase hyper-exponential distribution, denoted by H_2. The corresponding tuple of parameters is (p,ν,P), wherep = (p_1, p_2)^T, ν= (ν_1,ν_2)^T,andP = 0.Therefore, with probability p_i, the service time follows an exponential distribution with mean 1/ν_i.The second example is the Erlang-2 distribution, denoted by E_2. The corresponding tuple of parameters is (p,ν,P), wherep = (1,0)^T, ν= (θ, θ)^T,andP = [ 0 1; 0 0 ].An E_2 random variable is a sum of two i.i.d. exponential random variables, each having mean 1/θ. §.§ System Size Process and Diffusion ModelBefore we state the main results, we introduce the process we wish to approximate, as well as the approximating diffusion process – the piecewise OU process. Recall thatX = {X(t) ∈^d, t ≥ 0} is the system size process, where X(t) = (X_1(t), ... , X_d(t))^T,and X_i(t) is the number of customers of phase i in the system (queue + service) at time t. We emphasize that X is not a CTMC, but it is a deterministic function of a higher-dimensional CTMC, which will be described in Section <ref>. The process X depends on λ, n, α, p, P, and ν. However, in this chapter we keep α, p, P, and ν fixed, and allow λ and n to vary according to (<ref>). For the remainder of the chapter we write X^(λ) to emphasize the dependence of X on λ; the dependence of X^(λ) on n is implicit through (<ref>).Recall the definition of γ in (<ref>) and define the scaled random variableX̃^(λ)(∞) = δ (X^(λ)(∞) - γ n),where, for convenience, we let δ = 1/√(λ). To approximate X̃^(λ)(∞), we introduce the piecewise OU process Y = {Y(t), t≥ 0}. This is a d-dimensional diffusion process satisfyingY(t) = Y(0) - p β t - R ∫_0^t(Y(s) -p(e^TY(s))^+) ds - α p ∫_0^t (e^TY(s))^+ds + √(Σ) B(t).Above, B(t) is the d-dimensional standard Brownian motion and √(Σ) is any d× d matrix satisfying√(Σ)√(Σ)^T= Σ = diag(p) + ∑_k=1^d γ_k ν_k H^k + (I-P^T)diag(ν) diag(γ)(I-P),where the matrix H^k is defined as H^k_ii = P_ki(1-P_ki), H^k_ij = -P_kiP_kjfor j≠ i. Comparing the form of Σ above to (2.24) of <cit.> confirms that it is positive definite. Thus √(Σ) exists. Observe that Y depends only on β, α, p, P, and ν, all of which are held constant throughout this chapter. The diffusion process in (<ref>) has been studied by <cit.>. They prove that Y is positive recurrent by finding an appropriate Lyapunov function. In particular, this means that Y admits a stationary distribution.§ MAIN RESULTSWe now state our main results.For every integer m > 0, there exists a constant C_m = C_m(β,α,p,ν,P)>0 such that for all locally Lipschitz functions h : ^d → satisfying h(x) ≤x^2m for x∈^d,we haveh(X̃^(λ)(∞)) - h(Y(∞)) ≤C_m/√(λ) for allλ>0 satisfying (<ref>), which we recall below as n μ= λ+ β√(λ). Theorem <ref> will be proved inSection <ref>. As a consequence of the theorem, we immediately have the following corollary. There exists a constant C_1 = C_1(β,α,p,ν,P)>0 such that sup_h ∈𝒲^(d)h(X̃^(λ)(∞)) - h(Y(∞)) ≤C_1/√(λ) for allλ>0 satisfying (<ref>), where W^(d) is defined in (<ref>). In particular, X̃^(λ)(∞) ⇒Y(∞)as λ→∞. Suppose h ∈𝒲^(d). Without loss of generality, we may assume that h(0) = 0, otherwise we may simply consider h(x) - h(0). By definition of 𝒲^(d), h(x) ≤xforx∈^dand the result follows from Theorem <ref> with m=1.For any fixed β∈, there are only finitely many combinations of λ∈ (0, 4) and integer n≥ 1 satisfying (<ref>). Therefore, it suffices to prove Theorem <ref> by restricting λ≥ 4, a convenience for technical purposes.§ MARKOV REPRESENTATION The M/Ph/n+M system can be represented as a CTMCU^(λ) = { U^(λ)(t), t ≥0}taking values in 𝒰, the set of finite sequences {u_1, ... , u_k} . The sequence u={u_1, ... , u_k} encodes the service phase of each customer and their order of arrival to the system. For example, the sequence {5, 1, 4} corresponds to 3 customers in the system, with the service phases of the first, second and third customers (in the order of their arrival to the system) being 5, 1 and 4, respectively. We use u to denote the length of the sequence u. The irreducibility of the CTMC U^(λ) is guaranteed by (<ref>) and (<ref>). We remark here that U^(λ) is not the simplest Markovian representation of the M/Ph/n+M system. Another way to represent this system would be to consider a d+1 dimensional CTMC that keeps track of the total number of customers in the system, as well as the total number of customers in each phase that are currently in service; this d+1 dimensional CTMC is used in <cit.>. In this chapter we use the infinite dimensional CTMC U^(λ) because the system size process X^(λ) cannot be recovered sample path wise from the d+1 dimensional CTMC, it can only be recovered from U^(λ). Also, the CTMC U^(λ) will play an important role in our SSC argument in Section <ref>.In addition to the system size process X^(λ), we define the queue size process Q^(λ) = {Q^(λ)(t) ∈^d_+, t ≥ 0}, where Q^(λ)(t) = (Q^(λ)_1(t), ... , Q^(λ)_d(t))^T,and Q^(λ)_i(t) is the number of customers of phase i in the queue at time t. Then X^(λ)_i(t) - Q^(λ)_i(t) ≥ 0 is the number phase i customers in service at time t.To recover X^(λ)(t) and Q^(λ)(t) from U^(λ)(t), we define the projection functions Π_X : 𝒰→^dand Π_Q : 𝒰→^d. For each u∈ U and each phase i∈{1, …, d},(Π_X(u))_i = ∑_k=1^u 1_{ u_k = i}and(Π_Q(u))_i = ∑_k = n+1^u 1_{ u_k = i}.It is clear that on each sample pathX^(λ)(t) = Π_X(U^(λ)(t))andQ^(λ)(t) = Π_Q(U^(λ)(t))fort≥ 0. Because there is customer abandonment the Markov chain U^(λ) can be proved to be positive recurrent with a unique stationary distribution <cit.>. We use U^(λ)(∞) to denote the random element that has the stationary distribution. It follows that X^(λ)(∞)=Π_X(U^(λ)(∞)) has the stationary distribution of X^(λ), andX̃^(λ)(∞) in (<ref>) is given byX̃^(λ)(∞) = δ (Π_X(U^(λ)(∞))-γ n).For u ∈𝒰, we define x = δ(Π_X(u) - γ n), q = Π_Q(u)andz = Π_X(u) - q.When the CTMC is in state u, we interpret (Π_X(u))_i, q_i, and z_i as the number of the phase i customers in system, in queue, and in service, respectively. It follows that z≥ 0. Let G_U^(λ) be the generator of the CTMC U^(λ). To describe it, we introduce the lifting operator A. For any function f: ^d →, we define Af: 𝒰→ by Af(u) = f(δ (Π_X(u)-γ n)) = f(x).Hence, for any function f: ^d →, the generator acts on the lifted version Af as follows:G_U^(λ) Af(u)= ∑_i=1^d λ p_i( f(x + δ e^(i)) - f(x)) + ∑_i=1^d α q_i (f(x - δ e^(i)) - f(x)) + ∑_i=1^d ν_i z_i [∑_j=1^d P_ijf(x+δ e^(j)-δ e^(i)) + (1-∑_j=1^d P_ij)f(x-δ e^(i)) -f(x)].Observe that G_U^(λ) Af(u) does not depend on the entire sequence u; it depends on x, q, and the function f only.§ APPLYING STEIN'SMETHODIn this section, we prepare the ingredients needed to prove Theorem <ref> using the Stein framework introduced in Section <ref>. We prove Theorem <ref> in Section <ref>.§.§ Poisson EquationConsider the Poisson equation G_Y f_h(x) =h(Y(∞)) - h(x),where the generator G_Y of the diffusion process Y, applied to a function f(x) ∈ C^2(^d), is given by G_Y f(x) = ∑_i=1^d ∂_i f(x) [ p_i β - ν_i(x_i - p_i(e^Tx)^+) - α p_i (e^Tx)^+ + ∑_j=1^d P_jiν_j(x_j-p_j(e^Tx)^+)]+ 1/2∑_i,j=1^d Σ _ij∂_ij f(x)forx∈^d.Taking expected values in (<ref>) with respect to X̃^(λ)(∞), we focus on bounding the left hand side G_Y f_h(X̃^(λ)(∞)). The following lemma, based on the results of <cit.>,guarantees the existence of a solution to (<ref>) and provides gradient bounds for it. The proof of this lemma is given in Section <ref>.For any locally Lipschitz function h: ^d → satisfying h(x)≤x^2m, equation (<ref>) has a solution f_h(x). Moreover, there exists a constant C(m,1)>0 (depending only on (β,α,p,ν,P)) such thatfor x∈^df_h(x) ≤C(m,1)(1+x^2)^m,∂_i f_h(x) ≤C(m,1)(1+x^2)^m(1+x),∂_ij f_h(x) ≤C(m,1)(1+x^2)^m(1+x)^2,sup_y∈^d:y-x < 1∂_ij f_h(y)-∂_ijf_h(x)/y-x ≤ C(m,1) (1+x^2)^m(1+x)^3.§.§ Comparing GeneratorsThe following is an analogue of Lemma <ref>. Let h: ^d → satisfy h(x)≤x^2m. The function f_h(x) given by (<ref>) satisfiesG_U^(λ) Af_h(U^(λ)(∞)) = 0. To prove the lemma, we need finite moments of the steady-state system size. (a) LetL(u) =exp(e^T Π_X(u)) for u ∈𝒰. Then L(U^(λ)(∞))<∞.(b) all moments of e^TX^(λ)(∞) are finite. One may verify thatG_U^(λ) L(u) ≤λ(exp(1)-1)L(u) - α(e^T Π_X(u) -n)^+(1- exp(-1))L(u).It follows that there exist a positive constant C= C(λ, n, α) such that,whenever e^T Π_X(u) is large enough,G_U^(λ) L(u) ≤ -CL(u) + 1.Part (a) follows from <cit.>. Part (b) follows from (<ref>) and the equality e^T Π_X(U^(λ)(∞)) = e^T X^(λ)(∞).The function L(u) is said to be a Lyapunov function. Inequality (<ref>) is known as a Foster-Lyapunov condition and guarantees that the CTMC is positive recurrent; see, for example,<cit.>.A sufficient condition for (<ref>) to hold is given by <cit.> (alternatively, see <cit.>), namely[ G_U^(λ)(U^(λ)(∞),U^(λ)(∞))Af_h(U^(λ)(∞))]<∞.Above, G_U^(λ)(u,u) is the uth diagonal entry of the generator matrix G_U^(λ).In our case, the left side of (<ref>) is equal to = [ G_U^(λ)(U^(λ)(∞),U^(λ)(∞))f_h(X̃^(λ)(∞))]= λ + α(e^TX^(λ)(∞) - n)^+ + ∑_i=1^d ν_i (X^(λ)_i(∞) - Q^(λ)_i(∞) f_h(X̃^(λ)(∞))≤ λ + (α∨max_i {ν_i}) e^TX^(λ)(∞) f_h(X̃^(λ)(∞)),where the first equality follows from (<ref>) and (<ref>). One may apply (<ref>) and (<ref>) to see that the quantity above is finite. §.§ Taylor ExpansionTo provethath(X̃^(λ)(∞))) -h(Y(∞))= G_U^(λ) Af_h(U^(λ)(∞)) - G_Y f_h(X̃^(λ)(∞))is small, we perform Taylor expansion on G_U^(λ) Af_h(u), which is defined in (<ref>):G_U^(λ) Af_h(u) = ∑_i=1^d λ p_i( δ∂_i f_h(x) + δ^2/2∂_iif_h(ξ_i^+)) +α q_i(-δ∂_i f_h(x)+δ^2/2∂_iif_h(ξ_i^-))+ ∑_i=1^d ν_i z_i(1-∑_j=1^d P_ij)(-δ∂_i f_h(x)+ δ^2/2∂_iif_h(ξ_i^-)) +∑_i=1^d ∑_j=1^d ν_i z_i P_ij(-δ∂_i f_h(x)+ δ∂_j f_h(x) + δ^2/2∂_iif_h(ξ_ij)+ δ^2/2∂_jjf_h(ξ_ij)- δ^2 ∂_ijf_h(ξ_ij)),where ξ_i^+ ∈ [x, x+δ e^(i)], ξ_i^-∈ [x-δ e^(i), x] and ξ_ij lies somewhere between x and x-δ e^(i)+δ e^(j).Using the gradient bounds in Lemma <ref>, we have the following lemma, which will be provedin Section <ref>. There exists a constant C(m,2)>0 (depending only on (β,α,p,ν,P)) such that for any u ∈𝒰,G_U^(λ) Af_h(u) - G_Y f_h(x) = ∑_i=1^d ∂_i f_h(x)[(ν_i - α-∑_j=1^d P_jiν_j)( δ q_i - p_i(e^T x)^+)] + E(u),where q and x are as in (<ref>), δ as in (<ref>), and E(u) is an error term that satisfiesE(u) ≤δC(m,2) (1+x^2)^m (1+x)^4. § STATE SPACE COLLAPSEOne of the challenges we face comes from the fact that our CTMC U^(λ) is infinite-dimensional, while the approximating diffusion process is only d-dimensional.Recall the process (X^(λ),Q^(λ)) defined in (<ref>) and the lifting operator A acting on functions f:^d →, as defined in (<ref>). When acting on the lifted functions Af(U^(λ)(∞)), the CTMC generator G_U^(λ) depends on both X̃^(λ)(∞) and Q^(λ)(∞), but its approximation G_Y f(X̃^(λ)(∞)) only depends on X̃^(λ)(∞). This is captured in (<ref>) by the term∑_i=1^d ∂_i f_h(x)[(ν_i - α-∑_j=1^d P_jiν_j)( δq_i - p_i(e^T x)^+)].To bound this term, observe that for any 1 ≤ i ≤ d, (ν_i - α-∑_j=1^d P_jiν_j)∂_i f_h(x)( δ q_i - p_i(e^T x)^+) = (ν_i - α-∑_j=1^d P_jiν_j)(∂_i f_h(x)- ∂_i f_h(x - δ q + p(e^T x)^+) )( δ q_i - p_i(e^T x)^+)+ (ν_i - α-∑_j=1^d P_jiν_j)∂_i f_h(x - δ q + p(e^T x)^+)( δ q_i - p_i(e^T x)^+) = (ν_i - α-∑_j=1^d P_jiν_j)∑_k=1^d∂_ik f_h(ξ)( δ q_k - p_k(e^T x)^+)( δ q_i - p_i(e^T x)^+) + (ν_i - α-∑_j=1^d P_jiν_j)∂_i f_h(δ (z - γ n) + p(e^T x)^+)( δ q_i - p_i(e^T x)^+),where z, defined in (<ref>), is a vector that represents the number of customers of each type in service, and ξ is some point between x and x - δ q + p(e^T x)^+. In particular, there exists some constant C that doesn't depend on λ and n, such thatξ≤x+ δq + p(e^T x)^+ ≤ C x,because δ q_i ≤ (e^T x)^+ for each 1 ≤ i ≤ d (i.e. the number of phase i customers in queue can never exceed the queue size). In order to bound the expected value of (<ref>), we must prove a relationship between X̃^(λ)(∞) and Q^(λ)(∞). Intuitively, the number of customers of phase i waiting in the queue should be approximately equal to a fraction p_i of the total queue size. The following two lemmas bound the error caused by the SSC approximation. They are proved at the end of this section.Let Z^(λ)(∞) = X^(λ)(∞) - Q^(λ)(∞)be the vector representing the number of customers of each type in service in steady-state. Then conditioned on (e^T X̃^(λ)(∞))^+, the random vectors Q^(λ)(∞) and Z^(λ)(∞) are independent. Furthermore,[δ Q^(λ)(∞) - p(e^T X̃^(λ)(∞))^+ | (e^T X̃^(λ)(∞))^+ ] = 0,and for any integer m>0, there exists C(m, 3)>0 (depending only on (β,α,p,ν,P)) such that for all λ>0 and n≥ 1 satisfying (<ref>),[δ Q^(λ)(∞) - p(e^T X̃^(λ)(∞))^+^2m ] ≤δ^mC(m, 3) [(e^T X̃^(λ)(∞))^+]^m,where δ=1/√(λ) as in (<ref>). For any integer m>0, there exists C(m, 4)>0 (depending only on (β,α,p,ν,P)) such that for any locally Lipschitz function h: ^d → satisfying h(x)≤x^2m, and all λ>0 and n≥ 1 satisfying (<ref>)∑_i=1^d[∂_i f_h(X̃^(λ)(∞))[(ν_i - α-∑_j=1^d P_jiν_j)( δ Q^(λ)_i(∞) - p_i(e^T X̃^(λ)(∞))^+)]]≤ δ C(m, 4)[((e^T X̃^(λ)(∞))^+)^2]√([1 + X̃^(λ)(∞)^8])where f_h(x) is the solution to the Poisson equation (<ref>). We begin by proving (<ref>), for which it suffices to show that for all λ>0 and n≥ 1 satisfying (<ref>)[Q^(λ)(∞) - p(e^T X^(λ)(∞) - n)^+^2m] ≤C(m, 3)[(e^TX^(λ)(∞) - n)^+]^m.We first prove a version of (<ref>) for any finite time t ≥ 0. Then, (e^TX^(λ)(t) - n)^+ is the total number of customers waiting in queue at time t. Assume that the system is empty at time t = 0, i.e. X^(λ)(0) = 0. Fix a phase i.Upon arrival to the system, a customer is assigned to service phase i with probability p_i. Consider the sequence {ξ_j:j=1, 2, …}, where ξ_j is one if the jth customer to enter the system was assigned to phase i, and zero otherwise. Then {ξ_j:j=1, 2, …} is a sequence of iid Bernoulli random variables with (ξ_j=1)=p_i. For t > 0, define A(t) and B(t) to be the total number of customers to have entered the system, and entered service by time t, respectively. Also let ζ_j(t) be the indicator of whether customer j is still waiting in queue at time t. Then (e^TX^(λ)(t) - n)^+ = ∑_j=B(t)+1^A(t)ζ_j(t), Q^(λ)_i(t) = ∑_j= B(t) + 1^A(t)ξ_j ζ_j(t).Let Z^(λ)(t) = X^(λ)(t) - Q^(λ)(t) be the vector keeping track of the customer types in service at time t and let B(ℓ,p_i) be a binomial random variable with ℓ∈_+ trials and success probability p_i. Assuming X^(λ)(0)=0, by a sample path construction of the process U^(λ) one can verify that for any time t ≥ 0, the following three properties hold. First, for any z ∈_+^d, a,b ∈_+ with a≥ 1, and x_1, …, x_a, y_1, …, y_a ∈{0,1},(ξ_b+1 = x_1, …, ξ_b+a=x_a | A(t)= b+a, B(t)=b, Z^(λ)(t) = z,ζ_b+1=y_1, …, ζ_b+a=y_a)=(ξ_1 = x_1)(ξ_2 = x_2)…(ξ_a=x_a) =p_i^∑_i=1^a x_i(1- p_i)^a-∑_i=1^a x_i.The right side of (<ref>) is independent of b, z, y_1, …, y_a.It then follows from (<ref>), (<ref>)and (<ref>) thatfor any integer ℓ≥ 1, q_i∈_+, and z ∈_+^d,(Q^(λ)_i(t) = q_i | (e^TX^(λ)(t) - n)^+ = ℓ, Z^(λ)(t) = z) =(Q^(λ)_i(t) = q_i | (e^TX^(λ)(t) - n)^+ = ℓ) = (B(ℓ,p_i) =q_i ).Since (<ref>) holds for all t ≥ 0, it holds in stationarity as well.We now say a few words about how to construct U^(λ) and argue (<ref>)–(<ref>). One would start with four primitive sequences: a sequence of inter-arrival times, potential service times, patience times, and routing decisions. The sequence of potential service times would hold all the service information about each customer provided they were patient enough to get into service. The routing sequence would represent the phase each customer is assigned upon entering the system.To see why (<ref>) is true, we first observe that at any time t> 0, the random variable A(t) depends only on the inter-arrival time primitives; in particular, it is independent of the routing sequence {ξ_j, j≥ 1}. Second, any customer to arrive after customer number B(t)=b has no impact on any of the servers at any point in time during [0,t]. In particular, the primitives including {ξ_b+j, j≥ 1} associated to those customers are independent of B(t)=b and Z^(λ)(t). Lastly, the decisions of those customers whether to abandon or not by time t depends only on their arrival times, patience times, and the service history in the interval [0,t]. In particular, the sequence {ζ_b+j(t), j≥ 1} is independent of {ξ_b+j, j≥ 1}. This proves thethe first equality in (<ref>). We now move on to complete the proof of this lemma. We use (<ref>) to see that for any positive integer N, ( [Q^(λ)_i(t) - p_i(e^TX^(λ)(t) - n)^+]^2m1_{ (e^T X^(λ)(t) - n)^+ ≤ N}) = ∑_ℓ=1^N[(B(ℓ,p_i) - p_i ℓ)^2m]((e^TX^(λ)(t) - n) = ℓ)≤ ∑_ℓ=1^NC(m,6) ℓ^m((e^TX^(λ)(t) - n) = ℓ)=C(m,6) ([(e^TX^(λ)(t) - n)^+]^m1_{ (e^T X^(λ)(t) - n)^+ ≤ N}),where we have used the fact that there is a constant C(m,6)>0 such that[(B(ℓ,p_i) - p_i ℓ)^2m] ≤C(m,6) ℓ^mfor allℓ≥1;see, for example,(4.10) of <cit.>. Letting t→∞ in both sides of(<ref>), by the dominated convergence theorem, one has( [Q^(λ)_i(∞) - p_i(e^TX^(λ)(∞) - n)^+]^2m1_{ (e^T X^(λ)(∞) - n)^+ ≤ N})≤C(m,6) ([(e^TX^(λ)(∞) - n)^+]^m1_{ (e^T X^(λ)(∞) - n)^+ ≤ N}).Letting N→∞, by the monotone convergence theorem, one has (Q^(λ)_i(∞) - p_i(e^TX^(λ)(∞) - n)^+)^2m≤ C(m,6) [(e^TX^(λ)(∞) - n)^+]^m.Then (<ref>) follows from this inequality for each i and the fact that there is a constant B_m>0 such thatx^2m≤ B_m ∑_i=1^d (x_i)^2m for all x∈^d. One can check that (<ref>) can be obtained by an argument very similar to the one used to prove (<ref>). Recall that Z^(λ)(∞) = X^(λ)(∞) - Q^(λ)(∞) is the vector representing the number of customers of each type in service in steady-state. Then from (<ref>) we have[∂_i f_h(X̃^(λ)(∞))( δ Q^(λ)_i(∞) - p_i(e^T X̃^(λ)(∞))^+) ]= ∑_k=1^d[∂_ik f_h(ξ)( δ Q^(λ)_k(∞) - p_k(e^T X̃^(λ)(∞))^+)( δ Q^(λ)_i(∞) - p_i(e^T X̃^(λ)(∞))^+)]+ [∂_i f_h(δ (Z^(λ)(∞) - γ n) + p(e^T X̃^(λ)(∞))^+)( δ Q^(λ)_i(∞) - p_i(e^T X̃^(λ)(∞))^+)].By Lemma <ref>, the second expected value equals zero. For the first term, one can use the Cauchy-Schwarz inequality, together with the gradient bound (<ref>) and the SSC result (<ref>) to see that for all 1 ≤ i,k ≤ d,[∂_ik f_h(ξ)( δ Q^(λ)_k(∞) - p_k(e^T X̃^(λ)(∞))^+)( δ Q^(λ)_i(∞) - p_i(e^T X̃^(λ)(∞))^+)] ≤ ([(∂_ik f_h(ξ))^2])^1/2([( δ Q^(λ)_k(∞) - p_k(e^T X̃^(λ)(∞))^+)^4])^1/4 ×( [( δ Q^(λ)_i(∞) - p_i(e^T X̃^(λ)(∞))^+)^4 ])^1/4≤ δ C(2, 3)[(e^T X̃^(λ)(∞))^+]^2√([(∂_ik f_h(ξ))^2])≤ δ C(2, 3)[(e^T X̃^(λ)(∞))^+]^2C(m,1)√([(1+ξ^2)^2(1+ξ)^4]).We now combine everything together with the fact that ξ satisfies (<ref>) to conclude that there exists a constant C(m,4) that does not depend on λ or n, such that∑_i=1^d ∂_i [f_h(X̃^(λ)(∞))[(ν_i - α-∑_j=1^d P_jiν_j)( δ Q^(λ)_i(∞) - p_i(e^T X̃^(λ)(∞))^+)]]≤ δ C(m, 4)[(e^T X̃^(λ)(∞))^+]^2√([1 + X̃^(λ)(∞)^8]),which concludes the proof of the lemma.§ PROOF OF THEOREM <REF>To prove Theorem <ref>, we need an additional lemma on uniform bounds for moments of scaled system size.It will be proved in Section <ref>. For any integer m ≥ 0, there exists a constantC(m, 5)>0 (depending only on (β,α,p,ν,P)) such thatX̃^(λ)(∞)^m≤ C(m, 5). We remark that in the special case when the service time distribution is taken to be hyper-exponential, it is proved in <cit.> thatlim sup_λ→∞ exp(θX̃^(λ) (∞)) < ∞ for θ in a neighborhood around zero. The proof relies on a result that allows one to compare the system with an infinite-server system, whose stationary distribution isknown to be Poisson. It follows from Lemmas <ref> and <ref> that h(X̃^(λ)(∞)) -h(Y(∞)) =G_U^(λ) Af_h(U^(λ)(∞)) -G_Y f_h(X̃^(λ)(∞)) ≤∑_i=1^d[∂_i f_h(X̃^(λ)(∞))[(ν_i - α-∑_j=1^d P_jiν_j)( δ Q^(λ)_i(∞) - p_i(e^T X̃^(λ)(∞))^+)]]+δ C(m,2)[(1+X̃^(λ)(∞)^2)^m(1 + X̃^(λ)(∞))^4] ≤δ C(m, 4)[((e^T X̃^(λ)(∞))^+)^2]√([1 + X̃^(λ)(∞)^8])+δ C(m,2)[(1+X̃^(λ)(∞)^2)^m(1 + X̃^(λ)(∞))^4] .By Lemma <ref>, there are constants B_1(m), B_2(m)>0 (depending only on (β,α,p,ν,P)) such that [((e^T X̃^(λ)(∞))^+)^2]√([1 + X̃^(λ)(∞)^8])≤ B_1(m),[(1+X̃^(λ)(∞)^2)^m(1 + X̃^(λ)(∞))^4]≤ B_2(m).Therefore, the right side of (<ref>) is less than or equal to δ C(m,4) B_1(m) + δ C(m,2) B_2(m) ≤ (C(m,4) B_1(m) +C(m,2) B_2(m) ) 1/√(λ) for λ > 0.This concludes the proof of Theorem <ref>. § STATE DEPENDENT DIFFUSION COEFFICIENT In Chapter <ref>, we showed that using a state-dependent diffusion coefficient yields a much better approximation for the Erlang-C model. In this section we explore the use of a state-dependent diffusion coefficient for the M/Ph/n+M model. We perform a numerical study, as the multi-dimensional nature of the M/Ph/n+M model makes it difficult to prove any rigorous bounds.To understand which diffusion approximation to use, we first group the terms on the right hand side of (<ref>) by partial derivatives to see that G_U^(λ) Af(u) ≈∑_i=1^d ∂_i f(x)δ(λ p_i - α q_i - ν_i z_i) +∑_i=1^d ∂_ii f(x)1/2δ^2 ( λ p_i+ α q_i+ ν_i z_i + ∑_j=1^d P_jiν_j z_j ) - ∑_i=1^d∑_j=1^d∂_ij f(x) δ^2 ν_i z_i P_ij,u ∈𝒰,where z, qand x are defined in (<ref>). We wish to replace z and q by functions of x. We know that x_i = δ(z_i + q_i - γ_i n). Lemma <ref> tells us to use the approximationδ q_i ≈ p_i(e^T x)^+ = p_i(∑_i^d x_i)^+, which suggests that z_i =1/δ (x_i - q_i) + γ_i n ≈1/δ(x_i - p_i(e^T x)^+) + γ_i n.The state space of the CTMC makes it so z_i can never be negative, i.e. the number of customers in service is never negative. Therefore, z_i = (z_i)^+≈( 1/δ(x_i - p_i(e^T x)^+) + γ_i n)^+.We apply(<ref>) and (<ref>) to (<ref>) to arrive at the diffusion approximation with generator G f(x) =∑_i=1^d ∂_i f(x) (δλ p_i - α p_i(e^T x)^+ - ν_i (x_i - p_i(e^T x)^+ + δγ_i n)^+)+ ∑_i=1^d ∂_ii f(x)1/2( δ^2 λ p_i+ δα p_i(e^T x)^++ δν_i (x_i - p_i(e^T x)^+ + δγ_i n)^+ + δ∑_j=1^d P_jiν_j(x_j - p_j(e^T x)^+ + δγ_j n)^+ ) - ∑_i=1^d∑_j=1^d∂_ij f(x) δν_iP_ij(x_i - p_i(e^T x)^+ + δγ_i n)^+.Comparing the generator in (<ref>) to G_Y in (<ref>), we see that thecoefficients of the second derivatives are state-dependent in the former, but constant in the latter. Although we are not guaranteed that the diffusion process with generator given by (<ref>) is positive recurrent, we assume it is, and use a modified version of the finite element algorithm in <cit.> to compute its stationary distribution.In the rest of this section, we will be interested in approximating the steady-state total customer count in the system. For convenience, we defineT := X_1^(λ)(∞) + X_2^(λ)(∞)and T̃ := X̃_1^(λ)(∞) + X̃_2^(λ)(∞) = δ(T-n),where X̃^(λ)(∞) is defined in (<ref>).We set W := Y_1(∞) + Y_2(∞),where Y(∞) has the steady-state distribution of the diffusion process with generator G_Y. The random variable W is the constant diffusion coefficient approximation to T̃. Analogously to (<ref>), we let W_S be the approximation to T̃ based on the diffusion process with generator in (<ref>). The code used in the following numerical study is publicly available at <https://github.com/anton0824/mphnplusm>.§.§ M/C_2/n+M Model – No State Space CollapseWe first focus on the special case of the M/C_2/n+M model. The C_2 stands for a 2-phase Coxian distribution. The corresponding tuple of parameters is (p,ν,P), wherep = (1,0)^T, ν= (ν_1, ν_2)^T,andP = [0 P_12;00 ].It can be checked that in this case, 1/μ = 1/ν_1 + P_12/ν_1, γ_1 = μ/ν_1, and γ_2 = μ P_12/ν_2.All customers start out in phase 1, and after completing that phase they move on to phase 2 with probability P_12, or leave system with probability 1-P_12. Choosing the parameters ν_1, ν_2, P_12 is often done by first choosing the desired mean 1/μ andsquared coefficient ofvariation c_s^2; the squared coefficient of variation of a random variable Z equals Var(Z)/((Z))^2. After choosing 1/μ and c_s^2, we then set ν_1 = 2μ, P_12=1/(2c_s^2), ν_2 = P_12ν_1. In the following example, we choose μ = 1 and c_s^2 = 24. The algorithm of <cit.> that we use to compute the density of W and W_S require choosing a reference density, truncation rectangle, and a mesh resolution. To generate Table <ref>, and Figures <ref> and <ref>, we used a truncation rectangle of [-10, 35] × [-10, 35], and a lattice mesh in which all finite elements are 0.5 × 0.5 squares. The reference density used is similar to (3.21) and (3.23) of <cit.>, but with one exception. With a C_2 service time distribution, any customer in the buffer must be a type-1 customer, and therefore type-2 customers never abandon the system. Therefore, using the notation of <cit.>, we choose r_2(z) = exp( - z/μ(c_a^2 + c_s^2)- γ_j^2 β^2/1 + c_a^2),forz ≥ 0 . Since all customers start out in phase 1 of service, the M/C_2/n+M model can be represented by a 2-dimensional CTMC. Namely, {(X_1(t), X_2(t)), t ≥ 0} is a CTMC. This fact is important, because the diffusion approximation is also 2-dimensional, and no SSC is required. This means that (<ref>) and (<ref>) are actually equalities, not just approximations, and that the diffusion generator completely captures the first and second derivative terms of the Taylor expansion in (<ref>). We observed in Chapter <ref> that capturing the first and second derivative terms in the generator of the Erlang-C model gave us faster convergence rates. By similar logic, we expect the approximation in (<ref>) to have a faster convergence rate of 1/λ as opposed to 1/√(λ). Table <ref> is consistent with this expectation, and shows that when approximating |T̃|, the errors from using W and W_S shrink at rates 1/√(λ) and 1/λ, respectively. Similar results were observed for higher moments of T̃ as well. Another criterion by which we evaluate the diffusion approximations is how well they approximate the probability mass function (pmf) of T, the unscaled total customer count. Figure <ref> contains plots the pmf of T together with the constant and state-dependent coefficient approximations. We see that the benefit of the latter approximation is more pronounced for the smaller-sized system. We refer the reader to Figure <ref>, which plots the relative error of approximating (T ≥ k). We see from that figure that when approximating tail events, e.g. when (T ≥ k) ≤ 0.05,the state-dependent coefficient approximation performs significantly better.§.§ M/H_2/n+M ModelWe now focus on the M/H_2/n+M model, where the H_2 stands for a 2-phase hyper-exponential distribution. The corresponding tuple of parameters (p,ν,P) is p = (p_1, p_2)^T, ν= (ν_1,ν_2)^T,andP = 0.The starting service phase of each customer is random, and unlike how it was with the Coxian distribution, the process {X_1(t), X_2(t), t≥ 0} is not a CTMC. In particular, this means that the approximation in (<ref>) has non-zero approximation error. As a result, even though we use a state-dependent diffusion coefficient, we are unable to fully capture the first and second derivative terms in the Taylor expansion of G_U^(λ). We also have no reason to expect faster convergence rates because the error terms corresponding to the first derivatives are abottleneck of order 1/√(λ). Figures <ref> and <ref> compare the two diffusion approximations for a system with 100 servers. Due to the approximation error in (<ref>), using a state-dependent diffusion coefficient does not give us the improved accuracy we are accustomed to. In fact, we cannot conclude which approximation is better. To generate Figures <ref> and <ref>, we used the same reference density as in (3.21) and (3.23) of <cit.>, a truncation rectangle of[-15, 40]× [-15, 40], and a lattice mesh in which all finite elements are 0.5 × 0.5 squares; see <cit.> for more details. § CHAPTER APPENDIX§.§ Proof of Lemma <ref> (Generator Difference) The main idea here is that G_Y f_h(x) is hidden within G_U^(λ) Af_h(u), where the lifting operator A is in (<ref>). We algebraically manipulate the Taylor expansion of G_U^(λ) Af_h(u) to make this evident. First, we first rearrange the terms in the Taylor expansion (<ref>) to group them by partial derivatives. Thus, G_U^(λ) Af_h(u) equals ∑_i=1^d δ∂_i f_h(x)[p_i λ - α q_i - ν_i z_i + ∑_j=1^d P_jiν_j z_j] + ∑_i=1^d δ^2/2∂_iif_h(x)[p_i λ + α q_i + ν_i z_i + ∑_j=1^d P_jiν_j z_j] - ∑_i ≠ j^d δ^2∂_ijf_h(x)[P_ijν_i z_i ] + ∑_i=1^d δ^2/2(∂_iif_h(ξ_i^-)-∂_iif_h(x))[α q_i + (1-∑_j=1^d P_ij)ν_i z_i] + ∑_i=1^d δ^2/2(∂_iif_h(ξ_i^+)-∂_iif_h(x))[λ p_i ]- ∑_i ≠ j^d δ^2 (∂_ijf_h(ξ_ij)-∂_ijf_h(x))[P_ijν_i z_i] + ∑_i=1^d ∑_j=1^d δ^2/2(∂_iif_h(ξ_ij)-∂_iif_h(x))[ P_ijν_i z_i +P_jiν_j z_j] .To proceed we observe that (<ref>) gives us the identity-ν_i γ_i n + ∑_j=1^d P_jiν_jγ_j n= -n p_i.Recall the form of G_Y f_h(x) from (<ref>). From the form of Σ in (<ref>), we see thatΣ_ii = 2(p_i + ∑_j=1^d P_jiγ_j ν_j), Σ_ij = -(P_ijν_i γ_i + P_jiν_jγ_j)for j≠ i using (<ref>), (<ref>) and (<ref>), the difference G_U^(λ) Af_h(u) - G_Y f_h(x) becomes∑_i=1^d ∂_i f_h(x)[(ν_i - α-∑_j=1^d P_jiν_j)( δ q_i - p_i(e^T x)^+)] + ∑_i=1^d ∂_iif_h(x)[∑_j=1^d P_jiν_jγ_j ](n δ^2 - 1) - ∑_i ≠ j^d ∂_ijf_h(x)[P_ijν_iγ_i + P_jiν_jγ_j](nδ^2 - 1)- ∑_i=1^d δ^2/2∂_iif_h(x)[p_i (λ - n)- αq_i - ν_i (z_i- γ_i n) - ∑_j=1^d P_jiν_j (z_j- γ_j n)] - ∑_i ≠ j^d δ^2/2∂_ijf_h(x)[P_ijν_i (z_i- γ_i n) + P_jiν_j (z_j- γ_j n)]+ ∑_i=1^d δ^2/2 (∂_iif_h(ξ_i^-)-∂_iif_h(x))[α q_i + (1-∑_j=1^d P_ij)ν_i z_i] + ∑_i=1^d δ^2/2 (∂_iif_h(ξ_i^+)-∂_iif_h(x))[λ p_i ]- ∑_i ≠ j^d δ^2 (∂_ijf_h(ξ_ij)-∂_ijf_h(x))[P_ijν_i z_i] + ∑_i=1^d ∑_j=1^d δ^2/2 (∂_iif_h(ξ_ij)-∂_iif_h(x))[ P_ijν_i z_i +P_jiν_j z_j] .We remind the reader that our target is to prove thatG_U^(λ) Af_h(u) - G_Y f_h(x) = ∑_i=1^d ∂_i f_h(x)[(ν_i - α-∑_j=1^d P_jiν_j)( δ q_i - p_i(e^T x)^+)] + E(u),where E(u) is an error term that satisfies E(u) ≤δC(m,2) (1+x^2)^m (1+x)^4.We choose E(u) to be all the terms in (<ref>) except for the first line. We now describe how to bound E(u). Most of the summands in (<ref>) look as follows: a term in large square brackets multiplied by some partial derivative of f_h. The partial derivatives are very easy to bound; we simply use (<ref>) - (<ref>). We wish to point out that ξ_i^+, ξ_i^- and ξ_ij lie within distance 2δ of x. When 2δ < 1, (<ref>) implies ∂_ijf_h(ξ)-∂_ijf_h(x)≤ 2δ C (1+ x^2)^m (1+ x)^3 for some constant C>0 (i.e. an extra δ term is gained). When 2δ≥ 1 (by Remark <ref> this occurs in finitely many cases), we may use (<ref>) to obtain (<ref>) with a redefined C. From here on out, we shall let C>0 be a generic positive constant that will change from line to line, but will always be independent of λ and n.Now we shall list the facts needed to bound all the square bracket terms in (<ref>) except for the very first one. Recall that we are operating in the Halfin-Whitt regime as defined by (<ref>). Therefore, (nδ^2 - 1) = δβ andδ(λ- n) = -β. Furthermore, it must be true thatδq_i ≤(e^T x)^+ ≤C x, as the number of phase i customers may never exceed the total queue size. Next, δ(z_i - γ_i n) = x_i - δq_i ≤C xand lastly, δ^2 z_i ≤δ^2 γ_i n + δ^2 (z_i - γ_i n) ≤C(1+ x).It is now a simple matter to verify that the inequalities above, combined with the bounds on the partials of f_h are all that it takes to achieve our desired upper bound.The author thanks Jim Dai, Jiekun Feng, Shuangchi He, Josh Reed and John Pike for stimulating discussions. He also thanks the participants of Applied Probability & Risk Seminar in Fall 2014 at Columbia University for their feedback on this research, and the participants of the 2015 Workshop on New Directions in Stein's Method held at the Institute for Mathematical Sciences at the National University of Singapore andthey would like to thank the financial support from the Institute. This research is supported in part by NSF Grants CNS-1248117, CMMI-1335724, and CMMI-1537795.CHAPTER: MOMENT BOUNDS This appendix proves all of the moment bounds used in this document. Bounds for Chapters <ref>, <ref> and <ref> are proved in Sections <ref>, <ref>, and <ref>, respectively. § CHAPTER <REF> MOMENT BOUNDSWe first prove Lemma <ref> in Section <ref>, establishing the moment bounds for Erlang-C model. In Section <ref>, weprove Lemma <ref>,establishing the moment bounds for Erlang-A model.§.§ Erlang-C Moment BoundsWe first prove (<ref>), (<ref>), and (<ref>). Recalling the generator G_X̃ defined in (<ref>), we apply it to the function V(x) = x^2 to see that for k ∈_+ and x = x_k = δ(k - x(∞)), G_X̃ V(x) =λ( 2xδ + δ^2) + μ (k ∧ n)(-2xδ + δ^2) =2xδ(λ - nμ+ μ (k - n)^-) + μ + δ^2 μ (k ∧ n) =2xμ ( ζ + (x+ζ)^-) +μ+ δ^2μ (n - λ/μ + λ/μ - (k - n)^-) =2xμ ( ζ + (x+ζ)^-) + μ-δμζ + μ- δμ (x+ζ)^- =1(x ≤ -ζ)μ(-2x^2 + δ x ) + 1(x > -ζ) μ(2x ζ -δζ) + 2μ ≤1(x ≤ -ζ) μ(-3/2x^2 + δ^2/2) + 1(x > -ζ)μ(2x ζ -δζ) + 2μ.Instead of splitting the last two lines into the casesx ≤ -ζ and x > -ζ, we could have also consideredx < -ζ and x ≥ -ζ instead, and would have obtainedG_X̃ V(x)=1(x < -ζ)μ(-2x^2 + δ x ) + 1(x ≥ -ζ) μ(2x ζ -δζ) + 2μ ≤1(x < -ζ) μ(-3/2x^2 + δ^2/2) + 1(x ≥ -ζ)μ(2x ζ -δζ) + 2μ.We take expected values on both sides of (<ref>) with respect to X̃(∞), and apply Lemma <ref> to see that0 ≤-3/2μ[(X̃(∞))^2 1(X̃(∞) ≤ -ζ)] + μζ[(-2X̃(∞) +δ)1(X̃(∞) > -ζ)] + 2μ+ μδ^2/2.This implies that when ζ > δ/2,0 ≤-3/2μ[(X̃(∞))^2 1(X̃(∞) ≤ -ζ)] + 2μ+ μδ^2/2,and when ζ≤δ/2,0 ≤-3/2μ[(X̃(∞))^2 1(X̃(∞) ≤ -ζ)]+2μ+ μδ^2.Therefore, [(X̃(∞))^2 1(X̃(∞) ≤ -ζ)] ≤4/3 + 2δ^2/3,which proves (<ref>). Jensen's inequality immediately gives us[|X̃(∞)1(X̃(∞) ≤ -ζ)|] ≤√([(X̃(∞))^2 1(X̃(∞) ≤ -ζ)]),which proves (<ref>). Furthermore, (<ref>) also gives us [|X̃(∞)1(X̃(∞) > -ζ)| ] ≤1/ζ + δ^2/4ζ + δ/2,which is not quite (<ref>) because the inequality above has 1(X̃(∞) > -ζ) as opposed to 1(X̃(∞) ≥ -ζ) as in (<ref>). However, we can use (<ref>) to get the stronger bound [|X̃(∞)1(X̃(∞) ≥ -ζ)| ] ≤1/ζ + δ^2/4ζ + δ/2,which proves (<ref>). We now prove (<ref>), or [|X̃(∞)1(X̃(∞) ≤ -ζ)|] ≤ 2ζ.We use the triangle inequality to see that[|X̃(∞)1(X̃(∞) ≤ -ζ) |] ≤ζ +[|X̃(∞) +ζ| 1(X̃(∞) ≤ -ζ) ].The second term on the right hand side is just the expected number of idle servers, scaled by δ. We now show that this expected value equals ζ. Applying the generator G_X̃ to the test function f(x) = x, one sees that for all k ∈_+ and x = x_k = δ(k-x(∞)), G_X̃ f(x) = δλ - δμ (k ∧ n) = μ[ζ + (x + ζ)^-].Taking expected values with respect to X̃(∞) on both sides, and applying Lemma <ref>,we arrive at[|(X̃(∞) +ζ) 1(X̃(∞) ≤ -ζ)| ] = ζ,which proves (<ref>).We move on to prove (<ref>), or (X̃(∞) ≤ -ζ) ≤ (2+δ)ζ.Let I be the unscaled expected number of idle servers. Then by (<ref>),I = (X(∞) - n)^- =1/δ[|(X̃(∞) +ζ) 1(X̃(∞) ≤ -ζ)| ] = 1/δζ.Now let {π_k}_k=0^∞ be the distribution of X(∞). We want to prove an upper bound on the probability(X̃(∞) ≤ -ζ) = ∑_k=0^nπ_k ≤∑_k=0^⌊ n - √(R)⌋π_k +∑_k= ⌈ n - √(R)⌉^nπ_k.Observe that I = ∑_k=0^n (n-k) π_k ≥√(R)∑_k=0^⌊ n - √(R)⌋π_k.Now let k^* be the first index that maximizes {π_k}_k=0^∞, i.e.k^* = inf{k ≥ 0 : π_k ≥ν_j,for all j ≠ k}.Then(X̃(∞) ≤ -ζ) = ∑_k=0^⌊ n - √(R)⌋π_k +∑_k= ⌈ n - √(R)⌉^nπ_k ≤I/√(R) +(√(R)+1) π_k^*=ζ + (√(R)+1) π_k^*.Applying G_X̃ to the test function f(x) = (k ∧ k^*), we see that for all k ∈_+ and x = x_k = δ(k - x(∞)),G_X̃ f(x) = δλ 1(k < k^*) - δμ(k ∧ n) 1(k ≤ k^*).Taking expected values with respect to X(∞) on both sides and applying Lemma <ref>, we see that(X(∞) ≤ k^*) = μ/nμ-λ[(X(∞)-n)^-1(X(∞) ≤ k^*) ] - π_k^*λ/nμ-λ≥ 0.Using the inequality above, together with the fact that k^* ≤ n, we see that π_k^*≤μ/λ[(X(∞)-n)^-1(X(∞) ≤ k^*) ]≤μ/λ[(X(∞)-n)^-1(X(∞) ≤ n) ] = I/R = ζ/√(R).The fact that k^* ≤ n is a consequence of λ < nμ, and can be verified through the flow balance equations of the CTMC X. We combine the bound above with (<ref>) to arrive at (<ref>), which concludes the proof of this lemma.§.§ Erlang-A Moment Bounds Recall Lemma <ref> stated in Section <ref>. We outline the proof of it below. §.§.§ Proof Outline for Lemma <ref>: The Underloaded SystemThe proof of the underloaded case of Lemma <ref> is very similar to that of Lemma <ref>. Therefore, we only outline some key intermediate steps needed to obtain the results. We remind the reader that when R ≤ n, then ζ≤ 0. We first show how to establish (<ref>), which is proved in a similar fashion to (<ref>) of Lemma <ref> – by applying the generator G_X̃ to the Lyapunov function V(x) = x^2. The following are some useful intermediate steps for any reader wishing to produce a complete proof. The first step to prove (<ref>) is to get an analogue of (<ref>). Namely, when x ≤ -ζ, G_X̃ V(x) =-2μ x^2 + μδ x + 2μ≤ -3/2μ x^2 + μδ^2/2 + 2μ ,and when x ≥ -ζ,G_X̃ V(x) =-2α (x+ζ)^2 + αδ(x+ζ)-2μζ(x+ζ) - 2ζα(x+ζ) + μζ(δ - 2ζ) + 2μ ≤-3/2α (x+ζ)^2-2μζ(x+ζ) + δ^2α/2 + δ^2μ/8 + 2μ.From here, we use Lemma <ref> to get a statement similar to (<ref>), from which we can infer (<ref>) and by applying Jensen's inequality to (<ref>), we get (<ref>). Observe that this procedure yields (<ref>), (<ref>), and (<ref>) as well. We now describe how to prove (<ref>), which requires only a slight modification of (<ref>). Namely, for x ≥ -ζ,G_X̃ V(x) =2x(-α (x +ζ) + μζ) - δ(-α (x +ζ) + μζ) + 2μ.From this, we can deduce that since x ≥ -ζ, G_X̃ V(x) ≤ -2(μ∧α) x^2 - δ(-α (x +ζ) + μζ) + 2μ,and alsoG_X̃ V(x) ≤ -2μζ x - δ(-α (x +ζ) + μζ) + 2μ.Then Lemma <ref> can be applied as before to see that both 2μζ[ |X̃(∞)1(X̃(∞)≥ -ζ)|]and 2(μ∧α)[ (X̃(∞))^2 1(X̃(∞)≥ -ζ)]are bounded by2μ + μδ^2/2 - δ[ (-α (X̃(∞) +ζ) + μζ)1(X̃(∞) ≥ -ζ)].Applying the generator G_X̃ to the test function f(x) = x and taking expected values with respect to X̃(∞), we get b(X̃(∞)) = 0, or[ (-α (X̃(∞) +ζ) + μζ)1(X̃(∞) ≥ -ζ)] = μ[X̃(∞) 1(X̃(∞) < -ζ)].When combined with (<ref>), this implies that 2μ + μδ^2/2 - δ[ (-α (X̃(∞) +ζ) + μζ)1(X̃(∞) ≥ -ζ)]≤2μ + μδ^2/2 +μδ√(1/3(α/μδ^2 + δ^2 + 4 )),which proves (<ref>), because the quantity above is an upper bound for (<ref>). To prove (<ref>), we manipulate (<ref>) to get [| (X̃(∞) + ζ) 1(X̃(∞) ≤ -ζ)|]= ζ + α/μ[| (X̃(∞) + ζ)1(X̃(∞) > -ζ)|],to which we can apply the triangle inequality and (<ref>) to conclude (<ref>). Lastly, the proof of (<ref>) is nearly identical to the proof of (<ref>) in Lemma <ref>. The key step is to obtain an analogue of (<ref>). §.§.§ Proof Outline for Lemma <ref>: The Overloaded SystemThe proof of the overloaded case of Lemma <ref> is also similar to that of Lemma <ref>. Therefore, we only outline some key intermediate steps needed to obtain the results; the bounds in this lemma are not proved in the order in which they are stated. We remind the reader that when R ≥ n, then ζ≥ 0. We start by proving (<ref>). Although the left hand side of (<ref>) is slightly different from (<ref>) of Lemma <ref>, it is proved using the same approach – by applying the generator G_X̃ to the Lyapunov function V(x) = x^2. The following are some useful intermediate steps for any reader wishing to produce a complete proof. The first step to prove (<ref>) is to get analogue of (<ref>). Namely, whenx ≤ -ζ,G_X̃ V(x) =-2μ (x+ζ)^2 + μδ(x+ζ) + 2(μ +α)ζ(x+ζ) - 2αζ^2 - αδζ + 2μ ≤-2μ (x+ζ)^2+ 2(μ +α)ζ(x+ζ)+ 2μ, and when x ≥ -ζ, G_X̃ V(x) =-2α x^2 + αδ x + 2μ≤ -3/2α x^2 + αδ^2/2 + 2μ.From here, we use Lemma <ref> to get a statement similar to (<ref>), which implies(<ref>). Applying Jensen's inequality to (<ref>) yields (<ref>). The procedure used to get (<ref>) also yields (<ref>), (<ref>), and (<ref>). We now describe how to prove (<ref>) and (<ref>), which requires only a slight modification of (<ref>). Namely, we use the fact that for x ≤ -ζ,G_X̃ V(x) =2x(-μ (x +ζ) + αζ) - δ(-μ (x +ζ) + αζ) + 2μ.From this, one can deduce that since x ≤ -ζ, G_X̃ V(x) ≤ -2(μ∧α) x^2+ 2μ,and alsoG_X̃ V(x) ≤ -2αζx+ 2μ.Then Lemma <ref> and Jensen's inequality can be applied as before to get both (<ref>) and (<ref>).We now prove (<ref>). Observe that [| X̃(∞) 1(X̃(∞) ≥ -ζ) | ] =[| (X̃(∞)+ζ - ζ )1(X̃(∞) ≥ -ζ) | ]≥[| (X̃(∞)+ζ)1(X̃(∞) > -ζ) | - ζ 1(X̃(∞) > -ζ) ]≥[| (X̃(∞)+ζ)1(X̃(∞) > -ζ) | ] - ζ=μ/α[| (X̃(∞) + ζ) 1(X̃(∞) ≤ -ζ)|],where the last equality comes from applying the generator G_X̃ to the function f(x) = x and taking expected values with respect to X̃(∞) to see that b(X̃(∞)) = 0, or[ (-μ (X̃(∞) +ζ) + αζ)1(X̃(∞) ≤ -ζ)] = α[X̃(∞) 1(X̃(∞) > -ζ)].Therefore, [| (X̃(∞) + ζ) 1(X̃(∞) ≤ -ζ)|]≤α/μ[| X̃(∞) 1(X̃(∞) ≥ -ζ) | ],and we can invoke(<ref>)to conclude (<ref>).We now prove (<ref>), which requires additional arguments that we have not used in the proof of Lemma <ref>. We assume for now that λ≤ nμ + 1/2√(n)μ. Fix γ∈ (0, 1/2), and defineJ_1=∑_k=0^⌊ n-γ√(R)⌋π_k,J_2=∑^n_k=⌈ n-γ√(R)⌉π_k,where {π_k}_k=0^∞ is the distribution of X(∞). We note that by (<ref>), n/√(R)≥√(R)-1/2√(n/R)≥√(R)-1/2 ≥ 1/2,which implies that n-γ√(R) > 0. Then (X̃(∞) ≤ -ζ) = (X(∞) ≤ n) ≤ J_1 + J_2. To bound J_1 we observe that [|X(∞)+ζ|1_{X(∞)≤-ζ}] = 1/√(R)∑_k=0^n (n-k) π_k ≥γ∑_k=0^⌊ n-γ√(R)⌋π_k = γ J_1.Combining (<ref>)–(<ref>), we conclude thatJ_1 ≤1/γ2/√(3)(δ^2/4+1)(1/ζ∧√(α/μ∨ 1)∧α/μ√(μ/α∨ 1)) ≤1/γ2/√(3)(δ^2/4+1)(1/ζ∧√(α/μ)).Now to bound J_2, we apply G_X̃ to the test function f(x) = k∧ n, where x=δ (k-x(∞)), and take the expectation with respect to X̃(∞) to see that 0 = -λπ_n+(λ - nμ )(X(∞)≤ n)+μ[(X(∞)-n)^-1_{X(∞)≤ n}].Noticing that [(X(∞)-n)^-1_{X(∞)≤ n}] = 1/δ[|X(∞)+ζ|1_{X(∞)≤-ζ}],we arrive atπ_n ≤δ2/√(3)(δ^2/4+1) (1/ζ∧√(α/μ))+λ -nμ/λ(X(∞)≤ n).The flow balance equationsλπ_k-1 = kμπ_k, k=1,2,⋯,nimply that π_0<π_1<⋯<π_n-2<π_n-1≤π_n, and thereforeJ_2 ≤ (γ√(R) + 1)π_n≤(γ√(R) + 1)[δ2/√(3)(δ^2/4+1) (1/ζ∧√(α/μ))+λ -nμ/λ(X(∞)≤ n)] =(γ +δ)2/√(3)(δ^2/4+1) (1/ζ∧√(α/μ))+(γ√(R) + 1)λ -nμ/λJ_1 +(γ√(R) + 1)λ -nμ/λJ_2We use (<ref>), the fact that γ∈ (0,1/2), and that R ≥ n ≥ 1 to see that(γ√(R) + 1)λ -nμ/λ≤ (γ√(R) + 1)√(n)/2R≤1/2(γ + 1/√(R)) = 1/2(γ + 1)< 3/4. Then by rearranging terms in (<ref>) and applying (<ref>) we conclude that1/4 J_2 ≤ (γ +δ)2/√(3)(δ^2/4+1) (1/ζ∧√(α/μ))+3/41/γ2/√(3)(δ^2/4+1)(1/ζ∧√(α/μ)) =(γ +δ+3/41/γ) 2/√(3)(δ^2/4+1) (1/ζ∧√(α/μ)).Hence, we have just shown that under assumption (<ref>), (X̃(∞)≤ -ζ) ≤ J_1 + J_2 ≤1/γ2/√(3)(δ^2/4+1)(1/ζ∧√(α/μ)) + 4(γ +δ+3/41/γ) 2/√(3)(δ^2/4+1) (1/ζ∧√(α/μ))≤(3+δ)8/√(3)(δ^2/4+1) (1/ζ∧√(α/μ)),where to get the last inequality we fixed γ∈ (0,1/2) that solves γ + 1/γ = 3. We now wish to establish the same result without assumption (<ref>), i.e. when λ > nμ +1/2√(n)μ. For this, we rely on the following comparison result. Fix n,μ and α and let X^(λ)(∞) be the steady-state customer count in an Erlang-A system with arrival rate λ, service rate μ, number of servers n, and abandonment rate α. Then for any 0 < λ_1 < λ_2, (X^(λ_2)(∞) ≤ n) ≤(X^(λ_1)(∞) ≤ n).This says that with all other parameters being held fixed, an Erlang-A system with a higher arrival rate is less likely to have idle servers. For a simple proof involving a coupling argument, see page 163 of <cit.>. Therefore, for λ > nμ +1/2√(n)μ,(X^(λ)(∞)≤ n) ≤(X^(nμ +1/2√(n)μ )(∞)≤ n)≤(3+δ)8/√(3)(δ^2/4+1) (1/ζ^(nμ +1/2√(n)μ )∧√(α/μ))where ζ^(nμ +1/2√(n)μ ) is the ζ corresponding to X^(nμ +1/2√(n)μ )(∞), and satisfies 1/ζ^(nμ +1/2√(n)μ ) = 2α/μ√(n + √(n)/2/n)≤2α/μ√(3/2).This concludes the proof of (<ref>).§ CHAPTER <REF> MOMENT BOUNDS In this section we prove Lemmas <ref> and <ref>. To do so, we rely on the moment bounds in Lemma <ref> (from Section <ref>). However, the bounds from that lemma are not sufficient, and the following additional bounds are needed.For all n ≥ 1, λ > 0, and μ > 0 satisfying 0 < R < n,[(X̃(∞))^2 1(X̃(∞) ≤ -ζ) ] ≤( 5 + δ (1+δ/2) ) ζ^2 + (2+δ)ζ [(X̃(∞))^2 1(X̃(∞) ≥ -ζ) ] ≤δ^2 +8 + 4/ζ( 1/ζ + δ^2/4ζ + δ/2) +2(2δ + δ^3)/3ζ.We first prove (<ref>), or [(X̃(∞))^2 1(X̃(∞) ≤ -ζ) ] ≤( 5 + δ (1+δ/2) ) ζ^2 + (2+δ)ζ.Let x be of the form x = δ(k - R), where k ∈_+. Applying G_X̃ to the function f(x) = [ δ(k-n)^-]^2 =[ (x+ζ)^-]^2, and observing that 1(k ≤ n) = 1(x ≤ -ζ),we getG_X̃ f(x) =λ 1(x ≤ -ζ - δ)( 2δ(x + ζ) + δ^2) + μ (k ∧ n)1(x ≤ -ζ)( -2δ(x + ζ)+ δ^2) =λ 1(x ≤ -ζ)( 2δ(x + ζ) + δ^2) + μ k 1(x ≤ -ζ)( -2δ(x + ζ)+ δ^2) - δ^2λ 1(x = -ζ) =1(x ≤ -ζ)( 2δ(x + ζ)(λ - μ k) + δ^2(λ + μ k)) - δ^2λ 1(x = -ζ)=1(x ≤ -ζ)( 2δ(x + ζ)(λ -μ n) + 2μδ(x + ζ)(n-k) + δ^2(λ +μ k))- δ^2λ 1(x = -ζ) =1(x ≤ -ζ)( 2μ (x + ζ)ζ - 2μ (x + ζ)^2 + δ^2(λ + μ k))- δ^2λ 1(x = -ζ).Taking expected values on both sides and applying Lemma <ref>, we see that [(X̃(∞)+ζ)^2 1(X̃(∞) ≤ -ζ) ] ≤ζ[(X̃(∞)+ζ) 1(X̃(∞) ≤ -ζ) ] +δ^2/2μ(X̃(∞) ≤ -ζ)( λ +μ n)=ζ[(X̃(∞)+ζ) 1(X̃(∞) ≤ -ζ) ] + 1/2(X̃(∞) ≤ -ζ)(1 + δ^2 n). Recall (<ref>), which tells us that [(X̃(∞)+ζ) 1(X̃(∞) ≤ -ζ) ] = ζ, to see that[(X̃(∞)+ζ)^2 1(X̃(∞) ≤ -ζ) ] ≤ζ^2 + 1/2(X̃(∞) ≤ -ζ)(1 + δ^2 n) ≤ζ^2 + 2+δ/2ζ(1 + δ^2 n),where we used (<ref>) to get the last inequality. Since ζ = δ (n - λ/μ),δ^2 n=δ^2 ζ/δ + δ^2 λ/μ = δζ + 1,and hence,[(X̃(∞) + ζ)^2 1(X̃(∞) ≤ -ζ) ] ≤ζ^2 + 2+δ/2ζ(2 + δζ).By expanding the square inside the expected value on the left hand side and using (<ref>), we see that[(X̃(∞))^2 1(X̃(∞) ≤ -ζ) ]≤ζ^2 + 2+δ/2ζ(2 + δζ) + 2ζ[ | X̃(∞) 1(X̃(∞) ≤ -ζ)| ] ≤5ζ^2 + 2+δ/2ζ(2 + δζ) =( 5 + δ (1+δ/2) ) ζ^2 + (2+δ)ζ.This proves (<ref>). Now we prove (<ref>), or [(X̃(∞))^2 1(X̃(∞) ≥ -ζ) ] ≤δ^2 +8 + 4/ζ( 1/ζ + δ^2/4ζ + δ/2) +2(2δ + δ^3)/3ζ.Let x be of the form x = δ(k - R), where k ∈_+.Recall from (<ref>) that b(x) = μ[ζ + (x+ζ)^- ] = δ (λ - μ(k ∧ n)).Set a = δ(⌊ R ⌋ - R) < 0, and consider the function f(x) = x^3 1( x ≥ a+δ). Then G_X̃ f(x) =λ 1( x ≥ a+δ) ((x+δ)^3 - x^3 ) +λ 1( x = a) (x+δ)^3+ μ (k ∧ n)1( x > a+δ) ((x-δ)^3 - x^3) + μ (k ∧ n)1( x = a+δ) ( - x^3) =1( x ≥ a+δ) [ λ((x+δ)^3 - x^3 ) + μ (k ∧ n) ((x-δ)^3 - x^3)] + λ 1( x = a) (x+δ)^3 - μ (k ∧ n)1( x = a+δ) (x-δ)^3.Suppose x ≥ a+δ. Using the fact that 1( x = a+δ) = 1( k = ⌊ R ⌋ + 1), we see thatG_X̃ f(x) =λ (3δ x^2 + 3δ^2 x + δ^3) + μ (k ∧ n) (-3δ x^2 + 3δ^2 x - δ^3) - a^3 μ (k ∧ n)1( x = a+δ) =3δ x^2(λ - μ(k ∧ n)) + 3δ^2 x(λ + μ(k ∧ n)) + δ^3(λ - μ(k ∧ n)) - a^3 μ((⌊ R ⌋ + 1) ∧ n) 1( x = a+δ) =3x^2b(x) + 3δ^2 x(2λ - (λ - μ(k ∧ n))) + δ^2b(x) - a^3 μ((⌊ R ⌋ + 1) ∧ n) 1( x = a+δ) =3x^2b(x) + 6μ x - 3δ xb(x) + δ^2b(x)- a^3 μ((⌊ R ⌋ + 1) ∧ n) 1( x = a+δ).When x ∈ [a+δ,-ζ) (which is the empty interval if ⌊ R ⌋ + 1 = n), then b(x) = -μ x, andG_X̃ f(x) =-3μ x^3 + 6μ x + 3δμ x^2 - δ^2μ x - a^3 μ((⌊ R ⌋ + 1) ∧ n) 1( x = a+δ)≤-3μ(x^3 -δ x^2 + 1/3δ^2 x ) + 6μ x + δ^3 μ (⌊ R ⌋ + 1)≤-3μ(x^3 -δ x^2 + 1/3δ^2 x ) + 6μ x + δμ+ δ^3 μ ≤6μ x+ δμ+ δ^3 μ,where in the first inequality we used the fact that a≤δ, and in the last inequality we used the fact that g(x) := x^3 -δ x^2 + 1/3δ^2 x ≥ 0 for all x ≥ 0, which is true because g(0) = 0 and g'(x) ≥ 0 for all x ∈. Now when x ≥ -ζ, then b(x) = -μζ, and using (<ref>) we see thatG_X̃ f(x) =-3x^2μζ + 6μ x + 3δ xμζ - δ^2μζ - a^3 μ((⌊ R ⌋ + 1) ∧ n) 1( x = a+δ)≤-3μζ(x^2-δ x) + 6μ x - a^3 μ((⌊ R ⌋ + 1) ∧ n) 1( x = a+δ) ≤-3μζ(x^2-δ x) + 6μ x + δμ+ δ^3 μ ≤-3μζ(1/2x^2-1/2δ^2) + 6μ x + δμ+ δ^3 μ ,Combining (<ref>) and (<ref>) with (<ref>), we have just shown that G_X̃ f(x) ≤-3/2μζ(x^2-δ^2)1(x ≥ -ζ)+ 6μ x 1( x ≥ a+δ) + δμ+ δ^3 μ + λ 1( x = a) (x+δ)^3.Taking expected values on both sides above, and applying Lemma <ref>, we see that 3/2μζ[ (X̃(∞))^2 1(X̃(∞) ≥ -ζ) ]≤3/2μζδ^2 + 6μ[| X̃(∞) | ]+ δμ+ δ^3 μ + λ (a+δ)^3,and since λ (a+δ)^3 ≤λδ^3 = μδ, we have[ (X̃(∞))^2 1(X̃(∞) ≥ -ζ) ] ≤δ^2 + 4/ζ[| X̃(∞) | ] +2(2δ + δ^3)/3ζ.Using the moment bounds in (<ref>) and (<ref>), we conclude that [ (X̃(∞))^2 1(X̃(∞) ≥ -ζ) ] ≤δ^2 +8 + 4/ζ( 1/ζ + δ^2/4ζ + δ/2) +2(2δ + δ^3)/3ζ,which proves (<ref>).We now prove Lemma <ref>, followed by a proof of Lemma <ref>. Observe that (<ref>) is identical to (<ref>). Now assume that δ≤ 1. We begin by proving (<ref>). Using the moment bounds in (<ref>) and (<ref>), we see that(1 + 1/ζ)[|X̃(∞)1(X̃(∞) ≤ -ζ) |]≤(1 + 1/ζ)(2ζ∧√(4/3 + 2δ^2/3))≤(√(4/3 + 2δ^2/3) + 2) ≤(√(2) + 2).Next we prove (<ref>). Using the moment bounds in (<ref>) and (<ref>), we see that(1 + 1/ζ)[(X̃(∞))^21(X̃(∞) ≤ -ζ) ] ≤(1 + 1/ζ)((( 5 + δ (1+δ/2) ) ζ^2 + (2+δ)ζ)∧(4/3 + 2δ^2/3))≤2 + ((6.5ζ +3) ∧2/ζ) ≤ 2 + 7,where to get the last inequality we considered separately the cases when ζ≤ 1/2 and ζ≥ 1/2. To prove (<ref>), we use the moment bound (<ref>) to getζ( X̃(∞) ≥ -ζ) ≤ζ∧[| X̃(∞) 1(X̃(∞) ≥ -ζ)| ] ≤ζ∧(1/ζ + δ^2/4ζ + δ/2)≤2,where to get the last inequality we considered separately the cases where ζ≤ 1 and ζ≥ 1. The proof of (<ref>) is similar. We use the moment bound (<ref>) to see that ζ^2 ( X̃(∞) ≥ -ζ) ≤ζ^2 ∧[( X̃(∞))^2 1(X̃(∞) ≥ -ζ) ] ≤ζ^2 ∧(δ^2 +8 + 4/ζ( 1/ζ + δ^2/4ζ + δ/2) +2(2δ + δ^3)/3ζ) ≤ζ^2 ∧(1 +8 + 4/ζ( 1/ζ + 1/4ζ + 1/2) +2/ζ) ≤20,where to get the last inequality we considered separately the cases where ζ≤ 1 and ζ≥ 1.This concludes the proof of Lemma <ref>.We first prove (<ref>). From (<ref>), we know that (X̃(∞) ≤ -ζ) = (X(∞) ≤ n) = ∑_k=0^nπ_k≤ (2+δ)ζ.From the flow balance equations, one can see that π_⌊ R ⌋ maximizes {π_k}_k=0^∞. Now whenζ≤ 1,ζ = δ (n - R) = 1/√(R)(n - R) ≤ 1,which implies that R ≥ n - √(R)≥ n - √(n),where in the last inequality we used R < n. We use this inequality together with the fact that π_0 ≤π_1 ≤…≤π_⌊ R ⌋, which can be verified from the flow balance equations, to see that(2+δ)ζ≥∑_k=0^nπ_k ≥∑_k=0^⌊ R ⌋π_k ≥π_0⌊ R ⌋≥π_0⌊ n-√(n)⌋.Hence, for n ≥ 4, π_0 ≤n/⌊ n-√(n)⌋(2+δ)ζ/n≤n/n-√(n) - 1(2+δ)ζ/R≤4(2+δ)ζ/R = 4(2+δ)δ^2 ζ.To conclude the proof of (<ref>) we need to verify the bound above holds for n < 4, but this is simple to do. Observe that for n < 4, π_0 ≤ P(X(∞) ≤ n) ≤ (2+δ)ζ= (2+δ)Rδ^2ζ≤ (2+δ)n δ^2 ζ≤ 4(2+δ)δ^2ζ.This proves (<ref>), and we move on to prove (<ref>). From the flow balance equations corresponding to the CTMC X, it is easy to see that π_n = R^n/n!/∑_k=0^n-1R^k/k! + R^n/n!1/1-R/n≤ 1 - R/n = n-R/n = R/nn-R/R = R/nδζ≤δζ.This concludes the proof of the lemma. § CHAPTER <REF> MOMENT BOUNDS This section uses notation from Chapter <ref>.We first provide an intuitive roadmap for the proof. The goal is to show that a Lyapunov function for the diffusion process is also a Lyapunov function for the CTMC; this has two parts to it. In the first part of this proof, we compare how the two generators G_U^(λ) and G_Y act on this Lyapunov function, obtaining an upper bound for the difference G_U^(λ) - G_Y in (<ref>). One notesthat the right hand side of (<ref>) is unbounded. This is due to the difference in dimensions of the CTMC and diffusion process. To overcome this difficulty, we move on to the second part of the proof, which exploits our SSC result in Lemma <ref> to bound the expectation of the right hand side of (<ref>). We end up with a recursive relationship that guarantees the 2mth moment is bounded (uniformly in λ and n satisfying (<ref>)) provided that the mth moment is. Finally, we rely on prior results obtained in <cit.> for a uniform bound on the first moment.We remark that a version of this lemma was already proved <cit.> for the case where the dimension of the CTMC equals the dimension of the diffusion process. However, the difference in dimensions poses an additional technical challenge, which is overcome in the second part of this proof.Its enough to prove (<ref>) for the cases when m=2^j for some j≥ 0. Furthermore, we may assume that λ≥ 4 because by Remark <ref>, there are only finitely many cases when λ < 4. In all those cases, X̃^(λ)(∞)^m < ∞ by (<ref>). Throughout the proof, we shall use C,C_1,C_2,C_3,C_4 to denote generic positive constants that may change from line to line. They may depend on (m,β,α,p,ν,P), but will be independent of both λ and n. DefineV_m(x) = (1+V(x))^m,where V is as in (<ref>). By <cit.>, V_m also satisfiesG_Y V_m(x) ≤-C_1 V_m(x) + C_2as long as V ∈ C^3(^d) and satisfies condition (30) of <cit.>, which is easy to verify. To prove the lemma, we will show that for large enough λ, V satisfies G_U^(λ) AV_m(U^(λ)(∞)) ≤- C_1 V_m(X̃^(λ)(∞)) + C_2,where A is the lifting operator defined in (<ref>). We begin by observingG_U^(λ) AV_m ≤ G_U^(λ)AV_m - G_Y V_m + G_Y V_m ≤ G_U^(λ)AV_m - G_Y V_m -C_1 V_m + C_2.Using (<ref>), we write G_U^(λ)AV_m - G_Y V_m as∑_i=1^d ∂_i V_m(x)[(ν_i - α-∑_j=1^d P_jiν_j)( δ q_i - p_i(e^T x)^+)]+ ∑_i=1^d ∂_iiV_m(x)[∑_j=1^d P_jiν_jγ_j ](n δ^2 - 1) - ∑_i ≠ j^d ∂_ijV_m(x)[P_ijν_iγ_i + P_jiν_jγ_j](nδ^2 - 1)- ∑_i=1^d δ^2/2∂_iiV_m(x)[p_i (λ - n)- αq_i - ν_i (z_i- γ_i n) - ∑_j=1^d P_jiν_j (z_j- γ_j n)] - ∑_i ≠ j^d δ^2/2∂_ijV_m(x)[P_ijν_i (z_i- γ_i n) + P_jiν_j (z_j- γ_j n)]+ ∑_i=1^d δ^2/2 (∂_iiV_m(ξ_i^-)-∂_iiV_m(x))[α q_i + (1-∑_j=1^d P_ij)ν_i z_i] + ∑_i=1^d δ^2/2 (∂_iiV_m(ξ_i^+)-∂_iiV_m(x))[λ p_i ]- ∑_i ≠ j^d δ^2 (∂_ijV_m(ξ_ij)-∂_ijV_m(x))[P_ijν_i z_i] + ∑_i=1^d ∑_j=1^d δ^2/2 (∂_iiV_m(ξ_ij)-∂_iiV_m(x))[ P_ijν_i z_i +P_jiν_j z_j] .Now we wish to bound the derivatives of V_m. By <cit.>, V_m satisfies (16) and (30) of <cit.>, namely sup_y≤ 1V_m(x+y)/V_m(x)≤ Cand(∂_i V_m(x) + ∂_ij V_m(x) +∂_ijk V_m(x))(1+x) ≤ C V_m(x).For ξ being one of ξ_i^+, ξ_i^- or ξ_ij,∂_ijV_m(ξ) - ∂_ijV_m(x)(1+x) ≤δ∂_iji V_m(η) +∂_ijjV_m(η) (1+x) ≤ C δ V_m(x),where the first inequality comes from a Taylor expansion and the second inequality follows by (<ref>), the fact that η - x≤ 2δ < 1 and by (<ref>). Following the exact same argument that we used to bound (<ref>) in the proof of Lemma <ref> (with (<ref>) and (<ref>) replacing the gradient bounds of f_h there), we get G_U^(λ)AV_m - G_Y V_m ≤C δV_m(x) +C∑_i=1^d ∂_i V_m(x)[ q_i - p_i(e^T x)^+].Differentiating V, we see that(∇V(x))^T = 2(e^T x)e^T + 2 κ(x^T - p^T ϕ(e^T x)) Q̃ (I - p e^T ϕ'(e^Tx)). Combined with the fact that 0 ≤ϕ'(x)≤ 1, it is clear that∂_i V(x) ≤C(1+x).Therefore,G_U^(λ)AV_m - G_Y V_m ≤C δ V_m(x) +C∑_i=1^d mV_m-1(x)(1+x)[ q_i - p_i(e^T x)^+] .It remains to find an appropriate bound for V_m-1(x) (1+x)[ q_i - p_i(e^T x)^+] = δV_m-1(x)(1+x)[q_i - p_i(e^T x)^+/δ].We haveδ V_m-1(x) (1+x)[q_i - p_i(e^T x)^+/δ]≤ √(δ)V_m-1(x)(1+x)^2 + √(δ)V_m-1(x)[q_i - p_i(e^T x)^+^2/δ]≤C√(δ)V_m(x) + √(δ)V_m-2(x)V_2(x)+ √(δ)V_m-2(x)[q_i - p_i(e^T x)^+^2/δ]^2 ≤C√(δ)V_m(x) + √(δ)V_m(x)+ √(δ)V_m-4(x)V_4(x)+ √(δ) V_m-4(x)[q_i - p_i(e^T x)^+^2/δ]^4 ≤ …≤C√(δ)V_m(x) + √(δ)[q_i - p_i(e^T x)^+^2/δ]^m,where in the last inequality, we used the fact that m = 2^j. Using (<ref>), (<ref>) and (<ref>), G_U^(λ) AV_m(u) ≤- V_m(x)(C_1 - √(δ)C_3) + C_2 + √(δ)C_4∑_i=1^d [q_i - p_i(e^T x)^+^2/δ]^m,where x and q are related to u by (<ref>). The arguments in the proof of Lemma <ref> can be used to show G_U^(λ) A V_m(U^(λ)(∞)) = 0. Therefore, for δ small enough, EX̃^(λ)(∞)^2m≤C V_m(X̃^(λ)(∞))≤ C/(C_1 - √(δ)C_3)(C_2 + √(δ)C_4∑_i=1^d δ Q^(λ)_i(∞) - p_i(e^T X̃^(λ)(∞))^+^2m/δ^m).By (<ref>), it follows that X̃^(λ)(∞)^2m ≤C/C_1 - √(δ)C_3 (1 + √(δ)[(e^T X̃^(λ)(∞))^+]^m).Hence, we have a recursive relationship that guarantees sup_λ> 0 X̃^(λ)(∞)^2m < ∞ whenever sup_λ> 0 [(e^TX̃^(λ)(∞))^+]^m < ∞.To conclude, we need to verify thatsup_λ> 0 [(e^TX̃^(λ)(∞))^+] < ∞,but this was proved in equation (5.2) of <cit.>. CHAPTER: GRADIENT BOUNDS This appendix proves all of the gradient bounds used in this document. Section <ref> provides some generic tools for establishing gradient bounds in the case of a one-dimensional diffusion approximation, i.e. when the Poisson equation is an ordinary differential equation. Bounds for Chapters <ref>, <ref> and <ref> are proved in Sections <ref>, <ref>, and <ref>, respectively. § THE POISSON EQUATION FOR DIFFUSION PROCESSES To make this section self-contained, we begin by repeating Lemma <ref>. Let a̅:→_+ and b̅: → be continuous functions, and assume that inf_x ∈a̅(x) > 0. Assume that ∫_-∞^∞2/a̅(x)exp(∫_0^x2 b̅(u)/a̅(u) du)dx < ∞,and let V be a continuous random variable with density 2/a̅(x)exp(∫_0^x2 b̅(u)/a̅(u) du)/∫_-∞^∞2/a̅(x)exp(∫_0^x2 b̅(u)/a̅(u) du)dx,x ∈.Fix h:→ satisfying |h(V)| < ∞, and consider the Poisson equation 1/2a̅(x) f_h”(x) + b̅(x) f_h'(x) =h(V) - h(x),x ∈.There exists a solution f_h(x) to this equation satisfying f_h'(x) =e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) ( h(V) - h(y)) e^∫_0^y2b̅(u)/a̅(u)du dy=-e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) ( h(V) - h(y)) e^∫_0^y2b̅(u)/a̅(u)du dy , f_h”(x) = - 2b̅(x)/a̅(x) f_h'(x) +2/a̅(x)( h(V) - h(x) ). Provided h(x), a̅(x), and b̅(x)/a̅(x) are sufficiently differentiable, f_h(x) can have more than two derivatives. For example, f_h”'(x) =-(2b̅(x)/a̅(x))' f_h'(x) - 2b̅(x)/a̅(x) f_h”(x) - 2/a̅(x)h'(x) - 2a̅'(x)/a^2(x)(h(V) - h(x)).The biggest source of difficulty in bounding f_h'(x), f_h”(x), and f_h”'(x), are the integrals in (<ref>) and (<ref>). Before describing how to bound them, we give an alternative representation of f_h”(x), which is taken from the proof of <cit.>. The assumptions of the following lemma areonly slightly stronger than those in Lemma <ref>. Fix h:→ satisfying |h(V)| < ∞, and let f_h'(x) be as in (<ref>)–(<ref>). Assume a̅(x), b̅(x)/a̅(x), and h(x) are absolutely continuous. Iflim_x → -∞2b̅(x)/a̅(x) f_h'(x)e^∫_0^x2b̅(u)/a̅(u)du = 0,then f_h”(x) =e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x(2/a̅(y) h'(y) - 2a̅'(y)/a^2(y)( h(V) - h(y) )- (2b̅(y)/a̅(y))' f_h'(y))e^∫_0^y2b̅(u)/a̅(u)du dy.Similarly, if lim_x →∞2b̅(x)/a̅(x) f_h'(x)e^∫_0^x2b̅(u)/a̅(u)du = 0,then f_h”(x) = -e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞(2/a̅(y) h'(y) - 2a̅'(y)/a^2(y)( h(V) - h(y) )- (2b̅(y)/a̅(y))' f_h'(y))e^∫_0^y2b̅(u)/a̅(u)du dy. This lemma is proved at the end of this section.In practice, working with the representation in Lemma <ref> often yields better bounds on f_h”(x) than using (<ref>). Again, we see that both (<ref>) and (<ref>) contain integral term involving e^∫_0^y2b̅(u)/a̅(u)du. Tohelp bound these integrals, we make several assumptions on b̅(x). Assume that(a1)b̅(x) is a non-increasing function of x (a2)b̅(x) has at most one zero, and define x_0 = -∞, b̅(x) < 0, x ∈∞, b̅(x) > 0, x ∈ the zero of b̅(x),otherwise.It may be helpful to the reader to pretend that x_0 = 0, which will always be the case in this thesis. The following two lemmas present some useful inequalities that will be very helpful in getting the gradient bounds that we require. Due to their generality, they may also be of independent interest. We discuss assumptions ([eq:a1]a1) and ([eq:a2]a2) after their statements and proofs.Suppose b̅(x) satisfies both([eq:a1]a1) and ([eq:a2]a2), and let x_0 be as in (<ref>). Then e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤1/b̅(x),x < x_0, e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤1/b̅(x),x > x_0,where we adopt the convention that {x : x> ∞} = {x : x < -∞} = ∅. Furthermore, if x_0 is finite, then for any c_1 ∈ (-∞, x_0) and c_2 ∈ (x_0, ∞), e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤1/b̅(c_1)+2c_1 - x_0sup_y ∈ [c_1,x_0]1/a̅(y) ,x < x_0, e^-∫_x_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_x_0^y2b̅(u)/a̅(u)du dy ≤1/b̅(c_2) +2c_2-x_0sup_y ∈ [x_0,c_2]1/a̅(y),x > x_0.Suppose x_0 is finite. Since b̅(x_0) = 0, the bounds in (<ref>) and (<ref>) lose relevance for x near x_0. This is the reason for having (<ref>) and (<ref>).We first prove (<ref>). The assumption that b̅(x) is decreasing implies that b̅(y)/b̅(x) ≥ 1 for y ≤ x < x_0. Therefore,e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y)e^∫_0^y2b̅(u)/a̅(u)du dy = e^-∫_x_0^x2b̅(u)/a̅(u)du∫_-∞^x2 /a̅(y) e^∫_x_0^y2b̅(u)/a̅(u)du dy≤ e^-∫_x_0^x2b̅(u)/a̅(u)du∫_-∞^x2 b̅(y)/a̅(y)1/b̅(x) e^∫_x_0^y2b̅(u)/a̅(u)du dy = e^-∫_x_0^x2b̅(u)/a̅(u)du1/b̅(x)(e^∫_x_0^x2b̅(u)/a̅(u)du - e^-∫_x_0^-∞2b̅(u)/a̅(u)du) ≤1/b̅(x).One can justify (<ref>) using a symmetric argument. We now prove (<ref>). Fix c_1 < x_0 and suppose x ≤ c_1. Then (<ref>), together with the fact that b̅(x) ≥b̅(c_1) implies e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y)e^∫_0^y2b̅(u)/a̅(u)du dy ≤1/b̅(c_1).Now when x ∈ [c_1,x_0],e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y)e^∫_0^y2b̅(u)/a̅(u)du dy =e^∫_x_0^c_12b̅(u)/a̅(u)du/e^∫_x_0^x2b̅(u)/a̅(u)du e^-∫_x_0^c_12b̅(u)/a̅(u)du∫_-∞^c_12/a̅(y)e^∫_x_0^y2b̅(u)/a̅(u)du dy + e^-∫_x_0^x2b̅(u)/a̅(u)du∫_c_1^x2/a̅(y)e^∫_x_0^y2b̅(u)/a̅(u)du dy ≤e^-∫_c_1^x2b̅(u)/a̅(u)du1/b̅(c_1) +∫_c_1^x2/a̅(y)e^-∫_y^x2b̅(u)/a̅(u)du dy≤1/b̅(c_1) +sup_y ∈ [c_1,x_0]2c_1 - x_0/a̅(y),where in the last inequality we used the fact that b̅(u) ≥ 0 for u ∈ [c_1, x]. This proves (<ref>), and a symmetric argument can be used to prove (<ref>). The first inequality of (<ref>) was inspired by the well-known bound on the CDF of the normal distribution ∫_-∞^x e^-y^2/2 dy ≤∫_-∞^xy/x e^-y^2/2 dy= 1/x e^-x^2/2,x <0,which is used to prove gradient bounds for the normal distribution <cit.>. Suppose b̅(x) satisfies both([eq:a1]a1) and ([eq:a2]a2), and x_0, as defined in (<ref>), is finite. Suppose also that there exist ℓ≤ x_0 and r ≥ x_0 such that a̅(x) = a_ℓ for x < ℓ and a̅(x) = a_r for x > r. Then for any k ∈ℕ, e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2y^k/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤∑_j=0^kk!/(k-j)!(a_ℓ/2b̅(x))^j1/b̅(x)x^k-j,x < ℓ, e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2y^k/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤∑_j=0^kk!/(k-j)!(a_r/2b̅(x))^j1/b̅(x) x^k-j,x > r.Fix x < ℓ≤ x_0, thene^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2y^k/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy =∫_-∞^x2y^k/a̅(y) e^-∫_y^x2b̅(u)/a̅(u)du dy ≤∫_-∞^x2y^k/a̅(y) e^-b̅(x) ∫_y^x2/a̅(u)du dy=y^k/b̅(x) e^-b̅(x) ∫_y^x2/a̅(u)du|_y=-∞^x + 1/b̅(x)∫_-∞^xky^k-1 e^-b̅(x) ∫_y^x2/a̅(u)dudy.In the first inequality we used the fact that b̅(y) ≥ 0 for y ≤ x_0, and the last equality was obtained using integration by parts. At this point we invoke the assumption that a̅(x) = a_ℓ for x < ℓ to see that y^k/b̅(x) e^-b̅(x) ∫_y^x2/a̅(u)du|_y=-∞^x + 1/b̅(x)∫_-∞^xky^k-1 e^-b̅(x) ∫_y^x2/a̅(u)dudy =y^k/b̅(x) e^-2b̅(x)/a_ℓ (x-y)|_y=-∞^x + 1/b̅(x)∫_-∞^xky^k-1 e^-2b̅(x)/a_ℓ (x-y)dy =x^k/b̅(x) + 1/b̅(x)∫_-∞^xky^k-1 e^-2b̅(x)/a_ℓ (x-y)dy.Continuing to use integration by parts, we arrive at (<ref>). The case when x > r is handled symmetrically. In practice, the assumption that a̅(x) is constant for x < ℓ may be relaxed if we can establish some control over ∫_y^x2/a̅(u)du in order to bound the left hand side of (<ref>). Same goes for the case when x > r. Assumption ([eq:a2]a2) is made mostly for technical convenience. It is not hard to adapt the results above to the case when b̅(x) equals zero at more than one point. Assumption([eq:a1]a1) is quite reasonable when b̅(x) is the drift of a positive recurrent diffusion process on the real line. For the diffusion process to be positive recurrent, we expect its drift to be negative when the process is far to the right of zero, and to be positive when the diffusion is far to the left of zero; cf. the requirement in (<ref>). To further match this intuition, assumption([eq:a1]a1) can actually be weakened to say that b̅(x) is a non-increasing function outside some compact interval around zero, and the lemmas above could be modified accordingly to deal with this. One may compare assumption([eq:a1]a1), and its proposed relaxation, to the assumptions in <cit.>. We conclude this section with the proof of Lemma <ref>. Differentiating both sides of the Poisson equation (<ref>) yieldsf_h”'(x) = -(2b̅(x)/a̅(x))' f_h'(x) - 2b̅(x)/a̅(x) f_h”(x) - 2/a̅(x)h'(x) - 2a̅'(x)/a^2(x)(h(V) - h(x)).The derivative above exists almost everywherebecause we assumed that a̅(x), b̅(x)/a̅(x), and h(x) are all absolutely continuous. The latter assumption also implies that f_h”(x)e^∫_0^x2b̅(u)/a̅(u)du is absolutely continuous. Hence, for any x, ℓ∈,f_h”(x)e^∫_0^x2b̅(u)/a̅(u)du - f_h”(ℓ)e^∫_0^ℓ2b̅(u)/a̅(u)du=∫_ℓ^x2b̅(y)/a̅(y) f_h”(y)e^∫_0^y2b̅(u)/a̅(u)du + f_h”'(y)e^∫_0^y2b̅(u)/a̅(u)du dy =∫_ℓ^x(-2/a̅(y) h'(y) - 2a̅'(y)/a^2(y)[ h(V) - h(y) ] - (2b̅(y)/a̅(y))' f_h'(y))e^∫_0^y2b̅(u)/a̅(u)du dy.To conclude (<ref>),we wish to take ℓ→ -∞ and show that lim_ℓ→ -∞ f_h”(ℓ)e^∫_0^ℓ2b̅(u)/a̅(u)du = 0.Observe thatf_h”(ℓ)e^∫_0^ℓ2b̅(u)/a̅(u)du = 2b̅(ℓ)/a̅(ℓ)f_h'(ℓ)e^∫_0^ℓ2b̅(u)/a̅(u)du + ( h(V) - h(ℓ))2/a̅(ℓ)e^∫_0^ℓ2b̅(u)/a̅(u)du.By assumption, lim_ℓ→ -∞2b̅(ℓ)/a̅(ℓ)f_h'(ℓ)e^∫_0^ℓ2b̅(u)/a̅(u)du= 0. Furthermore, (<ref>) implies that lim_ℓ→ -∞2/a̅(ℓ)e^∫_0^ℓ2b̅(u)/a̅(u)du = 0. Lastly, our assumption that |h(Y)| < ∞ means that∫_-∞^∞ |h(y)| 2/a̅(y)e^∫_0^y2b̅(u)/a̅(u)du,which implies that lim_ℓ→ -∞h(ℓ)2/a̅(ℓ)e^∫_0^ℓ2b̅(u)/a̅(u)du = 0. This proves (<ref>). To prove (<ref>), we take ℓ→∞ and repeat the above arguments. § GRADIENT BOUNDS FOR CHAPTER <REF> In Section <ref>, we first prove Lemma <ref>, establishing the Wasserstein gradient bounds for Erlang-C model. In Section <ref>, we state and prove Lemma <ref>,establishing the Wasserstein gradient bounds for Erlang-A model. In Section <ref> we prove Lemmas <ref> and <ref>, establishing the Kolmogorov gradient bounds for both Erlang-C and Erlang-A models.§.§ Erlang-C Wasserstein Gradient Bounds Recall b(x) defined in (<ref>). For the remainder of Section <ref>, we seta̅(x) = 2μ,and b̅(x)= b(x) = -μ x,x ≤ -ζ, μζ,x ≥ -ζ, where ζ = δ(R - n) < 0. Observe that this b̅(x) satisfies both([eq:a1]a1) and ([eq:a2]a2), and that x_0 from (<ref>) equals zero. Furthermore,exp(∫_0^x2 b̅(u)/a̅(u) du) =e^-1/2x^2,x ≤ - ζ,e^-ζ (x+ζ)e^-1/2ζ^2,x ≥ -ζ.Fix h(x) ∈; without loss of generality we assume that h(0) = 0. The following lemma presents several bounds that will be used to prove Lemma <ref>.Let a̅(x) and b̅(x) be as in (<ref>). Thene^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤2/μ,x ≤ 0,1/μ e^ζ^2/2(2 + ζ),x ∈ [0,-ζ], e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤2/μ + 1/μζ,x ∈ [0,-ζ],1/μζ,x ≥ -ζ, e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤1/μ,x ≤ 0, 1/μ( 2e^1/2ζ^2 -1),x ∈ [0,-ζ], e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy ≤2/μ + 1/μζ^2,x ∈ [0,-ζ], x/μζ + 1/μζ^2,x ≥ -ζ, Y(∞)≤1/ζ + 1.We first prove (<ref>). When x ≤ 0, we can choose c_1 = -1 in(<ref>) to see thate^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤1/b̅(-1) +sup_y ∈ [-1,0]2/a̅(y) = 2/μ.For x ∈ [0,-ζ], e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy =e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^02/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy + e^-∫_0^x2b̅(u)/a̅(u)du∫_0^x2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy =e^x^2/2/e^-∫_0^02b̅(u)/a̅(u)du e^-∫_0^02b̅(u)/a̅(u)du∫_-∞^02/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy + e^x^2/2∫_0^x1/μ e^-y^2/2 dy≤e^ζ^2/22/μ + 1/μ e^ζ^2/2ζ=1/μ e^ζ^2/2(2 + ζ).We now prove (<ref>). When x ≥ -ζ, we use (<ref>) to see thate^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤1/b̅(-ζ) =1/μζ.When x ∈ [0,-ζ], e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy =e^-∫_0^x2b̅(u)/a̅(u)du∫_x^-ζ2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy + e^-∫_0^x2b̅(u)/a̅(u)du/e^-∫_0^-ζ2b̅(u)/a̅(u)du e^-∫_0^-ζ2b̅(u)/a̅(u)du∫_-ζ^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤e^-∫_0^x2b̅(u)/a̅(u)du∫_x^-ζ2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy + e^-∫_0^-ζ2b̅(u)/a̅(u)du∫_-ζ^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤e^-∫_0^x2b̅(u)/a̅(u)du∫_x^-ζ2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy + 1/μζ.We now bound the first term on the right hand side above. When ζ≥ 1, we use (<ref>) with c_2 = 1 to see that e^-∫_0^x2b̅(u)/a̅(u)du∫_x^-ζ2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤1/b̅(1) + 1/μ = 2/μ.When ζ≤ 1, e^-∫_0^x2b̅(u)/a̅(u)du∫_x^-ζ2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤e^-∫_0^12b̅(u)/a̅(u)du∫_0^12/a̅(y)dy ≤e^1/21/μ≤2/μ.Therefore, for x ∈ [0,-ζ],e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤2/μ + 1/μζ.To prove (<ref>), observe that when x ≤ 0,e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy =e^1/2x^2∫_-∞^x-y/μ e^-1/2y^2 dy = 1/μ,and when x ∈ [0,-ζ],e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy =e^1/2x^2∫_-∞^0-y/μ e^-1/2y^2 dy + e^1/2x^2∫_0^xy/μ e^-1/2y^2 dy=1/μ e^1/2x^2 + 1/μ e^1/2x^2(1 - e^-1/2x^2).We now prove (<ref>). Since a̅(x) ≡ 2μ, we can use (<ref>) to see that for x ≥ -ζ,e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy ≤x/b̅(x) + 2μ/2b̅(x)1/b̅(x) =x/μζ + 1/μζ^2.Furthermore, for x ∈ [0,-ζ],e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy =e^-∫_0^x2b̅(u)/a̅(u)du∫_x^-ζ2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy + e^-∫_0^x2b̅(u)/a̅(u)du/e^-∫_0^-ζ2b̅(u)/a̅(u)du e^-∫_0^-ζ2b̅(u)/a̅(u)du∫_-ζ^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy = e^1/2x^2∫_x^-ζy/μ e^-1/2y^2 dy + e^-∫_0^x2b̅(u)/a̅(u)du/e^-∫_0^-ζ2b̅(u)/a̅(u)du e^-∫_0^-ζ2b̅(u)/a̅(u)du∫_-ζ^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy=1/μ e^1/2x^2(e^-1/2x^2 - e^-1/2ζ^2) + e^-∫_0^x2b̅(u)/a̅(u)du/e^-∫_0^-ζ2b̅(u)/a̅(u)du e^-∫_0^-ζ2b̅(u)/a̅(u)du∫_-ζ^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy ≤1/μ + e^-∫_0^x2b̅(u)/a̅(u)du/e^-∫_0^-ζ2b̅(u)/a̅(u)du( ζ/μζ + 1/μζ^2)≤1/μ+( 1/μ + 1/μζ^2),where in the last inequality, we used the fact that e^-∫_0^x2b̅(u)/a̅(u)du≤ e^-∫_0^-ζ2b̅(u)/a̅(u)du.This proves (<ref>), and we move on to prove (<ref>). Letting V(x) = x^2, and recalling the form of G_Y from (<ref>), we consider G_Y V(x) =2xμ (ζ + (x + ζ)^-) + 2μ=-2μ x^2 1(x < -ζ) - 2xμζ 1(x ≥ -ζ) + 2μ.By the standard Foster-Lyapunov condition (see<cit.> for example), this implies that2[ (Y(∞))^2 1(Y(∞) < -ζ)] + 2ζ[Y(∞)1(Y(∞) ≥ -ζ)] ≤ 2,and in particular, [Y(∞)1(Y(∞) ≥ -ζ)] ≤1/ζ, [|Y(∞)1(Y(∞) < -ζ)|] ≤√([ (Y(∞))^2 1(Y(∞) < -ζ)])≤ 1,where we applied Jensen's inequality in the second set of inequalities. This concludes the proof of Lemma <ref>.Let b̅(x) and a̅(x) be as in (<ref>). We begin by bounding f_h'(x). Observe that since h(x) ∈ and h(0) = 0, then (<ref>) and (<ref>) imply that f_h'(x) ≤e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) ( |y| + Y(∞)) e^∫_0^y2b̅(u)/a̅(u)du dy, f_h'(x) ≤e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) ( |y| + Y(∞)) e^∫_0^y2b̅(u)/a̅(u)du dy.For x ≤ -ζ, we apply (<ref>), (<ref>), and (<ref>) to the first inequality above, and for x ≥ 0, we apply (<ref>), (<ref>), and (<ref>) to the second inequality above to see thatμf_h'(x)≤1 + 2 ( 1 + 1/ζ) ≤ 3 + 2/ζ,x ≤ 0,μf_h'(x)≤min{ 2e^1/2ζ^2 -1 + e^1/2ζ^2(2+ζ)( 1 + 1/ζ), 2 + 1/ζ^2 + (2 + 1/ζ)( 1 + 1/ζ) },x ∈ [0,-ζ],μf_h'(x)≤x/ζ + 1/ζ^2 + 1/ζ( 1 + 1/ζ) ≤1/ζ(x + 1 + 2/ζ),x ≥ -ζ.For x ∈ [0, -ζ], observe that when ζ≤ 1, then 2e^1/2ζ^2 -1 + (2+ζ)e^1/2ζ^2( 1 + 1/ζ) ≤ 3.3 - 1 + 5( 1 + 1/ζ) = 7.5 + 5/ζ,and when ζ≥ 1, then 2 + 1/ζ^2 + (2 + 1/ζ)( 1 + 1/ζ) ≤ 3 + 3( 1 + 1/ζ) = 6 + 3/ζ.Therefore, f_h'(x)≤1/μ(7.5 + 5/ζ),x ≤ -ζ, 1/μ1/ζ(x + 1 + 2/ζ),x ≥ -ζ.Before proceeding to bound f_h”(x) and f_h”'(x), we first note that both (<ref>) and (<ref>) are satisfied. This is because a̅(x) is constant, b̅(x) is piecewise linear,f_h'(x) is bounded as in (<ref>), but e^∫_0^x2b̅(u)/a̅(u)du decays exponentially fast as x →∞, and decays even faster as x → -∞. To bound f_h”(x), we use (<ref>) and (<ref>), together with the facts that a̅(x) is constant, h(x) ∈,andb̅'(x) = -μ 1(x < -ζ),x ∈,to see that f_h”(x)≤e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y)(1 + μf_h'(y) 1(y < -ζ)) e^∫_0^y2b̅(u)/a̅(u)du dyf_h”(x)≤e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y)(1 + μf_h'(y) 1(y < -ζ)) e^∫_0^y2b̅(u)/a̅(u)du dy .We know f_h'(x) is bounded as in (<ref>). For x ≤ -ζ, we apply (<ref>) to (<ref>) and for x ≥ 0 we apply (<ref>) to (<ref>) to conclude thatμf_h”(x)≤ 2 ( 1 + 7.5 + 5/ζ),x ≤ 0, min{(2+ζ)e^1/2ζ^2, 2 + 1/ζ}( 1 + 7.5 + 5/ζ),x ∈ [0,-ζ], 1/ζ,x ≥ -ζ.By considering separately the cases when ζ≤ 1/2 and ζ≥ 1/2, we see thatmin{(2+ζ)e^1/2ζ^2,2 + 1/ζ}≤ 4,and therefore,f_h”(x)≤34/μ( 1 + 1/ζ),x ≤ -ζ, 1/μζ,x ≥ -ζ.Lastly, we bound f_h”'(x), which exists for all x ∈ where h'(x) and b̅'(x) exist. Since a̅(x) is a constant and h(x) ∈, we know from (<ref>)that f_h”'(x)≤1/μ(1 + f_h”(x)b̅(x) + f_h'(x) b̅'(x)). For x ≥ -ζ, we use the forms of b̅(x) and b̅'(x) together with the bounds on f_h'(x) and f_h”(x) in (<ref>) and (<ref>) to see that f_h”'(x)≤1/μ( 1 + μζ1/μζ).Although tempting, it is not sufficient to use the bound on f_h”(x) in (<ref>) and the form of b̅(x)to bound f_h”(x)b̅(x) for all x ≤ -ζ. Instead, we multiply both sides of (<ref>) and (<ref>) by b̅(x) to see thatf_h”(x) b̅(x) ≤(1 + sup_y ≤ xμf_h'(y))b̅(x)e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy,x ≤ 0, f_h”(x) b̅(x) ≤(1 + sup_y ∈ [x,-ζ]μf_h'(y))b̅(x)e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy,x ∈ [0,-ζ].By invoking (<ref>) and (<ref>), together with the bound on f_h'(x) from (<ref>), we conclude thatf_h”'(x)≤1/μ(1 + (1 +7.5 + 5/ζ) + 7.5 + 5/ζ),x ≤ -ζ.Therefore, for those x ∈ where h'(x) and b̅'(x) exist,f_h”'(x)≤1/μ( 17 + 10/ζ) ,x ≤ -ζ, 2/μ,x ≥ -ζ.This concludes the proof of Lemma <ref>.§.§ Erlang-A Wasserstein Gradient Bounds Below we prove the Erlang-A gradient bounds, which were stated in Lemma <ref> of Section <ref>. Their proof is similar to that of Lemma <ref>. We only outline the necessary steps needed for a proof, and emphasize all the differences with the proof of Lemma <ref>. §.§.§ Proof Outline for Lemma <ref>: The Underloaded System In the Erlang-A model, b̅(x) = -μ x,x ≤ -ζ,-α(x+ζ)+ μζ,x ≥ -ζ,and a̅(x) = 2μ. To prove Lemma <ref>, we need the following version of Lemma <ref>. Consider the Erlang-A model (α > 0) with 0 < R ≤ n.Then there exists a constant C, independent of λ,μ,n, and α, such thate^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤C/μ,x ≤ 0 ,C/μe^ζ^2/2,x ∈ [0,-ζ], e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤C/μ(1+√(μ/α)∧1/ζ),x ∈ [0,-ζ],C/μ(√(μ/α)∧1/ζ),x ≥ -ζ, e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤C/μ,x ≤ 0, C/μe^ζ^2/2,x ∈ [0,-ζ], e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy ≤C/μ(1+1/ζ^2),x ∈ [0,-ζ], C/μ(1+μ/α),x ≥ -ζ, Y(∞)≤ 1+√(μ/α)∧1/ζ . To prove this lemma, we first observe thate^∫_0^x2b̅(u)/a̅(u)du =e^-1/2x^2, x≤ -ζ,e^-1/2ζ^2 e^μ/2αζ^2 e^-α/2μ(x+ζ-μ/αζ)^2, x≥ -ζ,By comparing (<ref>) to (<ref>) for the region x ≤ -ζ, we immediately see that (<ref>) and(<ref>) are restatements of (<ref>) and(<ref>), from Lemma <ref>, and hence have already been established. The proof of (<ref>) involves applying G_Y to the Lyapunov function V(x) = x^2 to see thatG_Y V(x) =-2μ x^2 1(x < -ζ) + 2(-α x^2 + xζ(μ - α) )1(x ≥ -ζ) + 2μ ≤-2μ x^2 1(x < -ζ) -2 (α∧μ ) x^2 1(x ≥ -ζ) + 2μ,and G_Y V(x) =-2μ x^2 1(x < -ζ) + 2(-α x(x+ζ) - μζx )1(x ≥ -ζ) + 2μ ≤-2μ x^2 1(x < -ζ) -2μζ x1(x ≥ -ζ) + 2μ.One can compare these inequalities to (<ref>) in the proof of Lemma <ref> to see that (<ref>) follows by the Foster-Lyapunov condition.We now go over the proofs of (<ref>) and (<ref>). We first prove (<ref>) when x ∈ [0,-ζ]. Just like in (<ref>) and the displays right below it, we can show thate^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤e^-∫_0^x2b̅(u)/a̅(u)du∫_x^-ζ2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy + e^-∫_0^-ζ2b̅(u)/a̅(u)du∫_-ζ^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤2/μ + 1/μζ.It can also be checked that e^-∫_0^-ζ2b̅(u)/a̅(u)du∫_-ζ^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy =e^1/2(x^2-ζ^2)e^α/2μ(μ/αζ)^2∫^∞_ζ1/μ e^-α/2μ(y+ζ-μ/αζ)^2dy =e^1/2(x^2-ζ^2)e^α/2μ(μ/αζ)^2∫^∞_μ/αζ1/μ e^-α/2μy^2dy≤e^α/2μ(μ/αζ)^2∫^∞_μ/αζ1/μ e^-α/2μy^2dy≤∫^∞_01/μ e^-α/2μy^2dy =1/μ√(π/2μ/α),where in the last inequality, we used the fact that for x ≥ 0,the function e^α/2μx^2∫^∞_x1/μ e^-α/2μy^2dy is maximized at x = 0 (this can be checked by differentiating the function).This proves the part of (<ref>) when x ∈ [0,-ζ]. The case when x ≥ -ζ is handled similarly.We now prove (<ref>). When x ∈ [0,-ζ], e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy =1/μ e^1/2x^2∫_x^-ζ y e^-1/2y^2dy + 1/μ e^α/2μ (μ/αζ)^2 e^1/2(x^2-ζ^2)∫_-ζ^∞ ye^-α/2μ(y+ζ-μ/αζ)^2 dy =1/μ(1 - e^1/2(x^2-ζ^2)) +1/μe^α/2μ (μ/αζ)^2 e^1/2(x^2-ζ^2)∫_-ζ^∞ ye^-α/2μ(y+ζ-μ/αζ)^2 dy≤1/μ + 1/μe^α/2μ (μ/αζ)^2∫_-ζ^∞ ye^-α/2μ( (y+ζ)^2 -2μ/α (y+ζ)ζ + (μ/αζ)^2) dy≤1/μ + 1/μ∫_-ζ^∞ ye^(y+ζ)ζ dy = 1/μ + 1/μ + 1/μζ^2,and when x ≥ -ζ,e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy=1/μe^α/2μ(x+ζ-μ/αζ)^2∫_x^∞ ye^-α/2μ(y+ζ-μ/αζ)^2 dy=1/μe^α/2μ(x+ζ-μ/αζ)^2∫_x+ζ-μ/αζ^∞ ye^-α/2μy^2 dy+ 1/μ(1-μ/α)ζ e^α/2μ(x+ζ-μ/αζ)^2∫_x+ζ-μ/αζ^∞ e^-α/2μy^2 dy≤1/μμ/α +1/μζ e^α/2μ(x+ζ-μ/αζ)^2∫_x+ζ-μ/αζ^∞y/ (x + ζ - μ/αζ) e^-α/2μy^2 dy=1/μ(μ/α + ζ1/α/μ (x + ζ - μ/αζ))≤1/μ( μ/α + 1).We now describe how to prove Lemma <ref>. To prove (<ref>), we repeat the procedure used to get (<ref>), except this time using the bounds in Lemma <ref> instead of those in Lemma <ref>. Using the resulting bounds on f_h'(x), we argue that (<ref>) and (<ref>) are true, just like we did in the proof of Lemma <ref>. We now describe how to prove (<ref>). When x ≤ 0,we apply (<ref>) and (<ref>) to (<ref>), and when x ≥ -ζ we apply (<ref>) and (<ref>) to (<ref>). The last region, when x ∈ [0,-ζ], has to be handled differently depending on the size of ζ. When ζ≤ 1, we just apply (<ref>) and (<ref>) to (<ref>). However, when ζ≥ 1, we manipulate (<ref>) to see thatf_h”(x) =-e^-∫_0^x2b̅(u)/a̅(u)du∫_x^-ζ1/μ(-h'(y) + μ f_h'(y)) e^-∫_0^y2b̅(u)/a̅(u)dudy-e^-∫_0^x2b̅(u)/a̅(u)du/e^-∫_0^-ζ2b̅(u)/a̅(u)du e^-∫_0^-ζ2b̅(u)/a̅(u)du∫_-ζ^∞1/μ(-h'(y) + α f_h'(y)) e^-∫_0^y2b̅(u)/a̅(u)du dy.We then apply (<ref>), (<ref>), and the fact that e^-∫_0^x2b̅(u)/a̅(u)du/e^-∫_0^-ζ2b̅(u)/a̅(u)du≤ 1 to conclude (<ref>). The proof of (<ref>) relies on (<ref>), which tells us that f_h”'(x)≤1/μ[ 1 + f_h”(x)b̅(x) + f_h'(x) b̅'(x)].Bounding f_h'(x) b̅'(x)only relies on (<ref>). The term f_h”(x)b̅(x) is bounded similarly to the way it is done in Lemma <ref>; see for instance (<ref>).This concludes the proof outline for Lemma <ref> when the system is underloaded.§.§.§ Proof Outline for Lemma <ref>: The Overloaded System For the overloaded case in Lemma <ref>, we again need the following version of Lemma <ref>.Consider the Erlang-A model (α > 0) with 0 < R ≤ n.Then there exists a constant C, independent of λ,μ,n, and α, such thate^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤C/μ(1∧μ/αζ),x ≤ -ζ ,C/μ(1+√(μ/α)∧ζ),x ∈ [-ζ,0], e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤C/μ√(μ/α) e^α/2μζ^2,x ∈ [-ζ,0],C/μ√(μ/α),x ≥ 0, e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤C/μ(1 +ζ∧μ/α),x ≤ -ζ, C/μ(μ/α+1),x ∈ [-ζ,0], e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy ≤C/μμ/αe^α/2μζ^2,x ∈ [-ζ,0], C/μμ/α,x ≥ 0 , Y(∞)≤√(μ/α) + 1. To prove this lemma, we first observe that a̅(x) = 2μ, b̅(x) = -μ (x+ζ)+ αζ,x ≤ -ζ,-α x,x ≥ -ζ,and e^-∫_0^x2b̅(u)/a̅(u)du =e^1/2(α/μζ)^2e^-α/2μζ^2e^-1/2(x+ζ-α/μζ)^2, x≤ -ζ,e^-α/2μx^2, x≥ -ζ.Observe thatin the region x ≥ -ζ, the form of (<ref>) is very similar to the(<ref>) in the region x ≤ -ζ. Hence, one can check that the arguments needed to prove Lemma <ref>'s (<ref>) and (<ref>) are nearly identical to the arguments used to prove Lemma <ref>'s (<ref>) and (<ref>).The proof of (<ref>) involves applying G_Y, where G_Y f(x) = 1/2a̅(x) f”(x) + b̅(x) f'(x)to the Lyapunov function V(x) = x^2 to see thatG_Y V(x) =-2α x^2 1(x > -ζ) + 2(-μ x^2 + xζ(α - μ ) )1(x ≤ -ζ) + 2μ ≤-2α x^2 1(x > -ζ) -2 (α∧μ ) x^2 1(x ≤ -ζ) + 2μ.One can compare this inequality to (<ref>) in the proof of Lemma <ref> to see that (<ref>) follows by the Foster-Lyapunov condition. We now describe how to prove (<ref>) and(<ref>). The proof of (<ref>) uses a series of arguments similar to those in the proof of (<ref>) of Lemma <ref>. We now prove (<ref>). When x ≤ -ζ, e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy=1/μ e^1/2(x+ζ-α/μζ)^2∫_-∞^x -ye^-1/2(y+ζ-α/μζ)^2 dy=1/μe^1/2(x+ζ-α/μζ)^2∫_-∞^x+ζ-α/μζ -ye^-1/2y^2 dy +1/μ (1-α/μ )ζ e^1/2(x+ζ-α/μζ)^2∫_-∞^x+ζ-α/μζ e^-1/2y^2 dy≤1/μ + ζ/μ( √(π/2)∧1/α/μζ -x - ζ) ≤ 1 +√(π/2)ζ∧μ/α,where the second last inequality uses logic similar to what was used in (<ref>). For x ∈ [-ζ, 0], e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy =1/μ e^-α/2μζ^2 e^1/2 (α/μζ)^2 e^α/2μx^2∫_-∞^-ζ -ye^-1/2(y+ζ-α/μζ)^2 dy +1/μe^α/2μx^2∫_-ζ^x -ye^α/2μy^2 dy.Repeating arguments from (<ref>), we can show that the first term above satisfies 1/μ e^-α/2μζ^2 e^1/2 (α/μζ)^2 e^α/2μx^2∫_-∞^-ζ -ye^-1/2(y+ζ-α/μζ)^2 dy ≤1/μe^α/2μ (x^2-ζ^2)(1 + μ/α),and by computing the second term explicitly, we conclude that e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤1/μe^α/2μ (x^2-ζ^2)(1 + μ/α) + 1/μμ/α( 1 - e^α/2μ (x^2-ζ^2))≤1/μ(1 + μ/α),which proves (<ref>).Having argued Lemma <ref>, we now use it to prove the bounds in (<ref>)–(<ref>). To prove (<ref>), we repeat the procedure used to get (<ref>), except this time using the bounds in Lemma <ref> instead of those in Lemma <ref>. Using the resulting bounds on f_h'(x), we argue that (<ref>) and (<ref>) are true, just like we did in the proof of Lemma <ref>.We now describe how to prove (<ref>). When x ≤ -ζ,we apply (<ref>) and (<ref>) to (<ref>).When x ≥ -ζ, instead of using the expressions for f_h”(x) in (<ref>) and (<ref>) like we would usually do, we instead apply (<ref>) to the bound f_h”(x)≤1/μf_h'(x)b̅(x) + 1/μ(x + Y(∞)),x ∈,which follows by rewriting the Poisson equation (<ref>) and using the Lipschitz property of h(x). We now prove (<ref>)–(<ref>). We recall (<ref>) to see thatf_h”'(x)≤1/μ[ 1 + f_h”(x)b̅(x) + f_h'(x) b̅'(x)].Bounding f_h'(x) b̅'(x) is simple, and only relies on (<ref>). The other term, f_h”(x)b̅(x), is bounded as follows. To prove (<ref>), i.e. when x ≤ -ζ, the term f_h”(x)b̅(x) is bounded similarly to the way it is done in Lemma <ref>; see for instance (<ref>). When x ≥ -ζ then f_h”(x)b(x) = αxf_h”(x),and the difference between (<ref>) and (<ref>) lies in the way that the quantity above is bounded. To get (<ref>), we simply apply the bounds on f_h”(x) from (<ref>) to the right hand side above. To prove (<ref>), we will first argue thatf_h”'(x)≤C/μ(α/μ+√(α/μ)+1) + C/μ(α/μ+√(α/μ)+1)^2 x, x∈ [-ζ,0], C/μ(α/μ+√(α/μ)+1), x≥ 0,where C is some positive constant independent of everything else; this will imply (<ref>). The only difference between the proof of (<ref>) and thebound on f_h”'(x) in (<ref>) is in how f_h”(x)b(x) is bounded; we now describe the different way to bound f_h”(x)b(x). When x ≥ 0, we bound f_h”(x)b̅(x) just like we did in Lemma <ref>; see for instance (<ref>). When x ∈ [-ζ,0], we want to prove thatf_h”(x)≤C/μ(α/μ+√(α/μ)+1)(1+√(μ/α)) + C/μ( ζ∧μ^2/α^2 ζ),which, after considering separately the cases when ζ≤μ /α and ζ≥μ /α, implies that f_h”(x)≤C/μ(α/μ+√(α/μ)+1)(1+√(μ/α)) + C/α.We can then use this fact to bound f_h”(x)b(x) =αxf_h”(x). To prove (<ref>) for ζ≤√(μ/α), we bound (<ref>) using (<ref>) and (<ref>). To prove (<ref>) for ζ≥√(μ/α), we bound (<ref>) using (<ref>) and (<ref>). We point out that to bound (<ref>) we need to perform a manipulation similar to the one in (<ref>).This concludes the proof outline for the overloaded case. §.§ Kolmogorov Gradient Bounds: Proof of Lemmas <ref> and <ref> Let a̅(x) and b̅(x) be as in (<ref>). Fix a ∈ and let h(x) = 1_(-∞,a](x). The thePoisson equation is b̅(x) f_a'(x) + 1/2a̅(x) f_a”(x) = F_Y(a) - 1_(-∞,a](x),where F_Y(x) = (Y(∞) ≤ x). Since 1_(-∞,a](x) is discontinuous, any solution to the Poisson equation will have a discontinuity in its second derivative, which makes the gradient bounds for it differ from the Wasserstein setting.Together, (<ref>) and (<ref>) both imply that f_a'(x)≤e^-∫_0^x2b̅(u)/a̅(u)dumin{∫_-∞^x2/a̅(y)e^∫_0^y2b̅(u)/a̅(u)du dy, ∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy }.Furthermore, f_a”(x) = 1/μ( F_Y(a) - 1_(-∞,a](x) - b̅(x) f_a'(x)).We now prove the Kolmogorov gradient bounds for the Erlang-C model.First of all, by (<ref>) and (<ref>), μf_a'(x)≤ 2,x ≤ 0,min{(2+ζ)e^1/2ζ^2,2 + 1/ζ},x ∈ [0,-ζ],1/ζ,x ≥ -ζ,and (<ref>) implies thatmin{(2+ζ)e^1/2ζ^2,2 + 1/ζ}≤ 4,which proves the bounds for f_a'(x). Second, (<ref>) and (<ref>) imply that for all x ∈, f_a”(x) ≤1/μ( 1 + b̅(x) e^-∫_0^x2b̅(u)/a̅(u)dumin{∫_-∞^x2/a̅(y)e^∫_0^y2b̅(u)/a̅(u)du dy, ∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy })≤2/μ,where f_a”(x) is understood to be the left derivative at the point x = a. The proof of this lemma is almost identical to the proof of Lemma <ref>. Its not hard to check that (<ref>) holds for the Erlang-A model as well. To prove the bounds on f_a'(x), we obtain inequalities similar to (<ref>) by using analogues of (<ref>) and (<ref>) from Lemmas <ref> and <ref>.These inequalities will imply (<ref>) and (<ref>) once we consider in them separately the cases when ζ≤ 1 and ζ≥ 1.§ GRADIENT BOUNDS FOR CHAPTERS <REF> AND <REF> In the setting of Chapter <ref> and <ref>,a̅(x)=μ ,x ≤ -1/δ,μ (2 + δ x),x ∈ [-1/δ, -ζ],μ (2 + δζ),x ≥ -ζ,andb̅(x) = -μ x,x ≤ -ζ, μζ,x ≥ -ζ, where ζ = δ(R - n) < 0. Observe that b̅(x) satisfies both([eq:a1]a1) and ([eq:a2]a2), and that x_0 from (<ref>) equals zero. Furthermore, exp(∫_0^x2 b̅(u)/a̅(u) du) = exp(1/δ^2+ 2/δ^2 - 4/δ^2log(2))exp(-x^2),x ≤ -1/δ, exp(- 4/δ^2log(2))exp[4/δ^2log(2 + δ x) - 2δ x/δ^2],x ∈ [-1/δ, -ζ],exp(- 4/δ^2log(2) + 2/δ^2(2log(2 + δζ) - δζ ) + 2ζ^2/2 + δζ)exp(-2ζ x /2+δζ),x ≥ -ζ.The following lemma presents several bounds that will be used to prove Lemma <ref> and <ref>. Let a̅(x) and b̅(x) be as in (<ref>). Thene^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤3/μ,x ≤ 0, 1/μe^ζ^2 (3 + ζ),x ∈ [0,-ζ], e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤1/μ( 2 + 1/ζ),x ∈ [0,-ζ],1/μζ,x ≥ -ζ, e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤1/μ,x ≤ 0, 2/μ e^ζ^2/2,x ∈ [0,-ζ], e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy ≤2/μ + 1/μζ^2 + δ/2μζ,x ∈ [0,-ζ], x/μζ + 1/μζ^2 + δ/2μζ,x ≥ -ζ, |Y(∞)| ≤√(δ^2 + 2) + √(2δ^2 + 4) + 2+δ^2/ζ + δ.To prove this lemma we verify (<ref>)–(<ref>) one at a time. We now prove (<ref>). Using (<ref>)with c_1 = -1, we see that for x ≤ 0, e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤1/b̅(-1)+sup_y ∈ [-1, 0]2/a̅(y)≤1/μ + 2/μ = 3/μ.Forx ∈ [0,-ζ],e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy =e^-∫_0^x2b̅(u)/a̅(u)du/e^-∫_0^02b̅(u)/a̅(u)du e^-∫_0^02b̅(u)/a̅(u)du∫_-∞^02/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy + e^-∫_0^x2b̅(u)/a̅(u)du∫_0^x2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤e^-∫_0^x2b̅(u)/a̅(u)du3/μ +e^-∫_0^x2b̅(u)/a̅(u)du∫_0^-ζ2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤e^-∫_0^x2b̅(u)/a̅(u)du3/μ +e^-∫_0^x2b̅(u)/a̅(u)duζ1/μ,where in the second last inequality we used (<ref>), and in the last inequality we used the fact that e^∫_0^y2b̅(u)/a̅(u)du≤ 1 and a̅(y) ≥ 2μ for y ∈ [0,-ζ]. From (<ref>), we know that e^-∫_0^x2b̅(u)/a̅(u)du =exp( -4/δ^2(log(2 + δ x) -log(2) -δ x/2 )),x ∈ [0,-ζ].Using Taylor expansion,log(2 + y) = log(2) + 1/2y - 1/2y^2/(2 + ξ(y))^2,y ∈ (-2, ∞),where ξ(y) is some point between 0 and y. Therefore, for x ∈ [0,-ζ],e^-∫_0^x2b̅(u)/a̅(u)du =exp( -4/δ^2(log(2 + δ x) -log(2) -δ x/2 ))=exp( 4/δ^21/2δ^2 x^2/(2 + ξ(δ x))^2) ≤exp(x^2/2),where in the last inequality we used the fact that ξ(δ x) ≥ 0 for x ≥ 0. Combining this with (<ref>), we conclude that e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤e^ζ^2/μ(3 + ζ),x ∈ [0,-ζ],which proves (<ref>). We now prove (<ref>). When x ≥ -ζ, (<ref>) implies that e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤1/b̅(x) =1/μζ.When x ∈ [0,-ζ], we can repeat the procedure in (<ref>) to see that e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤e^-∫_0^x2b̅(u)/a̅(u)du∫_x^-ζ2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy + 1/μζ.We now bound the first term on the right hand side above. When ζ≥ 1, we use (<ref>) with c_2 = 1 to see that e^-∫_0^x2b̅(u)/a̅(u)du∫_x^-ζ2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy≤1/b̅(1) + 1/μ = 2/μ.When ζ≤ 1, (<ref>) implies that e^-∫_0^x2b̅(u)/a̅(u)du∫_x^-ζ2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤e^-∫_0^-ζ2b̅(u)/a̅(u)du∫_0^12/a̅(y)dy ≤e^ζ^2/21/μ≤2/μ.Therefore, for x ∈ [0,-ζ],e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ≤2/μ + 1/μζ,which proves (<ref>). We now prove (<ref>). For x ≤ 0, e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy = e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x1/μ2b̅(y)/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy =1/μ(1 - e^-∫_-∞^x2b̅(u)/a̅(u)du) ≤1/μ.When x ∈ [0,-ζ],e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy = e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^0-2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy + e^-∫_0^x2b̅(u)/a̅(u)du∫_0^x2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy = e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^01/μ2b̅(y)/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy - e^-∫_0^x2b̅(u)/a̅(u)du∫_0^x1/μ2b̅(y)/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy =1/μe^-∫_0^x2b̅(u)/a̅(u)du( (e^∫_0^02b̅(u)/a̅(u)du - e^∫_0^-∞2b̅(u)/a̅(u)du) - (e^∫_0^x2b̅(u)/a̅(u)du - e^∫_0^02b̅(u)/a̅(u)du) )≤2/μe^-∫_0^x2b̅(u)/a̅(u)du≤2/μe^ζ^2/2,where in the last inequality we used (<ref>). This proves (<ref>), and now we prove (<ref>). Fix x ∈ [0,-ζ].We now prove (<ref>). Since a̅(x) = μ(2+δζ) for x ≥ -ζ, we can use (<ref>) to see that for x ≥ -ζ,e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy ≤x/b̅(x) + μ (2 + δζ)/2b̅(x)1/b̅(x)=x/μζ + 2 + δζ/2ζ1/μζ.Furthermore, for x ∈ [0,-ζ],e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy = - e^-∫_0^x2b̅(u)/a̅(u)du∫_x^-ζ1/μ2b̅(y)/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy + e^-∫_0^x2b̅(u)/a̅(u)du/e^-∫_0^-ζ2b̅(u)/a̅(u)du e^-∫_0^-ζ2b̅(u)/a̅(u)du∫_-ζ^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy≤ - e^-∫_0^x2b̅(u)/a̅(u)du∫_x^-ζ1/μ2b̅(y)/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy + e^-∫_0^-ζ2b̅(u)/a̅(u)du∫_-ζ^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy=1/μe^-∫_0^x2b̅(u)/a̅(u)du(-e^∫_0^-ζ2b̅(u)/a̅(u)du + e^∫_0^x2b̅(u)/a̅(u)du) + e^-∫_0^-ζ2b̅(u)/a̅(u)du∫_-ζ^∞2y/a̅(y) e^∫_0^y2b̅(u)/a̅(u)dudy ≤1/μ + ( ζ/μζ +2 + δζ/2ζ1/μζ),where in the first inequality, we used the fact that e^-∫_0^x2b̅(u)/a̅(u)du≤ e^-∫_0^-ζ2b̅(u)/a̅(u)du.This proves (<ref>), and we move on to verify (<ref>). Consider the Lyapunov function V(x) = x^2, and recall the form of G_Y from (<ref>) to see that G_Y V(x) =2xμ (ζ + (x + ζ)^-) + 2μ(1 + 1(x > -1/δ)(1 -δ(ζ + (x + ζ)^-))).Now when x < -ζ, G_Y V(x) =-2μ x^2 + 2μ(1 + 1(x > -1/δ)(1 +δ x) ) ≤-2μ x^2+ 2μδ x 1( x ∈ [0, -ζ))+ 4μ=-2μ x^2 1(x < 0) - 2μ(x^2 - δ x ) 1( x ∈ [0, -ζ)) + 4μ ≤-2μ x^2 1(x < 0) - μ(x^2 - δ^2 ) 1( x ∈ [0, -ζ)) + 4μ ≤-2μ x^2 1(x < 0) - μ x^2 1( x ∈ [0, -ζ)) + μδ^2 + 4μ,and when x ≥ -ζ, G_Y V(x) =- 2xμζ + 2 δμζ + 4μ=- 2μζ ( x - δ) 1( ζ < δ) - 2μζ ( x - δ) 1( ζ≥δ)+ 4μ ≤- 2μζx 1( ζ < δ) + 2μδ^21( ζ < δ) - 2μζ ( x - δ) 1( ζ≥δ) +4μ.Therefore, G_Y V(x) ≤-2μ x^2 1(x < 0) -μ x^2 1(x ∈ [0,-ζ))- 2μζx 1( ζ < δ)1(x ≥ -ζ) - 2μζ ( x - δ) 1( ζ≥δ)1(x ≥ -ζ) + 2μδ^21( ζ < δ) 1(x ≥ -ζ)+ μδ^2 1(x < -ζ) + 4μ,i.e. G_Y V(x) satisfies G_Y V(x) ≤ -f(x) + g(x),where f(x) and g(x) are functions from →_+. By the standard Foster-Lyapunov condition (see for example <cit.>), this implies thatf(Y(∞)) ≤ g(Y(∞)),or2[ (Y(∞))^2 1(Y(∞) < 0)] + [ (Y(∞))^2 1(Y(∞) ∈ [0,-ζ))] + 2ζ[Y(∞)1(Y(∞) ≥ -ζ)] 1( ζ < δ)+ 2ζ[(Y(∞)-δ )1(Y(∞) ≥ -ζ)] 1(ζ≥δ)≤2δ^2 + 4,from which we can see that[Y(∞)1(Y(∞) ≥ -ζ)] ≤δ^2/ζ +2/ζ +δ.Furthermore, by invoking Jensen's inequality we see that[ |Y(∞) 1(Y(∞) < 0)|] ≤√([ (Y(∞))^2 1(Y(∞) < 0)]) ≤√(δ^2 + 2),[ |Y(∞) 1(Y(∞) ∈ [0,-ζ))|] ≤√([ (Y(∞))^2 1(Y(∞) ∈ [0,-ζ))]) ≤√(2δ^2 + 4).Hence [ |Y(∞)|] =[ |Y(∞) 1(Y(∞) < 0)|] + [ |Y(∞) 1(Y(∞) ∈ [0,-ζ))|] + [Y(∞)1(Y(∞) ≥ -ζ)] ≤√(δ^2 + 2) + √(2δ^2 + 4) + 2+δ^2/ζ + δ.This proves (<ref>) and concludes the proof of this lemma. We are now ready to prove Lemma <ref> and <ref>.§.§ Proof of Lemma <ref> (W_2 Bounds) Recall our assumption that R ≥ 1, or equivalently, δ≤ 1. Throughout the proof we use C > 0 to denote a generic constant that does not depend on λ,n, and μ, and may change from line to line. We begin by bounding f_h'(x). Observe that since h(x) ∈ W_2 and h(0) = 0, then (<ref>) and (<ref>) imply that f_h'(x) ≤e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x2/a̅(y) ( |y| + Y(∞)) e^∫_0^y2b̅(u)/a̅(u)du dy,f_h'(x) ≤e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞2/a̅(y) ( |y| + Y(∞)) e^∫_0^y2b̅(u)/a̅(u)du dy.We apply (<ref>), (<ref>), and (<ref>) to the first inequality above when x ≤ -ζ to see thatμf_h'(x)≤C( 1 +1/ζ),x ≤ 0, μf_h'(x)≤ 2e^1/2ζ^2 + e^ζ^2 (3 + ζ)|Y(∞)|,x ∈ [0,-ζ],and apply (<ref>), (<ref>), and (<ref>) to the second inequality when x ≥ 0 to see thatμf_h'(x)≤2 + 1/ζ^2 + δ/2ζ + (2 +1/ζ)|Y(∞)|,x ∈ [0,-ζ],μf_h'(x)≤C/ζ(x + 1 + 1/ζ),x ≥ -ζ.Above, there are two possible bounds on μf_h'(x) when x ∈ [0, -ζ]. By considering separately the cases when ζ≤ 1 and ζ≥ 1, and using (<ref>) to bound | Y(∞) |, we conclude thatμf_h'(x)≤C( 1 +1/ζ),x ∈ [0,-ζ].Therefore,f_h'(x)≤C/μ( 1 +1/ζ),x ≤ -ζ, C/μζ(x + 1 + 1/ζ),x ≥ -ζ,which proves (<ref>). Using (<ref>), (<ref>), and (<ref>), the reader can verify that (<ref>) and (<ref>) are satisfied, which allows us to use the two forms of f_h”(x) in (<ref>) and (<ref>). We now bound f_h”(x). Since h(0) = 0 and h(x) ∈ W_2, we know that h(x)≤x and h'(x)≤ 1 for all x ∈. From (<ref>) and (<ref>), it follows thatf_h”(x)≤e^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x(2/a̅(y) +2a̅'(y)y/a^2(y)+ 2a̅'(y)/a^2(y)| Y(∞)|+ (2b̅(y)/a̅(y))' f_h'(y))e^∫_0^y2b̅(u)/a̅(u)dudy,f_h”(x)≤e^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞(2/a̅(y) +2a̅'(y)y/a^2(y) + 2a̅'(y)/a^2(y)| Y(∞)|+ (2b̅(y)/a̅(y))' f_h'(y))e^∫_0^y2b̅(u)/a̅(u)du dy.We now bound the terms inside the integrals above. By definition of a̅(x) in (<ref>), we see thata̅'(x) = μδ 1(x ∈ (-1/δ, -ζ]),where a̅'(x) is interpreted as the left derivative for x = -1/δ and x = -ζ. Therefore,a̅'(x)x/a̅(x) =μδx/μ (2 + δ x)1(x ∈ (-1/δ, -ζ]) ≤ 1(x ∈ (-1/δ, -ζ]),| Y(∞) | a̅'(x)/a̅(x) =| Y(∞) |μδ/μ (2 + δ x)1(x ∈ (-1/δ, -ζ])≤δ C(1 + 1/ζ) 1(x ∈ (-1/δ, -ζ]),where in the last inequality we used (<ref>) and the fact that δ≤ 1 to bound | Y(∞) |. Furthermore, 2b̅(x)/a̅(x) = -2x,x ≤ -1/δ, -2x/2 +δ x,x ∈ [-1/δ, -ζ],2ζ/2 + δζ,x ≥ -ζ, (2b̅(x)/a̅(x))' = -2,x ≤ -1/δ, -4/(2+δ x)^2,x ∈ (-1/δ, -ζ], 0,x > -ζ,where (2b̅(x)/a̅(x))' is interpreted as the left derivative at the points x = -1/δ and x = -ζ. Combining (<ref>) with the bound on f_h'(x) in(<ref>), we get(2b̅(x)/a̅(x))'f_h'(x) =2f_h'(x)1(x ≤-1/δ) + 4/(2+δ x)^2f_h'(x)1(x ∈(-1/δ, -ζ])≤2f_h'(x)1(x ≤-1/δ) + 4/2+δ xf_h'(x)1(x ∈(-1/δ, -ζ])≤C/1 + 1(x ∈(-1/δ, -ζ])(1+ δ x)1/μ(1 +1/ζ) 1(x ≤ -ζ)=C/a̅(x)(1 +1/ζ) 1(x ≤ -ζ).Therefore, when x ≤ -ζ we apply the bounds in (<ref>), (<ref>), and (<ref>) to (<ref>) to see thatf_h”(x)≤Ce^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x1/a̅(y)(1 +1(y ∈ (-1/δ, -ζ])+ δ(1 + 1/ζ) 1(y ∈ (-1/δ, -ζ])+ (1 +1/ζ) 1(y ≤ -ζ))e^∫_0^y2b̅(u)/a̅(u)dudy≤Ce^-∫_0^x2b̅(u)/a̅(u)du∫_-∞^x1/a̅(y)(1 +1/ζ)e^∫_0^y2b̅(u)/a̅(u)du dy,x ≤ -ζand when x ≥ 0 we apply the same bounds to (<ref>) to see that f_h”(x)≤Ce^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞1/a̅(y)(1 +1(y ∈ (-1/δ, -ζ])+ δ(1 + 1/ζ) 1(y ∈ (-1/δ, -ζ])+ (1 +1/ζ) 1(y ≤ -ζ))e^∫_0^y2b̅(u)/a̅(u)dudy≤Ce^-∫_0^x2b̅(u)/a̅(u)du∫_x^∞1/a̅(y)(1 +1/ζ)e^∫_0^y2b̅(u)/a̅(u)du dy,x ≥ 0. We apply (<ref>) to (<ref>) and (<ref>) to (<ref>) to getf_h”(x)≤C/μ(1 +1/ζ),x ≤ 0, min{e^ζ^2/2 (3 + ζ) , 2 + 1/ζ}C/μ(1 +1/ζ),x ∈ [0,-ζ], C/μζ,x ≥ -ζ,and by considering separately the cases when ζ≤ 1 and ζ≥ 1, we conclude thatf_h”(x)≤C/μ(1 +1/ζ),x ≤ -ζ, C/μζ,x ≥ -ζ,which proves (<ref>). Now we prove (<ref>). Recall the form of f_h”'(x) from (<ref>), which together with the facts that h(x)≤x and h'(x)≤ 1 implies that for all x ∈, f_h”'(x)≤(2b̅(x)/a̅(x))' f_h'(x) + 2b̅(x)/a̅(x) f_h”(x) + 2/a̅(x) + 2a̅'(x)/a^2(x)( x + | Y(∞)| ),where f_h”'(x) is interpreted as the left derivative at the points x = -1/δ and x = -ζ. We apply the bound on (2b̅(x)/a̅(x))' f_h'(x) from (<ref>), the bounds on a̅'(x) x/a̅(x) and | Y(∞)|a̅'(x)/a̅(x) from (<ref>) and (<ref>), and the fact that 1/a̅(x) ≤ 1/μ for all x ∈ to see that f_h”'(x)≤C/μ( 1 + 1/ζ) 1(x ≤ -ζ) + C/μ1(x > -ζ)+ 2b̅(x)/a̅(x) f_h”(x).It remains to bound 2b̅(x)/a̅(x) f_h”(x), but this term does not pose much added difficulty. Indeed, one can multiply both sides of (<ref>) and (<ref>) by 2b̅(x)/a̅(x) and invoke (<ref>) and (<ref>) to arrive at 2b̅(x)/a̅(x) f_h”(x)≤C/μ(1 +1/ζ),x ≤ -ζ, C/μ,x ≥ -ζ.This proves (<ref>) and concludes the proof of this lemma. §.§ Proof of Lemma <ref> (W_2 Fourth Derivative) This section is devoted to proving Lemma <ref>.In this entire section, we reserve the variable x to be of the form x = x_k = δ (k - R), where k ∈_+. Let a(x) and b(x) be as in (<ref>) and (<ref>), respectively, and let r(x) = 2b(x)/a(x), whose form can be found in (<ref>). The form of f_h”'(x) in (<ref>) implies that for any y ∈, f_h”'(y)-f_h”'(x-) ≤r'(y)-r'(x-)f_h'(y) + r'(x-)f_h'(x)-f_h'(y)+ r(y)-r(x)f_h”(y) + r(x)f_h”(x)-f_h”(y)+ 2/a(x) - 2/a(y)h'(y) + 2/a(x)h'(x-)-h'(y)+ 2a'(x-)/a^2(x) - 2a'(y)/a^2(y)(h(y) +h(Y(∞))) + 2a'(x-)/a^2(x)h(x)-h(y).We first state a few auxiliary lemmas that will help us prove Lemma <ref>. These lemmas are proved at the end of this section. The first lemma deals with the case when y ∈ (x-δ, x). Fix h(x) ∈ W_2 with h(0)=0, and let f_h(x) be the solution to the Poisson equation (<ref>) that satisfies the conditions of Lemma <ref>. There exists a constant C>0 (independent of λ, n, and μ), such that for all x = x_k = δ (k - R) with k ∈_+, all y ∈ (x-δ, x), and all n ≥ 1, λ > 0, and μ > 0 satisfying 1 ≤ R < n,r'(y)-r'(x-)f_h'(y) + r'(x-)f_h'(x)-f_h'(y)≤Cδ/μ(1 +1/ζ)1(x ≤ -ζ),r(y)-r(x)f_h”(y) + r(x)f_h”(x)-f_h”(y)≤Cδ/μ[(1+x)(1 +1/ζ)1( x≤ -ζ) + ζ1(x ≥ -ζ + δ) ],2/a(x) - 2/a(y)h'(y) + 2/a(x)h'(x-)-h'(y)≤Cδ/μ,2a'(x-)/a^2(x) - 2a'(y)/a^2(y) h(Y(∞)) +2a'(x-)/a^2(x)h(x)-h(y)≤Cδ/μ(1 + 1/ζ) 1(x ∈ [-1/δ + δ, -ζ])2a'(x-)/a^2(x) - 2a'(y)/a^2(y)h(y)≤Cδ/μ 1(x ∈ [-1/δ + δ, -ζ]) The second lemma deals with the case when y ∈ (x, x+δ). Consider the same setup as in Lemma <ref>, but this time let y ∈ (x, x + δ). Then r'(y)-r'(x-)f_h'(y) + r'(x-)f_h'(x)-f_h'(y)≤Cδ/μ[ (1 +1/ζ) 1(x ≤ -ζ-δ)+ 1/δ(1 +1/ζ)1(x∈{-1/δ, -ζ}) ],r(y)-r(x)f_h”(y) + r(x)f_h”(x)-f_h”(y)≤Cδ/μ[(1+x)(1 +1/ζ)1( x≤ -ζ-δ) + ζ1(x ≥ -ζ ) ],2/a(x) - 2/a(y)h'(y) + 2/a(x)h'(x-)-h'(y)≤Cδ/μ,2a'(x-)/a^2(x) - 2a'(y)/a^2(y) h(Y(∞)) +2a'(x-)/a^2(x)h(x)-h(y)≤Cδ/μ(1 + 1/ζ)1(x ∈ [-1/δ, -ζ])2a'(x-)/a^2(x) - 2a'(y)/a^2(y)h(y)≤Cδ/μ[ 1(x ∈ [-1/δ + δ, -ζ - δ]) + 1/δ 1(x ∈{-1/δ, -ζ})] With these two lemmas, the proof of Lemma <ref> becomes trivial.When y ∈ (x-δ, x), we just apply (<ref>)–(<ref>) from Lemma <ref> to (<ref>) to get (<ref>). Similarly, for y ∈ (x,x+δ) we apply (<ref>)–(<ref>) of Lemma <ref> to (<ref>) to get (<ref>). This concludes the proof of Lemma <ref>. §.§.§ Proof of Lemma <ref> Fix k ∈_+, let x = x_k = δ(k - R), and fix y ∈ (x-δ, x). Throughout the proof we use C> 0 to denote a generic constant that may change from line to line, but does not depend on λ, n, and μ. To prove this lemma we verify (<ref>)–(<ref>), starting with (<ref>). Using the form of r'(x) = (2b(x)/a(x))' in (<ref>), we see thatr'(y)-r'(x-) = r'(y)-r'(x-) 1(x ∈ [-1/δ + δ, -ζ]). Furthermore, r”(u) exists for all u ∈ (-1/δ, -ζ), and from (<ref>) one can see thatr”(u) = 8δ/(2 + δ u)^3≤ 8δ,u ∈ (-1/δ, -ζ).Therefore, r'(y)-r'(x-)f_h'(y)≤f_h'(y)1(x ∈ [-1/δ + δ, -ζ]) ∫_x-δ^xr”(u) du≤Cδ^2/μ(1 +1/ζ) 1(x ∈ [-1/δ + δ, -ζ ]),where in the last inequality we used the gradient bound (<ref>). Furthermore, we observe thatr'(x-)≤4 × 1(x ≤ -ζ), f_h'(x) - f_h'(y)≤∫_x-δ^xf_h”(u) du ≤Cδ/μ[(1 +1/ζ)1(x ≤ -ζ) + 1/ζ 1(x ≥ -ζ+δ)],where in the first line we used the form of r'(x) from (<ref>), and in the second line we used the gradient bound (<ref>). Recalling that δ≤ 1, we conclude thatr'(y)-r'(x-)f_h'(y) + r'(x-)f_h'(x)-f_h'(y)≤Cδ/μ(1 +1/ζ)1(x ≤ -ζ).This proves (<ref>), and we move on to show (<ref>). Observe thatr(x)≤2x1(x ≤ -ζ) + ζ 1(x ≥ -ζ+δ),r(x) - r(y)≤∫_x-δ^xr'(u) du ≤ 4δ 1(x ≤ -ζ),f_h”(y)≤C/μ[(1 +1/ζ)1(x ≤ -ζ) + 1/ζ 1(x ≥ -ζ+δ)],f_h”(x) - f_h”(y)≤∫_x-δ^xf_h”'(u) du ≤Cδ/μ[(1 + 1/ζ)1(x ≤ -ζ) +1(x ≥ -ζ+δ)],where the first two lines above are obtained using the form of r(x) in (<ref>), and in the last two lines we used the gradient bounds (<ref>) and (<ref>). Combining the bounds above proves (<ref>), and we move on to prove (<ref>). Observe that2/a(x)≤2/μ,2/a(x) - 2/a(y)≤2∫_x-δ^xa'(u)/a^2(u) du ≤2δ/μ 1(x ∈ [-1/δ + δ, -ζ]),h'(x-)≤1,and h'(x-) - h'(y)≤h”x-y≤δ,where in the first two lines we used the forms of a(x) and a'(x) from (<ref>) and (<ref>), and in the last line we used the fact that h(x) ∈ W_2. Combining these bounds proves (<ref>), and we move on to prove (<ref>). Observe that 2a'(x-)/a^2(x) =2δ/μ (2+δ x)^2 1(x ∈ [-1/δ + δ, -ζ]) ≤2δ/μ1(x ∈ [-1/δ + δ, -ζ]), 2a'(y)/a^2(y)≤2δ/μ1(x ∈ [-1/δ + δ, -ζ]),h(Y(∞))≤| Y(∞)|,and h(x)-h(y)≤h'x-y≤δ, where in the first line we used the forms of a(x) and a'(x) from (<ref>) and (<ref>), and in the last line we used the fact that h(x) ∈ W_2. We use the bounds above together with (<ref>) and the fact that δ≤ 1 to see that2a'(x-)/a^2(x) - 2a'(y)/a^2(y) h(Y(∞)) + 2a'(x-)/a^2(x)h(x)-h(y) ≤2a'(x-)/a^2(x)| Y(∞)| + 2a'(y)/a^2(y)| Y(∞)| + 2δ^2/μ1(x ∈ [-1/δ + δ, -ζ])≤Cδ/μ(1 + 1/ζ)1(x ∈ [-1/δ + δ, -ζ]) + 2δ^2/μ1(x ∈ [-1/δ + δ, -ζ])≤Cδ/μ(1 + 1/ζ)1(x ∈ [-1/δ + δ, -ζ]),which proves (<ref>). Lastly we show (<ref>). Observe that2a'(x-)/a^2(x) - 2a'(y)/a^2(y) = 2a'(x-)/a^2(x) - 2a'(y)/a^2(y) 1(x ∈ [-1/δ + δ, -ζ]), and that the derivative of 2a'(u-)/a^2(u) exists for all u ∈ (-1/δ, -ζ) and satisfies (2a'(u)/a^2(u))' = 4δ^2/μ (2 + δ u)^3,u ∈ (-1/δ, -ζ).Recalling that h(y)≤y, we see that2a'(x-)/a^2(x) - 2a'(y)/a^2(y)h(y)≤1(x ∈ [-1/δ + δ, -ζ]) ∫_x-δ^xy(2a'(u)/a^2(u))' du=1(x ∈ [-1/δ + δ, -ζ]) ∫_x-δ^x4δ/μ (2 + δ u)^2δ y/(2 + δ u) du≤1(x ∈ [-1/δ + δ, -ζ]) 4δ/μ∫_x-δ^xδ y/ (2 + δ u) du≤1(x ∈ [-1/δ + δ, -ζ]) 4δ/μδ(δ^2+1),where to obtain the last inequality, we used the fact that y-u≤δ and δ u ≥ -1 to see thatδ y/ (2 + δ u) = δ (y-u) + δ u/ (2 + δ u)≤δ^2 + δ u/ 2 + δ u≤δ^2 + 1.Recalling that δ≤ 1 establishes (<ref>), and concludes the proof of this lemma. §.§.§ Proof of Lemma <ref> Fix k ∈_+, let x = x_k = δ(k - R), and fix y ∈ (x,x+δ). Throughout the proof we use C> 0 to denote a generic constant that may change from line to line, but does not depend on λ, n, and μ. The proof for this lemma is very similar to the proof of Lemma <ref>. In most cases, the only adjustment necessary to the proof is to consider cases when x ≤ -ζ - δ and x ≥ -ζ, instead of x ≤ -ζ and x ≥ -ζ + δ. We now verify (<ref>)–(<ref>) in order, starting with (<ref>). Using the form of r'(x) in (<ref>), we see thatr'(y)-r'(x-) =r'(y)-r'(x-) 1(x ∈ [-1/δ + δ, -ζ- δ]) + (r'(y)+2) 1(x = -1/δ) + r'(x-) 1(x = -ζ). Therefore, r'(y)-r'(x-)f_h'(y)=r'(y)-r'(x-)f_h'(y) 1(x ∈ [-1/δ + δ, -ζ- δ]) + (r'(y)+2)f_h'(y)1(x = -1/δ) + r'(x-)f_h'(y)1(x = -ζ)≤Cδ^2/μ(1 +1/ζ) 1(x ∈ [-1/δ + δ, -ζ - δ]) + C/μ(1 +1/ζ) 1(x = -1/δ) + r'(x-)f_h'(y)1(x = -ζ),where in the last inequality, the first term is obtained just like in (<ref>), and the second term comes from the gradient bound (<ref>) and the fact that r'(y)≤ 4, which can be seen from (<ref>). Now using the gradient bounds (<ref>) and (<ref>), together with the facts that r'(ζ-)≤ 4 and δ≤ 1, we see thatr'(x-)f_h'(y)1(x = -ζ)≤r'(x-)f_h'(x)1(x = -ζ) + r'(x-)f_h'(x) - f_h'(y)1(x = -ζ)≤C/μ(1 +1/ζ) 1(x = -ζ) + r'(x-) 1(x = -ζ) ∫_-ζ^-ζ + δf_h”(u) du≤C/μ(1 +1/ζ) 1(x = -ζ),and thereforer'(y)-r'(x-)f_h'(y)≤Cδ/μ(1 +1/ζ) 1(x ∈ [-1/δ + δ, -ζ - δ]) + C/μ(1 +1/ζ) 1(x ∈{-1/δ, -ζ}).Furthermore, r'(x-)f_h'(x)-f_h'(y)≤r'(x-)∫_x^x+δf_h”(u) du ≤Cδ/μ(1 +1/ζ)1(x ≤ -ζ-δ) + Cδ/μζ1(x = -ζ) ,where in the second inequality we used that r'(x)≤ 4 and the gradient bound in (<ref>). Recalling that δ≤ 1, we can combine the bounds above to see that r'(y)-r'(x-)f_h'(y) + r'(x-)f_h'(x)-f_h'(y) ≤Cδ/μ[ (1 +1/ζ) 1(x ≤ -ζ-δ)+ 1/δ(1 +1/ζ)1(x∈{-1/δ, -ζ}) ],which proves (<ref>). The proofs for (<ref>), (<ref>), and (<ref>), are nearly identical to the proofs of (<ref>), (<ref>), and (<ref>) from Lemma <ref>, respectively, and we do not repeat them here. The only differences to note is that (<ref>) is separated into the cases x ≤ -ζ - δ and x ≥ -ζ, as opposed to (<ref>) which has x ≤ -ζ and x ≥ -ζ + δ. Likewise, (<ref>) contains 1(x ∈ [-1/δ, -ζ]), whereas (<ref>) contains 1(x ∈ [-1/δ + δ, -ζ]).Lastly we prove (<ref>). From the form of a'(x) in (<ref>), we see that2a'(x-)/a^2(x) - 2a'(y)/a^2(y) =2a'(x-)/a^2(x) - 2a'(y)/a^2(y) 1(x ∈ [-1/δ + δ, -ζ-δ]) + 2a'(y)/a^2(y) 1(x = -1/δ) + 2a'(x-)/a^2(x-) 1(x = -ζ).We can repeat the argument from (<ref>) to get2a'(x-)/a^2(x) - 2a'(y)/a^2(y)h(y) ≤4δ/μδ(δ^2+1)1(x ∈ [-1/δ + δ, -ζ - δ])+2a'(y)/a^2(y)y1(x=-1/δ) +2a'(x-)/a^2(x-)y1(x = -ζ).Then using (<ref>), the form of a'(x) in (<ref>), and the fact that a(x) ≤ 1/μ, we can bound the term above byCδ/μ1(x ∈ [-1/δ + δ, -ζ - δ]) +2/a(y)ya'(y)/a(y)1(x=-1/δ)+2/a(x-)(x+δ)a'(x-)/a(x-)1(x = -ζ)≤Cδ/μ1(x ∈ [-1/δ + δ, -ζ - δ]) + C/μ1(x=-1/δ) + C/μ1(x = -ζ).Hence,2a'(x-)/a^2(x) - 2a'(y)/a^2(y)h(y)≤Cδ/μ[ 1(x ∈ [-1/δ + δ, -ζ - δ]) + 1/δ 1(x ∈{-1/δ, -ζ})],which proves (<ref>) and concludes the proof of this lemma. §.§ Proof of Lemma <ref> (Kolmogorov Bounds)From (<ref>) and (<ref>) Its not hard to check thatf_z'(w) =(Y_S ≥ z)e^-∫_0^w2b̅(u)/a̅(u)du∫_-∞^w2/a̅(y)e^∫_0^y2b̅(u)/a̅(u)du dy ,w ≤ z,(Y_S ≤ z)e^-∫_0^w2b̅(u)/a̅(u)du∫_w^∞2/a̅(y) e^∫_0^y2b̅(u)/a̅(u)du dy ,w ≥ z,In fact, for w ≥ z ≥ -ζ, f_z'(w) =e^-∫_0^-ζ2b̅(y)/a̅(y)dye^-(w+ζ) 2b̅(-ζ)/a̅(-ζ)(Y_S ≤ z)∫_w^∞2/a̅(y) e^∫_0^-ζ2b̅(u)/a̅(u)due^(y+ζ) 2b̅(-ζ)/a̅(-ζ) dy =e^-w 2b̅(-ζ)/a̅(-ζ)2(Y_S ≤ z)/a̅(-ζ)∫_w^∞e^y 2b̅(-ζ)/a̅(-ζ) dy =-(Y_S ≤ z)/b̅(-ζ),and hence, f_z”(w) = 0 for w ≥ z. Applying (<ref>) to the form of f_z'(w) tells us that for x ≤ -ζ, f_z'(w)≤1/μe^ζ^2(3+ζ),which proves (<ref>). To prove the rest of the bounds on f_z”(w), we differentiate f_z'(w) to see that for w ≤ z, 1/(Y_S ≥ z)f_z”(w) = -2b̅(w)/a̅(w)e^-∫_0^w2b̅(u)/a̅(u)du∫_-∞^w2/a̅(y)e^∫_0^y2b̅(u)/a̅(u)du dy + 2/a̅(w).We claim that the right hand side above is bounded by 2/a̅(w) ≤ 2/μ when w ≤ 0. This is true for w = 0 because b̅(0) = 0. For w < 0, we use (<ref>) to see that 2b̅(w)/a̅(w)e^-∫_0^w2b̅(u)/a̅(u)du∫_-∞^w2/a̅(y)e^∫_0^y2b̅(u)/a̅(u)du dy ≤2/a̅(w).Combining this with the fact that b̅(w) > 0 for w < 0 verifies our claim. When w ∈ [0,-ζ], we apply (<ref>) and the fact that b̅(w) = μ w to (<ref>) to conclude that1/(Y_S ≥ z)f_z”(w)≤2/a̅(w)(μ w1/μe^ζ^2 (3 + ζ)+1) ≤C/μe^ζ^2(w(1+ζ)+1),where in the last inequality we used the fact that 2/a̅(w) ≤ 2/μ. This proves (<ref>), and it remains to prove the bound on f_z”(w) in the case when w ∈ [-ζ, z]. Observe that e^-∫_0^w2b̅(u)/a̅(u)du∫_-∞^w2/a̅(y)e^∫_0^y2b̅(u)/a̅(u)du dy =e^-∫_0^-ζ2b̅(u)/a̅(u)due^-∫_-ζ^w2b̅(u)/a̅(u)du∫_-∞^-ζ2/a̅(y)e^∫_0^y2b̅(u)/a̅(u)du dy + e^-∫_0^-ζ2b̅(u)/a̅(u)due^-∫_-ζ^w2b̅(u)/a̅(u)du∫_-ζ^w2/a̅(y)e^∫_0^-ζ2b̅(u)/a̅(u)due^∫_-ζ^y2b̅(u)/a̅(u)du dy ≤e^-∫_-ζ^w2b̅(u)/a̅(u)du1/μe^ζ^2 (3 + ζ)+ e^-∫_-ζ^w2b̅(u)/a̅(u)du∫_-ζ^w2/a̅(y)e^∫_-ζ^y2b̅(u)/a̅(u)du dy=e^-(w+ζ)2b̅(-ζ)/a̅(-ζ)1/μe^ζ^2 (3 + ζ)+ e^-(w+ζ)2b̅(-ζ)/a̅(-ζ)∫_-ζ^w2/a̅(-ζ)e^(y+ζ)2b̅(-ζ)/a̅(-ζ) dy=e^-(w+ζ)2b̅(-ζ)/a̅(-ζ)1/μe^ζ^2 (3 + ζ)+ 1/b̅(-ζ)(1 - e^-(w+ζ)2b̅(-ζ)/a̅(-ζ)) ≤e^-(w+ζ)2b̅(-ζ)/a̅(-ζ)1/μe^ζ^2 (3 + ζ)+ 1/b̅(-ζ) e^-(w+ζ)2b̅(-ζ)/a̅(-ζ).We combine the above inequality with (<ref>) to see that for w ∈ [-ζ, z],and therefore 1/(Y_S ≥ z)|f_z”(w)| =| -2b̅(-ζ)/a̅(-ζ)e^-∫_0^w2b̅(u)/a̅(u)du∫_-∞^w2/a̅(y)e^∫_0^y2b̅(u)/a̅(u)du dy + 2/a̅(-ζ)|≤2b̅(-ζ)/a̅(-ζ)(e^-(w+ζ)2b̅(-ζ)/a̅(-ζ)1/μe^ζ^2 (3 + ζ)+ 1/b̅(-ζ) e^-(w+ζ)2b̅(-ζ)/a̅(-ζ))+ 1/μ ≤1/μe^-(w+ζ)2b̅(-ζ)/a̅(-ζ)e^ζ^2 (3ζ + ζ^2)+ 1/μ e^-(w+ζ)2b̅(-ζ)/a̅(-ζ)+ 1/μ ≤1/μe^w2b̅(-ζ)/a̅(-ζ)e^ζ^2 (3ζ + ζ^2)+ 1/μ e^w2b̅(-ζ)/a̅(-ζ)+ 1/μ ≤1/μe^w2b̅(-ζ)/a̅(-ζ)e^ζ^2 (1+3ζ + ζ^2)+ 1/μ,where in the first inequality we used the fact that a(w) ≥ 2μ for w ∈ [-ζ, z]. This proves (<ref>) once we recall that 2b̅(-ζ)/a̅(-ζ) = 2ζ/(2+δζ).§ GRADIENT BOUNDS FOR CHAPTER <REF>In this section, we proveLemma <ref>. We adopt the notation from Chapter <ref>. Before proving the lemma, we introduce an important common quadratic Lyapunov function from <cit.>.This Lyapunov function plays a key role in the proof of this lemma. As in (5.24) of <cit.>, for x ∈^d, defineV(x) = (e^Tx)^2 + κ[x-p ϕ(e^Tx)]' M [x-p ϕ(e^Tx)],where κ>0 is some constant, M is some d× d positive definite matrix, and the function ϕ is a smooth approximation to x ⟼ x^+ and is defined byϕ(x) = x, if x ≥ 0,-1/2ϵ, if x ≤ -ϵ,smooth, if -ϵ < x < 0. In (5.24) of <cit.>, the authors use Q̃ to represent the positive definite matrix that we called M in (<ref>). We use M instead of Q̃ on purpose, to avoid any potential confusion with the queue size Q(t).For our purposes, “smooth" means that ϕ can be anything as long as ϕ∈ C^3(^d).We require that the “smooth" part of ϕ also satisfies -1/2ϵ < ϕ (x) < x and 0 ≤ϕ'(x)≤ 1. For example, ϕ can be taken to be a polynomial of sufficiently high degree on (-ϵ, 0) and this will satisfy our requirements.The vector p is as in (<ref>). The constant κ and matrix M are chosen just as in <cit.>; their exact values are not important to us. In their paper, they show that V(x) satisfiesG_YV(x) ≤-c_1V(x) + c_2for allx∈^dfor some positive constants c_1,c_2; this result requires α > 0, i.e. a strictly positive abandonment rate. Before proceeding to the proof of Lemma <ref>, we state two bounds on V(x) that shall be useful in the future. For some constant C>0,V(x) ≤ C(1+ x^2),x^2 ≤ C(1+V(x)).The first is immediate from the form of V(x), while the second is proved in <cit.>.Without loss of generality, we may assume that h(0) = 0, otherwise one may consider h(x) -h(0). This lemma is essentially a restatement of equation (22) and equation (40) from the discussion that follows after <cit.>. We verify that (22) and (40) are applicable in our case by first confirming that we have a function satisfying assumption 3.1 of <cit.>. Recalling the definition of V(x) from (<ref>), when ϕ is taken to be a polynomial (of sufficiently high degree to guarantee V(x) ∈ C^3(^d)), the function 1+V(x)satisfies assumption 3.1. To verify condition (17) of Assumption 3.1, one observes that X^(λ)(t) ≤X^(λ)(0) + n + A^(λ)(t),where A^(λ)(t) is the total number of arrivals to the system by time t and it is a Poisson random variable with mean λ t for each t≥ 0. The properties of Poisson processes then yield (17). By <cit.>,C(1+V(x))^malso satisfies assumption 3.1 for any constant C>0. Since we require that h(x)≤x^m, by (<ref>) we have h(x) - h(Y(∞)) ≤x^m + Y(∞)^m ≤C_m (1+V(x))^m.The finiteness of Y(∞)^m is guaranteed because one of the conditions of assumption 3.1 is thatG_Y (1+V(x))^m ≤-c_1 (1+V(x))^m + c_2for some positive constants c_1 and c_2. Therefore, equation (22) gives us (<ref>) and equation (40) gives us (<ref>) and (<ref>). We get (<ref>) by observing that in the discussion preceding (40), everything still holds if we replace B_x(l̅ / √(n)) by an open ball of radius 1 centered at x. We wish to point out that the constants in (40) and (22) do not depend on the choice of function h(x). | http://arxiv.org/abs/1704.08398v1 | {
"authors": [
"Anton Braverman"
],
"categories": [
"math.PR"
],
"primary_category": "math.PR",
"published": "20170427011031",
"title": "Stein's method for steady-state diffusion approximations"
} |
Long-wavelength deformations and vibrational modesin empty and liquid-filled microtubules and nanotubes:A theoretical study David Tománek December 30, 2023 ===================================================================================================================================================This work introduces a novel framework for quantifying the presence and strength of recurrent dynamics in video data. Specifically, we provide continuous measures of periodicity (perfect repetition) and quasiperiodicity (superposition of periodic modes with non-commensurate periods), in a way which does not require segmentation, training, object tracking or 1-dimensional surrogate signals. Our methodology operates directly onvideo data. The approach combines ideas from nonlinear time series analysis (delay embeddings) and computational topology (persistent homology), by translating the problem of finding recurrent dynamics in video data, into the problem of determining the circularity or toroidality of an associated geometric space. Through extensive testing, we show the robustness of ourscores with respect to several noise models/levels; we show that our periodicity score is superior to other methods when compared to human-generated periodicity rankings; and furthermore, we show that our quasiperiodicity score clearly indicates the presence of biphonation in videos of vibrating vocal folds, which has never before been accomplished end to end quantitatively.§ INTRODUCTIONPeriodicity characterizes many natural motions including animal locomotion (walking/wing flapping/slithering), spinning wheels, oscillating pendulums, etc.Quasiperiodicity, thought of as the superposition ofnon-commensurate frequencies, occursnaturally during transitions from ordinary to chaotic dynamics <cit.>.The goal of this work is to automate the analysis of videos capturing periodic and quasiperiodic motion. In order to identify both classes of motion in a unified framework, we generalize 1-dimensional (1D) sliding window embeddings <cit.> to reconstruct periodic and quasiperiodic attractors from videos[Some of the analysis and results appeared as part of the Ph.D. thesis of the first author <cit.>.][Code to replicate results: <https://github.com/ctralie/SlidingWindowVideoTDA>][Supplementary material and videos: <https://www.ctralie.com/Research/SlidingWindowVideoQuasi>].We analyze the resulting attractors usingpersistent homology, a technique which combines geometry and topology (Section <ref>), and we return scores in the range [0, 1] that indicate the degree of periodicity or quasiperiodicity in the corresponding video.We show that our periodicity measure compares favorable to others in the literature when ranking videos (Section <ref>).Furthermore, to our knowledge, there is no other method able to quantify the existence of quasiperiodicity directly from video data.Our approach is fundamentally different from most others which quantify periodicity in video.For instance, it is common to derive 1D signals from the video and apply Fourier or autocorrelation to measure periodicity.By contrast, our technique operates on raw pixels, avoiding common video preprocessing and tracking entirely.Using geometry over Fourier/autocorrelation also has advantages for our applications.In fact, as a simple synthetic example shows (Figure <ref>), the Fourier Transform of quasiperiodic signals is often very close to the Fourier transform of periodic signals.By contrast, the sliding window embeddings we design yield starkly different geometric structures in the periodic and quasiperiodic cases.We exploit this to devise a quasiperiodicity measurement, which we use to indicate the degree of “biphonation” in videos of vibrating vocal folds (Section <ref>), which is useful in automatically diagnosing speech pathologies.In the context of applied topology, our quasiperiodicity score is one of the first applications of persistent H_2 to high dimensional data, which is largely possible due to recent advancements in the computational feasibility of persistent homology <cit.>.§.§ Prior Work on Recurrence in Videos §.§.§ 1D Surrogate SignalsOne common strategy for detecting periodicity in video is to derive a 1D functionto act as a surrogate for its dynamics, and then to use either frequency domain (Fourier transform) or time domain (autocorrelation, peak finding) techniques. One of the earliest works in this genre finds level set surfaces in a spatiotemporal “XYT” volume of video (all frames stacked on top of each other), and then uses curvature scale space on curves that live on these “spatiotemporal surfaces” as the 1D function <cit.>.<cit.> use Fourier Transforms on pixels which exhibit motion, and define a measure of periodicity based on the energy around the Fourier peak and its harmonics.<cit.> extract contours and find eigenshapes from the contours to classify and parameterize motion within a period.Frequency estimation is done by using Fourier analysis and peak detection on top of other 1D statistics derived from the contours, such as area and center of mass.Finally, <cit.> derive a 1D surrogate function based on mutual information between the first and subsequent frames, and then look for peaks in the similarity function with the help of a watershed method.§.§.§ Self-Similarity Matrices Another class of techniques relies on self-similarity matrices (SSMs) between frames, where similarity can be defined in a variety of ways.<cit.> track a set of points on a foreground object and compare them with an affine invariant similarity.Another widely recognized technique for periodicity quantification <cit.>, derives periodicity measures based on self-similarity matrices of L1 pixel differences.This technique has inspired a diverse array of applications, including analyzing the cycles of expanding/contracting jellyfish <cit.>, analyzing bat wings <cit.>, and analyzing videos of autistic spectrum children performing characteristic repetitive motions such as “hand flapping” <cit.>.We compare to this technique in Section <ref>. §.§.§ Miscellaneous Techniques for Periodic Video QuantificationThere are also a number of works that don't fall into the two categories above.Some works focus solely on walking humans, since that is one of the most common types of periodic motion in videos of interest to people.<cit.> look at the “braiding patterns” that occur in XYT slices of videos of walking people.<cit.> perform blob tracking on the foreground of a walking person, and use the ratio of the second and first eigenvalues of PCA on that blob.For more general periodic videos, <cit.> make a codebook of visual words and look for repetitions within the resulting string.<cit.> take a deep learning approach to counting the number of periods that occur in a video segment.They use a 3D convolutional neural network on spatially downsampled, non-sequential regions of interest, which are uniformly spaced in time, to estimate the length of the cycle. Finally, perhaps the most philosophically similar work to ours is the work of <cit.>, who use cohomology to find maps of MOCAP data to the circle for parameterizing periodic motions, though this work does not provide a way to quantify periodicity. §.§.§ Our Work We showthat geometry provides a natural way to quantify recurrence (i.e. periodicity and quasiperiodicity) in video, by measuring the shape of delay embeddings. In particular, we propose several optimizations (section <ref>) which make this approach feasible. The resultingmeasure of quasiperiodicity, for which quantitative approaches are lacking, is used in section <ref> todetect anomalies in high-speed videos of vibrating vocal folds.Finally, in contrast to both frequency and time domain techniques, our method does not rely on the period length being an integer multiple of the sampling rate.§ BACKGROUND§.§ Delay Embeddings And Their Geometry Recurrence in video data can be captured via the geometry of delay embeddings; we describe this next.§.§.§ Video Delay EmbeddingsWe will regard a video as a sequence of grayscale[For color videos we can treat each channel independently, yielding a vector in ℝ^W × H × 3.In practice, there isn't much of a difference between color and grayscale embeddings in our framework for the videos we consider.] image frames indexed by the positive real numbers. That is, given positive integers W (width) andH (height), avideo with W× H pixels is afunction[ X: ℝ^+ ⟶ℝ^W× H ]In particular, a sequence ofimages X_1,X_2,…∈ℝ^W× H sampled at discrete times t_1 < t_2 < ⋯ yields one such function via interpolation. For an integer d ≥ 0, known as the dimension, a real number τ > 0, known as the delay, and a video X : ℝ^+ ⟶ℝ^W× H, we define the sliding window (also referred to as time delay) embedding of X – with parameters d and τ – at time t∈ℝ^+ as the vector SW_d, τ X(t) = [ [X(t);X(t + τ); ⋮; X(t + dτ) ]] ∈ℝ^W× H × (d+1) The subset of ℝ^W× H× (d+1) resulting from varying t will be referred to as the sliding window embedding of X. We remark that since the pixel measurement locations are fixed, the sliding window embedding is an “Eulerian” view into the dynamics of the video.Note that delay embeddings are generally applied to 1D time series, which can be viewed as 1-pixel videos (W = H = 1) in our framework. Hence equation (<ref>) is essentially the concatenation of the delay embeddings of each individual pixel in the video into one large vector. One of the main points we leverage in this paper is the fact that the geometry of the sliding window embedding carries fundamental information about the original video. We explore this next. §.§.§ Geometry of 1-Pixel Video Delay Embeddings As a motivating example, consider the harmonic (i.e. periodic) signalf_h(t) =cos(π/5 t) +cos(π/15 t )and the quasiperiodic signalf_q(t) =cos(π/5 t) +cos(1/5 t )We refer to f_has harmonic because its constitutivefrequencies, 1/10 and 1/30, are commensurate; that is, they are linearly dependent over the rational numbers ℚ⊂ℝ. By way of contrast, the underlying frequencies of thesignal f_q, 1/10 and 1/10π, are linearly independent over ℚ and hence non-commensurate. We use the term quasiperiodicity, as in the non-linear dynamics literature<cit.>, to denote the superposition of periodic processes whose frequencies are non-commensurate. This differs fromother definitions in the literature (e.g. <cit.>) which regard quasiperiodic as any deviation from perfect repetition.A geometric argument from<cit.> (see equation <ref> below and the discussion that follows) shows that given a periodic function f : [0,2π] ⟶ℝ with exactly N harmonics, ifd ≥ 2N and0 < τ < 2π/dthen the sliding window embedding SW_d,τf is atopological circle (i.e. a closed curve without self-intersections) which wraps around an N-dimensional torus𝕋^N = S^1 ×⋯× S^1_N-times, S^1 = {z∈ℂ: |z|= 1}As an illustration, we show in Figure <ref> a plot of f_h andof its sliding window embedding SW_d,τ f_h, via a PCA (Principal Component Analysis) 3-dimensional projection.However, if g: ℝ⟶ℝ is quasiperiodic with N distinct non-commensurate frequencies then, for appropriate d and τ, SW_d,τg is dense in (i.e. fills out) 𝕋^N <cit.>.Figure <ref> shows a plot of the quasiperiodic signal f_q(t) and a 3-dimensional projection, via PCA, of its sliding window embedding SW_d,τ f_q. The difference in geometry of the delay embeddings is stark compared to the difference between their power spectral densities, as shown in Figure <ref>.Moreover, as we will see next, theinterpretation of periodicity and quasiperiodicity as circularity and toroidality ofsliding window embeddings remains true for videos with higher resolution (i.e. max{W,H} > 1). The rest of the paper will show how one can use persistent homology, a tool from the field of computational topology, to quantify the presence of (quasi)periodicity in a video by measuring the geometry of its associated sliding window embedding. In short, we propose a periodicity score for a video X which measures the degree to which the sliding window embedding SW_d,τ X spans a topological circle, and a quasiperiodicity score which quantifies the degree to which SW_d,τ X covers a torus. This approach will be validated extensively: we show that our (quasi)periodicity detection method is robust under several noise models (motion blur, additive Gaussian white noise, and MPEG bit corruption); we compare several periodicity quantification algorithms and show that our approach is the most closely aligned with human subjects; finally, we provide an application to the automatic classification of dynamic regimes in high-speed laryngeal video-endoscopy.§.§.§ Geometry of Video Delay EmbeddingsThough it may seem daunting compared to the 1D case, the geometry of the delay embedding shares many similarities for periodic videos, as shown in <cit.>. Let us argue whysliding window embeddings from (quasi)periodic videos have the geometry we have described so far. To this end, consider an example video X that contains a set of N frequencies ω_1, ω_2, ..., ω_N.Let the amplitude of the n^th frequency and i^th pixel be a_in.For simplicity, but without loss of generality, assume that each is a cosine withzero phase offset.Then the time series at pixel i can be written asX_i(t) = ∑_n = 1^N a_incos(ω_n t) Grouping all of the coefficients together into a (W× H) × N matrix A, we can writeX(t) = ∑_n = 1^N A^n cos(ω_n t)where A^n stands for the n^th column of A.Constructing a delay embedding as in Equation <ref>:SW_d, τX(t) = ∑_n = 1^N [ [A^n cos(ω_n t); ⋮; A^n cos(ω_n (t + d τ) ]]and applying the cosine sum identity, we getSW_d, τX(t) = ∑_n = 1^N u⃗_ncos(ω_n t) - v⃗_n sin(ω_n t)where u⃗_n, v⃗_n ∈ℝ^W × H× (d+1) are constant vectors. In other words, the sliding window embedding of this video is the sum of linearly independent ellipses, which lie in the space of d+1 frame videos at resolution W × H. As shown in <cit.> for the case of commensurate frequencies, when the window length is just under the length of the period, all of the u⃗_n and v⃗_n vectors become orthogonal, and so they can be recovered by doing PCA on SW_d, τX(t).Figure <ref> shows the components of the first 8 PCA vectors for a horizontal line of pixels in a video of an oscillating pendulum. Note how the oscillations are present both temporally and spatially. §.§.§ The High Dimensional Geometry of Repeated PulsesUsing Eulerian coordinates has an important impact on the geometry of delay embeddings of natural videos.As Figure <ref> shows, pixels often jump from foreground to background in a pattern similar to square waves.These types of abrupt transitions require higher dimensional embeddings to reconstruct the geometry.To see why, first extract one period of a signal with period ℓ at a pixel X_i(t):f_i(t) = {[X_i(t)0 ≤ t ≤ℓ; 0 otherwise ]}Then X_i(t) can be rewritten in terms of the pulse asX_i(t) = ∑_m = -∞^∞ f_i(t - mℓ)Since X_i(t) repeats itself, regardless of what f_i(t) looks like, periodic summation discretizes the frequency domain <cit.>ℱ{X_i(t) }(k) ∝∑_m = -∞^∞ℱ(f_i(t)) ( m/ℓ) δ( m/ℓ - k )Switching back to the time domain, we can write X_i(t) asX_i(t) ∝∑_m = -∞^∞ℱ( f_i(t) ) (m/ℓ) e^i 2 π m/ℓ t In other words, each pixel is the sum of some constant offset plus a (possibly infinite) set of harmonics at integer multiples of 1/ℓ.For instance, applying Equation <ref> to a square wave of period ℓ centered at the origin is a roundabout way of deriving the Fourier Seriessin( 2 π/ℓ t ) + 1/3sin( 6 π/ℓ t ) + 1/5sin( 10 π/ℓ t ) +by sampling the sinc function sin(πℓ f)/(π f) at intervals of m/2ℓ (every odd m coincides with π/2 + k π, proportional to 1/k, and every even harmonic is zero conciding with π k).In general, the sharper the transitions are in X_i(t), the longer the tail of ℱ{f_i(t)} will be, and the more high frequency harmonics will exist in the embedding, calling for a higher delay dimension to fully capture the geometry, since every harmonic lives on a linearly independent ellipse.Similar observations about harmonics have been made in images for collections of patches around sharp edges (<cit.>, Figure 2).§.§ Persistent Homology Informally, topology is the study of properties of spaces which do not change after stretching without gluing or tearing. For instance, the number of connected componentsand the number of(essentially different) 1-dimensional loops which do not bound a 2-dimensional disk, are both topological properties of a space. It follows that a circle and a square are topologically equivalent since one can deform one onto the other, buta circle and a line segment are not because that would require either gluing the endpoints of the line segment or tearing the circle. Homology <cit.> is a tool from algebraic topology designed to measure these types of properties, and persistent homology <cit.> is an adaptation of these ideas to discrete collections of points (e.g., sliding window embeddings). We briefly introduce these concepts next.§.§.§ Simplicial ComplexesA simplicial complex is a combinatorial object used to represent and discretize a continuous space. With a discretization available, one can then compute topological properties by algorithmic means. Formally, a simplicial complex with vertices in a nonempty set V is a collection K of nonempty finite subsets σ⊂ V so that ∅≠τ⊂σ∈ K always implies τ∈ K. An element σ∈ K is called a simplex, and ifσ has (n+1) elements then it is called an n-simplex. The cases n=0,1,2 are special,0-simplices are called vertices, 1-simplices are called edges and 2-simplices are called faces. Here is an example to keep in mind: the circle S^1 = {z∈ℂ : |z| =1 } is a continuous space but its topology can be captured by a simplicial complex K with three vertices a,b,c, and three edges {a,b},{b,c}, {a,c}. That is, in terms of topological properties, the simplicial complexK ={{a},{b},{c}, {a,b}, {b,c}, {a,c}}can be regarded as a combinatorial surrogate forS^1: they both have 1 connected component,one 1-dimensional loop which does not bound a 2-dimensional region, and no other features in higher dimensions.§.§.§ Persistent Homology of Point CloudsThe sliding window embedding of a video X is, in practice,a finite set 𝕊𝕎_d,τX = {SW_d,τX(t):t∈ T} determined by a choice of T⊂ℝfinite. Moreover, since 𝕊𝕎_d,τX ⊂ℝ^W× H × (d+1) then the restriction of the ambient Euclidean distance endows 𝕊𝕎_d,τX with the structure of a finite metric space. Discrete metric spaces, also referred to as point clouds, are trivial from a topological point of view: a point cloud with Npoints simply has N connected components and no other features (e.g., holes) in higher dimensions. However, when a point cloud has been sampled from/around a continuous space with non-trivial topology (e.g., a circle or a torus), one would expect that appropriate simplicial complexes with vertices on the point cloud should reflect the topology of the underlying continuous space. This is what we will exploit next. Given a point cloud (𝕏, d_𝕏)– where 𝕏 is a finite set and d_𝕏: 𝕏×𝕏⟶ [0,∞) is a distance function – the Vietoris-Rips complex (or Rips complex for short) at scale ϵ≥0 is the collection of non-emptysubsets of 𝕏 with diameter less than or equal to ϵ:R_ϵ(𝕏) := {σ⊂𝕏 : d_𝕏(x_1, x_2) ≤ϵ,∀ x_i, x_j ∈σ}That is, R_ϵ(𝕏) is the simplicial complex with vertex set equal to 𝕏, constructed by addingan edge between any two vertices which are at most ϵ apart,adding all 2-dimensional triangular faces (i.e. 2-simplices) whose bounding edges are present, and more generally, adding all the k-simplices whose (k-1)-dimensional bounding facets have been included. We show in Figure <ref> the evolution of the Rips complex on a set of points sampled around the unit circle. The idea behind persistent homology is to track the evolution of topological features of complexes such asR_ϵ(𝕏), as the scale parameter ϵ ranges from 0 to some maximum value ϵ_𝗆𝖺𝗑≤∞. For instance, in Figure <ref> one can see that R_0 (𝕏) = 𝕏 has 40 distinct connect components (one for each point), R_0.30(𝕏) has three connected componentsand R_0.35(𝕏) has only one connected component; this will continue to be the case for every ϵ≥ 0.35. Similarly, there are no closed loops in R_0(𝕏) or R_0.30(𝕏) boundingempty regions, but this changeswhenϵ increasesto 0.35. Indeed, R_0.35(𝕏) has three 1-dimensional holes:the central prominent hole, and the two smallones to the left side.Notice, however, that as ϵ increases beyond 0.35 these holes will be filled by the addition of new simplices;in particular, for ϵ > 2 one has that R_ϵ(𝕏) will have only one connected component and no other topological features in higher dimensions.The familyℛ(𝕏) = {R_ϵ(𝕏)}_ϵ≥ 0 isknown as the Rips filtration of 𝕏, and the emergence/dissapearence of topologicalfeatures in each dimension (i.e., connected components, holes, voids, etc) as ϵ changes, can be codifiedin what arereferred to as the persistence diagrams of ℛ(𝕏). Specifically, for each dimension n = 0,1,… (0 = connected components, 1 = holes, 2 = voids, etc) one can record the value of ϵ for which a particular n-dimensional topological feature of the Rips filtration appears (i.e. its birth time), and when it disappears (i.e. its death time).The birth-death times (b,d)∈ℝ^2 of n-dimensional features for ℛ(𝕏) form a multiset 𝖽𝗀𝗆_n(ℛ(𝕏)) — i.e. a set whose elements can come with repetition — known as the n-dimensional persistence diagram of the Rips filtration on 𝕏. Since 𝖽𝗀𝗆_n(ℛ(𝕏)) is just a collection of points in the region {(x,y) ∈ℝ^2 : 0 ≤ x < y},we will visualize it asa scatter plot. The persistence of a topological feature with birth-death times(b,d) is the quantity d-b, i.e. its lifetime. We will also include the diagonal y= xin the scatter plot in order to visually convey the persistenceof eachbirth-death pair. In this setting, points far from the diagonal (i.e. with large persistence) represent topological features which are stable across scales and hence deemed significant, while points near the diagonal (i.e. with small persistence) are often associated with unstable features. We illustrate in Figure <ref> the process of going from a point cloud to the 1-dimensional persistence diagram of its Rips filtration. We remark that the computational task of determining all non-equivalent persistent homology classes of a filtered simplicial complex can, surprisingly, be reduced to computing the homology of a single simplicial complex <cit.>. Thisis in fact a problem in linear algebra that can be solved via elementary row and column operations on appropriate boundary matrices. The persistent homology ofℛ(𝕊𝕎_d,τX), and in particular its n-dimensional persistence diagrams for n=1,2, are the objects we will use to quantify periodicity and quasiperiodicity in a video X. Figures <ref> and <ref> show the persistence diagrams of the Rips filtrations, on the sliding window embeddings, for the commensurate and non-commensurate signals from Figures <ref> and <ref>, respectively.We use fast new code from the “Ripser” software package to make persistent H_2 computation feasible <cit.>.§ IMPLEMENTATION DETAILS§.§ Reducing Memory Requirements with SVD Suppose we have a video which has been discretely sampled at N different frames at a resolution of W × H, and we do a delay embedding with dimension d, for some arbitrary τ.Assuming 32 bit floats per grayscale value, storing the sliding window embedding requires 4WHN(d+1) bytes.For a low resolution 200 × 200 video only 10 seconds long at 30fps, using d = 30already exceeds 1GB of memory. In what follows we will address the memory requirements and ensuing computational burden to construct and access the sliding window embedding. Indeed, constructing the Rips filtrationonly requirespairwise distances between different delay vectors, this enables a few optimizations.First of all, for N points in ℝ^WH, where N ≪ WH, there exists an N-dimensional linear subspace which contains them. In particular let A be the (W × H) × N matrix with each video frame along a column. Performing a singular value decomposition A = USV^T, yields a matrixU whose columns form an orthonormal basis for the aforementioned N-dimensional linear subspace. Hence, by finding the coordinates of the original frame vectors with respect to this orthogonal basis = U^TA = U^TUSV = SVand using the coordinates of the columns of SV instead of the original pixels, we get a sliding window embedding of lower dimensionSW_d, τ (t)= [ [U^TX(t);⋮; U^TX(t + dτ) ]]for whichSW_d,τ(t) - SW_d,τ(t') = SW_d,τX(t) - SW_d,τX(t') Note that SV can be computed by finding the eigenvectors of A^TA; this has a cost of O(W^2H^2 + N^3) which is dominated by W^2H^2 if WH ≫ N.In our example above, this alone reduces the memory requirements from 1GB to 10MB.Of course, this procedure is the most effective for short videos where there are actually many fewer frames than pixels, but this encompasses most of the examples in this work.In fact, the break-even point for a 200x200 30fps video is 22 minutes.A similar approach was used in the classical work on Eigenfaces <cit.> when computing the principal components over a set of face images.§.§ Distance Computation via Diagonal ConvolutionsA different optimization is possible if τ = 1; that is, if delays are taken exactly on frames and no interpolation is needed.In this case, the squared Euclidean distance between SW_d, 1X(i) and SW_d, 1X(j) is||SW_d, 1X(i) - SW_d, 1X(j)||_2^2 = ∑_m = 0^d ||X(i+m) - X(j+m)||_2^2 Let D^2_X be the N × N matrix of all pairwise squared Euclidean distances between frames (possibly computed with the memory optimization in Section <ref>), and let D^2_Y be the (N-d) × (N-d) matrix of all pairwise distances between delay frames. Then Equation <ref> implies that D^2_Y can be obtained from D^2_X via convolution with a “rect function”, or a vector of 1s of length d+1, over all diagonals in D^2_X (i.e. a moving average).This can be implemented in time O(N^2) with cumulative sums. Hence,regardless of how d is chosen, the computation and memory requirements for computing D^2_Y depend only on the number of frames in the video.Also, D_Y can simply be computed by taking the entry wise square root of D^2_Y, another O(N^2) computation.A similar scheme was used in <cit.> when comparing distances of 3D shape descriptors in videos of 3D meshes.Figure <ref> shows self-similarity matrices on embeddings of the pendulum video with no delay and with a delay approximately matching the period. The effect of a moving average along diagonals with delay eliminates the anti-diagonals caused by the video's mirror symmetry. Even for videos without mirror symmetries, such as a video of a running dog (Figure <ref>), introducing a delay brings the geometry into focus, as shown in Figure <ref>. §.§ NormalizationA few normalization steps are needed in order to enable fair comparisons between videos with different resolutions, or which have a different range in periodic motion either spatially or in intensity.First, we perform a “point-center and sphere normalize” vector normalization which was shown in <cit.> to have nice theoretical properties.That is,SW_d, τ(t) =SW_d, τ(t) - (SW_d, τ(t)^T 1) 1/ ||SW_d, τ(t) - (SW_d, τ(t)^T 1) 1||_2where 1 is a WH(d+1) × 1 vector of all ones.In other words, one subtracts the mean of each component of each vector, andeach vector is scaled so that it has unit norm (i.e. lives on the unit sphere in ℝ^WH(d+1)).Subtractingthe mean from each component will eliminate additive linear drift on top of the periodic motion, while scaling addressesresolution / magnitude differences.Note that we can still use the memory optimization in Section <ref>, but we can no longer use the optimizations in Section <ref> since each window is normalized independently. Moreover, in order to mitigate nonlinear drift, we implement a simple pixel-wise convolution by the derivative of a Gaussian for each pixel in the original video before applying the delay embedding: X̂_i(t) = X_i(t) * -at exp^-t^2/(2σ^2) This is a pixel-wise bandpass filter which could be replaced with any other bandpass filter leveraging application specific knowledge of expected frequency bounds.This has the added advantage of reducing the number of harmonics, enabling a smaller embedding dimesion d.§.§ Periodicity/Quasiperiodicity Scoring Once the videos are normalized to the same scale, we can score periodicity and quasiperiodicity based on the geometry of sliding window embeddings. Let 𝖽𝗀𝗆_n be the n-dimensional persistence diagram for the Rips filtration on the sliding window embedding of a video, and definemp_i(𝖽𝗀𝗆_n) as the i-th largest difference d-b for (b,d)∈𝖽𝗀𝗆_n. In particularmp_1(𝖽𝗀𝗆_n) = max{d - b : (b,d) ∈𝖽𝗀𝗆_n}and mp_i(𝖽𝗀𝗆_n) ≥ mp_i+1(𝖽𝗀𝗆_n). We propose the following scores:* Periodicity Score (PS)PS = 1/√(3) mp_1(𝖽𝗀𝗆_1) Like <cit.>, we exploit the fact that forthe Rips filtration on S^1,the 1-dimensional persistence diagram has only one prominent birth-death pair with coordinates (0,√(3)). Since this is the limit shape of a normalized perfectly periodic sliding window video, the periodicity score is between 0 (not periodic) and 1 (perfectly periodic).* Quasiperiodicity Score (QPS)QPS = √( mp_2(𝖽𝗀𝗆_1) mp_1(𝖽𝗀𝗆_2)/3) This score is designed with the torus in mind.We score based on the second largest 1D persistence times the largest 2D persistence, since we want a shape that has two core circles and encloses a void to get a large score.Based on the Künneth theorem of homology, the 2-cycle (void) should die the moment the smallest 1-cycle dies. * Modified Periodicity Score (MPS)MPS = 1/√(3)( mp_1(𝖽𝗀𝗆_1) - mp_2(𝖽𝗀𝗆_1) ) We design a modified periodicity score which should be lower for quasiperiodic videos, than what the original periodicity score would yield. Note that we use ℤ_3 field coefficients for all persistent homology computations since, as shown by <cit.>, this works better for periodic signals with strong harmonics. Before we embark on experiments, let us explore the choice of two crucial parameters for the sliding window embedding: the delay τ > 0 and the dimension d∈ℕ. In practice we determine an equivalent pair of parameters: the dimension d and the window size dτ. §.§ Dimension and Window SizeTakens' embedding theorem is one of the most fundamental results in the theory of dynamical systems <cit.>. In short, it contends that (under appropriate hypotheses) there exists an integer D, so that for all d≥ D andgeneric τ >0 the sliding window embedding SW_d,τ X reconstructs the state space of the underlying dynamics witnessed by the signal X. One common strategy for determininga minimal such D is thefalse nearest-neighbors scheme <cit.>.The idea is to keep track of the k-th nearest neighbors of eachpoint in the delay embedding, and if they change as d is increased, then the prior estimates for d were too low.This algorithm was used in recent work on video dynamics <cit.>, for instance.Even if we can estimate d, however, how does one choose the delay τ?As shown in <cit.>, the sliding window embedding of a periodic signals is roundest (i.e. so that the periodicity score PS is maximized) when the window size, d τ, satisfies the following relation:d τ = π k/L( d/d+1)Here L is number of periods thatthe signal has in [0, 2π] and k∈ℕ. To verify this experimentally, we show in Figure <ref> how the periodicity score PS changes as a function of window size for the pendulum video, and how the choice of window size from Equation <ref> maximizes PS. To generate this figure we fixed a sufficiently large d andvaried τ. Let us now describe the general approach: Given a video we perform a period-length estimation step (see section <ref> next), which results in a positive real number ℓ. For a given d∈ℕ large enoughwe let τ > 0 be so that dτ = ℓ·d/d+1. §.§ Fundamental Frequency EstimationThough Figure <ref> suggests robustness to window size as long as the window is more than half of the period, we may not know what that is in practice.To automate window size choices, we do a coarse estimate using fundamental frequency estimation techniques on a 1D surrogate signal.To get a 1D signal, we extract the first coordinate of diffusion maps <cit.> using 10 % nearest neighbors on the raw video frames (no delay) after taking a smoothed time derivative.Note that a similar diffusion-based method was also used in recent work by <cit.> to analyze the frequency spectrum of a video of an oscillating 2 pendulum + spring system in a quasiperiodic state.Once we have the diffusion time series, we then apply the normalizedautocorrelation method of <cit.> to estimate the fundamental frequency.In particular, given a discrete signal x of length N, define the autocorrelation asr_t(τ) = ∑_j = t^t + N - 1 - τ x_j x_j+τ However, as observed by <cit.>, a more robust function for detecting periodicities is the squared difference functiond_t(τ) = ∑_j = t^t + N - 1 - τ (x_j - x_j+τ)^2which can be rewritten as d_t(τ) = m_t(τ) - 2r_t(τ) wherem_t(τ) = ∑_j = t^t + N - 1 - τ (x_j^2 + x_j+τ^2) Finally, <cit.> suggest normalizing this function to the range [-1, 1] to control for window size and to have an interpretation akin to a Pearson correlation coefficient: n_t(τ) = 1 - m_t(τ) - 2r_t(τ)/m_t(τ) = 2r_t(τ)/m_t(τ) The fundamental frequency is then the inverse period of the largest peak in n_t which is to the right of a zero crossing.The zero crossing condition helps prevent an offset of 0 from being the largest peak. Defining the normalized autocorrelation as in Equation <ref> has the added advantage that the value of n_t(τ) at the peak can be used to score periodicity, which the authors call clarity. Values closer to 1 indicate more perfect periodicities.This technique will sometimes pick integer multiples of the period, so we multiply n_t(τ) by a slowly decaying envelope which is 1 for 0 lag and 0.9 for the maximum lag to emphasize smaller periods.Figure <ref> shows the result of this algorithm on a periodic video, and Figure <ref> shows the algorithm on an irregular video.§ EXPERIMENTAL EVALUATIONNext we evaluate the effectiveness of the proposed (Modified) Periodicity and Quasiperiodicity scores onthree different tasks. First, we provide estimates ofaccuracyfor the binary classifications periodic/not-periodic or quasiperiodic/not-quasiperiodic in the presence of several noise models and noise levels. The results illustrate the robustness of our method. Second, we quantify the quality of periodicity rankings from machine scores, as compared to those generated by human subjects. In a nutshell, and after comparing with several periodicity quantification algorithms, our approach is shown to be the most closely aligned with the perception of human subjects. Third, we demonstrate thatour methodology can be used to automatically detect the physiological manifestations of certain speech pathologies (e.g., normal vs. biphonation), directly fromhigh-speed videos of vibrating vocal folds.§.§ Classification Under Varying Noise Levels/ModelsAs shown empirically in <cit.>, a common source of noise in videos comes from camera shake (blur); this is captured by point spread functions resemblingdirected random walks <cit.> and the amount of blur (i.e. noise level) is controlled by the extent in pixels of the walk. Other sources are additive white Gaussian noise (awgn), controlled by the standard deviation of the Gaussian kernel, and MPEG bit errorsquantified by the percentage of corrupted information.Figure <ref> shows examples of these noise types. For classification purposes we use three main recurrence classes. Three types of periodic videos (True periodic, TP): an oscillating pendulum, a bird flapping its wings, and an animation of a beating heart. Two types of quasiperiodic videos (True quasiperiodic, TQ): one showing two solid disks which oscillate sideways at non-commensurate rates, and the second showing two stationary Gaussian pulses with amplitudes non-commensurately modulated by cosine functions. Two videos without significant recurrence (True non-recurrent, TN): a video of a car driving past a landscape, and a video of an explosion. Each one of these seven videos is then corrupted by the threenoise models at three different noise levels (blur = 20, 40, 80, awgn σ = 1, 2, 3, bit error = 5, 10, 20%) as follows: given a particular video, a noise model and noise level, 600 instances are generated by sampling noise independently at random. .3cmResults: We report inTable <ref> the area under the Receiver Operating Characteristic (ROC) curve, or AUROC for short, for theclassification task TP vs. TN (resp. TQ vs. TN) and binary classifier furnished byPeriodicity (resp. Quasiperiodicity) Score. For instance, for the Blur noise model with noise level of 80× 80 pixels, the AUROC fromusing the Periodicity Score to classify the 600 instances of the Heartbeat video as periodic, and the 600 instances of the Driving video as not periodic is 0.91. Similarly, for the MPEG bit corruption model with 5% of bit error, the AUROC from using the Quasiperiodicity score to classify the 600 instances of the Quasiperiodic Sideways Disks as quasiperiodic and the 600 instances of the Explosions video as not quasiperiodic is 0.92. To put these numbers in perspective,AUROC = 1 is associated with a perfect classifier andAUROC = 0.5 corresponds toclassification by a random coin flip.Overall, the type of noise that degrades performance the most across videos is the bit error, which makes sense, since this has the effect of randomly freezing, corrupting, or even deleting frames, which all interrupt periodicity.The blur noise also affects videos where the range of motion is small.The pendulum video, for instance, only moves over a range of 60 pixels at the most extreme end, so an 80x80 pixel blur almost completely obscures the motion.§.§ Comparing Human and Machine Periodicity Rankings Next we quantify the extent to which rankings obtained from our periodicity score (Equation <ref>), as well as three other methods, agree with how humans rank videos by periodicity. The starting point is adataset of 20 different creative commons videos, each 5 seconds long at 30 frames per second.Some videos appear periodic, such as a person waving hands, a beating heart, and spinning carnival rides.Some of them appear nonperiodic, such as explosions, a traffic cam, and drone view of a boat sailing.And some of them are in between, such as the pendulum video with simulated camera shake.It is known that humans are notoriously bad at generating globally consistent rankings of sets with more than 5 or 7 elements <cit.>. However, when it comes to binary comparisons of the type “should A be ranked higher than B?” few systems are as effective as human perception, specially for the identification of recurrent patterns in visual stimuli.We will leverage this to generate a globally consistent ranking of the 20 videos in our initial data set.We use Amazon's Mechanical Turk (AMT) <cit.> to present each pair of videos in the set of 20, 202 = 190, each to three different users, for a total of 570 pairwise rankings. 15 unique AMT workers contributed to our experiment, using an interface as the one shown in Figure <ref>. In order to aggregate this information into a global ranking which is as consistent as possible with the pairwise comparisons, we implement a technique known as Hodge rank aggregation <cit.>. Hodge rank aggregation finds the closest consistent ranking to a set of preferences, in a least squares sense.More precisely, given a set of objects X, and given a set of comparisons P ⊂ X × X, we seek a scalar function s on all of the objects that minimizes the following sum∑_ (a, b) ∈ P |v_ab - (s_b - s_a)|^2where v_ab is a real number which is positive if b is ranked higher than a and negative otherwise.Thus, s is a function whose discrete gradient best matches the set of preferenceswith respect to an L^2 norm. Note that the preferences that we feedthe algorithm are based on the pairwise rankings returned from AMT.If video b is greater than video a, then we assign v_ab = 1, or -1 otherwise.Since we have 3 rankings for each video, we actually assign weights of +3, +1, -1, or -3.The +/- 3 are if all rankings agree in one direction, and the +/- 1 are if one of the rankings disagrees with the other two.Figure <ref> shows a histogram of all of the weighted scores from users on AMT.They are mostly in agreement, though there are a few +/- 1 scores.As comparison to the human scores, we use three different classes of techniques for machine ranking of periodicity. .3cmSliding Windows (SW): We sort the videos in decreasing order of Periodicity Score (Equation <ref>).We fix the window size at 20 frames and the embedding dimension at 20 frames (which is enough to capture 10 strong harmonics).We also apply a time derivative of width 10 to every frame. .3cmCutler-Davis <cit.>: The authors of this work present two different techniques to quantify periodicity from a self-similarity matrix (SSM) of video frames.The first is a frequency domain technique based on the peak of the average power spectral density over all columns (rows) of the SSM after linearly de-trending and applying a Hann window.To turn this into a continuous score, wereport the ratio of the peak minus the mean over the standard deviation. This method will be referred to as Frequency Score.As the authors warn, the frequency peak method has a high susceptibility to false positives.This motivated the design of a more robust technique in <cit.>, which works by finding peaks in the 2D normalized autocorrelation of the Gaussian smoothed SSMs.For videos with mirror symmetry, the peaks will lie on a diamond lattice, while for videos without mirror symmetry, they will lie on a square lattice. After peak finding within neighborhoods, one simply searches over all possible lattices at all possible widths to find the best match with the peaks.Since each lattice is centered at the autocorrelation point (0, 0), no translational checks are necessary. To turn this into a continuous score, let E be the sum ofEuclidean distances of the matched peaks in the autocorrelation image to the best fit lattice, let r_1 be the proportion of lattice points that have been matched, and let r_2 be the proportion of peaks which have been matched to a lattice point.Then we give the final periodicity score asCD_score = (1 + E/r_1)/(r_1 r_2)^3 A lattice which fits the peaks perfectly (r_1 = 1) with no error (E = 0) and no false positive peaks (r_2 = 1) will have a score of 1, and any video which fails to have a perfectly matched lattice will have a score greater than 1.Hence, we sort in increasing order of the score to get a ranking.As we will show, this technique agrees the second best with humans after our Periodicity Score ranking.One of the main drawbacks is numerical stability of finding maxes in non-isolated critical points around nearly diagonal regions in square lattices, which will erroneously inflate the score.Also, the lattice searching only occurs over an integer grid, but there may be periods that aren't integer number of frames, so there will always be a nonzero E for such videos.By contrast, our sliding window scheme can work for any real valued period length. .3cmDiffusion Maps + Normalized Autocorrelation “Clarity”: Finally, we apply the technique from Section <ref> to get an autocorrelation function, and we report the value of the maximum peak of the normalized autocorrelation to the right of a zero crossing, referred to as “clarity” by <cit.>.Values closer to 1 indicate more perfect repetitions, so we sort in descending order of clarity to get a ranking. .3cmFigure <ref> shows an example of these three different techniques on a periodic video.There is a dot which rises above the diagonal in the persistence diagram, a lattice is found which nearly matches the critical points in the autocorrelation image, and autocorrelation function on diffusion maps has a nice peak. By contrast, for a nonperiodic video (Figure <ref>), there is hardly any persistenthomology, there is no well matching lattice, and the first diffusion coordinate has no apparent periodicities..3cmResults: Once we have the global human rankings and the global machine rankings, we can compare them using the Kendall τ score <cit.>.Given a set of objects N objects X and two total orders >_1 and >_2, where >(x_a, x_b) = 1 if x_a > x_b and >(x_a, x_b) = -1 if x_a < x_b, the Kendall τ score is defined as τ = 1/N(N-1)/2∑_i < j (>_1(x_i, x_j)) (>_2(x_i, x_j)) For two rankings which agree exactly, the Kendall τ score will be 1.For two rankings which are exactly the reverse of each other, the Kendall τ score will be -1.In this way, it analogous to a Pearson correlation between rankings. Table <ref> shows the Kendall τ scores between all of the different machine rankings and the human rankings. Our sliding window video methodology (SW+TDA) agrees with the human ranking more than any other pair of ranking types.The second most similar are the SW and the diffusion clarity, which is noteworthy as they are both geometric techniques.Table <ref> also shows the average run times, in milliseconds, of the different algorithms on each video on our machine.This does highlight one potential drawback of our technique, since TDA algorithms tend to be computationally intensive.However, at this scale (videos with at most several hundred frames), performance is reasonable. §.§ Periodicity And Biphonation in High Speed Videos of Vocal Folds In this final task we apply our methodology to a real world problem of interest in medicine.We show that our method can automatically detect certain types of voice pathologies from high-speed glottography, or high speed videos (4000 fps) of the left and right vocal folds in the human vocal tract <cit.>.In particular, we detect and differentiate quasiperiodicity from periodicity by using our geometric sliding window pipeline.Quasiperiodicity is a special case of what is referred to as “biphonation” in the biological context, where nonlinear phenomena cause a physical process to bifurcate into two different periodic modes, often during a transition to chaotic behavior <cit.>.The torus structure we sketched in Figure <ref> has long been recognized in this context <cit.>, but we provide a novel way of quantifying it.Similar phenomena exist in audio <cit.>, but the main reason for studying laryngeal high speed video is understanding the biomechanical underpinnings of what is perceived in the voice. In particular, this understanding can potentially lead to practical corrective therapies and surgical interventions. On the other hand, the presence of biphonation in sound is not necessarily the result of a physiological phenomenon; it has been argued that it may come about as the result of changes in states of arousal <cit.>.In contrast with our work, the existing literature on video-based techniques usually employs an inherently Lagrangian approach, where different points on the left and right vocal folds are tracked, and coordinates of these points are analyzed as 1D time series (e.g. <cit.>, <cit.>).This is a natural approach, since those are the pixels where all of the important signal resides, and well-understood 1D signal processing technique can be used.However, edge detectors often require tuning, and they can suddenly fail when the vocal folds close <cit.>.In our technique, we give up the ability to localize the anomalies (left/right, anterior/posterior) since we are not tracking them, but in return we do virtually no preprocessing, and our technique is domain independent. .3cmResults: We use a collection of 7 high-speed videos for this analysis, drawn from a variety of different sources <cit.>, <cit.>, <cit.>, <cit.>.There are two videos which correspond to “normal” periodic vocal folds, three which correspond to biphonation <cit.>, and two which correspond to irregular motion[Please refer to supplementary material for an example video from each of these three classes].We manually extracted 400 frames per video (100milliseconds) and autotuned the window size based on autocorrelation of 1D diffusion maps (Section <ref>).We then chose an appropriate τandchose a time spacing so that each point cloud would have 600 points.As shown in Table <ref>, our technique is able to differentiate between the four classes.We also show PCA and persistence diagrams for one example for each class.In Figure <ref>, we see what appears to be a loop in PCA, and one strong 1D persistent dot confirms this.In Figure <ref>, we see a prominent torus in the persistence diagram.In Figure <ref>, we don't see any prominent structures in the persistence diagram, even though PCA looks like it could be a loop or a torus.Note, however, that PCA only preserves 13.7% of the variance in the signal, which is why high dimensional techniques are important to draw quantitative conclusions.§ DISCUSSIONWe have shown in this work howapplying sliding window embeddings to videos can be used to translate properties of the underlying dynamics into geometric features of the resulting point cloud representation. Moreover, we also showed how topological/geometric tools such as persistence homology can be leveraged to quantify the geometry of these embeddings. The pipeline was evaluated extensively showing robustness to several noise models, high quality in the produced periodicity rankings and applicability to the study of speech conditions form high-speed video data.Moving forward, an interesting avenue related to medical applications is the difference between biphonation which occurs from quasiperiodic modes and biphonation which occurs from harmonic modes.<cit.> shows that ℤ_3 field coefficients can be used to indicate the presence of a strong harmonic, so we believe a geometric approach is possible.This could be used, for example, to differentiate between subharmonic anomalies and quasiperiodic transitions <cit.>. § ACKNOWLEDGMENTSThe authors would like to thank Juergen Neubauer,Dimitar Deliyski, Robert Hillman, Alessandro de Alarcon, Dariush Mehta, and Stephanie Zacharias for providing videos of vocal folds.We also thank Matt Berger at ARFL for discussions about sliding window video efficiency, and we thank the 15 anonymous workers on the Amazon Mechanical Turk who ranked periodic videos. plain | http://arxiv.org/abs/1704.08382v2 | {
"authors": [
"Christopher J. Tralie",
"Jose A. Perea"
],
"categories": [
"cs.CV",
"I.2.10"
],
"primary_category": "cs.CV",
"published": "20170426235440",
"title": "(Quasi)Periodicity Quantification in Video Data, Using Topology"
} |
([ ) ⋰⋱Å𝔸 𝔹 ℂ 𝔻 𝔼 𝔽 𝔾 ℍ 𝕀 𝕁 𝕂 𝕃 𝕄 ℕ 𝕆 ℙ ℚ ℝ 𝕊 𝕋 𝕌 𝕍 𝕎 𝕏 𝕐 ℤA B C D E F G H̋ I J K ŁL M N ØO P Q R S T U V W X Y Zd e diag null rank rank span i Re ImtheoTheorem lemLemma corCorollary remRemark defiDefinition Proof a b c d e f g h i j k l m n o p q r s t u v w x y zu v λ_0 ρ̃ q̃ p̃ λ̃ 0 1α β γ δ η φ ψ λ ψ ρ τ ξ ζ ωåα β γ b̂ ĉ ε ĤLine Integral Solution of Hamiltonian Systems with Holonomic Constraints Luigi Brugnano ^a, Gianmarco Gurioli ^a, Felice Iavernaro ^b, Ewa B. Weinmüller ^c ^a Dipartimento di Matematica e Informatica “U. Dini”, Università di FirenzeViale Morgagni 67/a, I-50134 Firenze, Italy.^b Dipartimento di Matematica, Università di BariVia Orabona 4, I-70125 Bari, Italy.^c Institute for Analysis and Scientific Computing, Vienna University of TechnologyA-1040 Wien, Austria.– Dedicated to John Butcher, on the occasion of his 84-th birthday – ============================================================================================================================================================================================================================================================================================================================================================================================================================= In this paper, we propose a second-order energy-conserving approximation procedure for Hamiltonian systems with holonomic constraints.The derivation of the procedure relies on the use of the so-called line integral framework. We provide numerical experiments to illustratetheoretical findings.Keywords: constrained Hamiltonian systems; holonomic constraints; energy-conserving methods; line integral methods; Hamiltonian Boundary Value Methods; HBVMs. MSC: 65P10, 65L80, 65L06. § INTRODUCTIONWe consider the numerical approximation of a constrained Hamiltonian dynamics, described by the separable HamiltonianH(q,p) =1/2 p^⊤M^-1 p +U(q),q,p∈^m,where M is a symmetric and positive-definite matrix. The problem is completed by ν holonomic constraints,g(q) = 0∈^ν,where we assume that ν< m holds. Moreover, we also assume that all points are regular for the constraints, i.e., ∇ g(q)∈^m×ν has full column rank or, equivalently, ∇ g(q)^⊤ M^-1∇ g(q) is nonsingular. For simplicity, both U and g are assumed to be analytic. It is well-known that the problem defined by (<ref>)–(<ref>) can be cast in Hamiltonian form by defining the augmented Hamiltonian(q,p,λ) = H(q,p) + λ^⊤ g(q),where λ is the vector of Lagrange multipliers. The resulting constrained Hamiltonian system reads:q̇ =M^-1p, ṗ = -∇ U(q)-∇ g(q)λ,g(q)=0, t∈[0,T],and is subject to consistent initial conditions,q(0)=q_0, p(0)=p_0,such thatg(q_0)=0, ∇ g(q_0)^⊤M^-1 p_0 = 0.Note that the condition g(q_0)=0 ensures that q_0 belongs to the manifold= { q∈^m: g(q)=0},as required by the constraints, whereas the condition ∇ g(q_0)^⊤M^-1p_0 means that the motion initially stays on the tangent space toat q_0. On a continuous level, this condition is satisfied by all points on the solution trajectory, since, in order for the constraints to be conserved,ġ(q) = ∇ g(q)^⊤q̇ = ∇ g(q)^⊤M^-1 p = 0,holds. These latter constraints are sometimes referred to as hidden constraints.We stress that the condition (<ref>) can be conveniently relaxed for the numerical approximation. There, we only ask for ∇ g(q)^⊤M^-1p to be suitably small along the numerical solution. Consequently, when solving the problem on the interval [0,h], we require that the approximations,q_1≈ q(h),p_1≈ p(h), satisfy the conservation of both, the Hamiltonian and the constraints,H(q_1,p_1) = H(q_0,p_0),g(q_1) = g(q_0) = 0,and that the hidden constraints are relaxed to∇ g(q_1)^⊤M^-1p_1 = O(h^2).We recall that a formal expression for the vector λ is obtained by an additional differentiating of (<ref>), i.e.,g̈(q) = ∇^2 g(q) (M^-1p,M^-1p) - ∇ g(q)^⊤ M^-1[ ∇ U(q) + ∇ g(q)λ].Imposing the vanishing of this derivative yields [∇ g(q)^⊤ M^-1∇ g(q)]λ =∇^2 g(q) (M^-1p,M^-1p) - ∇ g(q)^⊤ M^-1∇ U(q).Consequently, the following result follows.The vector λ exists and is uniquely determined, provided that the matrix∇ g(q)^⊤ M^-1∇ g(q)is nonsingular. In fact, in such a case, from (<ref>), we obtainλ = [∇ g(q)^⊤ M^-1∇ g(q)]^-1[ ∇^2 g(q) (M^-1p,M^-1p) - ∇ g(q)^⊤ M^-1∇ U(q)] = : λ(q,p).Note that, for later use, we have introduced the notation λ(q(t),p(t)) in place of λ(t), to explicitly underline the dependence of the Lagrange multiplier on the state variables q and p at time t.We observe that an additional differentiation of (<ref>) provides a differential equation for the Lagrange multipliers, which can be solved together with the original problem. However, this procedure is cumbersome in general, since it requires the evaluation of higher order tensors. Numerical solution of Hamiltonian problems with holonomic constraints has been for a long time in the focus of interest. Many different approaches have been proposed such as the basic Shake-Rattle method <cit.>, which has been shown to be symplectic <cit.>, higher order methods obtained via symplectic PRK methods <cit.>, composition methods <cit.>, symmetric LMFs <cit.>. Further methods are based on discrete derivatives <cit.>, local parametrizations of the manifold containing the solution <cit.>, or on projection techniques <cit.>. See also <cit.> and the monographs <cit.>.In this paper we pursue a different approach, utilizing the so-called line integral, which has already been used when deriving the energy-conserving Runge-Kutta methods, for unconstrained Hamiltonian systems, cf. Hamiltonian Boundary Value Methods (HBVMs) <cit.> and the recent monograph <cit.>. Such methods have also been applied in a number of applications <cit.>, and here are used to cope with the constrained problem (<ref>)–(<ref>). Roughly speaking, the conservation of the invariant will be guaranteed by requiring that a suitable line integral vanishes. This line integral represents a discrete-time version of (<ref>). In fact, if we fix a stepsize h>0, then the conservation of the constraints (<ref>) at h, starting from the point q_0 defined in (<ref>), can be recast intog(q(h)) -g(q(0)) _=0 = ∫_0^h ∇ g(q(t))^⊤q̇(t) t = 0. For the continuous solution, this integral vanishes since the integrand is identically zero due to (<ref>) and (<ref>).However, we can relax this requirement in the context of a numerical method describing a discrete-time dynamics. In such a case, the conservation properties have to be satisfied only on a set of discrete times which are multiples of the stepsize h. Consequently, we consider a local approximation to q(t), say (t), such that(0) = q_0, (h) =: q_1≈ q(h),andg(q_1)-g(q_0) ≡ g((h))-g((0)) = ∫_0^h ∇ g((t))^⊤(t) t = 0,without requiring the integrand to be identically zero. This, in turn, enables a proper choice of the vector of the multipliers λ. As a result, we eventually obtain suitably modified HBVMs which enable to conserve both, the Hamiltonian and the constraints. We stress that the available efficient implementation of the original methods (see, e.g., <cit.>), which proved to be reliable and robust in the numerical solution of the unconstrained Hamiltonian problems, can now be adapted for dealing with the holonomic constraints. The paper is organized as follows. In Section <ref>, we provide the framework for devising the method via a suitable choice of the vector λ of the Lagrange multipliers, which we approximate by a piecewise-constant function. In Section <ref> further simplification towards numerical procedure is discussed. Then, in Section <ref>, we present a fully discrete method, resulting in a suitable modification of the original HBVMs. In Section <ref>, numerical experiments are shown to illustrate how the method works for a number of constrained Hamiltonian problems. Section <ref> contains the conclusions and possible future investigations.§ PIECEWISE-CONSTANT APPROXIMATION OF Λ In this section, we show that we can approximate the solution of problem (<ref>) on the interval [0,h], h=T/N, by looking for a constant vector λ∈^ν such that (<ref>) is satisfied. This is equivalent to require (q_1,p_1,λ) = (q_0,p_0,λ),g(q_1) = g(q_0) = 0,where the constant parameter λ is chosen in such a way that the constraints g(q_1)=0 hold. We will show that this procedure provides us with a second order approximation of the original problem, which becomes exact when the true multiplier is constant. Consequently, we approximate the problem (<ref>)–(<ref>), by the local problem=M^-1, = -∇ U()-∇ g(),t∈[0,h],subject to the initial conditions, cf. (<ref>),(0) = q_0, (0) = p_0,satisfying (<ref>). By settingq_1:=u(h),p_1 :=v(h),the constant parameteris chosen to guarantee the conservation of the Hamiltonian and the constraints, i.e., (<ref>). Starting from (<ref>), the procedure is then repeated on [h,2h] and the following intervals. The convergence result is now formulated in the following theorem. For all sufficiently small stepsizes h>0, the above procedure defines a sequence of approximations (q_n,p_n) such that, for all n=1,2,…:q_n = q(nh) + O(h^2),p_n = p(nh) + O(h^2),g(q_n)=0, ∇ g(q_n)^⊤ M^-1p_n = O(h^2).Moreover, (q_n+1,p_n+1) is obtained from (q_n,p_n) using a constant vector λ_n such thatλ_n = λ( q(nh), p(nh) ) + O(h),where λ(q,p) is defined in (<ref>) and, consequently, H(q_n+1,p_n+1) = H(q_n,p_n). The aim of this section is to show (<ref>)–(<ref>). Let us first consider the orthonormal basis on [0,1] given by the shifted and scaled Legendre polynomials {P_j},P_j∈Π_j, ∫_0^1 P_i(c)P_j(c) c=δ_ij, ∀ i,j=0,1,…,along with the expansions,M^-1(ch)= ∑_j≥0 P_j(c)γ_j(), ∇ U((ch)) = ∑_j≥0 P_j(c)ψ_j(), ∇ g((ch))= ∑_j≥0 P_j(c)ρ_j(),c∈[0,1],withγ_j()= M^-1∫_0^1 P_j(c)(ch) c, ψ_j() = ∫_0^1 P_j(c)∇ U((ch)) c, ρ_j()= ∫_0^1 P_j(c)∇ g((ch)) c,j≥ 0.Consequently, following the approach defined in <cit.>, the differential equations in (<ref>) can be rewritten as(ch) = ∑_j≥0 P_j(c) γ_j(), (ch) = -∑_j≥0 P_j(c)[ ψ_j() +ρ_j()], c∈[0,1].Moreover, using the initial conditions (<ref>), we formally obtain(ch) = q_0 + h∑_j≥0∫_0^cP_j(x) x γ_j(), (ch) = p_0 -h∑_j≥0∫_0^cP_j(x) x[ ψ_j() +ρ_j()], c∈[0,1].The following result is now cited from <cit.>.Let G:[0,h] → V, with V a vector space, admit a Taylor expansion at 0. Then∫_0^1 P_j(c) G(ch) c = O(h^j),j≥0. As a straightforward consequence, one has the following result. All coefficients specified in (<ref>) are O(h^j). Concerning the conservation properties of the approximations, the following result holds. For all ∈^ν, the solution of (<ref>)–(<ref>) satisfies(q_1,p_1,) = (q_0,p_0,).For any given ∈^ν, it follows from (<ref>), (<ref>), and (<ref>),(q_1,p_1,) - (q_0,p_0,) = ((h),(h),) - ((0),(0),)= ∫_0^h / t((t),(t),) t = ∫_0^h {_q((t),(t),)^⊤(t) + _p((t),(t),)^⊤(t)} t =h∫_0^1 {[∇ U((ch)) + ∇ g((ch))]^⊤(ch) + [ M^-1(ch)]^⊤(ch)} c=h∫_0^1 {[∇ U((ch)) + ∇ g((ch))]^⊤∑_j≥0P_j(c)γ_j() - . . [ M^-1(ch)]^⊤∑_j≥0P_j(c) [ ψ_j()+ρ_j()]} c= h∑_j≥0{( ∫_0^1 P_j(c)[∇ U((ch)) + ∇ g((ch))] c )^⊤γ_j() - . . ( M^-1∫_0^1 P_j(c)(ch) c)^⊤[ ψ_j()+ρ_j()]}= h∑_j≥0{ [ψ_j()+ρ_j()]^⊤γ_j() - γ_j()^⊤ [ψ_j()+ρ_j()]} = 0.As observed above, the conservation of the Hamiltonian (<ref>) is guaranteed, once the constraints are satisfied, i.e., g(q_1)=0. We now apply a line integral technique to determine the vectorand formulate the following result describing the very first step of the approximation procedure. Let us consider the problem (<ref>)–(<ref>) and assume that (q_0,p_0) is given such that, * ∇ g(q_0)^⊤M^-1∇ g(q_0) ∈^ν×ν is nonsingular;* g(q_0) = 0;* ∇ g(q_0)^⊤ M^-1 p_0 = 0.Then, for all sufficiently small h>0,∃!∈^ν such that the approximations in (<ref>) satisfy * g(q_1)=0 and, therefore, H(q_1,p_1)=H(q_0,p_0);* λ_0 = λ(q_0,p_0) + O(h);* q_1-q(h) = O(h^2), p_1-p(h)=O(h^2);* ∇ g(q_1)^⊤ M^-1 p_1 = O(h^2). Clearly, Theorem <ref> is the discrete counterpart of Theorem <ref>. Before showing Theorem <ref>, we have to state the following preliminary results. Let us consider the polynomial basis (<ref>). Then, we have ∫_0^1P_j(c)∫_0^c P_i(x) xc = ( X_s )_j+1,i+1,i,j=0,…,s-1,where ( X_s )_j+1,i+1 is the (j+1,i+1) entry of the matrixX_s:=rrrrξ_0 -ξ_1 ξ_1 0⋱⋱ ⋱-ξ_s-1 ξ_s-1 0, ξ_j = 1/2√(|4j^2-1|), j=0,…,s-1. Since the integrand on the left-hand side in (<ref>) is a polynomial of degree at most 2s-1, the integral can be computed exactly via the Gauss-Legendre formula of order 2s. Let c_1,…,c_s be the zeros of P_s and b_1,…,b_s be the corresponding weights. Then, introducing the matrices_s = ( P_j-1(c_i) ), _s = ( ∫_0^c_i P_j-1(x) x), Ω = (b_1,…,b_s) ∈^s× s,and setting e_i∈^s, the i-th unit vector, we have∫_0^1P_j(c)∫_0^c P_i(x) xc = ∑_ℓ=1^s b_ℓ P_j(c_ℓ)∫_0^c_ℓ P_i(x) x ≡ e_j+1^⊤_s^⊤Ω_s e_i+1.The result follows by observing that, due to the properties of Legendre polynomials <cit.>,_s = _s X_s, _s^⊤Ω_s=I_sfollows, where X_s is the matrix defined in (<ref>), and I_s∈^s× s is the identity matrix. This yieldse_j+1^⊤_s^⊤Ω_s e_i+1 = e_j+1^⊤_s^⊤Ω_s X_s e_i+1 = e_j+1^⊤ X_s e_i+1. We also need the following expansions. From (<ref>) and (<ref>), we conclude ρ_0()= ∇ g(q_0) + h/2∇^2 g(q_0) M^-1p_0 + O(h^2), ρ_0()^⊤ M^-1p_0= ∇ g(q_0)^⊤ M^-1 p_0 + h/2∇^2 g(q_0) (M^-1 p_0,M^-1p_0) + O(h^2). The first equality follows from Lemma <ref> and Corollary <ref>,ρ_0()= ∫_0^1 ∇ g ((ch)) c = ∫_0^1[∇ g((0)) + ch ∇^2 g((0))(0) + O((ch)^2)] c = ∇ g((0)) + h/2∇^2 g((0))(0) +O(h^2)= ∇ g(q_0) + h/2∇^2 g(q_0)∑_j≥0P_j(0)γ_j()+O(h^2)= ∇ g(q_0) + h/2∇^2 g(q_0)γ_0() +O(h^2)= ∇ g(q_0) + h/2∇^2 g(q_0)M^-1[ p_0 + O(h)] +O(h^2)= ∇ g(q_0) + h/2∇^2 g(q_0)M^-1p_0 +O(h^2).The second statement follows by transposition and multiplication from the right by M^-1p_0. We now show the results formulated in Theorem <ref>. (of Theorem <ref>). Let us assume thatg(q_0)=0 holds. Then, it follows from (<ref>)–(<ref>), g(q_1) = g(q_1)- g(q_0) = g((h)) - g((0)) = ∫_0^h / t g((t)) t= ∫_0^h ∇ g((t))^⊤(t) t = h∫_0^1 ∇ g((ch))^⊤(ch) c = h∫_0^1 ∇ g((ch))^⊤∑_j≥0 P_j(c) γ_j() c=h∑_j≥0ρ_j()^⊤γ_j() = h∑_j≥0ρ_j()^⊤M^-1∫_0^1 P_j(c)(ch) c=h∑_j≥0ρ_j()^⊤M^-1∫_0^1 P_j(c) {p_0-h∑_i≥0∫_0^c P_i(x) x[ ψ_i() +ρ_i()]} c=h∑_j≥0ρ_j()^⊤M^-1p_0 ∫_0^1 P_j(c)c - h^2∑_i,j≥0ρ_j()^⊤M^-1[ ψ_i() +ρ_i()] ∫_0^1 P_j(c)∫_0^c P_i(x) xc.Due to (<ref>),∫_0^1 P_j(c) c = δ_j0, and according to (<ref>)–(<ref>), we conclude g((h)) - g((0)) = hρ_0()^⊤M^-1{ p_0 -h[ ξ_0(ψ_0()+ρ_0()) - ξ_1(ψ_1()+ρ_1())] } - h^2∑_j≥1ρ_j()^⊤M^-1{[ ξ_j(ψ_j-1()+ρ_j-1()) - ξ_j+1(ψ_j+1()+ρ_j+1())]}=: Γ̂(,,,h) .By virtue of (<ref>) and Corollary <ref>, g((h))-g((0)) - hρ_0()^⊤ M^-1p_0/h^2 =-1/2{[ρ_0()^⊤ M^-1ρ_0() + O(h)] + ρ_0()^⊤ M^-1ψ_0() +O(h)}follows. Now, from (<ref>) and Lemma <ref>, we have g((h))-g((0)) - hρ_0()^⊤ M^-1p_0/h^2 = 1/2{g̈(q_0) -∇^2 g(q_0)(M^-1p_0,M^-1p_0)+ O(h)},ρ_0()^⊤ M^-1ρ_0()= ∇ g(q_0)^⊤ M^-1∇ g(q_0) + O(h), ρ_0()^⊤ M^-1ψ_0()= ∇ g(q_0)^⊤ M^-1∇ U(q_0) + O(h),and this means that (<ref>) tends to (<ref>), for h→ 0. Consequently,exists and is unique for all sufficiently small stepsizes h>0. On the other hand, g(q_1)-g(q_0) = g(q_1)=0, provided that (see (<ref>))Γ̂(,,,h) = 0.This means,ρ_0()^⊤M^-1p_0=h∑_j≥0ρ_j()^⊤M^-1{ξ_j[ψ_j-1+δ_j0()+ρ_j-1+δ_j0()] - ξ_j+1[ψ_j+1()+ρ_j+1()]}=h{(ξ_0 ρ_0()^⊤ M^-1ρ_0() +∑_j≥1ξ_j[ ρ_j()^⊤ M^-1ρ_j-1() - ρ_j-1()^⊤ M^-1ρ_j()]). .+ ξ_0ρ_0()^⊤ M^-1ψ_0() +∑_j≥1ξ_j[ ρ_j()^⊤ M^-1ψ_j-1() - ρ_j-1()^⊤ M^-1ψ_j()] },and can be formally recast into the following linear system:A(h) = b(h).Due to (<ref>) and Corollary <ref>, the coefficient matrix reads: A(h)=hξ_0 ρ_0()^⊤ M^-1ρ_0() +O(h^2) ≡ h/2∇ g(q_0)^⊤ M^-1∇ g(q_0) + O(h^2),and the right-hand side isb(h) = ρ_0()^⊤M^-1p_0 - ξ_0 h ρ_0()^⊤ M^-1ψ_0() + O(h^2)≡ ∇ g(q_0)^⊤M^-1p_0 + h/2[ ∇^2 g(q_0) (M^-1 p_0,M^-1p_0)- ∇ g(q_0)^⊤ M^-1∇ U(q_0)] + O(h^2).Consequently, (<ref>) is consistent with (<ref>), since∇ g(q_0)^⊤M^-1p_0=0, thus givingλ_0 = = [∇ g(q_0)^⊤ M^-1∇ g(q_0) + O(h)]^-1[ ∇^2 g(q_0) (M^-1 p_0,M^-1p_0)- ∇ g(q_0)^⊤ M^-1∇ U(q_0) + O(h)]≡ λ(q_0,p_0) + O(h).From Theorem <ref>, the conservation of energy follows.The third statement of the theorem can be shown using the nonlinear variation of constants formula, by noting that for t∈[0,h],λ(t)- ≡ λ(q(t),p(t))-λ(q(0),p(0))_=O(h) +O(h) = O(h).The last result follows from ∇ g(q_1)^⊤ M^-1 p_1= ∇ g(q(h)+O(h^2))^⊤ M^-1 (p(h)+O(h^2)) = ∇ g(q(h))^⊤ M^-1 p(h)_=0 + O(h^2) = O(h^2). Next, let us consider the mesht_n = nh,n=0,…,N,and the sequence of problems=M^-1, = -∇ U()-∇ g()λ_n,t∈[t_n,t_n+1],subject to initial conditions(t_n) = q_n, (t_n) = p_n,where λ_n is a suitable constant vector. Then, the following result follows.Consider the IVPs (<ref>)–(<ref>) and let us denote by (q(t),p(t))the solution of the problem (<ref>)–(<ref>). Moreover, let us assume that (q_n,p_n) satisfies the following conditions:* q_n-q(t_n) = O(h), p_n-p(t_n)=O(h);* ∇ g(q_n)^⊤M^-1∇ g(q_n) ∈^ν×ν is nonsingular;* g(q_n) = 0;* ∇ g(q_n)^⊤ M^-1 p_n = O(h^2).Then, for all sufficiently small h>0,∃!λ_n∈^ν such that the approximationsq_n+1:=u(t_n+1),p_n+1:=v(t_n+1),satisfy: * g(q_n+1)=0 and, therefore, H(q_n+1,p_n+1)=H(q_n,p_n);* λ_n = λ(q(t_n),p(t_n)) + O(h);* q_n+1-q(t_n+1) = O(h), p_n+1-p(t_n+1)=O(h);* ∇ g(q_n+1)^⊤ M^-1 p_n+1 = O(h^2). To show the first statement, we argue as we did to prove the first results in Theorem <ref>. This yields g(q_n+1)=0 provided that, cf. (<ref>)–(<ref>),A(h)λ_n = b(h)where A(h) and b(h) are defined as in (<ref>)–(<ref>) but with q_0 and p_0 replaced by q_n and p_n, respectively. Consequently, from (<ref>), we obtainλ_n = [∇ g(q_n)^⊤ M^-1∇ g(q_n) + O(h)]^-1 [ ∇^2 g(q_n) (M^-1 p_n,M^-1p_n)- ∇ g(q_n)^⊤ M^-1∇ U(q_n)-2/h∇ g(q_n)^⊤ M^-1 p_n^=O(h^2) + O(h)]= [∇ g(q_n)^⊤ M^-1∇ g(q_n) + O(h)]^-1[ ∇^2 g(q_n) (M^-1 p_n,M^-1p_n)- ∇ g(q_n)^⊤ M^-1∇ U(q_n) + O(h)]≡ λ(q_n,p_n) + O(h) = λ(q(t_n)+O(h),p(t_n)+O(h))+O(h) = λ(q(t_n),p(t_n))+O(h).Energy conservation follows, as before, from Theorem <ref>. Moreover, the nonlinear variation of constants formula, yields c q_n+1-q(t_n+1)p_n+1-p(t_n+1) = O(h) + O(h^2) = O(h),due to λ_n-λ(q(t),p(t))=O(h), for t∈[t_n,t_n+1], and the hypothesis q_n = q(t_n)+O(h), p_n=p(t_n)+O(h). In order to prove ∇ g(q_n+1)^⊤ M^-1 p_n+1 = O(h^2), we note that from the hypothesis∇ g(q_n)^⊤ M^-1p_n = O(h^2), the existence of _n∈^m such thatp_n-_n=O(h^2), ∇ g(q_n)^⊤ M^-1_n = 0follows. Using (q_n,_n) as local initial conditions for (<ref>), and repeating above steps to satisfy the constraints at t_n+1, we obtain_n = λ(q_n,_n) + O(h) ≡λ(q_n,p_n+O(h^2)) + O(h) = λ(q_n,p_n) + O(h) ≡λ_n+O(h),and corresponding approximations _n+1, _n+1 such thatg(_n+1)=0, ∇ g(_n+1)^⊤ M^-1_n+1 = O(h^2).From p_n-_n = O(h^2) and λ_n-_n=O(h), the nonlinear variation of constants formula yields, q_n+1-_n+1 = O(h^2),p_n+1-_n+1=O(h^2).Consequently, ∇ g(q_n+1)^⊤ M^-1 p_n+1 = ∇ g(_n+1+O(h^2))^⊤ M^-1 (_n+1+O(h^2)) = ∇ g(_n+1)^⊤ M^-1_n+1_=O(h^2) + O(h^2) = O(h^2). By means of Theorem <ref>, a straightforward induction argument enables to show the following relaxed version of Theorem <ref>. For all sufficiently small stepsizes h>0, the above procedure defines a sequence of approximations (q_n,p_n) such that, for all n=1,2,…,q_n = q(nh) + O(h),p_n = p(nh) + O(h),g(q_n)=0, ∇ g(q_n)^⊤ M^-1p_n = O(h^2).Moreover, (q_n+1,p_n+1) is obtained from (q_n,p_n) utilizing a constant vector λ_n such thatλ_n = λ( q(nh), p(nh) ) + O(h),where λ(q,p) is the function defined in (<ref>) and consequently, H(q_n+1,p_n+1) = H(q_n,p_n) holds. The fact that (<ref>) holds, in place of the weaker result (<ref>), is due to the symmetry of the proposed procedure. Note that a symmetric method is necessarily of even order <cit.>. The method is symmetric since we have shown that, forall sufficiently small h>0, there exists a unique λ_n such that from the solution of (<ref>)–(<ref>), with(q_n,p_n),g(q_n)=0,we arrive at a new point, where (q_n+1,p_n+1),g(q_n+1)=0,Since λ_n is unique, when we start from (<ref>) and solve backward in time (<ref>), we arrive at (<ref>), i.e. the procedure is symmetric. As a consequence of the symmetry of the method, the approximation order of (q_n,p_n) is even and therefore, (<ref>) holds in place of (<ref>). This completes the proof of Theorem <ref>.In the next theorem, we summarize in a more comprehensive form the statements derived previously in this section. Let us consider the problem (<ref>)–(<ref>), the mesh (<ref>), and the sequence of problems (<ref>)–(<ref>). The constant vector λ_n is chosen in such a way that for the new approximations defined by (<ref>), g(q_n+1)=0 follows. Then, for allsufficiently small stepsizes h>0, the above procedure defines a sequence of approximations (q_n,p_n,λ_n) satisfying [For the definition of λ(q,p) see (<ref>).]q_n=q(nh) + O(h^2), p_n=p(nh) + O(h^2), g(q_n)=0,∇ g(q_n)^⊤ M^-1p_n=O(h^2), λ_n= λ( q(nh), p(nh) ) + O(h),H(q_n,p_n)=H(q_0,p_0),n=0,1,…,N.Moreover, in case that λ is constant, λ(q(t),p(t))≡λ̅, ∀ t∈[0,T], the following statements hold: q_n=q(nh), p_n=p(nh), g(q_n)=0,∇ g(q_n)^⊤ M^-1p_n=0, λ_n= λ̅,H(q_n,p_n)=H(q_0,p_0),n=0,1,…,N.With other words, the discrete solution is exact, when the vector of the Lagrange multipliers is constant. Otherwise, it is second-order accurate for (q_n,p_n) and first order accurate for λ_n. In the latter case, the constraints and the Hamiltonian are conserved, while the hidden constraints remain O(h^2) close to zero. § POLYNOMIAL APPROXIMATION The first step towards the numerical solution of (<ref>)–(<ref>) is to truncate the series in the right-hand side of (<ref>),(ch) = ∑_j=0^s-1 P_j(c) γ_j(), (ch) = -∑_j=0^s-1 P_j(c)[ ψ_j() +ρ_j()], c∈[0,1].Here, the coefficientsγ_j, ψ_j, and ρ_j are as those defined in (<ref>). By imposing the initial conditions, the local approximation over the first step becomes(ch) = q_0 + h∑_j=0^s-1∫_0^cP_j(x) x γ_j(), (ch) = p_0 -h∑_j=0^s-1∫_0^cP_j(x) x[ ψ_j() +ρ_j()], c∈[0,1],with the new approximations which are formally still given by (<ref>). Again, the constant vector of the multipliers is uniquely determined by requiring that the constraints are satisfied at t_1=h. Consequently, we have the following expression, in place of (<ref>):ρ_0()^⊤M^-1p_0=h∑_j=0^s-2ρ_j()^⊤M^-1{ξ_j[ψ_j-1+δ_j0()+ρ_j-1+δ_j0()] - ξ_j+1[ψ_j+1()+ρ_j+1()]} + hξ_s-1ρ_s-1()^⊤ M^-1[ψ_s-2()+ρ_s-2()]=h{(ξ_0 ρ_0()^⊤ M^-1ρ_0() +∑_j=1^s-1ξ_j[ ρ_j()^⊤ M^-1ρ_j-1() - ρ_j-1()^⊤ M^-1ρ_j()]). .+ ξ_0ρ_0()^⊤ M^-1ψ_0() +∑_j=1^s-1ξ_j[ ρ_j()^⊤ M^-1ψ_j-1() - ρ_j-1()^⊤ M^-1ψ_j()] },which, in turn, yields equations which are formally similar to (<ref>)–(<ref>)[As before, this basic step defines a symmetric procedure.]. The process is then repeated by defining the mesh (<ref>) and considering the local problems_n(ch) = ∑_j=0^s-1 P_j(c) γ_j(_n), _n(ch) = -∑_j=0^s-1 P_j(c)[ ψ_j(_n) +ρ_j(_n)λ_n], c∈[0,1],u_n(0)=q_n,v_n(0) = p_n,n=0,…,N-1,where the coefficients γ_j(_n), ψ_j(_n), ρ_j(_n) are defined in (<ref>), withandreplaced by _n and _n, respectively. Consequently, we formally obtain the piecewise polynomial approximation,_n(ch)=q_n+h∑_j=0^s-1∫_0^cP_j(x) x γ_j(_n),_n(ch) =p_n -h∑_j=0^s-1∫_0^cP_j(x) x[ ψ_j(_n) +ρ_j(_n)λ_n], c∈[0,1],with the new approximations given by (see (<ref>))q_n+1 := _n(h) ≡ q_n+hγ_0(_n),p_n+1 := _n(h) ≡ p_n-h[ψ_0(_n) +ρ_0(_n)λ_n].As before, the constant vector λ_n∈^ν is chosen to satisfy the constraints g(q_n+1)=0 and it is implicitly defined by the equation,ρ_0(_n)^⊤M^-1p_n=h{(ξ_0 ρ_0(_n)^⊤ M^-1ρ_0(_n) +∑_j=1^s-1ξ_j[ ρ_j(_n)^⊤ M^-1ρ_j-1(_n) - ρ_j-1(_n)^⊤ M^-1ρ_j(_n)])λ_n . .+ ξ_0ρ_0(_n)^⊤ M^-1ψ_0(_n) +∑_j=1^s-1ξ_j[ ρ_j(_n)^⊤ M^-1ψ_j-1(_n) - ρ_j-1(_n)^⊤ M^-1ψ_j(_n)] }.This equation reduces to (<ref>) for n=0. Using arguments similar to those from the previous section (see also <cit.>), it is possible to show the following result. This result is a counterpart to Theorem <ref> for the piecewise polynomial approximation (<ref>) to the solution (q(t),p(t)) of problem (<ref>)–(<ref>). For all sufficiently small stepsizes h>0, the approximation procedure (<ref>)–(<ref>) is well defined and provides a sequence of approximations(q_n,p_n,λ_n) such thatq_n=q(nh) + O(h^2), p_n=p(nh) + O(h^2), g(q_n)=0,∇ g(q_n)^⊤ M^-1p_n=O(h^2), λ_n= λ( q(nh), p(nh) ) + O(h),H(q_n,p_n)=H(q_0,p_0),n=0,1,…,N.Moreover, in case that λ is constant, λ(q(t),p(t))≡λ̅, ∀ t∈[0,T], the following statements hold: q_n=q(nh) + O(h^2s), p_n=p(nh) + O(h^2s), g(q_n)=0,∇ g(q_n)^⊤ M^-1p_n=O(h^2s), λ_n= λ̅+O(h^2s),H(q_n,p_n)=H(q_0,p_0),n=0,1,…,N.§ FULL DISCRETIZATION In order to cast the above algorithm into a computational method, the integrals defining the coefficients γ_j(),ψ_j(),ρ_j(), j=0,…,s-1, in (<ref>), need to be approximated.[Since the method is a one-step method, we shall, as usual, only consider the first step.] To this aim, following the discussion in <cit.>, we use the Gauss-Legendre quadrature of order 2k (the interpolatory quadrature formula based at the zeros of P_k(c)), with nodes and weights (_i,_i), where k≥ s. Consequently, γ_j()≈ γ̂_j := M^-1∑_ℓ=1^k _ℓ P_j(_ℓ)(_ℓ h), ψ_j() ≈ ψ̂_j := ∑_ℓ=1^k _ℓP_j(_ℓ)∇ U((_ℓ h)), ρ_j()≈ ρ̂_j := ∑_ℓ=1^k _ℓP_j(_ℓ)∇ g((_ℓ h)),j=0,…,s-1.Formally, this is a k-stage Runge-Kutta method, whose computational cost depends on s rather than on k, since the actual unknowns are the 3s coefficients (<ref>) and the vector(see (<ref>)). We refer, to <cit.> for details. Let us now formulate the discrete problem to be solved in each integration step. We first define the matrices, cf. (<ref>),_s = ( P_j-1(_i)), _s = ( ∫_0^_i P_j-1(x) x) ∈^k× s, Ω̂= (_1,…,_k)∈^k× k,and the vectors and matricese = c 1 ⋮1∈^k, = cγ̂_0 ⋮ γ̂_s-1, = cψ̂_0 ⋮ ψ̂_s-1 ∈^sm, = cρ̂_0 ⋮ ρ̂_s-1∈^sm×ν.Recall that m is the dimension of the continuous problem and ν is the number of constraints. Then, the 3s equations from (<ref>), defining the discrete problem to be solved, amount to the system of equations, of (block) dimension s,= _s^⊤Ω̂⊗M^-1[ e⊗ p_0 - h_s⊗ I_m( +) ],= _s^⊤Ω̂⊗ I_m ∇ U( e⊗ q_0 +h_s⊗ I_m ),= _s^⊤Ω̂⊗ I_m ∇ g( e⊗ q_0 +h_s⊗ I_m ).We augment (<ref>) by the equation (<ref>) forwhich, by taking (<ref>) into account, can be rewritten ash[ξ_0 ρ̂_0^⊤ M^-1ρ̂_0 +∑_j=1^s-1ξ_j( ρ̂_j^⊤ M^-1ρ̂_j-1 - ρ̂_j-1^⊤ M^-1ρ̂_j)]= ρ̂_0^⊤M^-1(p_0-hξ_0ψ̂_0)-h∑_j=1^s-1ξ_j( ρ̂_j^⊤ M^-1ψ̂_j-1 - ρ̂_j-1^⊤ M^-1ψ̂_j).In (<ref>), ∇ U, when evaluated in a block vector of (block) dimension k, stands for the block vector made up of the k vectors resulting from the corresponding application of the function. The same straightforward notation is used for ∇ g. The new approximation is then given by, see (<ref>) and (<ref>),q_1 = q_0 + hγ̂_0,p_1 = p_0 -h[ ψ̂_0 +ρ̂_0].Note that the equations in (<ref>), together with (<ref>), formally coincide with those provided by a HBVM(k,s) method[Here, s is the degree of the polynomial approximation and k defines the order (actually equal to 2k) of the quadrature in the approximations (<ref>).] applied to solve the problem defined by the Hamiltonian (<ref>), where the vector of the multiplieris considered as a parameter,q̇ =M^-1p, ṗ = -∇ U(q)-∇ g(q), t≥0,q(0)=q_0, p(0)=p_0,cf. <cit.> for details. Consequently, equation (<ref>) defines the proper extension for handling the constrained Hamiltonian problem (<ref>)–(<ref>). For this reason, we continue to refer to the numerical method specified in (<ref>)–(<ref>) as to HBVM(k,s). Now, it is a ready to use numerical procedure. The discrete problem (<ref>)–(<ref>) can be solved via a straightforward fixed-point iteration, which converges under regularity assumptions, for all sufficiently small stepsizes h>0.[We refer to <cit.> for further details on procedures for solving the involved discrete problems. They are based on suitable Newton-splitting procedures, already implemented in computational codes <cit.>.] Moreover, for separable Hamiltonians, as it is the case in (<ref>), the last two equations in (<ref>) can be substituted into the first one, resulting in a single vector equation for . By settingΘ_(q) := ∇ U (q)+∇ g(q),we obtain= _s^⊤Ω̂⊗M^-1[ e⊗ p_0 - h_s_s^⊤Ω̂⊗ I_m Θ_( e⊗ q_0 +h_s⊗ I_m ) ],plus (<ref>) for .[Clearly,andcan be computed via the last two equations in (<ref>), onceandare known.]We skip further details, since they are exactly the same as for the original HBVMs, when applied to solve separable (unconstrained) Hamiltonian problems <cit.>. The following result follows from Theorem <ref> along with standard arguments from the analysis of HBVMs <cit.>.[For brevity, we omit the proof here.]For all sufficiently small stepsizes h>0, the HBVM(k,s) method (<ref>)–(<ref>) is well defined and symmetric. It provides a sequence of approximations(q_n,p_n,λ_n), n=0,1,…,N, such that q_n=q(nh) + O(h^2), p_n=p(nh) + O(h^2),∇ g(q_n)^⊤ M^-1p_n=O(h^2), λ_n= λ( q(nh), p(nh) ) + O(h),andg(q_n)= {[ 0, ; O(h^2k),, ]. H(q_n,p_n)-H(q_0,p_0)= {[ 0, ; O(h^2k),. ].Moreover, in case λ is constant, λ(q(t),p(t))≡λ̅, ∀ t∈[0,T], the following statements hold:q_n=q(nh) + O(h^2s), p_n=p(nh) + O(h^2s),∇ g(q_n)^⊤ M^-1p_n=O(h^2s), λ_n= λ̅+O(h^2s). We stress that by (<ref>), an exact or a (at least) practical conservation of both the constraints and the Hamiltonian can always be guaranteed. In fact, by choosing sufficiently large k, either the quadrature becomesexact, in the polynomial case, or the quadrature error is within the round-off error level, in the non polynomial case. This feature of the method will be always exploited in the numerical tests discussed in Section <ref>. Finally, we shall mention that for k=s, the HBVM(s,s) method reduces to the s-stage Gauss collocation method, <cit.>, which is symplectic. Moreover, in the limit k→∞, we retrieve the formulae studied in Section <ref>. This means that our approach can be also considered in the framework of Runge-Kutta methods with continuous stages <cit.>.§ NUMERICAL TESTSIn this section, to illustrate the theoretical properties of HBVM(k,s), we apply them to numerically simulate some Hamiltonian problems of the form (<ref>)–(<ref>) with holonomic constraints. In the focus of our attention are properties described in Theorem <ref>. §.§ PendulumWe begin with the planar pendulum in Cartesian coordinates, where a massless rod of length L connects a point of mass m to a fixed point (the origin). We assume a unit mass and length, m=1 and L=1, and normalize the gravity acceleration. Then, the Hamiltonian is given byH(q,p) = 1/2 p^⊤ p + e_2^⊤ q,e_2 := c 01,q := c xy,p:= q̇ ∈^2,and is subject to the constraintg(q) ≡ q^⊤ q -1 = 0.We also prescribe the initial conditions of the formq(0) = (0, -1)^⊤,p(0) = (1, 0)^⊤.Consequently, the constrained Hamiltonian problem reads: ẍ =-2xλ, ÿ = -1-2yλ,x^2+y^2=1, x(0)=0, y(0)=-1,ẋ(0) = 1,ẏ(0) = 0.In order to obtain a reference solution, we rewrite the problem in polar coordinates in such a way that θ=0 locates the pendulum at its stable rest position, so that x = sinθ,y = -cosθ.Thus, we arrive at the unconstrained Hamiltonian problemθ̈+sinθ = 0, θ(0) = 0, θ̇(0) = 1.Once this problem is solved, the solution of (<ref>) is recovered via the transformations (<ref>). Moreover, the Lagrange multiplier in (<ref>) turns out to be given byλ = 1/2( θ̇^2 +cosθ).To compute the reference solution for (<ref>), we solve (<ref>) by means of a HBVM(12,6) method [For unconstrained Hamiltonian problems.] of order 12 which is practically energy-conserving.According to (<ref>) and (<ref>), the Hamiltonian and the constraint are quadratic, so we expect HBVM(s,s) to conserve the energy and the constraint. In Table <ref>, we list the following quantities, obtained from the HBVM(s,s) methods for s=1,2,3, the stepsizes h=10^-12^-n and the interval of integration [0,10]: * the solution error (e_s),* the multiplier error (e_λ),* the Hamiltonian error (e_H),* the constraint error (e_g);* the hidden constraint error, defined by e_hc := max_n 2|x_nẋ_n+y_nẏ_n|. As predicted in Theorem <ref>, we can see that * all methods are second-order accurate in the space variables, with HBVM(1,1) less accurate than the others;* all methods are first-order accurate in the Lagrange multiplier;* all methods exactly conserve the Hamiltonian and the constraint;* all methods are second-order accurate in the hidden constraint. §.§ Conical pendulumNext, we consider the so-called conical pendulum, a particular case of the spherical pendulum, namely a pendulum of mass m, which is connected to a fixed point (i.e., the origin) by a massless rod of length L. For the conical pendulum, the initial condition is chosen in such a way that the motion is periodic with period T and occurs in the horizontal plane q_3 =z_0[Clearly, 0>z_0>-L.]. Again, assuming m=1 and L=1, and normalizing the acceleration of gravity, the Hamiltonian isH(q,p) := 1/2p^⊤ p + e_3^⊤ q,e_3 := c 001, q := c xyz,p := q̇ ∈^3,with the constraintg(q) := q^⊤ q-1 = 0.Here, we prescribe the consistent initial conditionsq(0) = r 2^-1/20-2^-1/2,p(0) = r 0 2^-1/40 ,generating a periodic motion withT = 2^3/4π,z_0 = -2^-1/2.Moreover, in such a case, the multiplier , which has the physical meaning of the tension on the rod, has to be constant and is given by λ_0 = 2^-1/2.Note that the Hamiltonian and the constraint are quadratic and according to Theorem <ref>, any HBVM(s,s) method conserves both of them and has order 2s. In Table <ref>, we list the errors in * the solution (e_s),* the multiplier (e_λ),* the Hamiltonian (e_H),* the constraints (e_g);* the hidden constraints, defined by e_hc := max_n ∇ g(q_n)^⊤ M^-1p_n. The problem is solved over 10 periods, with stepsizes h=T/n. As expected, the estimated rate of convergence for HBVM(s,s), s=1,2,3,4, is2s. Also, the Hamiltonian and the constraint are conserved up to round-off errors.Remarkably, also the error in the multiplier (e_λ) and in the hidden constraints (e_hc) appear to be within the round-off error level, whatever stepsize is used.In Figure <ref>,we plot the solution error from the computation over 100 periods using HBVM(2,2) with the stepsize h=T/100≈ 0.053, in Figure <ref>, the errors of the multiplier, Hamiltonian, constraint, and hidden constraints. One can see the linear growth of the solution error. The errors of the multiplier, constraint, Hamiltonian, and hidden constraints are negligible.§.§ Modified pendulum We now consider a modified version of the previous problem. With this simulation, we aim at exploiting the conservation of energy and constraints using a suitable high-order quadrature rule (<ref>). More precisely, we consider the following non-quadratic polynomial Hamiltonian:H(q,p) := 1/2p^⊤ p + (e_3^⊤ q)^4,q,p∈^3,with the non-quadratic polynomial constraintg(q) := ∑_i=1^3 (e_i^⊤ q)^2(4-i) -0.625 = 0.Here, we use the same initial condition as in (<ref>) but the vector of the multipliers is no more constant, so that the order of the method reduces to 2 (and 1 for the vector of the multipliers). Moreover, since the constraints and the Hamiltonian are polynomials of degree not larger than 6, any HBVM(3s,s) method will conserve both quantities. This is confirmed by Table <ref>, where the results for HBVM(3,1), HBVM(6,2), and HBVM(9,3) are listed. The interval of integration was again [0,10]. Table <ref> contains the errors in * the solution (e_s),* the multiplier (e_λ),* the Hamiltonian (e_H),* the constraints (e_g);* the hidden constraints (e_hc), formally defined via (<ref>).Again, as predicted in Theorem <ref>, we see that* all methods are second-order accurate in the state variables, with HBVM(3,1) less accurate than the others;* all methods are first-order accurate in the Lagrange multiplier;* all methods exactly conserve the Hamiltonian and the constraints;* all methods are second-order accurate in the hidden constraints.§.§ Tethered satellites system Finally, we discuss a closed-loop rotating triangular tethered satellites system,[This example can be found in <cit.>.] including three satellites (considered as mass-points) of masses m_i, i=1,2,3, joined by inextensible, tight, and massless tethers, of lengths L_i, i=1,2,3, respectively, which orbit around a massive body.[The Earth, in the original example.] As before, for sake of simplicity, we assume unit masses and lengths, and normalize the gravity constant. Consequently, if q_i:=(x_i,y_i,z_i)^⊤∈^3, i=1,2,3, are the positions of the three satellites, the constraints are given byg(q) := c (q_1-q_2)^⊤(q_1-q_2) - 1 (q_2-q_3)^⊤(q_2-q_3) - 1 (q_3-q_1)^⊤(q_3-q_1) - 1 = 0∈^3,and the Hamiltonian is specified by H(q,p) = ∑_i=1^3 (1/2p_i^⊤ p_i - 1/√(q_i^⊤ q_i)).The consistent initial conditions have the formq_1(0) = r01/2z_0,q_2(0) = r0 -1/2z_0,q_3(0) = c0 0 z_0-√(3)/2,andp_1(0)=p_2(0)= c 0 00, p_3(0) = c v_0 00,where z_0=20 and v_0 is such that the initial Hamiltonian is zero. This provides a configuration in which the first two satellites remain parallel to each other, moving in the planes y=1/2 and y=-1/2, respectively, and the third one moves around the tether joining the first two, in the plane y=0. In such a case, the Hamiltonian is non-polynomial. Nevertheless, using the HBVM(6,2) method with the stepsize h=0.1 over 10^4 steps, we obtain a qualitatively correct solution which conserves the Hamiltonian and the constraints within the round-off error level, see Figure <ref>. Here, we also plot the hidden constraints errors ∇ g(q_n)^⊤ M^-1p_n. At last, in Table <ref>, we list the following errors arising when solving the problem with HBVM(6,s) methods for s=1,2,3 and the stepsizes h=10^-12^-n, over the interval [0,10]: * the solution error (e_s),* the multipliers error (e_λ),* the Hamiltonian error (e_H),* the constraints error (e_g);* the hidden constraints errors (e_hc), formally defined by (<ref>).Again, as shown in Theorem <ref>,* all methods are second-order accurate in the state variables, with HBVM(6,1) much less accurate than the other two (which are almost equivalent);* all methods are first-order accurate in the Lagrange multipliers;* all methods exactly conserve the Hamiltonian and the constraints;* all methods are second-order accurate in the hidden constraints.To draw a general conclusion: it seems that using of HBVM(k,s), with s>1, in context of the numerical solution of the Hamiltonian problems with holonomic constraints can be recommended, although the method is only second-order accurate.§ CONCLUSIONSIn this paper, we have considered the numerical solution of Hamiltonian problems with holonomic constraints, by resorting to a line-integral formulation of the conservation of the constraints. 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Projected Runge-Kutta methods for constrained Hamiltonian systems. Appl. Math. Mech. – Engl. Ed. 37, No. 8 (2016) 1077–1094. ] | http://arxiv.org/abs/1704.08128v2 | {
"authors": [
"Luigi Brugnano",
"Gianmarco Gurioli",
"Felice Iavernaro",
"Ewa B. Weinmueller"
],
"categories": [
"math.NA",
"65P10, 65L80, 65L06"
],
"primary_category": "math.NA",
"published": "20170426140146",
"title": "Line Integral Solution of Hamiltonian Systems with Holonomic Constraints"
} |
Measurement of Galactic synchrotron emission] The angular power spectrum measurement of the Galactic synchrotron emission in twofields of the TGSS survey S. Choudhuri et al.]Samir Choudhuri^1,2Email:[email protected], Somnath Bharadwaj^2, Sk. Saiyad Ali^3, Nirupam Roy^4 Huib. T. Intema^5 and Abhik Ghosh^6,7^1 National Centre For Radio Astrophysics, Post Bag 3, Ganeshkhind, Pune 411 007, India^2 Department of Physics,& Centre for Theoretical Studies, IIT Kharagpur,Kharagpur 721 302, India^3 Department of Physics,Jadavpur University, Kolkata 700032, India^4 Department of Physics, Indian Institute of Science, Bangalore 560012, India^5 Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2333CA, Leiden, The Netherlands^6 Department of Physics and Astronomy, University of the Western Cape, Robert Sobukwe Road, Bellville 7535, South Africa^7 SKA SA, The Park, Park Road, Pinelands 7405, South Africa[ [===== Characterizing the diffuse Galactic synchrotron emission at arcminuteangular scales is needed to reliably remove foregrounds in cosmological 21-cm measurements. The study of this emission is also interesting in its own right. Here, we quantify the fluctuations of the diffuse Galactic synchrotron emission using visibility data for two of the fields observed by the TIFR GMRT Sky Survey (TGSS).We have used the 2D Tapered Gridded Estimator (TGE) to estimate the angular power spectrum (C_ℓ) from the visibilities. We find that the sky signal, after subtracting the point sources, is likely dominated by the diffuse Galactic synchrotron radiation across the angular multipole range 240 ≤ℓ≲ 500. We present a power law fit, C_ℓ=A×(1000/l)^β, to the measured C_ℓ over this ℓ range. We find that (A,β) have values(356±109 mK^2,2.8±0.3) and(54±26 mK^2,2.2±0.4) in the two fields.For the second field, however, there is indication of a significant residual point source contribution, and for this field we interpret the measured C_ℓ asan upper limit for thediffuse Galactic synchrotron emission. While in both fields the slopes areconsistent with earlier measurements, the second field appearsto have an amplitude which is considerably smaller compared to similar measurementsin other parts of the sky. methods: statistical, data analysis - techniques: interferometric- cosmology: diffuse radiation § INTRODUCTIONObservations of the redshifted 21-cm signal from the Epoch of Reionization (EoR) contain a wealth of cosmological and astrophysical information <cit.>. The Giant Metrewave Radio Telescope (GMRT, ) is currently functioning at a frequency band which corresponds tothe 21-cm signal from this epoch. Several ongoing and future experiments such as the Donald C. Backer Precision Array to Probe the Epoch of Reionization (PAPER, ), the Low Frequency Array (LOFAR, ), the Murchison Wide-field Array (MWA, ), the Square Kilometer Array (SKA1 LOW, ) and the Hydrogen Epoch of Reionization Array (HERA, ) are aiming to measure the EoR 21-cm signal. The EoR 21-cm signal is overwhelmed by different foregrounds which are four to five orders of magnitude stronger than the expected 21-cm signal <cit.>. Accurately modelling and subtracting the foregrounds from the data are the main challenges for detecting the EoR 21-cm signal. The diffuse Galactic synchrotron emission (hereafter, DGSE) is expected to be the most dominant foreground at10 arcminute angular scales after point source subtraction at 10-20 mJy level <cit.>.A precise characterization and a detailed understanding of the DGSEis needed to reliably remove foregrounds in 21-cm experiments. In this paper, we characterize the DGSE at arcminute angular scales which are relevant for the cosmological 21-cm studies. The study of the DGSEis also important in its own right. The angular power spectrum (C_ℓ) of the DGSE quantifies the fluctuations in the magnetic field and in the electron density of the turbulent interstellar medium (ISM) of our Galaxy (e.g. ).There are several observations towards characterizing the DGSE spanning a wide range of frequency. <cit.> have measured the all sky diffuse Galactic synchrotron radiation at 408 MHz. <cit.> and <cit.> have presented the Galactic synchrotron maps at a relatively higher frequency ( 1420 MHz). Using the 2.3 GHz Rhodes Survey, <cit.> have shown that the C_ℓ of the diffuse Galactic synchrotron radiation behaves like a power law (C_ℓ∝ℓ^-β) where the power law index β=2.43 in the ℓ range 2≤ℓ≤100.<cit.> have found that the value of β is 2.37 for the 2.4 GHz Parkes Survey in the ℓ range 40≤ℓ≤250. The C_ℓ measured from the Wilkinson Microwave Anisotropy Probe (WMAP) data show a slightly lower value of β (C_ℓ∝ℓ^2) for ℓ<200 <cit.>. <cit.> have analysed 150 MHz Westerbork Synthesis Radio Telescope (WSRT) observations to characterize the statistical properties of the diffuse Galactic emission and find that C_ℓ=A×(1000/l)^β mK^2 where A=253 mK^2 and β=2.2 for ℓ≤900. <cit.> have used GMRT 150 MHz observations to characterize the foregrounds for 21-cm experiments and find thatA=513 mK^2 and β=2.34 in the ℓ range 253≤ℓ≤800. Recently, <cit.> present the first LOFAR detection of the DGSEaround 160 MHz.They reported that the C_ℓ of the foreground synchrotron fluctuations is approximately a power law with a slope β≈1.8 up to angular multipoles of 1300.In this paper we study the statistical properties of the DGSE using two fields observed by the TIFR GMRT Sky Survey (TGSS[http://tgss.ncra.tifr.res.in]; ). We have used the data which was calibrated and processed by <cit.>. We have applied the Tapered Gridded Estimator (TGE; , hereafter Paper I) to the residual data to measure the C_ℓ of the background sky signal after point source subtraction. The TGE suppresses the contribution from the residual point sources in the outer region of the telescope's field of view (FoV) and also internally subtracts out the noise bias to give an unbiased estimate of C_ℓ <cit.>. For each field we are able to identify an angular multipole range where the measured C_ℓ is likely dominated by the DGSE, and we present power law fits for these. § DATA ANALYSIS The TGSS survey contains 2000 hours of observing timedivided on 5336 individual pointings on an approximate hexagonal grid. The observing time for each field is about 15 minutes.For the purpose of this paper, we have used only two data sets for two fields located at Galactic coordinates (9^∘,+10^∘; Data1) and (15^∘,-11^∘; Data2). We have selected these fields because they are close to the Galactic plane, and also the contributions from the very bright compact sources are much less in these fields. The central frequency of this survey is 147.5 MHz with an instantaneous bandwidth of 16.7 MHz which is divided into 256 frequency channels. All the TGSS raw data was analysed with a fully automated pipeline based on the SPAM package <cit.>. The operation of the SPAM package is divided into two parts: (a)Pre-processing and (b) Main pipeline. The Pre-processing step calculates good-quality instrumental calibration from the best available scans on one of the primary calibrators, and transfers these to the target field. In the Main pipeline the direction independent and direction dependent calibrationsare calculated for each field,and the calibrated visibilities are converted into“CLEANed” deconvolved radio images. The off source rms noise (σ_n) for the continuum images of these fields are 4.1 mJy/ Beam and 3.1 mJy/ Beam for Data1 and Data2 respectively, both values lie close to the median rms.noiseof 3.5 mJy/ Beam for the whole survey. The angular resolution of these observations is 25^”× 25^”. This pipeline applies direction-dependent gains toimage and subtractpointsources to a S_c=5σ_n flux threshold covering an angular regionofradius∼1.5 times the telescope's FoV (3.1^∘× 3.1^∘),and also includes afew bright sources even further away. Thesubsequent analysis here uses theresidual visibility data aftersubtracting out the discrete sources.We have used the TGE to estimate C_ℓ from the measuredvisibilities _i with _̆i referring to the corresponding baseline. As mentioned earlier, the TGE suppresses the contribution from the residual point sources in the outer region of the telescope's FoV and also internally subtracts out the noise bias to give an unbiased estimate of C_ℓ (details in , Paper I). The tapering is introduced by multiplying the sky with a Gaussian window function W(θ)=e^-θ^2/θ^2_w. The value of θ_w should be chosen in such a way that it cuts off the sky response well before the first null of the primary beam without removing too much of the signalfrom the central region. Here we have used θ_w=95^' which is slightly smaller than 114^', the half width at half maxima (HWHM) of the GMRT primary beamat 150 MHz. This is implemented by dividing the uv plane into a rectangular grid and evaluating the convolved visibilities _cg at every grid point g _cg = ∑_iw̃(_̆g-_̆i)_iwhere w̃()̆ is the Fourier transform of the taper window function W(θ) and _̆g refers to the baseline of different grid points. The entire datacontaining visibility measurements indifferent frequency channels that spans a 16 MHz bandwidthwas collapsed to a single grid after scaling each baseline to theappropriate frequency.The self correlation of the gridded and convolvedvisibilities (equation (10) and (13) of Paper I) can be written as, ⟨|_cg|^2 ⟩ = ( ∂ B/∂ T)^2 ∫ d^2 U|K̃(_̆g - ) |^2 C_2 π U_g+ ∑_i |w̃(_̆g-_̆i) |^2 ⟨|_i |^2 ⟩, where, ( ∂ B/∂ T) is the conversion factor from brightness temperature to specific intensity,_i is the noise contribution to the individual visibility _i and K̃(_̆g -) is an effective“gridding kernel” which incorporates the effects of (a) telescope's primary beam pattern (b) the tapering window function and (c) the baseline sampling in the uv plane. We have approximated the convolution in equation (<ref>) as, ⟨|_cg|^2 ⟩ =[( ∂ B/∂ T)^2 ∫ d^2 U|K̃(_̆g-)̆|^2 ] C_2 π U_g+ ∑_i |w̃(_̆g-_̆i) |^2 ⟨|_i |^2 ⟩,under theassumption that the C_ℓ (ℓ=2π||̆) is nearly constant across the width of K̃(_̆g - ). We define the Tapered Gridded Estimator (TGE) asÊ_g= M_g^-1 ( |_cg|^2 -∑_i |w̃(_̆g-_̆i) |^2 |_i |^2 ). where M_g is the normalizing factor which we have calculated by using simulated visibilities corresponding to an unit angular power spectrum (details in Paper I). We have ⟨Ê_g ⟩ = C_ℓ_g i.e. the TGE Ê_g provides an unbiased estimate of the angular power spectrum C_ℓ at the angular multipole ℓ_g=2 π U_g corresponding to the baseline _̆g. We have used the TGE to estimate C_ℓ and its variance in bins of equal logarithmic interval in ℓ (equations (19) and (25) in Paper I).§ RESULTS AND CONCLUSIONS The upper curves of the left and right panels of Figure <ref> show the estimated C_ℓ before point source subtraction for Data1 and Data2 respectively. We find that for both the data sets the measured C_ℓ is in the range 10^4-10^5 mK^2 across the entire ℓ range.Model predictions <cit.> indicate that the point source contribution is expected to be considerably larger than the Galactic synchrotron emission across much of the ℓ range considered here, however the two may be comparable at the smaller ℓ values of our interest. Further, the convolutionin equation (<ref>) is expected to be important at small ℓ, and it is necessary to also account for this.The lower curves of both the panels of Figure <ref> show the estimated C_ℓ after point source subtraction. We see that removing the point sources causes a very substantial drop in theC_ℓmeasured at large ℓ. This clearly demonstrates that theC_ℓat these angular scales was dominated by the point sources prior to their subtraction. We further believe that after point source subtraction the C_ℓmeasured at large ℓ continues to be dominated by theresidual point sources which are below the threshold flux.The residual fluxfrom imperfect subtraction of the bright sources possibly alsomakes a significant contribution in the measured C_ℓ at large ℓ. This interpretation is mainly guided by the modelpredictions (Figure 6 of ), and isalsoindicatedby the nearly flat C_ℓ which is consistent with the Poisson fluctuations of a random pointsource distribution.In contrast to this,C_ℓ shows a steep power-law ℓ dependence at small ℓ (≤ℓ_max) withℓ_max=580 and 440for Data1 and Data2 respectively. This steep power law is the characteristic of the diffuse Galactic emission and we believe that the measuredC_ℓ ispossibly dominated by the DGSE at thelarge angular scales corresponding to ℓ≤ℓ_max. As mentioned earlier,the convolutionin equation (<ref>) is expected to beimportant at large angular scalesand it is necessary toaccount for this inorderto correctly interpret the results at small ℓ.We have carried out simulations in order to assess the effect of theconvolution on the estimatedC_ℓ. GMRT visibility data was simulatedassuming that the sky brightness temperature fluctuations are a realization ofa Gaussian random field with input model angular power spectrum C^M_ℓ ofthe form given byeq. (<ref>). The simulations incorporate the GMRT primary beam pattern and the uv tracks corresponding to the actual observationunder consideration.The reader is referred to <cit.>for more details of the simulations.Figure <ref> shows the C_ℓ estimated from the Data1 simulationsforβ =3 and 1.5 which roughly encompasses the entire range ofthepower law index we expect for the Galactic synchrotron emission.We find that the effect of the convolution is important in the rangeℓ<ℓ_min=240, and we have excluded this ℓ range from our analysis. We are, however,able to recover the input model angular power spectrum quite accurately in the region ℓ≥ℓ_min which we have used for our subsequent analysis.We have also carried out the same analysis forData2 (not shown here) where we find that ℓ_min has a value that is almost the same as forData1. We have used the ℓ range ℓ_min≤ℓ≤ℓ_max to fit a power law of the formgiven in eq. (<ref>) to the C_ℓ measuredafter point source subtraction.The data points with 1-σ error bars and the best fit power law are shown in Figure <ref>. Note that we have identified one of the Data1 points asan outlier and excluded it from the fit. The best fit parameters (A,β),N the numberof data points used for the fit and χ^2/(N-2) thechi-squareper degree of freedom (reduced χ^2) are listed in Table <ref>. The rather low values of the reduced χ^2 indicate that the errors in the measuredC_ℓ have possibly been somewhat overestimated. In order to validate our methodology we have simulated the visibility data for an input model power spectrum with the best fit values of the parameters (A,β) and used this to estimate C_ℓ. The mean C_ℓ and 1-σ errors (shaded region) estimated from 128 realization of the simulation are shown in Figure <ref>. For the relevant ℓ range we find that the simulated C_ℓ is in very good agreement with the measured values thereby validating the entire fitting procedure.The horizontal lines in both the panels of Figure <ref> show the C_ℓ predicted from the Poisson fluctuations of residual point sources below a threshold flux density of S_c=50 mJy.The C_ℓ prediction here is based on the 150 MHz source counts of <cit.>.We find that for ℓ>ℓ_max the measured C_ℓ values are well in excess of this prediction indicating that (1.) there are significant residual imaging artifacts around the bright source (S > S_c) which were subtracted , and/or (2.) the actual source distribution is in excess of the predictions of the source counts. Note that the actual S_c values (20.5 and 15.5 mJy for Data1 and Data2 respectively) are well below 50 mJy, and the corresponding C_ℓ predictions will lie below the horizontal lines shown in Figure <ref>. For both the fields C_ℓ (Figure <ref>) is nearly flatat large ℓ (> 500)anditis well modeledby a power lawat smaller ℓ (240 ≤ℓ≲ 500). ForData1 thepower law rises above the flat C_ℓ, and the power law is likelydominated by the DGSE. However, for Data2 thepower lawfalls below the flat C_ℓ, and it is likely that in addition to the DGSE there is asignificant residual point sources contribution.For Data2 we interpret the best fit power law as an upper limit for the DGSE.The best fit parameters(A,β)=(356.23±109.5,2.8±0.3) and (54.6±26,2.2±0.4) for Data1 and Data2 respectivelyare compared withmeasurements from other 150 MHz observations such as <cit.>in Table <ref>. Further, we have also used an earlier work () at higher frequencies ( 408 and 1420 MHz) to estimate and compare the amplitude of the angular power spectrum of the DGSEexpected at our observing frequency. Using the best-fit parameters (tabulated at ℓ = 100) at 408 and 1420 MHz, we extrapolate the amplitude of the C_ℓat our observing frequency at ℓ = 1000for |b| ≥ 10^∘ and|b| ≥ 20^∘. In this extrapolation we use a mean frequency spectral index of α = 2.5 () (C_ℓ∝ν^2α). The extrapolated amplitude values are shown in Table <ref>.In Table <ref>, we note that the angular power spectra ofthe DGSEinthe northern hemisphere are comparatively larger thanthat of the southern hemisphere. The best fit parameter A forData1(Data2)agrees mostly with the extrapolated values obtained from b ≥ +10^∘ (b ≤ -10^∘) andb ≥ +20^∘ ( b ≤ -20^∘) within a factor of about 2 (4). The best fit parameter β forData1 and Data2 is within the range of 1.5-3.0 found by all the previous measurements at 150 MHz andhigher frequencies. The entire analysis here is based on the assumption that the DGSE is a Gaussian random field. This is possibly justified for thesmall patch of the sky under observation given that thediffuseemission is generated by a random processes likeMHD turbulence. The estimated C_ℓ remains unaffected even if this assumption breaks down, only the error estimates will be changed. Wenote that theparameters(A,β) are varying significantly from field to field across the different direction in the sky. We plan to extend this analysis for the whole sky and study the variation of the amplitude (A) and power law index (β) of C_ℓ using the full TGGS survey in future. § ACKNOWLEDGEMENTSWe thank an anonymous referee for helpful comments. S. Choudhuri would like to acknowledge the University Grant Commission, India for providing financial support. AG would like acknowledge Postdoctoral Fellowship from the South African Square Kilometre Array Project for financial support. We thank the staff of the GMRT that made these observations possible. GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research. We thank the staff of the GMRT that made these observations possible. GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.99 [Ali et al.(2008)]ali08Ali, S. S., Bharadwaj, S., & Chengalur, J. N. 2008, , 385, 2166[Bennett et al.2003]bennett03 Bennett C.L., Hill R.S., Hinshaw. G. et al., 2003, ApJS, 148, 97 [Bernardi et al.(2009)]bernardi09Bernardi, G., de Bruyn, A. G., Brentjens, M. A., et al. 2009, , 500, 965[Bharadwaj & Ali(2005)]bharadwaj05Bharadwaj, S., & Ali, S. S. 2005, , 356, 1519[Bowman et al. 2013]bowman13 Bowman J. D. et al., 2013, PASA, 30, e031 [Choudhuri et al.(2014)]samir14Choudhuri, S., Bharadwaj, S., Ghosh, A., & Ali, S. S. 2014, , 445, 4351[Choudhuri et al.(2016)]samir16bChoudhuri, S., Bharadwaj, S., Chatterjee, S.,Ali, S. 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"authors": [
"Samir Choudhuri",
"Somnath Bharadwaj",
"Sk. Saiyad Ali",
"Nirupam Roy",
"H. T. Intema",
"Abhik Ghosh"
],
"categories": [
"astro-ph.CO"
],
"primary_category": "astro-ph.CO",
"published": "20170427162552",
"title": "The angular power spectrum measurement of the Galactic synchrotron emission in two fields of the TGSS survey"
} |
Are chains of type I radio bursts generated by similar processes as drifting pulsation structures observed during solar flares? M. Karlický Received ; accepted=============================================================================================================================== The paper develops a new technique to extract a characteristic subset from a random source that repeatedly samples from a set of elements. Here a characteristic subset is a set that when containing an element contains all elements that have the same probability.With this technique at hand the paper looks at the special case of the tournament isomorphism problem that stands in the way towards a polynomial-time algorithm for the graph isomorphism problem. Noting that there is a reduction from the automorphism (asymmetry) problem to the isomorphism problem, a reduction in the other direction is nevertheless not known and remains a thorny open problem.Applying the new technique, we develop a randomized polynomial-time Turing-reduction from the tournament isomorphism problem to the tournament automorphism problem. This is the first such reduction for any kind of combinatorial object not known to have a polynomial-time solvable isomorphism problem. § INTRODUCTION.The graph automorphism problem asks whether a given input graph has a non-trivial automorphism. In other words the task is to decide whether a given graph is asymmetric. This computational problem is typically seen in the context of the graph isomorphism problem, which is itself equivalent under polynomial-time Turing reductions to the problem of computing a generating set for all automorphisms of a graph <cit.>. As a special case of the latter, the graph automorphism problem obviously reduces to the graph isomorphism problem. However, no reduction from the graph isomorphism to the graph automorphism problem is known. In fact, while many computational problems surrounding structural equivalence of combinatorial objects can all be Turing-reduced to one another, the relationship between the graph automorphism and the graph isomorphism problem remains a repeatedly posed open question (see for example <cit.>). With Babai's new ground-breaking algorithm <cit.> that solves the graph isomorphism problem and thereby also the graph automorphism problem in quasi-polynomial time, the question arises whether it is possible to go further and devise a polynomial-time algorithm. For such an endeavor to succeed, special cases such as the group isomorphism and the tournament isomorphism problem, for which the currently fastest algorithms have a running time of n^O(log n), should also be solvable in polynomial time. Tournaments, which are graphs in which between every pair of vertices there exists exactly one directed edge, also have an automorphism problem associated with them, asking whether a given tournament is asymmetric[Many publications in the context of graph isomorphism use the term rigid graph. However, the literature is inconsistent on the notion of a rigid graph, which can for example refer to having no non-trivial automorphism or no non-trivial endomorphism. We will use the notion asymmetric, which only ever means the former. Furthermore, we suggest the name graph asymmetry problem over graph automorphism problem,so as not to confuse it with the computational problem to compute the automorphism group.]. Again, for this problem the currently best running time is n^O(log n) and analogously to general graphs there is a simple reduction from the automorphism problem to the isomorphism problem, but no reverse reduction has been known.In this paper we show that there is a randomized polynomial-time Turing reduction from the tournament isomorphism problem to the tournament automorphism problem. This is the first such reduction for any kind of combinatorial object (apart from polynomial-time solvable cases of course).The main new technical tool that we develop in the first part of the paper is a technique to exploit an oracle to the graph automorphism problem in order to obtain a non-trivial automorphism-invariant partition of a graph that is finer than the orbit partition (Sections <ref>–<ref>). We call the parts of such a partition suborbits. This technique is essentially applicable to all graph classes, not just tournaments. It hinges on a method to extract a characteristic subset from a random source that repeatedly samples from a set of elements. Here we say that a set is characteristic if it is a union of level sets of the probability function. In the second part of the paper we show that, for tournaments, access to suborbits suffices to compute automorphism groups (Section <ref>). For this we adapt the group-theoretic divide and conquer approach of Luks <cit.> to our situation. In this second part we exploit that the automorphism group of tournaments is solvable and we leave it as an open question whether something similar can be forged that is applicable to the group isomorphism problem (see Section <ref>).It might be worth noting that the techniques actually do not use any of the new structural insights from the quasi-polynomial-time algorithm of <cit.>. Rather, the randomized sampling idea is heavily based on an older practical randomized algorithm designed to quickly detect non-isomorphism (<cit.>). It appears to be one of the few cases where randomization helps to derive a theoretical result for an isomorphism problem. We also borrow some ideas from a paper of Arvind, Das, and Mukhopadhyay concerned with tournament canonization <cit.>.The necessity for randomization to obtain theoretical results in the context of isomorphism checking appears to be quite rare. The earliest result exploiting randomization seems to go to back to Babai <cit.> and is a randomized algorithm for checking isomorphism of graphs of bounded color class size. However that algorithm is actually a Las Vegas algorithm (an algorithm that does not make errors), and in the meantime deterministic algorithms are available <cit.>. However, for the new reduction in this paper it seems unclear how to remove the use of randomization and even how to remove the possibility for errors.§.§ Related work:With respect to related work, we focus on results concerning graph automorphism as well as results concerning tournaments and refer the reader to other texts (for example <cit.>) for a general introduction to the graph isomorphism problem, current algorithms and overviews over complexity theoretic results. (Tournament automorphism) Let us start by highlighting two results specifically concerned with the tournament automorphism problem. Arvind, Das, and Mukhopadhyay <cit.> show that if tournament isomorphism is polynomial-time solvable then tournament canonization can be reduced in polynomial time to canonization of asymmetric tournaments. This implies now, with the result of the current paper, that from a canonization algorithm for asymmetric tournaments we can obtain a randomized canonization algorithm for tournaments in general. (In other words, the main theorem of our paper transfers to canonization.) On the hardness side, Wager <cit.> shows that tournament automorphism is hard for various circuit complexity classes (NL, C_=L, PL, DET, MOD_kL) under AC^0 reductions.(Graph automorphism) A lot of information on the complexity of graph automorphism can be found in the book by Köbler, Schöning, and Torán <cit.>. Concerning hardness of the automorphism problem, improving previous results of Torán <cit.>, Wagner shows hardness results for graphs of bounded maximum degree <cit.>. Agrawal and Arvind show truth table equivalence of several problems related to graph automorphism <cit.> and Arvind, Beigel, and Lozano study modular versions of graph automorphism <cit.> which for k∈ℕ ask whether the number of automorphisms of a given graph is divisible by k.The graph automorphism problem is of interest in quantum computing since it can be encoded as a hidden shift problem, as opposed to the graph isomorphism problem that is only known to be encodable as a hidden subgroup problem <cit.>.Recently, Allender, Grochow, and Moore <cit.> developed a zero-error randomized reduction from graph automorphism to MKTP, the problem of minimizing time-bounded Kolmogorov complexity, a variant of the minimum circuit size problem. In that paper they also extend this to a bounded-error randomized reduction from graph isomorphism to MKTP.(Tournament isomorphism) Concerning the tournament isomorphism problem, the currently fastest algorithm <cit.> has a running time of n^O(log n). With respect to hardness, Wagner's results for tournament automorphism also apply to tournament isomorphism <cit.>.Ponomarenko showed that isomorphism of cyclic tournaments can be decided in polynomial time <cit.>, where a cyclic tournament is a tournament that has an automorphism that is a permutation with a single cycle spanning all vertices. Furthermore he showed that isomorphism of Schurian tournaments can be decided in polynomial time <cit.>. § SAMPLING CHARACTERISTIC SUBSETS. Let M be a finite set. We define a sampler over M to be a probability measure _ M→ [0,1] on the elements of M. We think of a sampler as an oracle that we can invoke in order to obtain an element of M. That is, given a sampler, we can sample a sequence of elements m_1,…,m_t where each m_i is sampled independently from M according to _.We call a subset M' of M characteristic with respect to if for all m,m'∈ M it holds that m∈ M' and _(m') = _(m) implies m'∈ M'. Another way of formulating this condition is that M' is invariant under all probability-preserving bijections φ M → M, that is, those bijections that satisfy _(m) = _(φ(m)) for all m∈ M. When considering sampling algorithms we will not assume that we know the size of the set M.Our goal is to repeatedly invoke a sampler M so as to find a characteristic subset. The main difficulty in this is that we can never precisely determine the probability _(m) of an element m. Indeed, the only thing we can hope for is to get a good estimate for such a probability. The following lemma indicates that this might be helpful since the set of probabilities cannot be arbitrarily dense. Let _ be a discrete probability measure on the set M. Let P = {_(m)| m∈ M} be the set of probabilities that occur. For every positive integer i there is a j∈{6i+1,…,8i} such that [ (j-1/4)/(8i^2),(j+1/4)/(8i^2)]∩ P = ∅.Suppose for all j∈{6i+1,…,8i} there is some m_j with _(m_j) ∈ [ (j-1/4)/(8i^2),(j+1/4)/8i^2]. Then _(m_j)≠_(m_j') whenever j≠ j', implying in particular m_j ≠ m_j'. This yields 2i distinct elements m_j. Furthermore _(m_j)> 3/(4i)for all j∈{6i+1,…,8i}.Thus _({m_j| j∈{6i+1,…,8i}}) > 2i ·3/(4i) >1 yielding a contradiction. Using the lemma we can design an algorithm that, with high probability, succeeds at determining a characteristic set. There is a deterministicalgorithm that, given ε>0 and given access to a sampler S over an unknown set M of unknown size, runs in expected time polynomial in 1/(max_m∈ M_S(m)) ≤ |M| and ln1/ε and outputs a non-empty subset of M that is characteristic with probability 1-ε.Let p = max_m∈ M_S(m) andlet i = ⌈ 1/p⌉. First note that |M|≥ 1/p and that i ≤ 2/p ≤ 2|M|.Let P= {_S(m) | m∈ M} be the set of values that occur as probabilities of elements in M. The idea of the proof is to sample many times as to get good estimates for probabilities using Chernoff bounds and then to include in the output all elements with a probability above a certain threshold.The main difficulty of the Lemma arises from the fact that p is not known to the algorithm. We first describe an algorithm for the situation in which p is known which works in such a way so that it can be adapted in the end. We start by sampling T= max{⌈ i^3 2^17 (ln1/ε')⌉, ⌈ i^3 2^18 (ln1/ε')⌉^2}elements m_1,…,m_T from the sampler, where we set ε' = min{1/e,ε/8}. We then compute for each appearing element m_k a probability estimator #(m_k) for its probability by computing N(m_k)/T where N(m_k) is the number of times that element m_k has been sampled. Let Q= {#(m_k)| k∈{1,…, T}} be the set of probability estimators. Let ℓ be the smallest number in {6i+1,…,8i} such that [ (ℓ-1/8)/(8i^2),(ℓ+1/8)/(8i^2)]∩ Q=∅. If no such element exists, we declare the algorithm as failed.Otherwise, we output M' = {m_k |#(m_k)> ℓ/(8i^2)}. We call ℓ the cut-off.We analyze the probability that this algorithm succeeds in computing a characteristic subset. For this, let us define #(x) = 0 for x∈ M whenever x does not appear among the sampled elements. For each element x∈ M, the probability that |#(x)-_S(x)| ≥ 1/(2^7 i^2) is at most 2e^-T/2^17 i^3≤ 2ε'.Consider an experiment where we sample T elements according to S. We want to bound the probability that the observed T·#(x) deviates from its expected value μ T·_(x) by at least T/(2^7i^2). This deviation is at least δμ if we set δ1/2^7 i^2_(x) >0.We can thus use the Chernoff bound (see <cit.>) and conclude that the probability that|#(x)-_S(x)| ≥ 1/(2^7 i^2) is at most 2e^-μmin{δ^2/4,δ/2}≤ 2e^(-Tmin{1/2^16 i^4 _(x),1/2^8 i^2})≤ 2e^(-T/max{2^17 i^3,2^8 i^2 })≤ 2e^(-T/ 2^17 i^3),where the second inequality uses the fact _(x) ≤ p = 2/(2/p) ≤ 2/⌈ 1/p⌉ =2/i. Define A_k as the event that for the k-th sampled element m_k we have |#(m_k) - (m_k)| ≥ 1/(2^6 i^2). Thus the event A_k happens if #(m_k) deviates excessively from its expected value. [resume] The probability that there is a k∈{1,…,T} such that event A_k occursis at most 2ε'. To bound the probability of event A_k, we first consider (A_k | m_k = x), the probability of A_k under the condition that the k-th sampled element m_k is equal to x for some fixed element x∈ M. Considering that we already know that m_k = x we need to consider an experiment where we sample T-1 times independently from S and count the number of times we obtain element x. This number is then N(m_k)-1 since the item with number k itself adds one to the count of elements equal to x. If #(m_k)= N(m_k)/T ∉[ _S(m_k)-1/(2^6 i^2),_S(m_k)+1/(2^6 i^2)]then (N(m_k)-1)/(T-1) ∉ [ _S(m_k)-1/(2^7 i^2),_S(m_k)+1/(2^7 i^2)],as shown by the simple fact that for positive integers 2≤ a≤ b we have |a/b - (a-1)/(b-1)| ≤ 1 / (b-1) and 1/(2^7 i^2 ) ≥ 1/(T-1). Thus in our experiment with T-1 trials, the observed value N(m_k)-1 must deviate from its expected value μ (T-1) _(x) by at least (T-1)/(2^7 i^2). From the previous claim we obtain an upper bound of 2e^-(T-1)/ 2^17 i^3≤ 2e^-T/ 2^18 i^3,where the inequality uses the fact that T≥ 2.Since this bound is independent of x∈ M and since x was arbitrary, the bound is also an upper for (A_k).By the union bound and using T≥⌈ i^3 (ln1/ε') 2^18⌉^2, we obtain that the probability that there is a k∈{1,…,T} such that A_k happens is at most T ·2e^-T/2^18 i^3≤ T2e^-√(T)ε'≤ 2ε',where the last inequality follows since t^2e^-t <1 for t≥ 1. [resume] If the algorithm is declared as failed then A_k occurs for some k∈{1,…,T}. ByLemma <ref> there is an integer j∈{6i+1,…,8i} such that _(m)∉[(j-1/4)/(8i^2),(j+1/4)/(8i^2)] for all m∈ M. Define B_k as the event that for the k-th sampled element m_k we have #(m_k) ∈ [ (j-1/8)/(8i^2),(j+1/8)/(8i^2)]. The algorithm can only be declared a failure if event B_k happens for some k∈{1,…,T}. However, the event B_k implies the event A_k. [resume] Assuming the algorithm is not declared a failure, the probability that M' is empty is at most 2ε'.Since i≥⌈ 1/p⌉ there is an element x∈ M with _ (x) ≥ 1/i ≥ℓ/(8i^2). Then the probability that #(m_i) < (ℓ-1/8)/(8i^2)≤ (8i-1/8)/(8i^2) = (1-1/(64i))/i≤ (1-1/(64i)) _ (x) is at most 2ε' by Claim <ref>. [resume] If M' is not characteristic then A_k occurs for some k∈{1,…,T} with probability at least (1-ε').By Claim <ref> we can assume that the algorithm was not declared a failure. Note that, if M' is not characteristic then one of the following three things happens: there is an element m_k with #(m_k)> j but Pr_S(m_k)≤ℓ or there is an element m_k with #(m_k)≤ℓ but Pr_S(m_k)> ℓ, or #(x) = 0 for an element with Pr_S(x)> ℓ.However, by the choice of ℓ, we know that #(m_k)∉ [ (ℓ-1/8)/(8i^2),(ℓ+1/8)/(8i^2)]. Thus in the first two cases we conclude that event A_k occurs. The third option is that #(x) = 0 for an element with Pr_S(x)> ℓ. There are at most 1/ℓ<8i^2/(6i)= 4i/3 elements x with such a probability and for each the probability for #(x) = 0 is at most (1-ℓ)^T≤ (1-3/(4i))^T≤ε' 3/(4i). So by the union bound we obtain a total probability of at most ε'.Combining the claims we obtain that the algorithm fails with probability at most 2ε'+ 2ε' +ε≤ 5ε' ≤ε.Until this point we have assumed that the value of p is known to the algorithm. To remedy this we repeatedly run the algorithm with a simple doubling technique. In each iteration we run the described algorithm assuming that 1/p∈ [i,2i]. Here we sample T elements of M. In the next iterations we replace i by 2i and repeat. We also replace ε' by ε' /2. Since p≥ 1/|M|, The number of iterations is logarithmic in 1/p. The total number of sampled items is at most twice the number of items sampled in the last round. Thus, overall we obtain an algorithm with expected polynomial time. To ensure that we obtain a suitable error bound it suffices to note that the probabilities of Claims <ref>, <ref> and <ref> actually decrease when i is replaced by an arbitrary smaller number. Skipping the first round, we obtain an error of at most 5ε'/2 + 5ε'/4 +5ε'/8 +…≤ε. (Note that this argument in particular comprises the fact that if in an iteration a set is being output by the algorithm it is still characteristic with sufficiently high probability.) We note several crucial observations about any algorithm solving the problem just described. There is no algorithm that for every set M and sampler S always outputs the same set M' with high probability.Indeed, consider the set M = {a,b}. Choosing _S (a) = _S(b) = 1/2 means that M' must be {a,b}. Choosing _S(a) = 1 and _(b) = 0 implies that M' must be {a}. However, there is a continuous deformation between these two samplers, while possibilities for the set M' are discrete. It is not difficult to see that the probability distribution of the output set M' must be continuous in the space of samplers, and thus, whatever the algorithm may be, there must be samplers for which the algorithm sometimes outputs {a} and sometimes outputs {a,b}.Let us also remark that the analysis of the running time of the algorithm is certainly far from optimal. In particular a large constant of (2^18)^2 arises only from the goal to keep the computations simple and the desire to have a bound that also holds for small values of |M|. Once one is interested in small running times, one might even ask whether it is possible to devise an algorithm running in time sublinear in |M|. However, recalling the coupon collector theorem and considering uniform samplers one realizes that one cannot expect to make do with o(|M|log |M|) samplings. However, if the set M is of algebraic nature, for example forms a group, then there might be meaningful ways to sample characteristic substructures (see Section <ref>).§ GADGET CONSTRUCTIONS FOR ASYMMETRIC TOURNAMENTSThere are several computational problems fundamentally related to the graph isomorphism problem. This relation manifests formally as polynomial-time Turing (or even many-one) reductions between the computational tasks. Such reductions are typically based on gadget constructions which we revisit in this section.While the graph isomorphism problem asks whether two given graphs are isomorphic, in the search version of this decision problem an explicit isomorphism is to be found, whenever one exits. The graph automorphism problem asks whether a given graph has a non-trivial automorphism (i.e., an automorphism different from the identity). In other words the task is to decide whether the given graph is asymmetric. Two other related problems are the task to determine generators for the automorphism group (G) and the task to determine the size of the automorphism group |(G)|. For all named problems there is a colored variant, where the given graphs are vertex colored and isomorphisms are restricted to be color preserving. We denote the respective problems by , and .It is well known that between all these computational problems – except – there are polynomial-time Turing reductions (we refer for example to <cit.>, <cit.>, <cit.>). Concerning the special case of , while there is a reduction from to the other problems, a reverse reduction is not known.The reductions are typically stated for general graphs, but many of the techniques are readily applicable to restricted graph classes. By a graph class we always mean a collection of possibly directed graphs closed under isomorphism. The isomorphism problem for graphs in 𝒞, denoted _𝒞, is the computational task to decide whether two given input graphs from 𝒞 are isomorphic. If one of the input graphs is not in 𝒞 the answer of an algorithm may be arbitrary, in fact the algorithm may even run forever.Analogously, for each of the other computational problems that we just mentioned, we can define a problem restricted to 𝒞 giving us for example _𝒞, and _𝒞and the colored versions _𝒞, _𝒞, and _𝒞.As remarked in <cit.>, most of the reduction results for general graphs transfer to the problems for a graph class 𝒞 if one has, as essential tool, a reduction from _𝒞 to _𝒞. Suppose that for a graph class 𝒞 there is a polynomial-time many-one reduction from _𝒞 to _𝒞 (i.e., _𝒞≤_m^p _𝒞)[Let us remark for completeness that a Turing reduction assumption _𝒞≤_T^p _𝒞 actually suffices for the theorem.]. Then * _𝒞 polynomial-time Turing-reduces to _𝒞 (i.e., _𝒞≤_T^p _𝒞),* The search version of _𝒞 polynomial-time Turing-reduces to the decision version of _𝒞, and* _𝒞polynomial-time Turing-reduces to _𝒞 (i.e., _𝒞≤_T^p _𝒞).In this paper we are mainly interested in two classes of directed graphs, namely the class of tournaments and the class of asymmetric tournaments . For the former graph class, a reduction from the colored isomorphism problem to the uncolored isomorphism problem is given in <cit.>. The colored tournament isomorphism problem is polynomial-time many-one reducible to the (uncolored) tournament isomorphism problem (i.e., _≤_m^p _). However, for our purposes we also need the equivalent statement for asymmetric tournaments.Taking a closer look at the reduction described in <cit.> yields the desired result. In fact it also shows that the colored asymmetry problem reduces to the uncolored asymmetry problem. Denoting for a graph class 𝒞 by 𝒞 the class of those graphs in 𝒞 that are asymmetric (i.e., have a trivial automorphism group), we obtain the following.* The isomorphism problem for colored asymmetric tournaments is polynomial-time many-one reducible to the isomorphism problem for (uncolored) asymmetric tournaments (i.e., _≤_m^p _). * The colored tournament asymmetry problem is polynomial-time many-one reducible to the (uncolored) tournament asymmetry problem (i.e., _≤_m^p _).In <cit.> given two colored tournaments T_1 and T_2, a gadget construction is described that adds new vertices to each tournament yielding T_1' and T_2' so that T_1≅ T_2 ⇔T'_1≅ T'_2.The authors show that every automorphism of T'_i fixes the newly added vertices. However, from the construction it is clear that T_1 is asymmetric if and only if T'_i is asymmetric, since all vertices that are added must be fixed by every automorphism. This demonstrates both parts of the lemma.We sketch a gadget construction that achieves these properties and leave the rest to the reader. For each i∈{1,2} the construction is as follows. Suppose without loss of generality that the colors of T_i are {1,…,ℓ} with ℓ≥ 2. We add a directed path u_1→…→ u_ℓ to the graph. A vertex v∈ V(T_i) has u_j as in-neighbor if j is the color of v. Otherwise u_j is an out-neighbor of v. We add two more vertices a and ,b to the graph. The only out-neighbor of vertex a is b. The in-neighbors of b are the vertices in {a,u_1,…,u_ℓ}. It can be shown that a is the unique vertex with maximum in-degree. This implies that b and thus all u_j are fixed by all automorphisms. As mentioned above, reductions for computational problems on general graphs can often be transferred to the equivalent problems restricted to a graph class 𝒞. However, let us highlight a particular reduction where this is not the case. Indeed, it is not clear how to transfer the reduction from to (which involves taking unions of graphs) to a reduction from _𝒞 to _𝒞, even when provided a reduction of _𝒞 to _𝒞.For the class of tournaments however, we can find such a reduction, of which we can make further use. * The isomorphism problem for tournaments polynomial-timeTuring-reduces to the task to compute a generating set for the automorphism group of a tournament (i.e., _≤_T^p _).* The isomorphism problem for colored asymmetric tournaments is polynomial-time many-onereducible to tournament asymmetry (i.e., _≤_m^p _). * The search version of the isomorphism problem for colored asymmetric tournaments Turing-reduces to tournament asymmetry. Suppose we are given two tournaments T_1 and T_2 on the same number of vertices n for which isomorphism is to be decided. By Theorem <ref> we can assume that the tournaments are uncolored.Let Tri(T_1,T_2) be the tournament obtained by forming the disjoint union of the three tournaments T_1, T_1' and T_2 where T_1≅ T_1' . We add edges from all vertices of T_1 to all vertices of T_1', from all vertices of T_1' to all vertices of T_2 and from all vertices of T_2 to all vertices of T_1 (see Figure <ref>). We observe that two vertices that are contained in the same of the three sets V(T_1), V(T_2), V(T_1') have n common out-neighbors. However, two vertices that are not contained in the same of these three sets have at most n-1 common out-neighbors. We conclude that an automorphism of Tri(T_1,T_2) preserves the partition of V(Tri(T_1,T_2)) into the three sets V(T_1), V(T_1') and V(T_2). Given a generating set for (Tri(T_1,T_2)) it holds that there is some generator that maps a vertex from V(T_1) to a vertex from V(T_2) if and only if T_1 and T_2 are isomorphic. This proves the first part of the lemma. Suppose additionally that T_1 and T_2 are asymmetric. We then further conclude that the tournament Tri(T_1,T_2) has a non-trivial automorphism if and only if T_1 and T_2 are isomorphic. This shows that the decision version of asymmetric tournament isomorphism reduces to tournament asymmetry. Since the search version is Turing-reducible to the decision version of isomorphism (Theorem <ref>) this finishes the proof.For Turing reductions, the converse of the previous lemma also holds.In fact the converse holds for arbitrary graph classes.Let 𝒞 be a graph class. * The task to compute a generating set for the automorphism group of graphs in 𝒞 Turing-reduces to the isomorphism problem for colored graphs in 𝒞 (i.e., _𝒞≤_T^p _𝒞). * Asymmetry checking for graphs in 𝒞 polynomial-time Turing-reduces to isomorphism checking of asymmetric colored graphs in 𝒞 (i.e., _𝒞≤_T^p _𝒞).The proof of the first part is a well known reduction that already appears in <cit.>. We can also see it by applying Part <ref> of Theorem <ref> to the class of colored graphs in 𝒞. For the second part, assume we have an oracle O_1 for isomorphism checking of colored asymmetric graphs in 𝒞. Then we also have an oracle O_2 for the search-version of isomorphism checking of colored asymmetric graphs in 𝒞. Indeed, we can find an isomorphism by individualizing more and more vertices in both graphs while keeping the graphs isomorphic. When all vertices are singletons, there is only one option for the isomorphism.Now let G be a graph in 𝒞. Without loss of generality assume that V(G)= {v_1,…,v_n}. For every t,t'∈{1,…,n} with t>t' we call O_2(G_(v_1,…,v_t-1,v_t),G_(v_1,…,v_t-1,v_t')).Here the notation G_(u_1,…,u_ℓ) denotes the graph G colored such that the color of u_i is i and vertices not in {u_1,…,u_ℓ} have color 0. (With respect to the partition of the vertices into color classes this is the same as constructing the graph obtained from G by individualizing u_1,…,u_ℓ one after the other.) If we find an isomorphism among the calls then this isomorphism is non-trivial since it maps v_t to v_t' and thus G is not asymmetric. Conversely if G is not asymmetric, then let j be the least integers for which G_(v_1,…,v_j) is asymmetric. Then j<n, (since G_(v_1,…,v_n-1) is always asymmetric) and there is a t'>j such that G_(v_1,…,v_j-1,v_j) and G_(v_1,…,v_j-1,v_t') are isomorphic. This isomorphism will be found by the oracle.While the oracle O_2 can sometimes output incorrect answers, namely when one of the inputs is not asymmetric, O_2 is certifying in the sense that we can check whether a given answer is really an isomorphism. Thus, we avoid making any errors whatsoever. § INVARIANT AUTOMORPHISM SAMPLERS FROM ASYMMETRY TESTS As discussed before, the asymmetry problem of a class of graphs reduces to the isomorphism problem of graphs in this class. However, whether there is a reduction in the reverse, or whether the asymmetry problem may actually be computationally easier than the isomorphism problem is not known. To approach this question, we now explore what computational power we could get from having available an oracle for the asymmetry problem.An invariant automorphism sampler for a graph G is a sampler over (G)∖{𝕀} which satisfies the property that if _(φ) = p then _(ψ^-1∘φ∘ψ) = p for all ψ∈(G). We first show how to use an oracle for asymmetry to design an invariant automorphism sampler for a tournament T. Given an oracle for asymmetry of tournaments (_) we can construct for every given colored (or uncolored) tournament T that is not asymmetric an invariant automorphism sampler. The computation time (and thus the number of oracle calls) required to sample once from is polynomial in |V(T)|. Let O_1 be an oracle for uncolored tournament asymmetry. By Lemma <ref>, we can transform the oracle O_1 for the asymmetry of uncoloredtournaments into an oracle O_2 for asymmetry of colored tournaments. By Lemma <ref> Part <ref>, we can also assume that we have an oracle O_3 that decides the isomorphism problem of colored asymmetric tournaments. More strongly, Lemma <ref> Part <ref> makes a remark on the search version, thus we can assume that O_3 also solves the isomorphism search problem for asymmetric tournaments.To obtain the desired sampler we proceed as follows. In the given tournament T we repeatedly fix (by individualization, i.e., giving it a special color) uniformly, independently at random more and more vertices until the resulting tournament is asymmetric. This gives us a sequence of colored tournaments T = T_0, T_1, …, T_t such that (T_t)= {𝕀}, (T_t-1)≠{𝕀}and such that T_t = (T_t-1)_(v) for some vertex v. In other words, T_t is obtained from T_t-1 by individualizing v which makes the graph asymmetric. Using the available oracle O_2, we can compute the set V” of those vertices v” in V(T)∖{v} that have the same color as v such that ((T_t-1)_(v”))= {𝕀}. There must be at least one vertex in V” since T_t-1 is not asymmetric. Using the oracle O_3, we can then compute the subset V”'⊆ V” of those vertices v”' for which (T_t-1)_(v”') and T_t are isomorphic.Next, we pick a vertex u∈ V”' uniformly at random. Since both (T_t-1)_(u)and T_t are asymmetric, using the oracle O_3 for the isomorphism search problem we can compute an isomorphism φ from (T_t-1)_(u) to T_t. This isomorphism φ is unique and it is a non-trivial automorphism of (T). Algorithm <ref> gives further details.(Invariance) The invariance follows directly from the fact that all steps of the algorithm either consist of choosing a vertex uniformly at random or computing an object that is invariant with respect to all automorphisms fixing all vertices that have been randomly chosen up to this point.(Running time) Concerning the running time, one call of Algorithm <ref> uses less than 2n calls to oracle O_2 and at most n calls to oracle O_3. The overall running time is thus polynomial.Let us comment on whether the technique of the lemma can be applied to graph classes other than tournaments. For the technique to apply to a graph class 𝒞, we require the oracle O_2, which solves colored asymmetry 𝒞, and the oracle O_3 which solves the isomorphism search problem for asymmetric colored objects in 𝒞. (The oracle O_1 is a special case of O_2.) In the case of tournaments, having an oracle O_1 (i.e., an oracle for uncolored asymmetry) is sufficient to simulate the oracles O_2 and O_3, but this is not necessarily possible for all graph classes 𝒞. It is however possible to simulate such oracles for every graph class that satisfy some suitable (mild) assumptions, as can be seen from the discussion in Section <ref>. In particular, given an oracle for asymmetry of all graphs we can construct an invariant automorphism sampler for all graphs that are not asymmetric. § INVARIANT SUBORBITS FROM INVARIANT AUTOMORPHISM SAMPLERSLet G be a directed graph.Let be an invariant automorphism sampler for G. We now describe an algorithm that, given access to an asymmetry oracle, constructs a non-discrete partition of V(G) which is finer than or at least as fine as the orbit partition of G under (G) and invariant under (G). Here, a partition π is invariant under (G) if π = ψ(π) for all ψ∈(G). (A partition is discrete if it consists only of singletons.)For every c∈ℕ, there is a randomized polynomial-time algorithm that, given a graph G and an invariant automorphism sampler for G constructs with error probability at most 1/|G|^c a non-discrete partition π of V(G) such that * π is finer than or at least as fine as the orbit partition of V(G) under (G) and* π is invariant under (G).The algorithm also provides a set of certificates Φ = {φ_1,…, φ_m}⊆(G) such that for every pair of vertices v,v'∈ V(G) that lie in the same class of π there is some φ_i with φ_i(v) = v'. Let M = {(v,w)| v,w∈ V(G), v≠ w, ∃φ∈(G)φ(v) = w } be the set of pairs of two distinct vertices lying in the same orbit. With the sampler we can simulate a sampler ' over M invariant under (G) as follows. To create an element for ' we sample an element φ from and uniformly at random choose an element v from the support (φ) ={x∈ V(G)|φ(x)≠ x} of φ. Then the element for ' is (v,φ(v)).It follows form the construction that ' is a sampler for M. Moreover, since all random choices are independent and uniform, ' is invariant under automorphisms.Using the algorithm from Theorem <ref> we can thus compute a characteristic subset M' of M. Since ' is (G)-invariant, the fact that M' is characteristic implies that it is also (G)-invariant.For the given c∈ℕ, to obtain the right error bound, we choose ε to be 1/|G|^c for the algorithm from Theorem <ref>. Then the error probability is at most ε= 1/|G|^c and the running time is polynomial in |M| = O(|G|^2) and ln |G|^c = O(|G|) and thus polynomial in the size of the graph.Regarding M' as a binary relation on V(G) we compute the transitive closure andlet π be the partition of V(G) into equivalence classes of said closure, where vertices that do not appear at all as entries in M' form their own class. By construction, elements that are in the same class of π are in the same orbit under (G). Moreover π is (G)-invariant since M' is (G)-invariant.To provide certificates for the elements in M' we can store all elements given to us by S. For each (v,w)∈ M' we can thus compute an automorphism of φ_v,w∈(G) with φ_v,w(v) = w.For pairs in the transitive closure of M' we then multiply suitable automorphisms.If a partition π satisfies the conclusion of the lemma, we call it an invariant collection of suborbits. We call the elements of Φ the certificates. Let us caution the reader that the set Φ returned by the algorithm is not necessarily characteristic. Moreover, the orbits of the elements in Φ might not necessarily be contained within classes of π.We call an algorithm an oracle for invariant suborbits if, given a tournament T, the algorithm returns a pair (π,Φ) constituting invariant suborbits and certificates, in case T is not asymmetric, and returns the discrete partition π and Φ ={𝕀} whenever T is asymmetric.§ COMPUTING THE AUTOMORPHISM GROUP FROM INVARIANT SUBORBITS To exploit invariant suborbits we make use of the powerful group-theoretic technique to compute stabilizer subgroups.There is an algorithm that,given a permutation group Γ on {1,…,n} and subset B⊆{1,…,n}, computes (generators for) the setwise stabilizer of B. If Γ is solvable, then this algorithm runs in polynomial time. We will apply the theorem in the following form: Let G be a graph and Γ a solvable permutation group on V(G). Then Γ∩(G) can be computed in polynomial time. This follows directly from the theorem by considering the induced action of Γ on pairs of vertices from V(G) and noting that Γ∩(G) consists of those elements that stabilize the edge set.In our algorithm we will also use the concept of a quotient tournament (that can for example implicitly be found in <cit.>, see also <cit.>). Let T be a tournament and let π be a partition of V(T) in which all parts have odd size. We define T/π, the quotient of T modulo π, to be the tournament on π (i.e., the vertices of T/π are the parts of π) where for distinct C,C'∈ V(T/π) = π there is an edge from C to C' if and only if in T there are more edges going from C to C' than edges going from C' to C. Note that since both |C| and |C'| are odd there are either more edges going from C to C' or more edges going from C' to C. This implies that T/π is a tournament.Suppose we are given as an oracle a randomized Las Vegas algorithm that computes invariant suborbits for tournaments in polynomial time. Then we can compute the automorphism group of tournaments in polynomial time.We describe an algorithm that computes the automorphism group of a colored tournament given a randomized oracle that provides invariant suborbits.(Description of the algorithm)Let T be a given colored tournament. (Case 0: T is not monochromatic.) If T is not monochromatic then we proceed as follows:Let Col be the set of colors that appear in T. For c∈Col, let V^c be the set of vertices of color c and let T^c = T[V^c] be the subtournament induced by the vertices in V^c.We recursively compute (T^c) for all c∈Col. Let Ψ^c be the set of generators obtained as an answer. We lift every generator to a permutation of V(T) by fixing all vertices outside of V^c. Let Ψ^c be the set of lifted generators of Ψ^c and let Ψ = ⋃_c∈ColΨ^c be the set of all lifted generators. Since (T^c) =⟨Ψ^c⟩ is solvable, we conclude that ⟨Ψ⟩ is a direct product of solvable groups and thus solvable. We can thus compute ⟨Ψ⟩∩(T) using Theorem <ref> and return the answer. This concludes Case 0. In every other case we first compute a partition π into suborbits using the oracle and a corresponding set of certificates Φ. For a partition π of some set V we denote for v∈ V by [v]_π the element of π containing v. We may drop the index when it is obvious from the context. If |T|=1 then we simply return the identity.(Case 1: π is trivial). In case π is trivial (i.e., π={V(T)}), we know that T is transitive. We choose an arbitrary vertex v∈ V(T).Let λ be the coloring of V(T) satisfyingλ(u) =1ifu=v2if(u,v)∈ E(T)3otherwise.We recursively compute a generating set Ψ for (T'), where T' is T recolored with λ. We then return Ψ∪Φ. (Case 2: not all classes of π have the same size.)We color every vertex with the size of the class of π in which it is contained. Now T is not monochromatic anymore and we recursively compute (T) with T having said coloring. (In other words, we proceed as in Case 0.) (Case 3: all classes of π have the same size but π is non-trivial.) We compute for each pair of distinct equivalence classes C and C' of π an isomorphism φ_(C,C') from T[C] to T[C'] or determine that no such isomorphism exists, as follows: We choose for each C an arbitrary vertex v∈ C. We let T_C be the tournament obtained from T[C] by coloring v with 1, all in-neighbors of v with 2 and other vertices with 3. We let T_C,C' = Tri(T_C,T_C') be the triangle tournament of T_C and T_C' where (T_C)' is an isomorphic copy of T_C (as defined in Section <ref> in the proof of Lemma <ref>).Using recursion we compute (T_C,C'). From the result we can extract an isomorphism from T[C] to T[C'] since V(T[C]) and V(T[C']) are blocks of T_C,C'. (Case 3a:) If it is not the case that for every pair C,C' of color classes there is an isomorphism from T[C] to T[C'] then we color the vertices of T so that v,v' have the same color if and only if there is an isomorphism from T[([v])] to T[([v'])], where as before for every vertex u we denote by [u] the class of π containing u. With this coloring, T is not monochromatic anymore and we recursively compute (T) with T having said coloring. (In other words, we proceed as in Case 0.)(Case 3b:) Otherwise, for every pair C,C' of color classes, there is an isomorphism from T[C] to T[C']. Note that all color classes are of odd size since T[C] is transitive (as dictated by π). Thus, we can compute the quotient tournament T/π. We recursively compute a generating set Ψ = { g_1,…,g_t } for the automorphism group of T/π. We lift each g_i to a permutation g_i of V(T) as follows. The permutation g_i maps each vertex v to φ_([v],g_i([v]))(v).Since g_i is a permutation and each φ_(C,C') is a bijection, the map g_i is a permutation of V(T). Let Ψ= {g_1,…,g_t} be the set of lifted generators. As next step, for each class C we recursively compute a generating set Υ_C for (T[C]). We lift each generator in Υ_C to a permutation of V(T) by fixing all vertices outside of C obtaining the set Υ_C of lifted generators.Consider the group Γ generated by the set Ψ∪⋃_C∈πΥ_C. As a last step, using Theorem <ref> we compute the subgroup Γ' =Γ∩(T). The details of this algorithm are given in Algorithm <ref>.(Running time)We first argue that all work performed by an iteration of the algorithm apart from the recursive calls is polynomial in n, say O(n^c) for some constant c. This is obvious for all instructions of the algorithm except the task to compute the intersection of ⟨Ψ⟩∩(T) in Case 0 (Line <ref>) and the task to compute ⟨Ψ⟩∩(T)in Case 3b (Line <ref>). However, in Case 0, the group generated by ⟨Ψ⟩ is a direct product of solvable groups, thus solvable, and in Case 3b, the group ⟨Ψ∪⋃_C∈πΥ_C⟩ is a subgroup of a wreath product of a solvable group with a solvable group and is thus solvable. (Alternatively we can observe that the natural homomorphism from the group ⟨Ψ∪⋃_C∈πΥ_C⟩ to Ψ has kernel ⟨⋃_C∈πΥ_C⟩, a direct product of solvable groups.) In either case, using the algorithm from Theorem <ref>, the group intersection can be computed in polynomial time.It remains to consider the number of recursive calls. We will bound the amount of work of the algorithm in terms of t, the maximum size of a color class of T, and the number of vertices n. Denote by R(t,n) the maximum number of nodes in the recursion tree over all tournaments for which the color classes have size at most t andthe number of vertices is at most n. Note that R(t,n) is monotone increasing in both components. First note that if t<n the algorithm will end up in Case 0.The recursive bound in Case 0 (Line <ref>) is then R(t,n) ≤ 1+∑_i = 1^ℓ R(a_i,a_i) for some positive integers a_1,…,a_ℓ∈ℕ (the color class sizes)that sum up to n but are smaller than n.In Case 1, we have t=n. The tournament T' is colored into three color classes.Since T is transitive (and thus every vertex has in- and out-degree (t-1)/2), in T' there is one color class of size 1 and there are two classes of size (t-1)/2. The recursive call will lead to Case 0, which then yields one trivial recursive call on a tournament of size 1 and two calls with tournaments of size (t-1)/2. We obtain a recursive bound (for Line <ref>) of R(t,n)≤2+ 2 R((t-1)/2,(t-1)/2)≤3 R(t/2,t/2).In Case 2, we have t=n and observe that the recursive call is for a tournament that is not monochromatic. Thus the recursive call will end up in Case 0. We thus obtain a recursive bound (for Line <ref>) of R(t,n) ≤2+ ∑_i = 1^ℓ R(a_i,a_i) for some positive integers a_1,…,a_ℓ∈ℕ that sum up to n but are smaller than n.In Case 3, we have t=n. Note that if the classes of π have size t' then t'≤ n/3 (elements of π are all equally large and there are at least 2 but there is an odd number) and there are (n/t')^2=(t/t')^2 recursive calls in Line <ref>. In the graph T_C,C' the color classes have size at most 3 (t'-1)/2 and there are at most 3t' vertices. (The increase of a factor 3 comes from the Tri() operation.)Thus the cost for such calls is bounded by (t/t')^2 · R(3(t'-1)/2,3t')≤ (t/t')^2 · R(3t'/2,3t')R_3, where 3t'/2≤ n/2 = t/2.Using the same arguments as before, in Case 3a we thus get a recursive bound for Line <ref> of ∑_i = 1^ℓ R(b_i,b_i) and thus for Case 3a in total a bound of R(t,n) ≤ 1 + R_3+ ∑_i = 1^ℓ R(b_i,b_i) for some positive integers b_1,…,b_ℓ∈ℕ that sum up to n but are smaller than n-t'= t-t'.In Case 3b we need to additionally consider the cost for the recursive call in Line <ref>. This cost is at most R(t/t',t/t') where t'≥ 2 since the coloring is not discrete. Also there is a recursive cost of t/t'·R(t',t') coming from Line <ref>.Thus in this case we end up with R(t,n) ≤ 1 + R(t/t',t/t') + R_3 + t/t'·R(t',t').Summarizing we get that R(t,n)is bounded by1if n=12+∑_i = 1^ℓ R(a_i,a_i)in Cases 0 and 2, with ∑_i=1^ℓa_i = n and a_i≤ n-13 R(t/2,t/2) in Case 11 + R_3+ ∑_i = 1^ℓ R(b_i,b_i) in Case 3a, with ∑_i=1^ℓb_i = n and b_i≤ t-t'1 + R(t/t',t/t') + R_3 + t/t'·R(t',t')in Case 3b, where R_3 = (t/t')^2 · R(3t'/2,3t') and t' satisfies 3t'/2 ≤ t/2 and t/ t'≤ t/2 and 3t'≤ t. Let us define S(m) as the maximum of R(t,n) over all pairs of positive integers (t,n) with t+n≤ m and t≤ n. Then we get from the above considerations that S(m) is bounded by one of the following 1if m=22+∑_i = 1^ℓ S(a_i)∑_i=1^ℓa_i ≤ m and a_i≤ m-13 S(m/2) 1 + (m/t')^2 · S(9/2 · t')+ ∑_i = 1^ℓ S(b_i) ∑_i=1^ℓb_i = mandb_i≤ m-t'1 + S(m/t') + (m/t')^2 · S(9/2 · t')+ m/t'·S(t'),where t' satisfies 9/2 · t'≤ 3/4· m and t/ t'≤ m/2. It is now simply a calculation to show that for d sufficiently large, the function F(m)= m^d satisfies all the recurrence bounds for S (of course as lower bounds rather than upper bounds). We show the calculation for the most interesting case, the Case 3a.Let x = m-t'. Then 5t'≤ x. Furthermore the equation for Case 3a says S(x+t')≤ 1+ ((x+t')/t')^2 S(9/2 t') + ∑_i = 1^ℓ S(b_i) where b_i≤ x and ∑_i=1^ℓb_i = x+t'. Note for the function F that ∑_i = 1^ℓ F(b_i), under the conditions b_i≤ x and ∑_i=1^ℓb_i = x+t', gets maximized as x^d + (t')^d. For the right hand side we get 1+ ((x+t')/t')^2(9/2)^d (t')^d + x^d + (t')^d ≤ 1+ x^d+ (6/5)^2 (9/2)^d x^2(t')^d-2 +(t')^d which is certainly bounded by (x+t')^d for d sufficiently large since the expansion of (x+t')^d contains the summands x^d, (t')^d and d x^d-1 t'≤ d 5^d-3 x^2 (t')^d-2. Thus F is an upper bound for S.Overall we obtain a polynomial-time algorithm from this recursive bound. This in particular implies that the algorithm halts. (Correctness)For the correctness proof we analyze the different cases one by one. By induction we can assume that recursive calls yields correct answers.For Case 0, since the last instruction intersects some group with the automorphism group it is clear that the algorithm can only return automorphisms of T. Let us thus assume that φ∈(T). Then, for each color class c, the set V_c is invariant under φ and φ|_V_c∈(T[V^c]). This implies that φ∈⟨Ψ⟩.For Case 1, T is transitive since π = V(T) shows that V(T) is an orbit. Thus, (T) is generated by the point stabilizer (T)_v {ψ∈(T)|ψ(v) = v}and an arbitrary transversal (i.e., a subset of elements of containing a representative from each coset of (T)_v in (T)). Since Φ contains a certificate for all pairs of distinct vertices (v,v') and since Φ⊆(T) we conclude that (T) = ⟨Φ∪(V(T'))⟩.For Case 2, it suffices to note that for every integer i∈ℕ the set {v ∈ V(T) | |[v]_π| = i } is invariant under (T). For Case 3, Line <ref> note that similar to Case 1, the graphs T[C] and T[C'] are transitive and thus the individualization in Line <ref> does not make isomorphic graphs non-isomorphic.For Case 3a, again note that for v∈ V(T) the set {v' ∈ V(T) | T[([v])] ≅ T[([v'])] } is invariant. For Case 3b we argue similarly to Case 0. Since the lastinstruction intersects some group with the automorphism group it is clear that the algorithm can only return automorphisms of T. Let us thus assume that φ∈(T).Then φ induces an automorphism ψ of T/π. Note that there is some ψ in ⟨Ψ⟩ that also induces ψ on T/π. It suffices now to show that the map ψ^-1∘φ is in ⟨⋃_C∈πΥ_C⟩. Consider C∈π. Then ψ^-1∘φ maps C to C and more strongly it induces an automorphism of T[C] which must be contained in ⟨Υ_C⟩. We conclude that ψ^-1∘φ is in ⟨⋃_C∈πΥ_C⟩ finishing the proof. We have now assembled all the required parts to prove the main theorem of the paper.* There is a randomized (one-sided error) polynomial-time Turing reduction from tournament isomorphism to asymmetry testing of tournaments (i.e., _≤_r,T^p _).* There is a randomized polynomial-time Turing reduction from the computational task to compute generators of the automorphism group of a tournament to asymmetry testing of tournaments (i.e., _≤_r,T^p _).Recall that a two-sided error algorithm for an isomorphism search problem can be readily turned into a one-sided error algorithm by checking the output isomorphism for correctness. Thus, by Lemma <ref> Part <ref> it suffices to prove the second part of the corollary. Combining Lemma <ref> and Theorem <ref>, from an oracle to tournament asymmetry we obtain a randomized Monte Carlo (i.e., with possible errors) algorithm that computes invariant suborbits. Given a Las Vegas algorithm (i.e., no errors) for suborbits, the previous theorem provides us with a computation of the automorphism group of tournaments. It remains to discuss the error probability we get from using a Monte Carlo algorithm instead of a Las Vegas algorithm. Since there is only a polynomial number of oracle calls, and since the error bound in Theorem <ref> can be chosen smaller than 1/|G|^c for every fixed constant c, the overall error can be chosen to be arbitrarily small.§ DISCUSSION AND OPEN PROBLEMS This paper is concerned with the relationship between the asymmetry problem _𝒞 and isomorphism problem _𝒞. While under mild assumptions there is a reduction from the former to the latter, a reduction in the other direction is usually not known. However, for tournaments we now have such a randomized reduction.The first question that comes to mind is whether the technique described in this paper applies to other graph classes. While the sampling techniques from Sections <ref> to <ref> can be applied to all graph classes that satisfy mild assumptions (e.g., _𝒞≤_t^p _𝒞 and _𝒞≤_t^p _𝒞) the algorithm described in Section <ref> crucially uses the fact that automorphism groups of tournaments are solvable. This is not the case for general graphs, so for the open question of whether reduces to this may dampen our enthusiasm. However, what may bring our enthusiasm back up is that there are key classes of combinatorial objects that share properties similar to what we need.In particular, this brings us to the question whether the techniques of the paper can be applied to group isomorphism. Just like for tournament isomorphism, finding a faster algorithm for group isomorphism (given by multiplication table) is a bottleneck for improving the run-time bound for isomorphism of general graphs beyond quasi-polynomial. Since outer-automorphism groups of simple groups are solvable, we ask: Can we reduce the group isomorphism problem to the isomorphism problem for asymmetric groups? This question is significant since an asymmetry assumption on groups is typically a strong structural property and may help to solve the entire group isomorphism problem. However, here one has to be careful to find the right notion of asymmetry since all groups have inner automorphisms. For such notions different possibilities come to mind. A second natural open question would be whether there is a deterministic version of the algorithms given in this paper. As a last open problem recall that it was shown in Section <ref>that one can extract a characteristic subset for a sampler over a set M in time that depends polynomially on M. Since the automorphism group of a graph can be superpolynomial in the size of the graph, we had to take a detour via suborbits in Section <ref>. There can be no general way to extract a characteristic subset of M in polynomial time if |M| is not polynomially bounded, since we might never see an element twice. However, if M has an algebraic structure, in particular if M is a permutation group over a polynomial size set, this is not clear.Thus we ask: Is there a polynomial-time (randomized) algorithm that extracts a characteristic subgroup using a sampler Γ over a permutation group?plainurl | http://arxiv.org/abs/1704.08529v1 | {
"authors": [
"Pascal Schweitzer"
],
"categories": [
"cs.DM",
"cs.DS",
"math.CO",
"F.2.2; F.1.3"
],
"primary_category": "cs.DM",
"published": "20170427121827",
"title": "A polynomial-time randomized reduction from tournament isomorphism to tournament asymmetry"
} |
Bohmian MechanicsRoderich Tumulka[Fachbereich Mathematik, Eberhard-Karls-Universität, Auf der Morgenstelle 10, 72076 Tübingen, Germany. E-mail: [email protected]]August 15, 2019 ======================================================================================================================================================================== In this paper, we investigate an optimal design problem motivated by some issues arising in population dynamics. In a nutshell, we aim at determining the optimal shape of a region occupied by resources for maximizing the survival ability of a species in a given box and we consider the general case of Robin boundary conditions on its boundary. Mathematically, this issue can be modeled with the help of an extremal indefinite weight linear eigenvalue problem. The optimal spatial arrangement is obtained by minimizing the positive principal eigenvalue with respect to the weight, under a L^1 constraint standing for limitation of the total amount of resources. The specificity of such a problem rests upon the presence of nonlinear functions of the weight both in the numerator and denominator of the Rayleigh quotient. By using adapted rearrangement procedures, a well-chosen change of variable, as well as necessary optimality conditions, we completely solve this optimization problem in the unidimensional case by showing first that every minimizer is unimodal and bang-bang. This leads to investigate a finite dimensional optimization problem. This allows to show in particular that every minimizer is (up to additive constants) the characteristic function of three possible domains: an interval that sticks on the boundary of the box, an interval that is symmetrically located at the middle of the box, or, for a precise value of the Robin coefficient, all intervals of a given fixed length.principal eigenvalue, population dynamics, optimization, calculus of variations, rearrangement/symmetrization, bang-bang functions.49J15, 49K20, 34B09, 34L15. § INTRODUCTION§.§ The biological model In this paper, we consider a reaction-diffusion model for population dynamics. We assume that the environment is spatially heterogeneous, and present both favorable and unfavorable regions. More specifically, we assume that the intrinsic growth rate of the population is spatially dependent. Such models have been introduced in the pioneering work of Skellam <cit.>, see also <cit.> and references therein.We also assume that the population tends to move toward the favorable regions of the habitat, that is, we add to the model an advection term (or drift) along the gradient of the habitat quality. This model has been introduced by Belgacem and Cosner in <cit.>.More precisely, we assume that the flux of the population density u(x,t) is of the form -∇ u + α u ∇ m, where m(·) represents the growth rate of the population, and will be assumed to be bounded and to change sign. From a biological point of view, the function m(x) can be seen as a measure of the access to resources at a location x of the habitat. The nonnegative constant α measures the rate at which the population moves up the gradient of the growth rate m. With a slight abuse of language, we will also say that m( · ) stands for the local rate of resources or simply the resources at location x.This leads to the following diffusive-logistic equation{[ ∂_t u =(∇ u - α u ∇ m) + λ u(m-u) in Ω× (0,∞),; e^α m(∂_n u -α u∂_n m) +β u=0on ∂Ω× (0,∞), ].where Ω is a bounded region of ^n (n=1,2,3) which represents the habitat, β≥0, and λ is a positive constant. The case β=0 in (<ref>) corresponds to the no-flux boundary condition: the boundary acts as a barrier for the population. The Dirichlet case, where the boundary condition on ∂Ω is replaced by u=0, corresponds to the case when the boundary is lethal to the population, and can be seen as the limit case when β→∞. The choice 0<β<∞ corresponds to the case where a part of the population dies when reaching the boundary, while a part of the population turns back.Plugging the change of function v=e^-α m u into Problem (<ref>) yields to{[ ∂_t v = Δ v + α∇ v ·∇ m + λ v(m-e^α mv) in Ω× (0,∞),;e^α m∂_n v + β v=0on ∂Ω× (0,∞). ].The relation v=e^-α mu ensures that the behavior of models (<ref>) and (<ref>) in terms of growth, extinction or equilibrium is the same. Therefore, we will only deal with Problem (<ref>) in the following.It would be natural a priori to consider weights m belonging to L^∞(Ω) without assuming additional regularity assumption. Nevertheless, for technical reasons that will be made clear in the following, we will temporarily assume that m∈ C^2(Ω). Moreover, we will also make the following additional assumptions on the weight m, motivated by biological reasons. Given m_0∈ (0,1) and κ>0, we will consider that * the total resources in the heterogeneous environment are limited:∫_Ω m ≤ -m_0|Ω|, * m is a bounded measurable function which changes sign in Ω, i.e. |{x∈Ω, m(x)>0}| >0 ,and using an easy renormalization argument leads to assume that-1≤ m ≤κ a.e. in Ω . Observe that the combination of (<ref>) and (<ref>) guarantees that the weight m changes sign in Ω.In the following, we will introduce and investigate an optimization problem in which roughly speaking, one looks at configurations of resources maximizing the survival ability of the population. The main unknown will be the weight m and for this reason, it is convenient to introduce the set of admissible weights= {m∈L^∞(Ω), msatisfies assumptions (<ref>), (<ref>) and (<ref>)}. A principal eigenvalue problem with indefinite weight It is well known that the behavior of Problem (<ref>) can be predicted from the study of the following eigenvalue problem with indefinite weight (see <cit.>){[ -Δ - α∇ m ·∇ = Λ m in Ω,;e^α m∂_n+β =0 on ∂Ω , ].which also rewrites{[ -(e^α m∇) = Λ m e^α min Ω,; e^α m∂_n+β =0 on ∂Ω. ].Recall that an eigenvalue Λ of Problem (<ref>) is said to be a principal eigenvalue if Λ has a positive eigenfunction. Using the same arguments as in <cit.>, the following proposition can be proved. For sake of completeness, we propose a sketch of the proof in Appendix <ref>. * In the case of Dirichlet boundary condition, there exists a unique positive principal eigenvalue denoted λ_1^∞(m), which is characterized by λ_1^∞ (m) = inf_∈_0∫_Ω e^α m|∇|^2/∫_Ω m e^α m^2,where _0 = {∈H^1_0(Ω), ∫_Ω m e^α m^2>0}.* In the case of Robin boundary condition with β > 0, the situation is similar to the Dirichlet case, and λ_1^β(m) is characterized by λ_1^β (m) = inf_∈∫_Ω e^α m|∇|^2 + β∫_∂Ω^2/∫_Ω m e^α m^2,where = {∈H^1(Ω), ∫_Ω m e^α m^2>0}.*In the case of Neumann boundary condition (β=0), * if ∫_Ω m e^α m < 0, then the situation is similar as the Robin case, and λ_1^β(m)>0 is given by (<ref>) with β=0,* if ∫_Ω m e^α m≥0, then λ_1^β(m) =0 is the only non-negative principal eigenvalue.Following <cit.> (applied in the special case where the operator coefficients are periodic with an arbitrary period), one has the following time asymptotic behavior characterization of the solution of the logistic equation (<ref>):* if λ>λ_1^β (m), then (<ref>) has a unique positive equilibrium, which is globally attracting among non-zero non-negative solutions,* if λ_1^β (m) > 0 and 0<λ<λ_1^β (m), then all non-negative solutions of (<ref>) converge to zero as t→∞. According to the existing literature (see e.g. <cit.>), the existence of λ_1^β(m) defined as the principal eigenvalue of Problem (<ref>) for C^2 weights follows from the Krein Rutman theory. Nevertheless, one can extend the definition of λ^β_1(m) to a larger class of weights by using Rayleigh quotients, as done in Proposition <ref> (see Remark <ref>).From a biological point of view, the above characterization yields a criterion for extinction or persistence of the species.A consequence is that the smaller λ_1^β (m) is, the more likely the population will survive. This biological consideration led Cantrell and Cosner to raise the question of finding m such that λ_1^β (m) is minimized, see <cit.>. This problem writesinf_m∈λ_1^β(m).or respectivelyinf_m∈λ_1^∞(m).in the case of Dirichlet conditions.Biologically, this corresponds to finding the optimal arrangement of favorable and unfavorable regions in the habitat so the population can survive. It is notable that, in the Neumann case (β=0), if we replace Assumption (<ref>) with ∫_Ω m ≥ 0 in the definition of , then λ_1^0(m)=0 for every m∈. Biologically, this means that any choice of distribution of the resources will ensure the survival of the population. §.§ State of the art Analysis of the biological model (with an advection term) Problem (<ref>) was introduced in <cit.>, and studied in particular in <cit.>, where the question of the effect of adding the drift term is raised. The authors investigate if increasing α, starting from α=0, has a beneficial of harmful impact on the population, in the sense that it decreases or increases the principal eigenvalue of Problem (<ref>).It turns out that the answer depends critically on the condition imposed on the boundary of the habitat. Under Dirichlet boundary conditions, adding the advection term can be either favorable or detrimental to the population, see <cit.>. This can be explained by the fact that if the favorable regions in the habitat are located near the hostile boundary, this could result in harming the population. In contrast, under no-flux boundary conditions, it is proved in <cit.> that a sufficiently fast movement up the gradient of the ressources is always beneficial. Also, according to <cit.>, if we start with no drift (α=0), adding the advection term is always beneficial if the habitat is convex. The authors however provide examples of non-convex habitats such that introducing advection up the gradient of m is harmful to the population.Optimal design issues The study of extremal eigenvalue problems with indefinite weights like Problem (<ref>), with slight variations on the parameter choices (typically α=0 or α>0) and with different boundary conditions (in general Dirichlet, Neumann or Robin ones) is a long-standing question in calculus of variations. In the survey <cit.>, results of existence and qualitative properties of optimizers when dealing with non-negative weights are gathered. In the survey article <cit.>, the biological motivations for investigating extremal problems for principal eigenvalue with sign-changing weights are recalled, as well as the first existence and analysis properties of such problems, mainly in the 1D case.A wide literature has been devoted to Problem (<ref>) (or close variants) without the drift term, i.e. with α=0.Monotonicity properties of eigenvalues and bang-bang properties of minimizers[It means that the L^∞ constraints on the unknown m are saturated a.e. in Ω, in other words that every optimizer m^* satisfies m^*(x)∈{-1,κ} a.e. in Ω.] were established in <cit.>, <cit.> and <cit.> for Neumann boundary conditions (β=0) in the 1D case. In <cit.>, the same kind of results were obtained for periodic boundary conditions. We also mention <cit.>, for an extension of these results to principal eigenvalues associated to the one dimensional p-Laplacian operator.In this article, we will investigate a similar optimal design problem for a more general model in which a drift term with Robin boundary conditions is considered. In the simpler case where no advection term was included in the population dynamics equation, a fine study of the optimal design problem <cit.> allowed to emphasize existence properties of bang-bang minimizers, as well as several geometrical properties they satisfy.Concerning now the drift case model with Dirichlet or Neumann boundary conditions, the existence of principal eigenvalues and the characterization of survival ability of the population in terms of such eigenvalues has been performed in <cit.>.However and up to our knowledge, nothing is known about the related optimal design problem (<ref>) or any variant.Outline of the article This article is devoted to the complete analysis of Problem (<ref>) in the 1D case, that is Ω=(0,1). In Section <ref>, we discuss modeling issues and sum up the main results of this article. The precise (and then more technical) statements of these results are provided in Section <ref> (Theorems <ref>, <ref>, <ref> and <ref>), as well as some numerical illustrations and consequences of these theorems. The whole section <ref> is devoted to proving Theorem <ref> whereas the whole section <ref> is devoted to proving Theorems <ref>, <ref> and <ref>. It is split into four steps that can be summed up as follows: (i) proof that one can restrict the search of minimizers to unimodal weights, (ii) proof of existence, (iii) proof of the bang-bang character of minimizers. The consequence of these three steps is that there exists a minimizer of the form m^*=κ_E-_Ω\ E, where E is an interval. The fourth step hence writes: (iv) optimal location of E whenever E is an interval of fixed length. Finally, we gather some conclusions and perspectives for ongoing works in Section <ref>. §.§ Modeling of the optimal design problem and main results From now on, we focus on the 1D case n=1. Hence, for sake of simplicity, we will consider in the rest of the article that Ω = (0,1) .In the whole paper, if ω is a subset of (0,1), we will denote by χ_ω the characteristic function of ω.As mentioned previously (see Section <ref>), we aim at finding the optimal m (whenever it exists) which minimizes the positive principal eigenvalue λ_1^β (m) of Problem (<ref>).For technical reasons, most of the results concerning the qualitative analysis of System (<ref>) (in particular, the persistence/survival ability of the population as t→ +∞, the characterization of the principal eigenvalue λ_1^β(m), and so on) are established by considering smooth weights, say C^2. The following theorem emphasizes the link between the problem of minimizing λ_1^β(m) over the class ∩ C^2(Ω) and a relaxed one (as will be shown in the following), where one aims at minimizing λ_1^β over the larger class .The following theorem will be made more precise in the following, and its proof is given at the end of Section <ref> below. When α is sufficiently small, the infimum inf { λ_1^β(m) , m∈ ∩ C^2(Ω)} is not attained for any m∈∩ C^2(Ω). Moreover, one hasinf_m∈∩ C^2(Ω)λ_1^β(m) = min_m∈λ_1^β(m),and every minimizer m^* of λ_1^β overis a bang-bang function, i.e. can be represented as m^*=κ_E - _Ω∖ E, where E⊂Ω is a measurable set. As a consequence, throughout the paper, we consider the following optimization problem.Optimal design problem.Fix β∈ [0,∞]. We consider the extremal eigenvalue problemλ_*^β=inf{λ_1^β (m),m∈},whereis defined by (<ref>) and where λ_1^β (m) is the positive principal eigenvalue of{[-(e^α m' )'=λ me^α m in (0,1),; e^α m(0)'(0)=β(0),e^α m(1)'(1)=-β(1). ].Problem (<ref>) above is understood in a weak sense, that is, in the sense of the variational formulation:Find φ in H^1(0,1) such that for all ψ∈H^1(0,1),∫_0^1e^α mφ' ψ'+β (φ(0)ψ(0)+φ(1)ψ(1))= λ_1^β(m) ∫_0^1 m e^α mψ. §.§ Solving of the optimal design problem (<ref>) Let us first provide a brief summary of the main results and the outline of this article. Brief summary of the main results In a nutshell, we prove that under an additional smallness assumption on the non-negative parameter α, the problem of minimizing λ_1^β(·) overhas a solution writingm^*=κ_E^*-_Ω\ E^*,where E^* is (up to a zero Lebesgue measure set) an interval. Moreover, one has the following alternative: except for one critical value of the parameter β denoted β_α,δ, either E^* is stuck to the boundary, or E^* is centered at the middle point of Ω. More precisely, there exists δ∈ (0,1) such that: –1mm 1mm* for Neumann boundary conditions, one has E^*=(0,δ) or E^*=(1-δ,1);* for Dirichlet boundary conditions, one has E^*=((1-δ)/2,(1+δ)/2); * for Robin boundary conditions, there exists a threshold β_α,δ>0 such that, if β<β_α,δ then the situation is similar to the Neumann case, whereas if β>β_α,δ the situation is similar to the Dirichlet case.Figure <ref> illustrates different profiles of minimizers. The limit case β=β_α,δ is a bit more intricate. For a more precise statement of these results, one refers to Theorems <ref>, <ref> and <ref>.In this section, we will say that a solution m_*^β (whenever it exists) of Problem (<ref>) is of Dirichlet type if m_*^β=(κ+1)_((1-δ)/2,(1+δ)/2)-1 for some parameter δ>0.We first investigate the Neumann and Robin cases. The Dirichlet case is a byproduct of our results on the Robin problem.Neumann boundary conditionsIn the limit case where Neumann boundary conditions are imposed (i.e. β=0), one has the following characterization of persistence, resulting from the Neumann case in Proposition <ref> (see <cit.>). Let m∈. There exists a unique α^⋆(m)>0 such that–1mm 0mm* if α <α^⋆(m), then ∫_0^1 m e^α m <0 and λ_1^0(m)>0,* if α≥α^⋆(m), then ∫_0^1 m e^α m≥ 0 and λ_1^0(m)=0.As a consequence, in order to analyze the optimal design problem (<ref>) which minimizes the positive principal eigenvalue λ_1^β(m), it is relevant to consider (at least for the Neumann boundary conditions) α uniformly small with respect to m. This is the purpose of the following theorem which is proved in Section <ref> below.The infimumα̅= inf_m∈α^⋆(m)is attained at every function m_*∈ having the bang-bang property and such that ∫_Ω m_*=-m_0. In other words, the infimum is attained at every m_*∈ which can be represented as m_*=κ_E -_Ω∖ E, where E is a measurable subset of Ω of measure (1-m_0)/(κ+1). Moreover, one computes α̅= 1/1+κln(κ+m_0/κ(1-m_0)) >0. A consequence of the combination of Theorem <ref> and Proposition <ref> is that ∫_Ω me^α m<0 for every m∈ whenever α <α̅. Let β=0and α∈ [0,α̅). The optimal design problem (<ref>) has a solution. If one assumes moreover that α∈ [0,min{1/2,α̅}), then the inequality constraint (<ref>) is active, and the only solutions of Problem (<ref>) are m=(κ+1)_(0,δ^*)-1 and m=(κ+1)_(1-δ^*,1)-1,where δ^* =1-m_0/κ +1.Robin boundary conditionsThe next result is devoted to the investigation of the Robin boundary conditions case, for an intermediate value of β in (0,+∞). For that purpose, let us introduce the positive real number β_α,δ such thatβ_α,δ={[e^-α/√(κ)δarctan(2√(κ)e^α(κ+1)/κ e^2α(κ+1)-1) if κ e^2α(κ+1)>1 ,;π e^-α/2√(κ)δ if κ e^2α(κ+1)=1 ,; e^-α/√(κ)δarctan(2√(κ)e^α(κ+1)/κ e^2α(κ+1)-1)+π e^-α/√(κ)δif κ e^2α(κ+1)<1. ].We also introduce δ^*= 1-m_0/1+κ and ξ^*=κ+m_0/2(1+κ),and we denote by β_α^* the real number β_α,δ^*.Note that the particular choice |{m=κ}|=δ^* corresponds to choosing ∫_0^1 m = -m_0 if m is bang-bang. It is also notable that if E^*=(ξ^*,ξ^*+δ^*) in (<ref>), then {m=κ} is a centered subinterval of (0,1). Letβ≥ 0, and α∈ [0,α̅). The optimal design problem (<ref>) has a solution m_*^β. Defining δ =1-m_0/κ +1, where m_0=-∫_0^1 m_*^β and assuming moreover that α∈ [0,min{1/2,α̅}), one has the following. * If β <β_α,δ, then ∫_0^1 m_*^β=-m_0 and the solutions of Problem (<ref>) coincide with the solutions of Problem (<ref>) in the Neumann case.* If β >β_α,δ, then the solutions of Problem (<ref>) are of Dirichlet type. Moreover, if we further assume that α<sinh^2(β_1/2^*ξ^*)/1+2sinh^2(β_1/2^*ξ^*),then ∫_0^1 m_*^β=-m_0 and the solutions of Problem (<ref>) coincide with the solutions of Problem (<ref>) in the Dirichlet case.* If β =β_α,δ, then ∫_0^1 m_*^β = -m_0 and every function m=(κ+1)_(ξ,ξ+δ^*)-1 where ξ∈ [0,1-δ^*] solves Problem (<ref>).This result is illustrated on Figure <ref>. It can be seen as a generalization of <cit.>, where the case α=0 is investigated. Let us comment on these results. It is notable that standard symmetrization argument cannot be directly applied. Indeed, this is due to the presence of the term e^α m at the same time in the numerator and the denominator of the Rayleigh quotient defining λ_1^β(m). The proofs rest upon the use of a change of variable to show some monotonicity properties of the minimizers, combined with an adapted rearrangement procedure as well as a refined study of the necessary first and second order optimality conditions to show the bang-bang property of the minimizers.Let us now comment on the activeness of the inequality constraint (<ref>). In the case α=0, one can prove that a comparison principle holds (see <cit.>, Lemma 2.3). A direct consequence is that the constraint (<ref>) is always active. In our case however, it can be established that the comparison principle fails to hold, and the activeness of the constraint has to be studied a posteriori. Note that under the assumptions of Theorem <ref>, with the additional assumption (<ref>), Theorem <ref> rewrites: –1mm 0mm* if β<β_α^*, then the only solutions of Problem (<ref>) are the Neumann solutions;* if β>β_α^*, then the only solution of Problem (<ref>) is the Dirichlet solution;* if β=β_α^*, then every function m=(κ+1)_(ξ,ξ+δ^*)-1 where ξ∈ [0,1-δ^*] solves Problem (<ref>). We can prove that, if assumption (<ref>) fails to hold, then there exist sets of parameters such that ∫_0^1m_*^β < m_0.Dirichlet boundary conditionsFinally, as a byproduct of Theorem <ref>, we have the following result in the case of Dirichlet boundary conditions.Let β=+∞and α≥ 0. The optimal design problem (<ref>) has a solution.If one assumes moreover that α∈ [0, 1/2), then any solution of Problem (<ref>) writes m=(κ+1)_((1-δ)/2,(1+δ)/2)-1 for some δ∈ (0,1).§.§ Qualitative properties and comments on the results It is interesting to notice that, according to the analysis performed in Section <ref> (see (<ref>) and (<ref>)) the optimal eigenvalue λ^β_* is the first positive solution of an algebraic equation, the so-called transcendental equation. More precisely,* in the case β <β_α,δ, the optimal eigenvalue λ_*^β is the first positive root of the equation (of unknown λ)tan(√(λκ)δ)=√(κ)e^α (κ+1)(λ+β^2e^2α)tanh(√(λ)(1-δ))+2β e^α√(λ)/β e^α√(λ)(κ e^2 α (κ+1)-1)tanh(√(λ)(1-δ))+e^2α(λκ e^2ακ-β^2),* in the case β >β_α,δ, the optimal eigenvalue λ_*^β is the first positive root of the equation (of unknown λ)tan(√(λκ)δ)=√(κ)e^α (κ+1)(λ+β^2e^2α)sinh(√(λ)(1-δ))+2β√(λ)e^αcosh(√(λ)(1-δ))/𝒟_α(β,λ) , where𝒟_α(β,λ) = 1/2(κ e^2α (1+κ)-1)(β^2e^2α+λ)cosh (√(λ)(1-δ)) + β e^α√(λ)(κ e^2 α(κ+1)-1) sinh (√(λ)(1-δ))+1/2(1+κ e^2α (1+κ))(λ-β^2e^2α).These formulae provide an efficient way to compute the numbers λ_*^β since it comes to the resolution of a one-dimensional algebraic equation. On Figure <ref>, we used this technique to draw the graph of β↦λ_*^β for a given choice of the parameters α, κ and m_0. From a practical point of view, we used a Gauss-Newton method on a standard desktop machine.It is notable that one can recover from this figure, the values λ_*^0 (optimal value of λ_1 in the Neumann case) as the ordinate of the most left hand point of the curve and λ_*^∞ (optimal value of λ_1 in the Dirichlet case) as the ordinate of all points of the horizontal asymptotic axis of the curve. Finally, the concavity of the function β↦λ_*^β can be observed on Figure <ref>. This can be seen as a consequence of the fact that λ_*^β writes as the infimum of linear functions of the real variable β.§ PROOF OF THEOREM <REF> In view of Proposition <ref>(<ref>), we start by maximizing ∫_0^1 me^α m over . The supremumsup_m∈∫_0^1 me^α mis attained at some m∈. Moreover, if m is a maximizer of (<ref>), then m is bang-bang, i.e. can be represented as m=κ_E - _(0,1)∖ E, where E is a measurable set in (0,1), and ∫_0^1 m = -m_0. We first consider a problem similar to (<ref>), where we remove the assumption that m should change sign in (0,1), namely we consider the maximization problemsup_m∈ℳ_m_0,κ∫_0^1 me^α mwhere ℳ_m_0,κ = {m∈L^∞(0,1), msatisfies assumptions (<ref>) and (<ref>)}. Step 1. Restriction to monotone functions We claim that the research of a maximizer for Problem (<ref>) can be restricted to the monotone non-increasing functions of ℳ_m_0,κ. Indeed, if m∈ℳ_m_0,κ, we introduce its monotone non-increasing rearrangement m^↘ (see e.g. <cit.> for details). By the equimeasurability property of monotone rearrangements, one has ∫_0^1 m^↘ = ∫_0^1 m. Since it is obvious that m^↘ also satisfies Assumption (<ref>), one has m^↘∈ℳ_m_0,κ. Moreover, the equimeasurability property also implies that ∫_0^1 m^↘ e^α m^↘ = ∫_0^1 me^α m, which concludes the proof of the claim. Step 2. Existence of solutions Let us now show that there exists a maximizer for Problem (<ref>). To see this, we consider a maximizing sequence m_k associated with Problem (<ref>). By the previous point, we may assume that the functions m_k are non-increasing. Helly's selection theorem ensures that, up to a subsequence, m_k converges pointwise to a function m^*. Hence, -1≤ m^* ≤κ a.e. in (0,1), and ∫_0^1 m^* ≤ -m_0 by the dominated convergence theorem, which implies that m^*∈ℳ_m_0,κ. Using the dominated convergence theorem again, we obtain that ∫_0^1 m_k e^α m_k→∫_0^1 m^* e^α m^* as k→∞. Therefore, m^* is a maximizer of (<ref>). Step 3. Optimality conditions and bang-bang properties of maximizers We now prove that every maximizer m^* of Problem (<ref>) is bang-bang. Note that since m^* is bang-bang if and only if its monotone non-increasing rearrangement is bang-bang, we may assume that m^* is non-increasing. As a consequence, we aim at proving that m^* can be represented as m^*=(κ +1) _(0,γ)-1 for some γ∈ (0,1). We assume by contradiction that |{-1<m^*<κ}|>0. We will reach a contradiction using the first order optimality conditions. Introduce the Lagrangian function associated to Problem (<ref>), defined by: (m,μ)∈ℳ_m_0,κ×↦∫_0^1 m e^α m-η(∫_0^1 m(x)dx +m_0 ).Denote by η^* the Lagrange multiplier associated to the constraint ∫_0^1 m ≤ -m_0. Since we are dealing with an inequality constraint, we have η^*≥ 0. If x_0 lies in the interior of the interval {-1<m^*<κ} and h = _(x_0-r,x_0+r), then we observe that m^*+rh∈ℳ_m_0,κ and m^*-rh∈ℳ_m_0,κ if r>0 is small enough. The first order optimality conditions then yield that ⟨ d_mℒ(m^*,μ^*),h⟩ = 0, that is∫_0^1 h(e^α m^*(1+α m^*)-η^*) = 0.Consequently, the Lebesgue Density Theorem ensures that e^α m^*(1+α m^*)=η^* a.e. in {-1<m^*<κ}. Studying the function y↦ e^α y(1+α y) yields that m^* is equal to a constant ζ∈ [-1/α,κ) in {-1<m^*<κ}. Therefore, m^* can be represented as m^*=κ_[0,γ_1] + ζ_(γ_1,γ_2) - _[γ_2,1], where 0≤γ_1 < γ_2≤ 1. Let us show that one has necessarily γ_1=γ_2, by constructing an admissible perturbation which increases the cost function whenever γ_1<γ_2. For θ>0, we introduce the function m^*_θ defined bym^*_θ = κ_[0,γ_1^θ] + ζ_(γ_1^θ,γ_2^θ) - _[γ_2^θ,1],where γ_1^θ = γ_1+(1+ζ)θ and γ_2^θ = γ_2-(κ-ζ)θ. Note that ∫_0^1 m^*_θ = ∫_0^1 m^* and m^*_θ∈ [-1,κ] a.e. in (0,1), which implies that m^*_θ∈ℳ_m_0,κ if θ is sufficiently small. One computes∫_0^1 (m^*_θ e^α m^*_θ - m^*e^α m^*) = θ( (1+ζ)(κ e^ακ - ζ e^αζ) - (κ-ζ)(e^-α+ζ e^αζ) ).Setting ψ : ζ↦ (1+ζ)(κ e^ακ - ζ e^αζ) - (κ-ζ)(e^-α+ζ e^αζ), one has ψ”(ζ)=-α e^αζ (1+κ)(2+αζ), from which we deduce that ψ is strictly concave in [-1/α,κ]. Since ψ'(-1/α)=0, ψ'(κ)<0 and ψ(κ)=0, we obtain that ψ(ζ)>0 for all ζ∈ [-1/α,κ). As a consequence, if θ is small enough, then m^*_θ∈ℳ_m_0,κ and ∫_0^1 m^*_θ e^α m^*_θ> ∫_0^1 m^* e^α m^*, which is a contradiction.We have then proved that m^* writes m^*=(κ +1) _E -1 for some measurable set E⊂ (0,1). As a consequence, ∫_0^1 m^* e^α m^* is maximal when |E| is maximal, that is, when |E| = (1-m_0)/(κ+1), which corresponds to ∫_0^1 m^* = -m_0. Since ∫_0^1 m^* e^α m^* does not depend on the set E in the representation m^*=(κ+1)_E -1, we deduce that every bang-bang function in ℳ_m_0,κ satisfying ∫_0^1 m^*=-m_0 is a maximizer of Problem (<ref>).To conclude, observe that because of Assumption (<ref>), every bang-bang function m^* in ℳ_m_0,κ satisfying ∫_0^1 m^*=-m_0 changes sign, which implies that one has in fact m^*∈. This concludes the proof.We can now prove Theorem <ref>. Denote by m any bang-bang function ofsatisfying ∫_0^1 m=-m_0 and by m^* their decreasing rearrangement. According to the first step of the proof of Lemma <ref>, there holds α^⋆(m)=α^⋆(m^*), where α^⋆ is defined in Proposition <ref>. Moreover, using that m^*=(κ+1)_(0,γ) -1 with γ=(1-m_0)/(κ+1), a quick computation shows that α^⋆(m^*) = 1/1+κlnκ+m_0/κ(1-m_0). Indeed,α^⋆(m^*) is reached. Consider α<α^⋆(m^*), which implies that ∫_0^1 m^* e^α m^* <0. We can apply Lemma <ref> which yields that ∫_0^1 m e^α m <0 for every m∈. We then deduce that α<α^⋆(m) for every m∈. Therefore, one has α^⋆(m^*) ≤α^⋆(m) for all m∈, which proves that α̅= α^⋆(m^*) and concludes the proof of Theorem <ref>.§ PROOFS OF THEOREMS <REF>, <REF> AND <REF> Since the proof of Theorem <ref> can be considered as a generalization of the proofs of Theorems <ref> and <ref>, we will only deal with the general case of Robin boundary conditions (i.e. β∈ [0,+∞]) in the following. The proofs in the Neumann and Dirichlet cases become simpler since the rearrangement to be used is standard (monotone rearrangement in the Neumann case and Schwarz symmetrization in the Dirichlet case). This is why in such cases, the main simplifications occur in Section <ref> where one shows that a minimizer function m_*^β is necessary unimodal (in other words, m_*^β is successively non-decreasing and then non-increasing on (0,1)). §.§ Every minimizer is unimodal We will show that the research of minimizers can be restricted to unimodal functions of . Take a function m∈. We will construct a unimodal function m^R ∈ such that λ_1^β(m^R)≤λ_1^β(m), where the inequality is strict if m is not unimodal. We denote bythe eigenfunction associated to m, in other words the principal eigenfunction solution of Problem (<ref>).According to the Courant-Fischer principle, there holdsλ_1^β(m)=_m^β[φ]=min_∈H^1(0,1) ∫_0^1 m e^α m^2 >0_m^β[φ] ,where _m^β[φ]=∫_0^1 e^α m(x)'(x) ^2dx+β(0)^2+β (1)^2/∫_0^1 m(x) e^α m(x)(x)^2dx. §.§.§ A change of variableLet us consider the change of variable y=∫_0^x e^-α m(s)ds,x∈ [0,1].The use of such a change of variable is standard when studying properties of the solutions of Sturm-Liouville problems (see e.g. <cit.>).Noting that y seen as a function of x is monotone increasing on [0,1], let us introduce the functions c, u and m̃ defined byc(x)=∫_0^x e^-α m (s)ds, u(y)=(x),andm̃(y)=m(x),for x∈ [0,1] and y∈ [0,c(1)].Notice that ∫_0^c(1)m̃(y)e^αm̃ (y)dy=∫_0^1m (x)dx≤ -m_0. Let us introduce x^+=minx∈ [0,1]argmax φ (x),in other words x^+ denotes the first point of [0,1] at which the function φ reaches its maximal value. We will also need the point y^+ as the range of x^+ by the previous change of variable, namely y^+=∫_0^x^+ e^-α m(s)ds. §.§.§ Rearrangement inequalities Using the change of variable (<ref>) allows to writeλ_1^β(m) = _m^β [φ]=N_1+N_2/D_1+D_2with [N_1 = ∫_0^y^+u'(y)^2dy+β u(0)^2,N_2 = ∫_y^+^c(1)u'(y)^2dy+β u(c(1))^2,;D_1 = ∫_0^y^+m̃(y)e^2αm̃(y)u(y)^2dy, D_2 = ∫_y^+^c(1)m̃(y)e^2αm̃(y)u(y)^2dy. ]Step 1. Unimodal rearrangements Introduce the functionu^R defined on (0,c(1)) byu^R(y)={[ u^↗(y)on (0,y^+),; u^↘(y) on (y^+,c(1)), ].where u^↗ denotes the monotone increasing rearrangement[Recall that, for a given function v∈(0,L) with L>0, one defines its monotone increasing rearrangement v^↗ for a.e. x∈ (0,L) by v^↗(x)=sup{c∈| x∈Ω_c^*}, where Ω_c^*=(1-|Ω_c|,1) with Ω_c={v>c}.] of u on (0,y^+) andu^↘ denotes the monotone decreasing rearrangement[Similarly, v^↘ is defined by v^↘(x)=v^↗(1-x).] of u on (y^+,c(1)) (see for instance <cit.> for details and see Figure <ref> for an illustration of this procedure). Thanks to the choice of y^+, it is clear that this rearrangement does not introduce discontinuities, and more precisely that u^R∈H^1(0,c(1)).Similarly, we also introduce the rearranged weight m̃^R, defined by m̃^R(y)={[m̃^↗(y)on (0,y^+),;m̃^↘(y) on (y^+,c(1)), ].with the same notations as previously.Observe that, by the equimeasurability property of monotone rearrangements the intervals (0,y^+) and (y^+,c(1)), one has∫_0^c(1)m̃^R(y)e^αm̃^R(y)dy=∫_0^c(1)m̃(y)e^αm̃ (y)dy≤ -m_0. Step 2. The rearranged function m̃^R decreases the Rayleigh quotient Let us now show that u^R decreases the previous Rayleigh quotient. First, one has by property of monotone rearrangements that u^R is positive. Writing∫_0^c(1)m̃^R(y)e^2αm̃^R(y)u^R(y)^2dy=aaaaaaaa∫_0^c(1)(m̃^R(y)e^2αm̃^R(y)+e^-2α)u^R(y)^2dy-e^-2α∫_0^c(1)u^R(y)^2dyto deal with a positive weight and combining the Hardy-Littlewood inequality with the equimeasurability property of monotone rearrangements on (0,y^+) and then on (y^+,c(1)), we obtainD_1≤∫_0^y^+m̃^R(y)e^2αm̃^R(y)u^R(y)^2dyand D_2≤∫_y^+^c(1)m̃^R(y)e^2αm̃^R(y)u^R(y)^2dyand therefore∫_0^c(1)m̃^R(y)e^2αm̃^R(y)u^R(y)^2dy≥ ∫_0^c(1)m̃(y)e^2αm̃(y)u(y)^2dy= ∫_0^1 m(x) e^α m(x)(x)^2dx>0.Indeed, we used here that the function η↦η e^2αη is increasing on [-1,κ] whenever α≤ -1/2. Therefore, we claim that the rearrangement of the function m̃e^2αm̃ according to the method described above coincides with the function m̃^Re^2αm̃^R, whence the inequality above. Roughly speaking, we will use this inequality to construct an admissible test function in the Rayleigh quotient (<ref>) from the knowledge of u^R. Also, we easily see that (u^R)^2(0)=min_[0,y^+](u^R)^2≤ u^2(0)and(u^R)^2(c(1))=min_[y^+,c(1)](u^R)^2≤ u^2(c(1)).Using now Polyà's inequality twice providesN_1≥∫_0^y^+ ((u^R)')^2+β (u^R)^2(0) andN_2≥∫_y^+^c(1) ((u^R)')^2+β (u^R)^2(c(1))As a result, by combining Inequalities (<ref>), (<ref>) and (<ref>), one getsλ_1^β(m) ≥∫_0^c(1)(u^R)'(y)^2dy+β u^R(0)^2+β u^R(c(1))^2/∫_0^c(1)m̃^R(y)e^2αm̃^R(y)u^R(y)^2dy . Consider now the change of variable z=∫_0^y e^αm̃^R(t)dt, as well as the functions m^R and φ^R defined bym^R(z)=m̃^R(y)andφ^R(z)=u^R(y),for all y∈ [0,c(1)] and z∈ [0,1][Indeed, notice that, according to the equimeasurability property of monotone rearrangements, one has∫_0^c(1)e^αm̃^R(y)dy=∫_0^c(1)e^αm̃ (y)dy=∫_0^1dx=1. ].Observe that m^R is admissible for the optimal design problem (<ref>). Indeed∫_0^1 m^R(z)dz = ∫_0^c(1)m̃^R(y)e^αm̃^R(y)dy ≤ -m_0by (<ref>). Since it is obvious that -1≤m̃^R≤κ and that m̃^R changes sign, we deduce immediately that m^R satisfies Assumptions (<ref>) and (<ref>).Note that one has also u^R(0)=φ^R(0), u^R(c(1))=φ^R(1) and ∫_0^c(1)(u^R)'(y)^2dy = ∫_0^1 e^α m^R(z)(φ^R')^2(z)dz,∫_0^c(1)m̃^R(y)e^2αm̃^R(y)u^R(y)^2dy = ∫_0^1m^R(z)e^α m^R(z)φ^R(z)^2dz.In particular and according to (<ref>) and the standard properties of rearrangements, there holds∫_0^1m^R(z)e^α m^R(z)φ^R(z)^2dz>0, and φ^R∈H^1(0,1) so that the function φ^R is admissible in the Rayleigh quotient _m^R^β. Hence, we infer from (<ref>) that λ_1^β(m) ≥_m^R^β [φ^R] ≥λ_1^β(m^R).Finally, investigating the equality case of Polyà's inequality, it follows that the inequality (<ref>) is strict if u is not unimodal, that is, if φ is not unimodal (see for example <cit.> and references therein). We have then proved the following result. Every solution m_*^β of the optimal design problem (<ref>) is unimodal, in other words, there exists x_* such that m_*^β is non-decreasing on (0,x_*) and non-increasing on (x_*,1). Moreover, the associated eigenfunction φ_*^β, i.e. the solution of System (<ref>) with m=m_*^β, is non-decreasing on (0,x_*) and non-increasing on (x_*,1).By using the change of variable (<ref>) as well as the same reasonings and notations as above, it is notable that for every m∈, the principal eigenvalue λ_1^β(m) solves the eigenvalue problem-u”(y)=λ_1^β(m)m̃(y)e^2αm̃(y)u(y), on (0,c(1)).An easy but important consequence of this remark is the following: applying the Krein-Rutman theory to this problem yields existence, uniqueness and simplicity of λ_1^β(m). §.§ Existence of minimizers We start by stating and proving a Poincaré type inequality. The proof of the existence of a solution for the optimal design problem (<ref>) relies mainly on Lemmas <ref> and <ref>. (Poincaré type inequality). Assume that α∈ [0,min{1/2,α̅}). There exists a constant C>0 such that for every ∈H^1(0,1) and m∈ satisfying ∫_0^1 m e^α m^2 >0, one has ∫_0^1 ^2 ≤ C ∫_0^1 '^2. Assume that the inequality does not hold. Therefore, for every k ∈, there exist _k∈H^1(0,1) and m_k ∈ such that ∫_0^1 m_k e^α m_k_k^2>0 and∫_0^1 _k^2 > k ∫_0^1 _k'^2. First notice that we may assume that the functions m_k and _k are non-increasing in (0,1). Indeed, if we introduce the monotone non-increasing rearrangements m_k^↘ and _k^↘ of m_kand _k, we have∫_0^1 m_k e^α m_k_k^2 = ∫_0^1 (m_k e^α m_k+e^-α) _k^2 - ∫_0^1 e^-α_k^2 ≤∫_0^1 m_k^↘ e^α m_k^↘_k^↘^2,where we have used the Hardy-Littlewood inequality and the equimeasurability property of the monotone rearrangements. Note that since the function η↦η e^αη is increasing on [-1,κ] whenever α≤ 1/2, one has (m_k e^α m_k)^↘ = m_k^↘ e^α m_k^↘. Moreover, (<ref>) implies that ∫_0^1 _k^↘^2 > k ∫_0^1 _k^↘'^2, where we have used the equimeasurability property and the Polyá inequality.We may further assume that for each k, ∫_0^1 _k^2=1. Since the sequence _k is bounded in H^1(0,1), there is a subsequence _k such that _k ⇀ weakly in H^1 and _k → strongly in L^2. As a consequence, ∫_0^1 '^2 ≤lim inf∫_0^1 _k'^2 = 0, which implies thatis contant in (0,1). Note that since ∫_0^1 ^2=1,must be positive in (0,1).Since the functions m_k are non-increasing, Helly's selection theorem ensures that, up to an extraction, m_k converges pointwise to a function m. We infer that -1≤ m≤κ a.e., and that ∫_0^1 m ≤ -m_0, by dominated convergence. We also obtain that m_k e^α m_k→ m e^α m in L^2. A consequence is that ∫_0^1 m_k e^α m_k_k^2 →∫_0^1 m e^α m^2 as k→∞. Indeed, ∫_0^1 m e^α m^2 - ∫_0^1 m_k e^α m_k_k^2 = ∫_0^1 (m e^α m-m_k e^α m_k) ^2 + ∫_0^1 m_k e^α m_k (^2-_k^2) k→∞⟶ 0.Sinceis constant, we deduce that ∫_0^1 m e^α m≥ 0 . We also have that ∫_0^1 m e^α m≤ 0 since ∫_0^1 m_k e^α m_k<0 for every k (recall that the inequality holds true for every function inwhenever α <α̅). We have finally proved that ∫_0^1 m e^α m = 0. We claim that m cannot change sign. Indeed, otherwise, m would lie in the set , which would imply that ∫_0^1 m e^α m <0. We then deduce that m=0 a.e. in (0,1), which is impossible since ∫_0^1 m≤ -m_0<0.If β<+∞ (Neumann and Robin cases) and α∈ [0,min{1/2,α̅}) (resp. β=+∞(Dirichlet case) and α∈ [0,1/2)), then the infimum λ_*^β of λ_1^β over is achieved at some m_*^β∈.In this proof, we only deal with the case where β<+∞. Indeed, we claim that all the lines can be easily adapted in the Dirichlet case since, in this case, the Poincaré inequality is satisfied without the assumption α<α̅.Consider a minimizing sequence m_k for Problem (<ref>). By Lemma <ref>, one can assume that the functions m_k are unimodal. As in the proof of Lemma <ref>, we may assume that m_k converges pointwise[Indeed, the proof of Helly's selection theorem extends easily to the case of unimodal functions.], and in L^2 to a function m^*∈L^∞(0,1). Moreover, m^* satisfies -1≤ m^*≤κ a.e. in (0,1), and ∫_0^1 m^* ≤ -m_0.For each k, let _k be the eigenfunction associated to λ_1^β(m_k) with _k>0. That is, the functions _k satisfy the variational formulation: for all ψ∈H^1(0,1),∫_0^1 e^α m_k_k' ψ' + β (_k(0)ψ(0)+_k(1)ψ(1)) = λ_1^β(m_k) ∫_0^1 m_k e^α m_k_k ψ.We may assume that for each k, ∫_0^1 m_k e^α m_k_k^2 =1, which implies that λ_1^β(m_k) = ∫_0^1 e^α m_k_k'^2 +β(_k(0)^2+_k(1)^2). We deduce from Lemma <ref> that the sequence _k is bounded in H^1(0,1). Hence, there is a subsequence _k such that _k → in L^2, and _k ⇀ in H^1. By Lemma <ref>, we can assume that the functions _k are unimodal. Write ψ_k = _k-. Taking ψ=ψ_k in (<ref>) yields∫_0^1 e^α m_kψ_k'^2 = - ∫_0^1 e^α m_kψ_k' ' -β(ψ_k(0)(ψ_k(0)+(0)) + ψ_k(1)(ψ_k(1)+(1) ) + λ_1^β(m_k) (∫_0^1 m_k e^α m_kψ_k^2 + ∫_0^1 m_k e^α m_kψ_k ). Since _k ⇀ in H^1(0,1), one has ψ_k(0)→ 0 and ψ_k(1)→ 0 as k→∞. Therefore, (<ref>) implies that e^-α∫_0^1 ψ_k'^2 ≤∫_0^1 e^α m_kψ_k'^2 → 0 as k→∞. As a consequence, the sequence _k converges in fact strongly to the functionin H^1.As in the proof of Lemma <ref>, one has ∫_0^1 m^* e^α m^*^2=lim_k→∞∫_0^1 m_k e^α m_k_k^2 = 1. Firstly, this forces m^* to change sign, which implies that m^*∈. Secondly, one hasλ_1^β(m^*) ≤∫_0^1 e^α m^*'^2 + β((0)^2+(1)^2) = lim_k→∞λ_1^β(m_k).Therefore, the infimum λ_*^β is attained at m^*∈.§.§ Every minimizer is bang-bang At this step, we know according to Lemma <ref> that any minimizer m_*^β is unimodal. Let us show moreover that it is bang-bang, in other words equal to -1 or κ a.e. in [0,1].Step 1. A new optimal design problem The key point of the proof is the following remark: the function m_*^β solves the optimal design probleminf_m∈ ∫_0^1 m(x) e^α m(x)_*^β(x)^2dx>0_m^β[φ_*^β]where _*^β denotes the eigenfunction associated to λ_*^β = λ_1^β(m_*^β). Indeed, assume by contradiction the existence of m∈ such that ∫_0^1 m(x) e^α m(x)_*^β(x)^2dx>0 and _m^β[φ_*^β]< _m_*^β^β[φ_*^β]. This would hence imply that λ_*^β >λ_1^β (m) whence the contradiction. Notice that this also implies in particular the existence of a solution for Problem (<ref>) and therefore that the constraint ∫_0^1 m(x) e^α m(x)_*^β(x)^2dx>0 is not active at m=m_*^β. In other words, ∫_0^1 m_*^β (x) e^α m_*^β (x)_*^β(x)^2dx>0.Let us now introduce the set given by ℐ=(0,1)\ ({m_*^β=-1}∪{m_*^β=κ}). Note that ℐ is an element of the class of subsets of [0,1] in which -1<m_*^β(x)<κ a.e. Notice that ℐ also writesℐ= ⋃_k=1^+∞ℐ_ kwhere ℐ_ k={x∈ (0,1) | -1+1/k< m_*^β(x)<κ-1/k}.We will prove that the set ℐ has zero Lebesgue measure. To this end, we argue by contradiction: we assume in the following of the proof that |ℐ|>0.Step 2. The range of m_*^β lies in {-1,0,κ}In this step of the proof, we will prove that, up to a zero Lebesgue measure set, range(m_*^β)⊂{-1,0,κ}. To see this, we will use the previous remark and write the first order optimality conditions for Problem (<ref>). For that purpose, let us introduce the Lagrangian functional ℒ associated to Problem (<ref>), defined byℒ:×∋ (m,η)↦_m^β [φ_*^β]+η(∫_0^1m(x)dx+m_0).Note that we do not take into account the inequality constraint in the definition of the Lagrangian functional. Indeed, we aim at writing the first order optimality conditions at m=m_*^β and we know that the inequality constraint is not active, according to the remark above. In the following, we will denote by η^* the Lagrange multiplier associated to the (integral) equality constraint for Problem (<ref>). In particular, m_*^β minimizes the functional ∋ m↦ℒ(m,η^*). Notice that since we are dealing with inequality constraints, one has necessarily η^*≥ 0.Since |ℐ|>0 by assumption, ℐ_k is of positive measure when k is large enough. If |ℐ_k|>0, take x_0∈ℐ_k and let (G_k,n)_n∈ be a sequence of measurable subsets with G_n,k included in ℐ_k and containing x_0. Choosing h=_G_k,n, note that m_*^β +th∈ and m_*^β-th∈ when t small enough. Writingℒ(m_*^β± th,η^* )≥ℒ(m_*^β ,η^* ), dividing this inequality by t and letting t go to 0, it follows that ⟨ d_mℒ(m_*^β,η^*),h⟩=0.Moreover, one computes⟨ d_mℒ(m_*^β,η^*),h⟩=∫_G_n,kh(x) e^α m_*^β (x)(αφ_*^β '(x)^2-λ_*^β(α m_*^β (x)+1)φ_*^β(x)^2)/∫_0^1 m_*^β (x) e^α m_*^β (x)_*^β(x)^2dxdx +η^*|G_n,k|.Assume without loss of generality that φ_*^β is normalized such that ∫_0^1 m_*^β e^α m_*^β(_*^β)^2=1. Dividing the equality (<ref>) by |G_k,n| and letting G_k,n shrink to {x_0} as n→ +∞ shows that ψ_0(x_0)=-η^*e^-α m_*^β (x_0)for almost every x_0∈ℐ_k,according to the Lebesgue Density Theorem, whereψ_0(x)=αφ_*^β '(x)^2-λ_*^β(α m_*^β (x)+1)φ_*^β(x)^2. The set ℐ (and therefore ℐ_k) is either an open interval or the union of two open intervals, and the restrictions of the functions m_*^β and φ_*^β to ℐ belong to H^2(ℐ).The first point is obvious and results from the unimodal character of m_*^β stated in Lemma <ref>. Let us show that m_*^β is continuous on each connected component of ℐ.Let us consider the change of variable (<ref>) introduced in Section <ref>, namely y=∫_0^x e^-α m_*^β (s)ds, for all x∈ [0,1]. Introduce also the functions c:[0,1]∋ x↦∫_0^x e^-α m_*^β (s)ds and m̃_*^β defined on [0,c(1)] by m̃_*^β(y)=m_*^β(c^-1(y)). The crucial argument rests upon the fact that c(ℐ)=c({-1<m_*^β<κ})={-1<m̃_*^β<κ},since c is in particular continuous. Furthermore, it follows from (<ref>) that the function m̃_*^β satisfies α e^-2αm̃_*^β(y_0)u_*^β '(y_0)^2-λ_*^β(αm̃_*^β (y_0)+1)u_*^β(y_0)^2=-η^*e^-αm̃_*^β (y_0)on c(ℐ),where u_*^β is defined by u_*^β(y)=φ_*^β(c^-1(y)) for all y∈ [0,c(1)]. A simple computation shows that the function u_*^β solves in a distributional sense the o.d.e.-u_*^β”(y)=λ_*^βm̃_*^β(y)e^2αm̃_*^β(y)u_*^β(y)in (0,c(1)].By using standard elliptic regularity arguments (see e.g. <cit.>), we infer that u_*^β belongs to H^2(0,1) and is in particular C^1 on [0,c(1)]. According to (<ref>) and applying the implicit functions theorem, we get that the function m̃_*^β is necessarily itself C^1 on ℐ. Using the regularity of m̃_*^β and c, and since the derivative of c is pointwisely bounded by below by e^-ακ, we infer that the restriction of the function m_*^β=m̃_*^β∘ c^-1 to ℐ belongs to H^1(ℐ). Furthermore, consider one connected component, say (x^1_ℐ,x^2_ℐ) of ℐ. Since for all x∈ (x^1_ℐ,x^2_ℐ) there holdsc(x)=∫_0^x e^-α m_*^β(s)ds, one infers that for all x∈ (x^1_ℐ,x^2_ℐ), one has c'(x)=e^-α m_*^β(x) and thus, c∈H^2(ℐ) by using that m_*^β∈H^1(ℐ). As a result, since m_*^β=m̃_*^β∘ c^-1, one gets successively that m_*^β and φ_*^β are H^2 on ℐ (by using in particular (<ref>) for φ_*^β). According to Lemma <ref>, the function m_*^β is H^2 on each interval of ℐ (and hence of ℐ_k). Therefore, using that φ_*^β”(x)=-αm_*^β'φ_*^β'-λ_*^β m_*^βφ_*^β on ℐ_k, this last equality being understood in L^2(ℐ_k), one computes ψ_0'(x) = -2φ_*^β'(x)(λ_*^β(2α m_*^β+1)φ_*^β+α^2m_*^β'(x)φ_*^β'(x))-αλ_*^βm_*^β'(x)φ_*^β(x)^2for every x∈ℐ_k. According to Lemma <ref>, we claim thatφ_*^β'(x) and m_*^β'(x) have the same sign (with the convention that the number 0 is at the same time of positive and negative sign) for a.e. x∈ (0,1) and therefore m_*^β'(x)φ_*^β'(x)≥ 0 for a.e. x∈ (0,1). Since α≤ 1/2, one has 2α m_*^β+1≥ -2α+1≥ 0 and with the notations of Lemma <ref>, it follows that ψ_0'(x) is nonpositive on (0,x_*) and nonnegative on (x_*,1), implying that ψ_0 is non-increasing on (0,x_*) and non-decreasing in (x_*,1). Moreover and according to the previous discussion, it is obvious that -η^*e^-α m_*^β is non-decreasing on (0,x_*) and non-increasing in (x_*,1), since η^*≥ 0. We then infer from the previous reasoning and since the integer k was chosen arbitrarily that there exist x_0, y_0, x_1, y_1 such that ℐ=(x_0,y_0)∪ (x_1,y_1) with 0<x_0≤ y_0≤ x_*≤ x_1≤ y_1 and the equalityψ_0(x)=-η^*e^-α m_*^β (x)holds true on ℐ. If x_0<y_0 (resp. x_1<y_1), notice that one has necessarily m_*^β=0 on (x_0,y_0) (resp. on (x_1,y_1)). Indeed, it follows from (<ref>) and the monotonicity properties of ψ_0 and m_*^β on (x_0,y_0) and (x_1,y_1) that ψ_0 and m_*^β are constant on (x_0,y_0) and (x_1,y_1). According to (<ref>) and since α∈ [0,1/2], it follows that φ_*^β is also constant on (x_0,y_0) and (x_1,y_1). Using the equation solved by φ_*^β, one shows that necessarily, m_*^β=0 on (x_0,y_0) and (x_1,y_1). This achieves the proof that m_*^β is equal to -1, 0 or κ a.e.Step 4. The minimizer m_*^β is bang-bang In this last step, we will use the second order optimality conditions to reach a contradiction. Since by hypothesis, ℐ has positive Lebesgue measure, it is not restrictive to assume that x_0<y_0. We will reach a contradiction with an argument using the second order optimality conditions. Introduce the functional :∋ m↦_m^β[φ_*^β] as well as its first and second order derivative in an admissible direction h denoted respectively ⟨ d(m),h⟩ and d^2 (m)(h,h). One has ⟨ d(m_*^β),h⟩= α∫_0^1 h e^α m_*^β (φ_*^β')^2-λ_*^β∫_0^1 h(1+α m_*^β)e^α m_*^β(φ_*^β)^2.Consider an admissible[For every m∈, the tangent cone to the setat m, denoted by 𝒯_m, is the set of functions h∈L^∞(0,1) such that, for any sequence of positive real numbers ε_n decreasing to 0, there exists a sequence of functions h_n∈L^∞(0,1) converging to h as n→ +∞, and m+ε_nh_n∈ for every n∈ (see for instance <cit.>).] perturbation h supported by (x_0,y_0). The first order optimality conditions yield that ⟨ d(m_*^β),h⟩=0 and one has therefored^2 (m_*^β)(h,h) = α^2 ∫_0^1 h^2 e^α m_*^β (φ_*^β')^2-αλ_*^β∫_0^1h^2 e^α m_*^β (2+α m_*^β)(φ_*^β)^2= -2αλ_*^β∫_0^1h^2 (φ_*^β)^2<0whenever ∫ h^2>0, since _*^β is constant and m_*^β=0 on (x_0,y_0). It follows that for a given admissible perturbation h as above, we have (m_*^β+ε h)<(m_*^β) provided that ε>0 is small enough.We have reached a contradiction, which implies that x_i=y_i, i=0,1. We have then proved the following lemma.Every solution m_*^β of the optimal design problem (<ref>) is bang-bang, in other words equal to -1 or κ a.e. in (0,1). Proof of Theorem <ref> We end this section with providing the proof of Theorem <ref>. Assume that 0≤α < min{1/2,α̅} and consider a solution m_*^β of the optimal design problem (<ref>). Introduce the principal eigenfunction _*^β associated with m_*^β, normalized in such a way that ∫_0^1 m_*^β e^α m_*^β(_*^β)^2 = 1.By Lemmas <ref> and <ref>, the function m_*^β is unimodal and bang-bang. On can easily construct a sequence of smooth functions m_k insuch that m_k converges a.e. to m_*^β in (0,1). The dominated convergence theorem yields that ∫_0^1 m_k e^α m_k(_*^β)^2 →∫_0^1 m_*^β e^α m_*^β(_*^β)^2=1 as k→∞. Hence, the following inequality holds when k is large enoughλ_1^β(m_k) ≤_m_k^β[_*^β]k→∞⟶_m_*^β^β[_*^β] = λ_1^β(m_*^β),by dominated convergence. We deduce that lim supλ_1^β(m_k) ≤λ_1^β(m_*^β), which yields that λ_1^β(m_k) →λ_1^β(m_*^β) as k→∞ and proves (<ref>). As a consequence, Lemma <ref> implies that λ_1^β does not reach its infimum over ∩ C^2(Ω).§.§ Conclusion: end of the proofAccording to Lemmas <ref> and <ref>, any minimizer m_*^β for the optimal design problem (<ref>) is unimodal and bang-bang. We then infer that it remains to investigate the case where the admissible design m writesm_*^β=(κ+1)_I-1, where I is a subinterval of (0,1) whose length is δ = 1-m_0/κ+1 with m_0≤ -m_0 (note that -m_0 plays the role of the optimal amount of resources ∫_0^1m^* for m^* solving Problem (<ref>)). This is the main goal of this section.For that purpose, let us introduce the optimal design probleminf{λ_1^β (m),m=(κ+1)_(ξ,ξ+δ)-1,ξ∈ [0, (1-δ)/2]}. In the formulation of the problem above, we used an easy symmetry argument allowing to reduce the search of ξ to the interval [0, (1-δ)/2] instead of [0,1-δ]. The following propositions conclude the proof of Theorems <ref>, <ref> and <ref>. Their proofs are given respectively in Appendices <ref> and <ref> below. Let κ>0, β≥ 0, α∈ [0,α̅), m_0∈ [m_0,1) and δ be defined as above. The optimal design problem (<ref>) has a solution. Moreover, * if β <β_α,δ, then m=(κ+1)_(0,δ)-1 and m=(κ+1)_(1-δ,1)-1 are the only solutions of Problem (<ref>),* if β >β_α,δ, then m=(κ+1)_((1-δ)/2,(1+δ)/2)-1 is the only solution of Problem (<ref>),* if β =β_α,δ, then every function m=(κ+1)_(ξ,ξ+δ)-1 with ξ∈ [0,1-δ] solves Problem (<ref>). Under the assumptions of Proposition <ref>, if one assumes moreover that (<ref>) holds true in the case β≥β_α,δ, then one has ∫_0^1 m_*^β=-m_0. § PERSPECTIVES The same issues as those investigated in this work remain relevant in the multi-dimensional case, from the biological as well as the mathematical point of view. Indeed, the same considerations as in Section <ref> lead to investigate the problem inf_m∈λ_1^β(m)withλ_1^β (m) = inf_∈∫_Ω e^α m|∇|^2 + β∫_∂Ω^2/∫_Ω m e^α m^2.Such a problem needs a very careful analysis. It is likely that such analysis will strongly differ from the one led in this article. Indeed, we claim that except maybe for some particular sets Ω enjoying symmetry properties, we cannot use directly the same kind of rearrangement/symmetrization techniques. Furthermore, the change of variable introduced in Section <ref> is proper to the study of Sturm-Liouville equations. We used it to characterize persistence properties of the diffusive logistic equation with an advection term and to exploit the first and second order optimality conditions of the optimal design problem above, but such a rewriting has a priori no equivalent in higher dimensions.We plan to investigate the following issues: * (biological model) existence, simplicity of a principal eigenvalue for weights m in the class , without additional regularity assumption;* (biological model) time asymptotic behavior of the solution of the logistic diffusive equation with an advection term, and characterization of the alternatives in terms of the principal eigenvalue;* (optimal design problem) existence and bang-bang properties of minimizers;* (optimal design problem) development of a numerical approach to compute the minimizers.It is notable that, in the case where α=0, several theoretical and numerical results gathered in <cit.> suggest that properties of optimal shapes, whenever they exist, strongly depend on the value of m_0.Another interesting issue (relevant as well in the one and multi-D models) concerns the sharpness of the smallness assumptions on α made in Theorems <ref>, <ref> and <ref>. From these results, one is driven to wonder whether this assumption can be relaxed or even removed.§ SKETCH OF THE PROOF OF PROPOSITION <REF> In this appendix, we briefly sketch the proof of Proposition <ref>, for sake of completeness. The proof follows a method proposed by Hess and Kato in <cit.> (see also <cit.>).We start by considering the eigenvalue problem{[ -(e^α m∇)-λ m e^α m = μ in Ω,; B = 0 on ∂Ω , ].where λ is a real number, and B is defined by B= in the case of Dirichlet boundary conditions, and B=e^α m∂_n+ β in the case of Neumann or Robin conditions. A standard application of Krein-Rutman theory implies that the eigenvalue problem (<ref>) has a unique principal eigenvalue μ(λ). The eigenvalue μ(λ) is simple, and it is the smallest eigenvalue of Problem (<ref>) (see for example <cit.>). As a consequence, λ is a principal eigenvalue of Problem (<ref>) if and only if μ(λ)=0.It is also known that the principal eigenvalue μ(λ) can be characterized byμ(λ) = inf {∫_Ω e^α m|∇|^2 - λ∫_Ω me^α m^2, ∈H^1_0(Ω), ∫_Ω^2 = 1 }in the case of Dirichlet boundary conditions, and byμ(λ) = inf {∫_Ω e^α m|∇|^2 + β∫_∂Ω^2 - λ∫_Ω me^α m^2, ∈H^1(Ω), ∫_Ω^2 = 1}in the case of Neumann or Robin conditions.Notice that since the function λ↦μ(λ) is defined as an infimum of affine then concave functions of λ, it is itself concave. Moreover, considering well-chosen test functions in the Rayleigh quotient, we see that μ(λ)→ -∞ as |λ|→∞. Indeed, the assumption ∫_Ω m<0 ensures that there are admissible test functions _1 and _2 such that ∫_Ω m e^α m_1^2 >0 and ∫_Ω m e^α m_2^2 < 0.If the boundary conditions are of Dirichlet type, or of Robin type with β≠0, then it is obvious that μ(0)>0. Therefore, the function λ↦μ(λ) has exactly two zeros: one positive and one negative. As a consequence Problem (<ref>) has a unique positive principal eigenvalue.In the case of Neumann boundary conditions, that is when β=0, it is clear that μ(0)=0. Moreover, differentiating m with respect to λ yields that μ'(λ) = -(∫_Ω m e^α m v^2) / (∫_Ω v^2), where v is any eigenfunction associated with the eigenvalue μ(λ). As a consequence, μ'(0)= -1/|Ω|∫_Ω m e^α m, and we deduce that: * if ∫_Ω m e^α m<0, then there exists a unique positive principal eigenvalue;* if ∫_Ω m e^α m≥ 0, then 0 is the only non-negative principal eigenvalue. § OPTIMAL LOCATION OF AN INTERVAL (PROOF OF PROPOSITION <REF>)This section is devoted to the proof of Proposition <ref>. For that purpose, let us assume that m=(κ+1)_(ξ,ξ+δ)-1 with δ=1-m_0/κ+1. Notice that δ is chosen in such a way that ∫_0^1m=-m_0 and that one has necessarily ξ∈ [0,1-δ]. In what follows, we will restrict the range of values for ξ to the interval [0,(1-δ)/2] by noting that λ_1^β (m)=λ_1^β (m̂), with m̂= (κ+1)_(1-ξ-δ,1-ξ)-1.Step 1. Explicit solution of System (<ref>) Assume temporarily that ξ>0.In that case, according to standard arguments of variational analysis, System (<ref>) becomes{[ -”=-λ in (0,ξ),; -”=λκ in (ξ,ξ+δ),; -”=-λ in (ξ+δ,1),; (ξ^-)=(ξ^+), ((ξ+δ)^-)=(ξ+δ)^+) ,; e^-α'(0)=β(0),e^-α'(1)=-β(1). ].completed by the following jump conditions on the derivative of e^α(κ+1)'(ξ^+)='(ξ^-), and'((ξ+δ)^+)=e^α(κ+1)'((ξ+δ)^-).According to (<ref>), there exists a pair (A,B)∈^2 such that(x)={[ A√(λ)cosh(√(λ) x)+β e^αsinh(√(λ) x)/√(λ)cosh(√(λ)ξ)+β e^αsinh(√(λ)ξ)in (0,ξ),;Ccos (√(λκ)x)+Dsin (√(λκ)x)in (ξ,ξ+δ),; B√(λ)cosh(√(λ) (x-1))-β e^αsinh(√(λ) (x-1))/√(λ)cosh(√(λ)(ξ+δ-1))-β e^αsinh(√(λ)(ξ+δ-1)) in (ξ+δ,1) , ].where the expression of the constants C and D with respect to A and B is determined by using the continuity ofat x=ξ and x=ξ+δ, namelyC = Asin (√(λκ)(ξ+δ))-Bsin (√(λκ)ξ )/sin(√(λκ)δ), D= -Acos (√(λκ)(ξ+δ))-Bcos (√(λκ)ξ )/sin(√(λκ)δ) .Plugging (<ref>) into (<ref>), the jump condition (<ref>) rewrites M[ A; B ]=[ 0; 0 ]with M=[ m_11 m_12; m_21 m_22 ],wherem_11=√(κ)e^α(κ+1)(√(λ)cosh(√(λ)ξ)+β e^αsinh(√(λ)ξ))cos(√(λκ)δ)+(√(λ)sinh(√(λ)ξ)+β e^αcosh(√(λ)ξ))sin(√(λκ)δ) , m_12= - √(κ)e^α(κ+1) (√(λ)cosh(√(λ)ξ)+β e^αsinh(√(λ)ξ)) , m_21= -√(κ)e^α(κ+1)(√(λ)cosh(√(λ)(ξ+δ-1))-β e^αsinh(√(λ)(ξ+δ-1))) , m_22=√(κ)e^α(κ+1)(√(λ)cosh(√(λ)(ξ+δ-1))-β e^αsinh(√(λ)(ξ+δ-1)))cos (√(λκ)δ) - (√(λ)sinh(√(λ)(ξ+δ-1))-β e^αcosh(√(λ)(ξ+δ-1)))sin (√(λκ)δ) .Step 2. A transcendental equation Since the pair (A,B) is necessarily nontrivial (else, the functionwould vanish identically which is impossible by definition of an eigenfunction), one has necessarilyM=m_11m_22-m_12m_21=0. This allows to obtain the so-called transcendental equation. After lengthly computations, this equation can be recast in the simpler formsin (√(λκ)δ)F_α(ξ,β,λ)=0,where F_α(ξ,β,λ)= -F_α^s(ξ,β,λ)sin(√(λκ)δ) +√(κ)e^α(κ+1)F^c_α(β,λ)cos (√(λκ)δ)withF^s_α(ξ,β,λ) = β e^α√(λ)(κ e^2 α(κ+1)-1) sinh (√(λ)(1-δ))+1/2(1+κ e^2α (1+κ))(λ-β^2e^2α)cosh(√(λ)(1-2ξ-δ))+1/2(κ e^2α (1+κ)-1)(β^2e^2α+λ)cosh (√(λ)(1-δ)),F^c_α(β,λ)=(λ+β^2e^2α)sinh(√(λ)(1-δ))+2β√(λ)e^αcosh(√(λ)(1-δ)) . In the sequel, we will denote by λ^β_* (resp. m^β_*) the minimal value for Problem (<ref>) (resp. a minimizer), i.e.λ^β_*=λ_1^β (m^β_*)=inf{λ_1^β (m),m=(κ+1)_(ξ,ξ+δ)-1,ξ∈ [0, (1-δ)/2]}.The existence of such a pair follows from the continuity of [0,1-δ]∋ξ↦λ^β_1((κ+1)_(ξ,ξ+δ)-1) combined with the compactness of [0,(1-δ)/2].In the Dirichlet case (corresponding formally to take β=+∞) and for the particular choice ξ=0, the transcendental equation rewritestan(√(λκ)δ)=-√(κ)e^α(κ+1)tanh(√(λ)(1-δ)).It is then easy to prove that the first positive root of this equation λ_D,0 is such that √(λ_D,0)∈ (π/(2√(κ)δ)),π/(√(κ)δ))). We thus infer thatinf_ξ∈ [0, (1-δ)/2]λ_1^β ((κ+1)_(ξ,ξ+δ)-1)≤inf_ξ∈ [0, (1-δ)/2]lim_β→ +∞λ_1^β ((κ+1)_(ξ,ξ+δ)-1) <π^2/κδ^2,by noting that the mapping _+∋β↦λ_1^β (m) is non-decreasing. Indeed, this monotonicity property follows from the fact that λ_1^β (m) writes as the infimum of affine functions that are increasing with respect to β. As a consequence, there holds sin(√(λ^β_*κ)δ)> 0.According to Remark <ref>, one can restrict the study to the parameters λ such that (<ref>) holds true and in particular sin (√(λκ)δ)≠ 0. Hence, the transcendental equation (<ref>) simplifies intoF_α (ξ,β,λ)=0.A standard application of the implicit functions theorem using the simplicity of the principal eigenvalue yields that the mapping [0,(1-δ)/2]∋ξ↦λ_1^β ((κ+1)_(ξ,ξ+δ)-1) is differentiable, and in particular, so is the mapping [0,(1-δ)/2]∋ξ↦λ^β_*. Let ξ^* denote the optimal number ξ minimizing [0,(1-δ)/2]∋ξ↦λ_1^β ((κ+1)_(ξ,ξ+δ)-1).Step 3. Differentiation of the transcendental equation Let us assume that ξ^*≠ 0. Then, we claim that .∂λ_1^β/∂ξ|_ξ=ξ^*=0. This claim follows immediately from the necessary first order optimality conditions if ξ^*∈ (0,(1-δ)/2). If ξ^*=(1-δ)/2, this is still true by using the symmetry property (<ref>) enjoyed by λ_1^β. Therefore, assuming that ξ^*≠ 0, it follows that0=.∂λ_1^β/∂ξ|_ξ=ξ^*∂ F_α/∂λ(ξ^*,β,λ^β_*)+∂ F_α/∂ξ(ξ^*,β,λ^β_*)=∂ F_α/∂ξ(ξ^*,β,λ^β_*)by differentiating (<ref>) with respect to ξ. Let us compute ∂ F_α/∂ξ(ξ^*,β,λ^β_*). According to (<ref>), one has0=∂ F_α/∂ξ(ξ^*,β,λ^β_*) =∂ F^s_α/∂ξ(ξ^*,β,λ^β_*)sin(√(λκ)δ)= -√(λ^β_*)(1+κ e^2α (1+κ))(λ^β_*-β^2e^2α)sinh(√(λ^β_*)(1-2ξ^*-δ))sin(√(λ^β_* κ)δ) .Since sin(√(λ_*^βκ)δ)> 0 and 1-δ-2ξ^*≥ 0, one has either sinh(√(λ^β_*)(1-2ξ^*-δ))=0, which yields ξ^*=1-δ/2, or λ^β_*=β^2e^2α.The next result is devoted to the investigation of the equality λ^β_*=β^2e^2α.The equality (<ref>) holds true if, and only if β=β_α,δ, where β_α,δ is defined by (<ref>). Moreover, if β <β_α,δ, thenλ^β_*>β^2e^2α whereas if β >β_α,δ, then λ^β_*<β^2e^2α. First notice thatλ_1^β (m_*^β)/β^2 = 1/βmin_m∈min_∈{∫_Ω e^α m|∇|^2/β∫_Ω m e^α m^2+∫_∂Ω^2/∫_Ω m e^α m^2}.where = {∈H^1(Ω), ∫_Ω m e^α m^2>0} (see Eq. (<ref>)). For φ∈ and m∈, the function _+^* ∋β↦∫_Ω e^α m|∇|^2/β∫_Ω m e^α m^2+∫_∂Ω^2/∫_Ω m e^α m^2 is non-increasing, and therefore, so is the function _+^* ∋β↦min_m∈min_∈∫_Ω e^α m|∇|^2/β∫_Ω m e^α m^2+∫_∂Ω^2/∫_Ω m e^α m^2. As a product of a non-increasing and a decreasing positive functions, we infer that the mapping _+∋β↦λ_1^β (m_*^β)/β^2 is decreasing, by using (<ref>). Notice also that its range is (0,+∞).Then, since Eq. (<ref>) also rewrites λ_1^β (m_*^β)/β^2 = e^2α, it has a unique solution β_α,δ in _+^*. Let us compute β_α,δ. One hasF^s_α(ξ,β_α,δ,λ_*^β) = λ_*^β (κ e^2α(κ+1)-1) (sinh (√(λ_*^β)(1-δ))+cosh (√(λ_*^β)(1-δ)))F^c_α(β_α,δ,λ_*^β)=2λ_*^β(sinh(√(λ_*^β)(1-δ))+cosh(√(λ_*^β)(1-δ)))for every ξ∈ [0,(1-δ)/2]. It is notable that the previous quantities do not depend on ξ.By plugging (<ref>) into the transcendental equation (<ref>), one gets thatβ_α,δ satisfies (κ e^2α(κ+1)-1) sin(√(κλ_*^β)δ)=2√(κ)e^α (κ+1)cos(√(κλ_*^β)δ) and in particulartan(√(κλ_*^β)δ)=2√(κ)e^α (κ+1)/κ e^2α(κ+1)-1,whenever κ e^2α (κ+1)≠ 1.The expected result hence follows easilyfrom the uniqueness of β_α,δ and the fact that _+∋β↦λ_1^β (m_*^β)/β^2 is decreasing. Step 4. Conclusion of the proof We thus infer that, except if β=β_α,δ one has the following alternative: either ξ^*=0 or ξ^*=1-δ/2.In order to compute ξ^*, we will compare the real numbers F_α(0,β,λ) and F_α((1-δ)/2,β,λ). Let us introduce the function Δ_α,β defined by Δ_α,β(λ) =F_α(0,β,λ)-F_α((1-δ)/2,β,λ)/sin(√(λκ)δ).One computes Δ_α,β(λ)= -1/2(λ-β^2e^2α)(κ e^2α(κ+1)+1) (cosh(√(λ)(1-δ) )-1) according to (<ref>). According to Lemma <ref>, one infers that * if β <β_α,δ, then λ_*^β >β^2e^2α and Δ_α,β(λ_*^β)<0,* if β >β_α,δ, then λ_*^β <β^2e^2α and Δ_α,β(λ_*^β)>0. Since λ_*^β is the first positive zero of the transcendental equation for the parameter choice ξ=ξ^*, we need to know the sign of F_α(0,β,λ) and F_α((1-δ)/2,β,λ) on the interval [0,λ_*^β] to determine which function between F_α(0,β,·) and F_α((1-δ)/2,β,·) vanishes at λ_*^β. For that purpose, we will compute the quantity ∂ F_α(ξ,β,λ)/∂√(λ) at λ=0. According to (<ref>), one has. ∂ F_α(ξ,β,λ)/∂√(λ)|_λ=0 = √(κ)δ(- F_α^s (ξ,β,λ) cos (√(λκ)δ)+ e^α (κ+1).∂ F^c_α(ξ,β,λ)/∂√(λ)cos (√(λκ)δ))|_λ=0 =√(κ)δβ^2e^2α +√(κ)δ e^α(κ+2)β ( β e^α(1-δ) +2 )>0. As a result, since F_α(ξ,β,0)=0 for every ξ∈ [0,(1-δ)/2], the functions F_α(0,β,·) and F_α((1-δ)/2,β,·) are both positive on (0,λ_*^β) and according to the discussion on the sign of Δ_α,β(λ_*^β) above, we infer that–0mm 0mm* if β <β_α,δ, then Δ_α,β(λ_*^β)<0 and ξ^*=0, * if β >β_α,δ, then Δ_α,β(λ_*^β)>0 and ξ^*=(1-δ)/2.§ PROOF OF PROPOSITION <REF> In this section, we give the proof of Proposition <ref>. For this purpose, let m_*^β be a solution of Problem (<ref>), and assume by contradiction that ∫_0^1 m_*^β < -m_0. Note that, as a consequence, the first order optimality conditions imply that η^*=0, and ψ_0(x)≥ 0 for every x∈{m_*^β=-1},where we use the same notations as those of Section <ref>.We first assume that β≤β_α,δ. As a consequence, Lemma <ref> implies that λ_*^β≥β^2 e^2α. By Lemmas <ref> and <ref>, we know that m_*^β is bang-bang, and a neighborhood of either 0 or 1 lies in {m_*^β=-1}. Assume that the former is true, and observe that since '(0)=β e^α(0), one has ψ_0(0) = (αβ^2e^2α -λ_*^β(1-α))(0)^2. The assumption α<1/2 implies that αβ^2e^2α -λ_*^β(1-α) ≤λ_*^β(2α-1)<0. As a consequence, (<ref>) yields that (0)=0, and therefore '(0)=0. Sincesatisfies ”=λ_*^β in a neighborhood of 0,we deduce that =0 in this neighborhood, which is a contradiction. The case when 1∈{m_*^β=-1} is similar.We now assume that α<α_0 and β_α,δ < β < ∞. Theorem <ref> implies that m_*^β writes m_*^β=(κ+1)_(ξ,1-ξ)-1 for some ξ∈ (ξ^*,1/2], since ∫_0^1 m_*^β < -m_0. Observe that ifis an eigenfunction of (<ref>) associated to λ_*^β, then ”=λ_*^β on (0,ξ), and '(0)=β e^α(0). As a consequence, for some constant A> 0 and for x∈ (0,ξ),one has (x) = A(√(λ_*^β)cosh(√(λ_*^β)x ) + β e^αsinh(√(λ_*^β)x )). An easy computation shows that for every x∈ (0,ξ),ψ_0(x) =λ_*^β A^2 ( (2α -1)((λ_*^β+β^2e^2α) sinh^2 (√(λ_*^β)x ) .. . .+2β√(λ_*^β) e^αcosh(√(λ_*^β)x ) sinh(√(λ_*^β)x ) ) . + αβ^2e^2α -(1-α)λ_*^β).We aim at proving that ψ_0(ξ^*)<0, which is in contradiction with (<ref>). Noting that the terms λ_*^β, 2β√(λ_*^β) e^αcosh(√(λ_*^β)ξ^* ) sinh(√(λ_*^β)ξ^* ) and (1-α)λ_*^β are all non-negative, we deduce from (<ref>) that it is enough to prove thatα < (1-2α) sinh^2 (√(λ_*^β)ξ^* ). In the following, we note λ_1,α^β(m) (resp. λ_*,α^β), instead of λ_1^β(m) (resp. λ_*^β), in order to emphasize the dependency on α.Since β↦λ_1,α^β(m_*^β) is non-decreasing (see Appendix <ref>), one hasλ_*,α^β = λ_1,α^β(m_*^β)≥ λ_1,α^β_α^* (m_*^β) ≥ λ_*,α^β_α^*. Note that it is easily proved that the function α↦β_α^* is decreasing.Consequently, one has β_α^*>β_1/2^*, and therefore λ_*,α^β_α^*≥λ_*,α^β_1/2^*. Moreover, Lemma <ref> yields that λ_*,α^β_1/2^* > (β_1/2^*)^2 e^2α≥(β_1/2^*)^2. Combining this last inequality with (<ref>), we obtain that √(λ_*,α^β)≥β_1/2^*. As a consequence, since ξ^*< ξ, one has α < α_0 < sinh^2(√(λ_*,α^β)ξ)/1+2sinh^2(√(λ_*,α^β)ξ), which yields (<ref>) and achieves the proof of the result when β>β_α^*.We are left with dealing with the case β=∞. Observe that in this case the function ψ_0 takes the simpler form ψ_0(x) =λ_*^∞ A^2 ( (2α -1) sinh^2 (√(λ_*^∞)x ) +α). As a consequence, the assumption α<α_0 still implies that ψ_0(ξ^*)<0, and the previous reasoning holds, which concludes the proof.abbrv | http://arxiv.org/abs/1704.08016v2 | {
"authors": [
"Fabien Caubet",
"Thibaut Deheuvels",
"Yannick Privat"
],
"categories": [
"math.AP",
"math.OC"
],
"primary_category": "math.AP",
"published": "20170426084957",
"title": "Optimal location of resources for biased movement of species: the 1D case"
} |
definitionDefinition proposition[definition]Proposition lemma[definition]Lemma algorithm[definition]Algorithm fact[definition]Fact theorem[definition]Theorem corollary[definition]Corollary conjecture[definition]Conjecture postulate[definition]Postulate axiom[definition]Axiom remark[definition]Remark example[definition]Example question[definition]Question⊓⊔ 501em=0pt=0501em=0pt=0 .6exto 0pt -.23ex"16D<@>leostyle ifundefinedselectfontleo αβ̱γδ̣ϵεζηθ κ̨łλμνξπρ̊στῠφχ̧ψøωΓΔΘŁΛΞΠΣΥΦΨØΩ A B C D E F G H I J K L M N O P Q R S T U V W X Z A B C D E F G H I J K L M N O P Q R S T U V W X ZÂB̂ĈD̂ÊF̂ĜĤÎĴK̂L̂M̂N̂ÔP̂R̂ŜT̂ÛV̂ŴX̂Ẑ CDCD_RSCDSCD_ROCDSOCD rkrk_Rsrksrk_R rrkrsrk grktrkarkbrksbrk orksork birankCMI Co CPSCSSCSDdiag Dim EPR EVGHZ GHZg GH GZ HZ I iso span LocLOCC LUmax maxmin min mspec per PPT prPROrank SDSEP SLOCC sr osrsupp Tr W werner spX_X_^̊X_X_^̊⊗†∅←⇐↔⇔⊕⊗→⇒⊂⊆∖∧.6ex to 0pt -.23ex"16DJoint Center for Quantum Information and Computer Science, University of Maryland, College Park, Maryland, USA Department of Physics, Florida Atlantic University, FL 33431, USA Institut für Quantengravitation, Universität Erlangen-Nürnberg, Staudtstr. 7/B2, 91058 Erlangen, GermanyAuthor to whom correspondence should be addressed: [email protected] Department of Physics, Tsinghua University, Beijing, People's Republic of China Collaborative Innovation Center of Quantum Matter, Beijing 100190, People's Republic of China Department of Mathematics & Statistics, University of Guelph, Guelph, Ontario, Canada Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, CanadaPerimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada Let V=⊗_k=1^N V_k be an N-particle Hilbert space, whose individual single-particle space is the one with spin j and dimension d=2j+1.Let V_(w) be the subspace of V with constant weight w, consisting of vectors whose total spins are w. We show that the combinatorial properties of the constant weight condition impose strong constraints on the reduced density matrices for any vector |ψ⟩ in the constant weight subspace V_(w), which limit the possibility of the entanglement structures of |ψ⟩. Our results find applications in the overlapping quantum marginal problem, quantum error-correcting codes, and the spin-network structures in quantum gravity. Local density matrices of many-body statesin the constant weight subspaces Jie Zhou December 30, 2023 ============================================================================addtoresetequationsection§ INTRODUCTION Consider a system of N particles, each with spin j and dimension d=2j+1. The Hilbert space of the system is then V=⊗_k=1^N V_k, where all of the V_k's are identical. For any vector ψ∈V, its entanglement structure can be analyzed by the reduced density matrices (RDMs) of ψ⊗ψ^*∈ V⊗ V^* <cit.>. There are, however, always restrictions on these RDMs, given by, e.g. the principle of entanglement monogamy <cit.>. In a more general setting, the quantum marginal problem considers the consistency of a set of given local density matrices. This problem turns out to be a hard problem even with the existence of quantum computers. On the other hand, generic states are always highly entangled, in the large-j limit <cit.>.Besides those general analyses, there are also physical considerations that may restrict the form of ψ, hence the entanglement structures of ψ. For instance, ground states of local Hamiltonians would satisfy the entanglement area law, hence may be well-approximated by the tensor network representation <cit.>. States with special symmetry are also discussed such as the Dicke states and their generalizations <cit.>. Stabilizer/graph states are considered in the scenario of quantum error correction and one-way quantum computing <cit.>. Very recently, states that may be represented by (restricted) Boltzman machine are considered to apply machine learning techniques to study many-body ground states <cit.>.In this work, we consider the restriction to constant weight subspaces. A state ψ in V possesses a constant weight w, if it lies in the subspace which has an orthonormal basis {|m_1,…,m_N⟩} satisfying ∑_i m_i=w. In the spin language, the state has a fixed J_z-component of the total spin. These subspaces arise naturally as decoherence free subspaces under collective dephasing of the system <cit.>– that is, since each qubit gets a phase factor that only depends on its own weight, any constant weight state obtains a global phase for the collective dephasing. In this sense, a constant weight state is invariant under the collective dephasing noise, hence is decoherence free. Also, constant weight states are discussed in many other contexts, such as the atomic Dicke states and its generalizations. When w=0, this subspace contains the invariant subspace of zero angular momentum, which is widely discussed in loop quantum gravity. Despite that the constant-weight condition arise naturally in these many places, the entanglement structure of these spaces has not been studied systematically.We discuss the properties of the RDMs of constant weight states in a very general setting. We show that there exist linear conditions between the elements of RDMs, for any j, N and w, which can be written down explicitly. Our key idea is that the combinatorial properties given by the constant weight constraint, which is mathematically a partition of w, lead to such linear structures of the reduced density matrices. These conditions could find many applications. For instance, it implies that there is no perfect tensor in a constant weight subspace, for any j when N≥ 4, which is a concept that has recently received attention from understanding quantum gravity from the quantum information viewpoint <cit.>. Also, given the intimate connections between perfect tensors and quantum error-correcting codes, our results give restrictions on the achievable distance on constant-weight quantum codes. In practice, our conditions can also be used as good certificates for decoherence-free subspaces.We organize our paper as follows. In Sec. <ref>, we define our notions and provide background information on constant weight subspaces in the SU(2) case. In Sec. <ref> we discuss the combinatorial structure of the constant weight condition that leads to our main theorem on linear relations of RDMs. In Sec. <ref>, we discuss further the relationships between these linear structures. In Sec. <ref>, we discuss the application of our main results to the quantum marginal problem and the nonexistence of perfect tensors. In Sec. <ref>, we discuss the generalization to the SU(n) case. Finally, in Sec. <ref>, we discuss the generalization for relaxing the constant weight condition by introducing the notion of frequency matrix. § PRELIMINARIES According to the standard representation theory <cit.>, the study of representations of the Lie group SU(2) is essentially equivalent to that of the Lie algebra 𝔰𝔲(2). In this work, we shall use the language of the latter for simplicity. The Lie algebra 𝔰𝔲(2) is generated by the Chevalley-Serre basis {H,X,Y}, whose Lie algebra structure is given by[H,X]=2X ,[H,Y]=-2Y ,[X,Y]=H .They act on the standard representation ℂ^2 of 𝔰𝔲(2)by multiplication by the following matrices:H= [10;0 -1 ] ,X= [ 0 1; 0 0 ] ,Y= [ 0 0; 1 0 ] .These matrices are related to the Pauli matricesJ_z= [10;0 -1 ] ,J_x= [ 0 1; 1 0 ] ,J_y= [0 -i;i0 ]byX= 1/2(J_x+iJ_y)=J_+ , Y=1/2 (J_x-i J_y)=J_- ,H= J_z=J_3 . The finite dimensional irreducible representations of 𝔰𝔲(2) are classified by the dimension d∈ℤ_+. One has for each d∈ℤ_+ an irreducible representation Sym^⊗ 2jℂ^2 of dimension d=2j+1, where ℂ^2 is the standard representation. Each representation W of dimension D, not necessarily irreducible, can be decomposed into a direct sum of irreducible representations. Moreover, there exists a Hermitian metric ⟨-,- ⟩ on W such that the decomposition is orthogonal. We shall denote the dual of W with respect to the Hermitian metric by W^*, and the dual of a vector v∈ W by v^*.According to the weight decomposition of irreducible representations, one can then find an orthonormal basis of W ℬ={e_1,e_2,⋯, e_D}whose weights, the eigenvalues under the action of H, areα_1,α_2,⋯, α_D ,respectively. Note that here the weight is 2 times the usual notion of spin. See Figure. <ref> for an illustration. We label each of the vectors in the basis ℬ for W by its sub-index r∈{1,2,⋯, D}, and vice versa. We shall adapt this convenient convention throughout this work. We consider in this work the tensor productV=⊗_k=1^N V_k ,where all of the components V_k are identicalto some given representation, not necessarily irreducible, say W. In our following discussion, we shall consider the non-trivial case N≥2. A basis of V is then provided by ℬ^⊗ N, whose elements are indexed by the multi-indices I=(i_1,i_2,⋯, i_N) corresponding to the vectore_I:=e_i_1⊗ e_i_2⋯⊗ e_i_N .The weight of this vector is easily seen to beweight(I):=α_i_1+α_i_2+⋯α_i_N .Any vector ψ in V is then represented byψ=∑_e_I∈ℬ^⊗ N a_Ie_I ,a_I∈ℂ . We now discuss the notion of partial trace. Choose a subset of components Λ⊆{1,2,⋯ N} for V,and defineV_Λ=⊗_k∈Λ V_k ,V_Λ^c=⊗_k∉Λ V_k .The identity operator 𝕀_V_Λ∈End(V_Λ) is equivalently represented by a unique tensor Δ_V_Λ∈ V_Λ⊗V̌_Λ, or alternatively a unique tensor Δ̌_V_Λ∈V̌_Λ⊗ V _Λ, where the notationmeans the linear dual in the category of vector spaces. The explicit formula for Δ_V_Λ is displayed in (<ref>) below.The Hermitian metric gives an identification between the Hermitian dual V^* and the linear dual V̌. This identification will be assumed frequently in this work. By this identification, the element ψ⊗ψ^*∈ V⊗ V^* then determines an element in End (V)=V ⊗V̌. Thus one can contract it with the tensor Δ_V_Λ. This is equivalent to the following pairing using the Hermitian metric⟨Δ_V_Λ ,ψ⊗ψ^*⟩∈End (V_Λ^c) .As priori, ⟨Δ_V_Λ ,ψ⊗ψ^*⟩ is only an element in V_Λ^c⊗ V_Λ^c^*, but in (<ref>), we have used such identification to identify it with an element in End (V_Λ^c)=V_Λ^c⊗V̌_̌Λ̌^̌č. The partial trace of ψ⊗ψ^* over the vector space V_Λ is defined to beTr_V_Λ( ψ⊗ψ^* ):=⟨Δ_V_Λ ,ψ⊗ψ^*⟩ . The above definition can be applied to a general element in V⊗ V^* which is not necessarily of the form ψ⊗ψ^*.Writing I=(L; K), where K runs over the index set for the orthonormal basis ℬ^⊗ |Λ|={e_K} of V_Λ and L over that forV_Λ^c, we then haveΔ_V_Λ =∑_e_K∈ℬ^⊗ |Λ| e_K⊗ e_K^* .The notation |Λ| stands for the cardinality of the index set Λ.Hence (hereafter Θ_D :={1,2,⋯, D})Tr_V_Λ( ψ⊗ψ^* )= ∑_L, L'∑_K∈Θ_D^⊗ |Λ|a_(L;K) a^*_(L';K) e_L⊗ e_L'^*∈End (V_Λ^c) .Consider the D=2, N=4 case. Given a state |ψ⟩=∑_i_1,i_2,i_3,i_4 a_i_1,i_2,i_3,i_4|i_1,i_2,i_3,i_4⟩, taking partial trace over V_1⊗ V_4 can be calculated by_V_1⊗ V_4(|ψ⟩⟨ψ|) =∑_i_2,i_3,i_2',i_3'=1,2|i_2i_3⟩⟨i_2'i_3'|∑_i_1,i_4=1,2a_i_1i_2i_3i_4a^*_i_1i_2'i_3'i_4 . With respect to the basis we have chosen, Tr_V_Λ( ψ⊗ψ^* ) is naturally represented by its entries (Tr_V_Λ( ψ⊗ψ^* ))_L, L'= ∑_K∈Θ_D^⊗ |Λ|a_(L;K) a^*_(L';K) ,L, L'∈Θ_D^⊗|Λ^c| .The tensor ψ⊗ψ^*∈End(V_Λ^c⊗ V_Λ) has rank one, hence the dimension of its kernel is dim (V_Λ^c⊗ V_Λ) -1. Taking the partial trace over V_Λ would at most increase the rank of the resulting partial trace by dimV_Λ: forgetting about the component V_Λ in V_Λ^c⊗ V_Λ would at most decrease the dimension of the kernel by dimV_Λ. Therefore, the rank of Tr_V_Λ( ψ⊗ψ^* ) has an upper boundrank(Tr_V_Λ( ψ⊗ψ^* ))≤ 1+ dim V_Λ .In order that the partial trace, as an element in V_Λ^c⊗ V_Λ^c^*, has full rank, the following condition has to be metdim V_Λ^c≤1+ dim V_Λ .In the present case, all of the components are isomorphic representations. Hence the above condition reduces to|Λ|≥ [N 2] .Intuitively, one must contract enough components in order for the resulting partial trace to have possibly maximal rank.§ COMBINATORICS IN PARTIAL TRACE ON THE CONSTANT WEIGHT SUBSPACE We shall discuss in this section some combinatorial structure of partial trace and of the constant weight subspace, based on which we shall discuss some applications in Section <ref> . Among the entries in the partial trace (<ref>), of particular interest are the diagonal ones ρ_L^Λ^c := ∑_K∈Θ_D^⊗ |Λ||a_(L;K) |^2,L∈Θ_D^⊗|Λ^c| . We fix Λ^c={1,2,⋯, M, M+1} for some 0≤ M≤ N-1. Writing the index set L as (I_0; i_M+1), where I_0=(i_1, i_2,⋯,i_M) is a multi-index and, 1 ≤ i_k ≤ D, k=1,2,⋯ M, M+1. Then the diagonal pieces of the partial trace over V_Λ are represented by the entriesρ_(I_0;i_M+1 )^Λ^c = ∑_K∈Θ_D^⊗ |Λ| |a_(I_0;i_M+1;K) |^2, ∀(I_0;i_M+1)∈Θ_D^⊗|Λ^c| .An element K takes the form (i_M+2, ⋯, i_N). By moving the position of i_M+1 from M+1 to a position valued in the set {M+1,M+2,⋯, N} which symbolically is denoted by *, we get similarly the quantities. To be more precise, denoteΛ^c_*={1,2,⋯, M, *} , Λ_*= {1,2,⋯ N}-Λ^c_* ,then we haveρ_(I_0; i_*)^Λ^c_*={1,2,⋯, M, *}= ∑_(i_M+1,⋯, i_*, ⋯, i_N)∈Θ_D^⊗ |Λ_*| |a_(I_0; i_M+1,⋯, i_*, ⋯, i_N)|^2, ∀(I_0;i_*)∈Θ_D^⊗|Λ_*^c| .Here the notation i_* means the index i_* is omitted. For simplicity, we denote K_*=(i_M+1,⋯, i_*, ⋯, i_N) and K'=( i_M+1,⋯, i_*, ⋯, i_N). Since the sub-index k of i_k already contains the information i_k∈ V_k, we can denote for convenience K'=(i_*; K_*).Later we shall consider ∑_*∈{M+1,M+2,⋯, N}∑_i_*=1 e_i_*∈ V_*^Dρ_(I_0;i_* )^{1,2,⋯ , M, *} = ∑_*∈{M+1,M+2,⋯, N}∑_i_*=1 e_i_*∈ V_*^D∑_K_*∈Θ_D^⊗ |Λ_*| |a_(I_0; i_*; K_*) |^2,where e_i_*∈ V_* indicates that the index i_* labels different basis vectors in V_*. §.§ Combinatorial identity in the constant weight subspace After restricting to the constant weight subspace V_(w) of V consisting of vectors of weight w, an index K' that appear in the sum (<ref>) corresponds to a (N-M)-tuple x=(x_M+1, ⋯ ,x_*, ⋯, x_N), where x_k is the weight of the vector labeled by i_k for k=M+1,⋯, *, ⋯ , N. The constant weight condition is translated to the condition that x is a solution to the equation∑_k=M+1^Nx_k=-weight(I_0)+w ,x_k∈{α_1,α_2,⋯ ,α_D} ,where the subindex k of x_k labels different components V_k, and the subindex r of α_r labels different basis vectors in each V_k. We denote the latter set of solutions by 𝒳.Note that different elements in 𝒳 can correspond to the same partition: permuting an (N-M)-tuple can give a different (N-M)-tuple but they correspond to the same partition. After modulo the action by the symmetry group 𝔖_N-M, the set 𝒳/𝔖_N-M of cosets is then in one-to-one correspondence with the set of partitions (with all elements in a partition valued in {α_1,α_2,⋯ ,α_D}) of the following function S of weight(I_0) and w S=-weight(I_0)+w .More precisely, any element in the set 𝒳/𝔖_N-M, denoted by [x], is given by a partitionα_1· n_1([x])+α_2· n_2([x])+⋯ + α_D n_D([x]) .Here n_r([x]), r=1,2,⋯, D is the frequency of α_r appearing in the partition, which is independent of the choice of the representative of the coset [x]. They are subject to the conditions that∑_r=1^Dα_rn_r([x])=S , ∑_r=1^D n_r([x])=N-M . Therefore, we get the following result. There exists a nonzero solution {b_r}_r=1^D to the equation∑_r=1^Db_rn_r([x])=0 ,∀[x]∈𝒳/𝔖_N-M .An explicit one is given byb_r=α_r-S N-M ,r=1,2,⋯, D .A natural question is about the uniqueness of the solution. This will be addressed in Section <ref> below using the notion of frequency matrix.Take N=5, j=1. Then D=3 and α_r/2∈{-1,0,1} for r∈{1,2,3}. Consider the case M=1,w=0. Labeling the basis in V_k by spin, which is half of the eigenvalue of H acting on V_k, we then have the followingweight(I_0)/2 {[x]} (n_r([x])) b=(b_r)^t-1 [1,0,0,0][1,0,-1,1] [ 0 3 1; 1 1 2 ] [ -5/4; -1/4;3/4 ]0 [1,-1,1,-1][1,-1,0,0] [0,0,0,0] [ 2 0 2; 1 2 1; 0 4 0 ] [ -1/4;0; 1/ 4 ]1 [-1,0,-1,1][-1,0,0,0] [ 2 1 1; 1 3 0 ] [ -3/ 4;1/ 4; 5/4 ]We can now prove the following theorem. Consider the constant weight subspace V_(w) of V consisting of vectors of weight w. Fix an integer 1≤ M≤ N-1 and an index I_0∈Θ_D^⊗ M such that the set 𝒳 of solutions to (<ref>) is non-empty. Then there exists a nonzero solution {b_r}_r=1^D to the equation∑_r=1^D b_r∑_ *∈{M+1,M+2,⋯, N}e_r∈ V_*ρ^{1,2,⋯, M,*}_(I_0;r)=0 ,where e_r∈ V_* indicates that r labels different basis vectors in V_*. Straightforward computation as in (<ref>) shows that∑_ *∈{M+1,M+2,⋯, N}∑_ i_*=1e_i_*∈ V_*^D b_i_*ρ^{1,2,⋯, M,*}_(I_0;i_*) = ∑_*∈{M+1,M+2,⋯, N}∑_i_*=1 e_i_*∈ V_*^Db_i_*∑_K_*∈Θ_D^⊗ |Λ_*| |a_(I_0; i_*; K_*)|^2.After interchanging the order of summation on i_* and *, one would obtain∑_r=1^Db_r∑_*∈{M+1,M+2,⋯, N}e_r∈ V_*∑_K_*∈Θ_D^⊗ |Λ_*| |a_(I_0; r; K_*)|^2 = ∑_r=1^Db_r∑_*∈{M+1,M+2,⋯, N}e_r∈ V_*ρ^{1,2,⋯, M,*}_(I_0;r) .Now we restrict ourselves to the constant weight subspace and hence replace the multi-index (i_*;K_*) by a solution x∈𝒳 given in (<ref>). Then the LHS in (<ref>) gives∑_r=1^Db_r ∑_x ∈𝒳∑_ *∈{M+1,M+2,⋯,N}x_*= α_r |a_(I_0; x)|^2 .Theorem <ref> is then equivalent to the vanishing of (<ref>) above.We define Supp(x) to be the set of distinct values in the entries of x. This gives a function on the set 𝒳 of solutions. It is obvious that it is invariant under the action by the group 𝔖_N-M and hence descends to a function on the set 𝒳/𝔖_N-M of partitions. We then define Supp([x]) to be Supp(x) for any representation x of [x].Therefore,∑_r=1^Db_r ∑_x ∈𝒳∑_ *∈{M+1,M+2,⋯,N}x_*= α_r |a_(I_0; x)|^2= ∑_x ∈𝒳∑_r=1^Db_r∑_ *∈{M+1,M+2,⋯,N}x_*= α_r |a_(I_0; x)|^2= ∑_x∈𝒳∑_r: α_r∈Supp(x)b_r n_r(x) |a_(I_0; x)|^2= ∑_x∈𝒳/𝔖_N-M∑_x∈ [x]∑_r:α_r∈Supp([x])b_r n_r([x]) |a_(I_0; x)|^2= ∑_x∈𝒳/𝔖_N-M∑_ r:α_r∈Supp(x)b_r n_r([x])∑_x∈ [x] |a_(I_0; x)|^2= ∑_[x]∈𝒳/𝔖_N-M(∑_ r:α_r∈Supp([x])b_r n_r([x]) )(∑_x∈ [x]|a_(I_0; x)|^2 ).For any partition [x], if α_r∉Supp([x]), then n_r([x])=0 automatically. It follows that∑_[x]∈𝒳/𝔖_N-M(∑_ r: α_r∈Supp([x])b_r n_r([x]) )(∑_x∈ [x]|a_(I_0; x)|^2 ) = ∑_[x]∈𝒳/𝔖_N-M(∑_ r=1^Db_r n_r([x]) )(∑_x∈ [x]|a_(I_0; x)|^2 ).This is vanishing due to the equation ∑_r=1^Db_rn_r([x])=0 for any [x]∈𝒳/𝔖_N-M, as proved in Lemma <ref>. The above results exhibit only part of the combinatorial properties in partial trace. The actual combinatorial structure in partial trace is much richer. For example, the quantity considered in (<ref>) is closely related to Tr_V_*⊗ V_Λ(ψ⊗ψ^*) ,Λ∪{*}={M+1,M+2,⋯ N} ,whose (I_0,I_0)-diagonal entry is ∑_i_*=1^D∑_e_i_*∈ V_*ρ_(I_0;i_* )^{1,2,⋯ , M, *} .In particular, if we take M=1, then Tr_V_*⊗ V_Λ defines an element in End (V_1) and we have ∑_*∈{2,3,⋯, N}∑_i_*=1^D∑_e_i_*∈ V_*ρ_(I_0;i_* )^{1, *} =(N-1)·( Tr_V_*⊗ V_Λ(ψ⊗ψ^*))_(I_0,I_0) 。The summation of the above over I_0 gives the further trace over V_1. We therefore have∑_I_0∑_*∈{2,3,⋯, N}∑_i_*=1^D∑_e_i_*∈ V_*ρ_(I_0;r )^{1, *} =(N-1) ·Tr_V(ψ⊗ψ^*) .When combined with Theorem <ref>, this will be useful in theapplications discussed in Section <ref> below where we shall prove that the converse statement is also true. §.§ The relation between different M'<M For different choices of V_Λ, the patterns in the combinatorics of the partial trace shown in Lemma <ref> in fact only depends on the cardinality N-M-1 of Λ. For different values of M, Theorem <ref> givesdifferent sets of relations. We shall show in this section that the most informative one is the one with largest possible M subject to the condition M+1≤ [N 2] (the least possible number of components being traced out according to (<ref>)), the others are its consequences.Fixing M, consider another value M' such that M'<M. Our argument is by induction. Hence we shall assume for now that M'=M-1. We single out the component in M-M'. Assume it is the first component, by permutation or relabeling if necessary. Recall that the relations in Lemma <ref> is about the combinatorics of 𝒳/𝔖_N-M. We now show that the solution {b_r} given in (<ref>) implies the solution {b_r'}. We start with the fact that each of the partitions [x] satisfies∑ b_rn_r([x])=0 , ∑ n_r([x])=N-M , ∑α_r n_r([x])=S .Here the existence of {b_r} is guaranteedby induction hypothesis. Later we shall see that the solution given in (<ref>) is a natural one consistent with the induction procedure.The goal is then to prove the existence of {b_r'} such that the '-version of the above equations are satisfied, for any [x'].Choose a value s for the first component in the process of taking partial trace over the N-M' components. We can then classify x' into two sets: one involves s and the other one does not. For those not involving the specified s, it must involve some other value s̃. Then we apply the following same reasoning to [x']=[x]+s̃.If we can prove the result for any possible value of s, then by exhausting all the possible values for s, we are done with the checking for any [x'], as any x' must be of the form [x']=[x]+s for some s. Hence it suffices to consider those involving any fixed value s, for which we have [x']=[x]+s, [x]∈ 𝒳/𝔖_N-M, with∑ n_r([x'])=N-M'=N-M+1 , ∑α_r n_r([x'])=S' .We setb_r'=b_r+δ_r ,for some δ_r. We want it to depend only on the numbers S≤ S' being partitioned and M=M'+1 so that we can proceed by induction. Now we compute∑_r=1^D b_r' n_r([x'])= ∑_r=1^D b_r n_r([x]) +∑_r=1^Dδ_r n_r([x]) +b_s+δ_r ,= (N-M)δ_r +b_s+δ_r= b_s +(N-M+1)δ_r = b_s + (N-M')δ_r .We setb_r= α_r-S N-M ,∀r=1,2,⋯, D ,andδ_r=S N-M-S' N-M' , ∀r=1,2,⋯, D .This then does the job ∑_r=1^D b_r' n_r([x'])=0. In fact, from this one can see that δ_r is independent of r. Furthermore, one has from the above and(<ref>) thatb_r'= α_r-S' N-M' .Hence it keeps the pattern for b_r shown in (<ref>) unchanged. Therefore, one can proceed by induction. § APPLICATIONSTheorem <ref> is a strong structural condition on the partial trace Tr_V_Λ( ψ⊗ψ^* )∈End(V_Λ^c). One immediate application is for the overlapping quantum marginal problem when restricting to the constant weight subspace. For overlapping quantum marginal problem, very few results were known <cit.> and most of them can only be applied to small systems. To the best of our knowledge, no further conditions are known if we restrict the pre-image to lie in a given subspace.To make things precise, we first give the definitions of density operator and density matrix, which are the practical notions in talking about distributions in probability theory.Suppose E is a Hermitian vector space. A density operator ϱ is an element in End(E) satisfying*It is normalized in the sense that Tr ϱ=1.*It is a self-adjoint, positive definite operator.Fixing an orthonormal basis {f_L} for E, then the density operator ϱ is represented by a matrix (ϱ_LL' ) called the density matrix. The self-adjoint property translates into the property that the density matrix is Hermitian. We denote its diagonal entries byρ_L:=ϱ_LL . For example, for any unit norm vector v∈ E, the operator v⊗v^* gives a density operator. §.§ Quantum marginal problemThe quantum marginal problem is formulated in the following way. Consider the Hermitian vector space V=⊗_k=1^N V_k. For each subset {i,j}⊆{1,2,⋯, N}, defineΛ_ij^c:={i,j} , Λ_ij:= {1,2,⋯, N}-{i,j} .Given a collection of density operators {ϱ^Λ^c_ij}, consisting of one density operator(called two-body below) ϱ^Λ^c_ij for each Λ^c_ij, we want to ask whether there exists a density operator ϱ^{1,2,⋯, N}=ψ⊗ψ^* on V, supported on the subspace V_(w) of constant weight w, such that its partial trace over V_Λ_ij satisfies the following relationTr_V_Λ_ij ϱ^{1,2,⋯, N}=ϱ^Λ^c_ij . ∀ {i,j}⊆{1,2,⋯, N} ,When there exists such a ϱ^{1,2,⋯, N}, then ψ is a state in the constant weight subspace V_(w).If not, then there could be two possibilities:1. there does not exist ϱ^{1,2,⋯, N} at all, either on the constant weight subspace or not;2. there exist some global states but none of them is in a constant weight subspace. Our results of Theorem <ref> directly give necessary conditions for this problem. With respect to some given orthonormal basis of V of the form (<ref>), which is induced by those on the components V_k, the diagonal entries of ϱ^Λ^c_ij are given by {ρ^Λ^c_ij_L}_L ∈Θ_D^|Λ^c|. If (<ref>) is true, then these diagonal entries {ρ^Λ^c_ij_L}_L ∈Θ_D^|Λ^c| must coincide with the ones defined in (<ref>) in Section <ref>. Hence one must have, for the solution {b_r} given in (<ref>) (which in particular depends on I_0), the following relations provided in (<ref>):∑_r=1^D b_r∑_ *∈{M+1,M+2,⋯, N}e_r∈ V_*ρ^{1,2,⋯, M,*}_(I_0;r)=0 , ∀I_0 ,as well as the relations obtained by moving the index set {1,2,⋯, M} inside {1,2,⋯, N}. For instance, we can take M=1, our result then leads to a new necessary condition for the set of the density operators (ϱ^{1,2},ϱ^{1,3},⋯,ϱ^{1,N}) having a lift into a constant weight subspace:∑_r=1^D b_r ∑_*∈{2,⋯,N}e_r∈ V_*ρ_(I_0:r)^{1,*}=0 , ∀I_0 .These linear constraints cannot be obtained from the trivial conditionsTr_V_p(ϱ^{1,p})=Tr_V_q(ϱ^{1,q}) , ∀2≤ p< q≤ N . Here is another closely related problem. Assuming that {ϱ^Λ^c_ij} indeed descends from a density operator ϱ^{1,2,⋯ N}=ψ⊗ψ^*, we want to know to what extent we can know the property of ψ (e.g., deviation from being supported on a constant weight subspace) from the condition (<ref>) and its permutations.We now show that the necessary condition (<ref>) and its permutations provided by Theorem <ref> is actually sufficient, provided that the above assumption that {ϱ^Λ^c_ij} descends from a density operator ϱ^{1,2,⋯ N}=ψ⊗ψ^* is met. Note that the condition (<ref>) and its permutations are much weaker than the set of relations obtained by permuting (<ref>). To see this, we assume that (<ref>) and its permutations are met for a set of {b_r} given by (<ref>) for some w_0,b_r=α_r--weight(I_0)+w_0 N-1 ,r=1,2,⋯, D .Then we get∑_r=1^D (α_r--weight(I_0)+w_0 N-1)∑_*∈{2,⋯,N}e_r∈ V_*ρ_(I_0:r)^{1,*}=0 , ∀ I_0 . We first decompose ψ into a sum of its projections ψ_w to the constant weight subspaces V_(w) ψ=∑_w∑_e_I∈ V_(w) a_Ie_I :=∑_wψ_w , a_I∈ℂ .Then since different V_(w)'s are orthogonal, we haveTr_V_Λ_1* (ψ⊗ψ^*) = ∑_wTr_V_Λ_1* (ψ_w⊗ψ_w^*) .Denote the diagonal matrices of Tr_V_Λ_1* (ψ_w⊗ψ_w^*) by ρ_(I_0:r)^{1,*}(w). Then we have for the diagonal entries thatρ_(I_0:r)^{1,*}=∑_wρ_(I_0:r)^{1,*} (w) . Applying Theorem <ref> to each component ψ_w, we get∑_r=1^D (α_r--weight(I_0)+w N-1)∑_*∈{2,⋯,N}e_r∈ V_*ρ_(I_0:r)^{1,*} (w)=0 , ∀ I_0 .This gives∑_w∑_r=1^D (α_r--weight(I_0)+w N-1)∑_*∈{2,⋯,N}e_r∈ V_*ρ_(I_0:r)^{1,*} (w)=0 , ∀ I_0 . On the other hand, from (<ref>), (<ref>), we also have∑_r=1^D (α_r--weight(I_0)+w_0 N-1)∑_w∑_*∈{2,⋯,N}e_r∈ V_*ρ_(I_0:r)^{1,*} (w)=0 , ∀ I_0 .One can change the order of summation on r and w and get∑_w∑_r=1^D (α_r--weight(I_0)+w_0 N-1) ∑_*∈{2,⋯,N}e_r∈ V_*ρ_(I_0:r)^{1,*} (w)=0 , ∀ I_0 . Taking the difference between (<ref>) and (<ref>), we obtain∑_w∑_r=1^D w-w_0 N-1∑_*∈{2,⋯,N}e_r∈ V_*ρ_(I_0:r)^{1,*} (w)=0 , ∀ I_0 .Simplifying this relation a little further, we get∑_ww-w_0 N-1∑_*∈{2,⋯,N}(∑_i_*=1^D∑_e_i_*∈ V_*ρ_(I_0:i_*)^{1,*} (w))=0 , ∀ I_0 .The combinatorics in (<ref>) tells that ∑_i_*=1^D∑_e_i_*∈ V_*ρ_(I_0;i_* )^{1, *}(w) = ( Tr_V_*⊗ V_Λ(ψ_w⊗ψ_w^*))_(I_0,I_0) ,∑_*∈{2,⋯,N}∑_i_*=1^D∑_e_i_*∈ V_*ρ_(I_0;i_* )^{1, *}(w) =(N-1) ( Tr_V_{2,3,⋯, N}(ψ_w⊗ψ_w^*))_(I_0,I_0) .Then it follows from (<ref>) that∑_w(w-w_0) ( Tr_V_{2,3,⋯, N}(ψ_w⊗ψ_w^*))_(I_0,I_0) =0 , ∀ I_0 . In particular, a consequence of this says that the expectation value of the operator 𝐇=∑_kH_k on the density operator ϱ^{1,2,⋯,N} is the same as that of the constant operator w_0. Here H_k=Id⊗⋯⊗ H⊗⋯Id acts nontrivially on V_k only, hence 𝐇 must be diagonal since H is in the Cartan subalgebra.0 = ∑_I_0∑_w(w-w_0) ( Tr_V_{2,3,⋯, N}(ψ_w⊗ψ_w^*))_(I_0,I_0)= ∑_I_0∑_w w( Tr_V_{2,3,⋯, N}(ψ_w⊗ψ_w^*))_(I_0,I_0) -∑_I_0∑_w w_0( Tr_V_{2,3,⋯, N}(ψ_w⊗ψ_w^*))_(I_0,I_0)= ∑_ww∑_I_0( Tr_V_{2,3,⋯, N}(ψ_w⊗ψ_w^*))_(I_0,I_0) -w_0∑_w∑_I_0( Tr_V_{2,3,⋯, N}(ψ_w⊗ψ_w^*))_(I_0,I_0)= ∑_ww·Tr_V(ψ_w⊗ψ_w^*) -w_0∑_wTr_V(ψ_w⊗ψ_w^*)= Tr_V (𝐇ϱ^{1,2,⋯, N} ) -w_0·Tr_V( ϱ^{1,2,⋯, N} ).We remark that (<ref>) is in fact stronger than this. From (<ref>) we can see that Tr_V_{2,3,⋯, N}(ψ_w⊗ψ_w^*) only have diagonal terms: this is special as the leftover after the partial trace has only one component. Therefore, (<ref>) actually means that the following two are equivalent operators (contrasting (<ref>))∑_ww( Tr_V_{2,3,⋯, N}(ψ_w⊗ψ_w^*)) =∑_ww_0( Tr_V_{2,3,⋯, N}(ψ_w⊗ψ_w^*)).Taking further the partial trace over V_1 then yields (<ref>). We have shown in (<ref>) thatTr_V_{2,3,⋯, N}(𝐇-H_1)ψ⊗ψ^* +(H_1-w_0)Tr_V_{2,3,⋯, N} ψ⊗ψ^*=0 .Permuting the index from 1 to k givesTr_V_{1,2,3,⋯, N}-{k}(𝐇-H_k)ψ⊗ψ^* +(H_k-w_0)Tr_V_{1,2,3,⋯, N}-{k} ψ⊗ψ^*=0 , ∀ k=1,2,⋯ N .Multiplying this by a polynomial operator f(H_k), and summing over k, we then getTr_V ∑_k=1^N( (𝐇-w_0) f(H_k)) ψ⊗ψ^*=0 . Taking f(H_k)=w_0 givesTr_V (N 𝐇 w_0-N w_0^2) ψ⊗ψ^*=0 .Taking f(H_k)=H_k yieldsTr_V (𝐇^2-𝐇 w_0) ψ⊗ψ^*=0 .Combining the above two, we getTr_V (𝐇-w_0)^2 ψ⊗ψ^*= ∑_wTr_V_(w) (w-w_0)^2 ψ_w⊗ψ^*_w=0.This can be true only when ψ⊗ψ^* is supported on the constant weight subspace V_(w_0). Hence we have shown that one can determine whether a state is supported on a constant weight subspace by all of its two-body local information. The proof above also shows that the vanishing of fluctuation of H would give another necessary and sufficient condition to this problem. However, in the case leakage exists, our method gives a more practical and powerful criteria than the mean and fluctuation method.§.§ Perfect tensor As another concrete example of our applications, we now use our conditions to study the notion of perfect tensor, which is recently widely studied in the theory of AdS/CFT, and is understood as an interesting proposal to realize the holographic principle in many-body quantum systems. Perfect tensors can build tensor network state exhibiting interesting holographic correspondence <cit.>. In particular, the tensor network made by perfect tensors derives the Ryu-Takayanagi formula of holographic entanglement entropy, namely, the entanglement entropy of the boundary quantum system equals the minimal surface area in the bulk <cit.>.Furthermore, recently it has been shown that perfect tensors can represent quantum channels which are of strongest quantum chaos <cit.>. The quantum transition defined by perfect tensors turns out to maximally scramble the quantum information such that the initial state cannot be recovered by local measurements. It has also been suggested that a perfect tensor should represent the holographic quantum system dual to the bulk quantum gravity with a black hole. A vector ψ∈ V is called a perfect tensor if for all possible choices Λ satisfying |Λ|≥N/2, the conditionTr_V_Λ( ψ⊗ψ^* )=c_|Λ|·𝕀_V_Λ^cis satisfied for some non-vanishing constant c_|Λ|. The following result follows from Theorem <ref>. Fixing N≥ 4, then for any w, there does not exist a perfect tensor in the constant weight subspace with weight w.We prove by contradiction. Suppose there exists a perfect tensor ψ. We can then take Λ^c with cardinality M+1 such that the condition in (<ref>) is fullfilled. That is,M+1≤ N-[N+1 2]=[N 2] .Then according to Definition <ref>, one must haveTr_V_Λ( ψ⊗ψ^* )=c_|Λ|·𝕀_V_Λ^c ,for some non-vanishing constant c_|Λ|. It is easy to see that c_|Λ| only depends on the cardinality of Λ: the further trace over V_Λ^c should give a multiple of the identity endomorphism which is independent of the choice of Λ.We now consider the entries ρ_(I_0;r )^Λ^c constructed in (<ref>). All of them are equal to c_|Λ| which without loss of generality can be normalized to 1. Then we haveρ_(I_0;r )^Λ^c = 1, ∀(I_0;r)∈Θ_D^Λ^c . We now show that if M≠ 0, that is, the set I_0 is nonempty, then there always exists I_0 such that∑_r=1^D b_r≠ 0 .The condition M≥ 1 requires N≥ 4 according to (<ref>). To check the condition (<ref>), we compute∑_r=1^Db_r=∑_r=1^Dα_r- D·S N-M .Due to the structure theory of representations, one has ∑_r=1^Dα_r=0. Hence the condition boils down toS=-weight(I_0)+w≠ 0 .This can always be satisfied by choosing a suitable I_0, which is contradictory with the claim in Theorem <ref>. Hence there does not exist such a perfect tensor. Perfect tensors also have an intimate connection to quantum error-correcting codes <cit.>. An N-spin perfect tensor can be equivalently viewed as a length-N quantum error-correcting code encoding a single quantum state,with the code distance δ=⌊ N/2⌋+1. Our results hence indicates in the constant weight subspace, there is no such code exist.We will now use our results to further understand the existence of invariant perfect tensors. Invariant tensors are the tensors in V with vanishing total angular momentum. They play an important role in the theory of loop quantum gravity <cit.>, and particularly the structure of Spin-Networks <cit.>. Spin-network states, as quantum states of gravity, are networks of invariant tensors, and represent the quantization of geometry at the Planck scale. Classically an arbitrary three-dimensional geometry can be discretized and built piece by piece by gluing polyhedral geometries. The spin-network state built by invariant tensors quantizes the geometry made by polyhedra. As the building block of spin-network, the n-valent invariant tensor represents the quantum geometry of a polyhedron with n faces. The reason in brief is that the quantum constraint equation ∑_i=1^nJ_iψ=0 (vanishing total angular momentum) is a quantum analog of the polyhedron closure condition ∑_i=1^nA⃗_i=0 in three-dimensional space (see e.g. Appendix A in <cit.> for details). Given that both invariant tensors and perfect tensors relate to quantum gravity from different perspectives, it is interesting to incorporate the idea of perfect tensors with that of invariant tensors, and define a new concept that we call invariant perfect tensor. A nonzero vector ψ∈ V is called an invariant tensor ifHψ=0, Xψ=0, Yψ=0 .A nonzero vector ψ∈ V is called an invariant perfect tensor if it is both perfect and invariant. A partial study of invariant perfect tensors has been carried out in <cit.>, which shows that at N=2,3 invariant tensors are always perfect, but strictly there is no invariant perfect tensor at N=4, although invariant tensors generically approximate perfect tensors asymptotically in large j.The result in Theorem <ref> generalizes the conclusion for invariant perfect tensor to arbitrary N≥4. Since invariant tensors live in the constant weight w=0 subspace, we obtain the following. There does not exist invariant perfect tensor for any j, for N≥ 4. N=3 invariant tensors are employed in spin-network states for 2+1 dimensional gravity, while N≥4 invariant tensors build spin-network states for 3+1 dimensional gravity <cit.>. The above results show that the entanglement exhibited by the local building block of quantum gravity (at Planck scale) is not as much as a perfect tensor. So the holographic property of quantum states is obscure at the Planck scale. The holography displayed by quantum gravity at semi-classical level then suggests that in order to understand quantum gravity using tensor networks, the entanglement of perfect tensor, as being important to understand holography, should be a large scale effect coming from coarse-graining the Planck scale microstates. Namely although the perfect tensor is missing at the Planck scale, it may emerge approximately at the larger scale, and makes tensor networks demonstrate holography. This idea is very much consistent with the recent proposal in <cit.>, which shows the spin-network states in 3+1 dimensions can indeed give tensor networks exhibiting holographic duality at the larger scale. Then it is interesting to understand how (approximate) perfect tensors emerge from non-perfect invariant tensors via coarse-graining from the Planck scale to larger scale. The research in this perspective will be reported in a future publication.§ GENERALIZATIONS AND EXTENSIONS§.§ Generalization to SU(n)We have considered in the above the case whereV is the tensor product of N copies of a not necessarily irreducible representation W of SU(2). We now generalize this to the SU(n) case.Consider V=⊗_k=1^NV_k, where all of the V_k's are given by the same irreducible representation W of SU(n) of dimension D. Suppose a basis of the Cartan subalgebra of SU(n) is given by H^(1),H^(2),⋯, H^(n-1). One then has the weight space decompositionW=⊕_α⃗∈Δ W_α⃗ ,where Δ is the weight space and W_α⃗ is the eigenspace with the weight vector α⃗= (α^(1),α^(2),⋯α^(n-1)), that isW_α⃗={v∈ W | H^(i) v=α^(i) v ,∀ i=1,2,⋯, n-1} .In particular, the action of any element in the Cartan subalgebra, symbolically denoted by H^(*), is diagonal on W_α⃗ and hence on W. The above decomposition is orthogonal. We choose an orthonormal basis e_1,e_2,⋯, e_D whose eigenvalues under H^(*) are given by α⃗_1^(⋆),α⃗_2^(⋆),⋯, α⃗_D^(⋆). The constant-weight condition becomes the condition that under the action of the Cartan subalgebra generated by H^(1),H^(2),⋯, H^(n-1), the weight vector is constant, say w⃗=(w^(1),w^(2),⋯, w^(n-1)). In particular, the weight under H^(*) is the fixed number w^(*).Then everything discussed in the SU(2) case follows. The same reasoning also works when W is not irreducible, in which case a similar orthogonal decomposition in (<ref>) still exists, thanks to the structure theory for finite dimensional representations of the Lie group SU(n).This then shows that there is no perfect tensor in a constant weight subspace for the group G=SU(n) when N≥ 4.§.§ Relaxing the constant weight subspace condition We now discuss to what extent one can relax the constant weight condition.Recall that the combinatorics in partial trace allows one to pass from the space 𝒳 to its quotient 𝒳/𝔖_N-M. What makes the proof in Theorem <ref> work is the relation (<ref>) in Lemma <ref> ∑_r=1^Db_rn_r([x])=0 ,∀[x]∈𝒳/𝔖_N-M ,with the condition in (<ref>) ∑_r=1^D b_r≠ 0 .The condition∑_r=1^D n_r([x])=N-M ,∀[x] ∈𝒳/𝔖_N-M is automatically satisfied, according to (<ref>) which follows from the definition of 𝒳.Suppose we impose a certain constraint which is not necessarily the constant weight condition. Assume that the set of vectors satisfying this constraint, required to be independent of the ordering of the N components, is indexed by the set 𝒴. Denote the cardinality of the quotient 𝒴/𝔖_N-M by P. Then the non-existence of perfect tensors in the space 𝒴 would follow if the following conditions are satisfied∑_r=1^Db_rn_r([y])=0 ,∑_r=1^D b_r≠ 0 , ∀ [y]∈𝒴/𝔖_N-M .We fix a set of representatives {[y_i], i=1,2,⋯, P} for 𝒴/𝔖_N-M, and denote the matrix of frequencies byA=(A_ir)=(n_r([y_i]))_i=1,2⋯ P;r=1,2⋯ D .Then the above two equations become the conditions for the vector b=(b_1,⋯, b_D)^t Ab=0 ,(1,1,⋯, 1) b≠ 0 .We denote à to be the matrix obtained by adding a row of 1's below the P-th row of A. The existence of such a vector b is equivalent to the condition thatrankA<D , rank Ã= rankA+1 .Consider the case where each V_k in V=⊗_k=1^N V_k is the irreducible representation SU(2) of dimension D=2j+1. We put the constant weight condition. This is the main interest in this work. In this case, it is straightforward to show that the cardinality P of 𝒳/𝔖_N-M isP=Coeff_S+D(N-M)∏_r=1^D1 1-t_r ,where t_r, r=1,2,⋯ D are formal parameters with grading 2r-1 respectively, and Coeff_kf(t_1,t_2,⋯,t_D) is the sum of the coefficients of the grading-k terms in the Taylor expansion of f(t_1,t_2,⋯,t_D) near (t_1,t_2,⋯,t_D)=(0,0,⋯,0). Lemma <ref> applies to this case.The existence of a non-trivial solutions tells that rankA ≤ D-1. In fact, more is known and we can show that rankA =D-1 for generic D and N-M. For simplicity we consider the case where each V_k in V=⊗_k=1^N V_k is the irreducible representation of SU(2) with dimension D=2j+1. The general case where V_k is not irreducible is similar.Now according to (<ref>), we see that if [x] is a certain partition in 𝒳/𝔖_N-M, then the following is also a partition if N-M≥ 2 [x]+α_a· 1+ α_b· 1 ,where α_a, α_b are subject to the condition that α_a+ α_b=0. Furthermore, one also has the combinations[x]+ α_a· 2+α_b-1· 1 + α_b+1· 1, [x]+ α_a-1· 1 +α_a+1· 1 +α_b· 2.The former gives a genuine partition provided the relation 1≤ b-1, b+1≤ D can be satisfied, which is the case when D≥ 3, N-M≥ 3. The latter is similar. We shall refer to the case D≥ 3, N-M≥ 3 the generic case, the others are isolated cases which can be dealt with easily.We then take the frequency vectors corresponding to the partitions in the set[x] ,[x]+ α_a· 1 + α_b· 1,[x]+ α_a· 2 + α_b-1· 1+ α_b+1·1,b≤ D-1,[x]+α_1· 1+α_3· 1+α_D-1· 2 .By computing the determinants inductively, it is easy to see that the corresponding frequency vectors span a vector space of dimension at least D-1. Combing the relation rankA ≤ D-1 implied by Lemma <ref>, we are then led to the conclusion that rankA = D-1 for generic D, N-M.As another example, we put the constraint∑_r=1^Dα_r^2 n_r=S ,where α_r,r=1,2,⋯, D, are the weights of the basis ℬ={e_1,e_2,⋯, e_D}. The solution {b_r}_r=1^D to (<ref>) then exists if we choose I_0, S suitably. § ACKNOWLEDGMENTS We thank Markus Grassl for helpful discussions. J. C. was supported by the Department of Defence. M. H. acknowledges support from the US National Science Foundation through grant PHY-1602867, and Start-up Grant at Florida Atlantic University, USA. Y. L. acknowledges support from Chinese Ministry of Education under grants No.20173080024. B. Z. is supported by NSERC and CIFAR. J. Z. was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.28 natexlab#1#1 bibnamefont#1#1 bibfnamefont#1#1 citenamefont#1#1 url<#>1 urlprefixURL [Tóth(2007)]toth2007detectionauthorG. 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"authors": [
"Jianxin Chen",
"Muxin Han",
"Youning Li",
"Bei Zeng",
"Jie Zhou"
],
"categories": [
"quant-ph",
"gr-qc",
"math-ph",
"math.MP"
],
"primary_category": "quant-ph",
"published": "20170427133636",
"title": "Local density matrices of many-body states in the constant weight subspaces"
} |
Spreading law on a completely wettable spherical substrate: The energy balance approach Masao Iwamatsu December 30, 2023 ========================================================================================= We introduce a neural semantic parser which converts natural language utterances to intermediate representations in the form of predicate-argument structures, which are induced with a transition system and subsequently mapped to target domains.The semantic parser is trained end-to-end using annotated logical forms or their denotations. We achieve the state of the art on and and obtain competitive results on and .The induced predicate-argument structures shed light on the types of representations useful for semantic parsing and how these are different from linguistically motivated ones.[Our code will be available at <https://github.com/cheng6076/scanner>.] § INTRODUCTION Semantic parsing is the task of mapping natural language utterances to machine interpretable meaning representations. Despite differences in the choice of meaning representation and model structure, most existing work conceptualizes semantic parsing following two main approaches. Under the first approach, an utterance is parsed and grounded to a meaning representation directly via learning a task-specific grammar <cit.>. Under the second approach, the utterance is first parsed to an intermediate task-independent representation tied to a syntactic parser and then mapped to a grounded representation <cit.>.A merit of the two-stage approach is that it creates reusable intermediate interpretations, which potentially enables the handling of unseen words and knowledge transfer across domains <cit.>.The successful application of encoder-decoder models <cit.> to a variety of NLP tasks has provided strong impetus to treat semantic parsing as a sequence transduction problem where an utterance is mapped to a target meaning representation in string format <cit.>.Such models still fall under the first approach, however, in contrast to previous work <cit.> they reduce the need for domain-specific assumptions, grammar learning, and more generally extensive feature engineering. But this modeling flexibility comes at a cost since it is no longer possible to interpret how meaning composition is performed. Such knowledge plays a critical role in understand modeling limitations so as to build better semantic parsers. Moreover, without any task-specific prior knowledge, the learning problem is fairly unconstrained, both in terms of the possible derivations to consider and in terms of the target output which can be ill-formed (e.g., with extra or missing brackets).In this work, we propose a neural semantic parser that alleviates the aforementioned problems. Our model falls under the second class of approaches where utterances are first mapped to an intermediate representation containing natural language predicates. However, rather than using an external parser <cit.> or manually specified CCG grammars <cit.>, we induce intermediate representations in the form of predicate-argument structures from data.This is achieved with a transition-based approach which by design yields recursive semantic structures, avoiding the problem of generating ill-formed meaning representations.Compared to existing chart-based semantic parsers <cit.>, the transition-based approach does not require feature decomposition over structures and thereby enables the exploration of rich, non-local features.The output of the transition system is then grounded (e.g., to a knowledge base) with a neural mapping model under the assumption that grounded and ungrounded structures are isomorphic.[We discuss the merits and limitations of this assumption in Section <ref>.] As a result, we obtain a neural network that jointly learns to parse natural language semantics and induce a lexicon that helps grounding. The whole network is trained end-to-end on natural language utterances paired with annotated logical forms or their denotations.We conduct experiments on four datasets, including (which has logical forms; ), <cit.>, <cit.>, and <cit.> (which have denotations).Our semantic parser achieves the state of the art on and , while obtaining competitive results on and .A side-product of our modeling framework is that the induced intermediate representations can contribute to rationalizing neural predictions <cit.>.Specifically, they can shed light on the kinds of representations (especially predicates) useful for semantic parsing.Evaluation of the induced predicate-argument relations against syntax-based ones reveals that they are interpretable and meaningful compared to heuristic baselines, but they sometimes deviate from linguistic conventions.§ PRELIMINARIES Problem Formulation Let 𝒦 denote a knowledge base or more generally a reasoning system, and x an utterance paired with a grounded meaning representation G or its denotation y. Our problem is to learn a semantic parser that maps x to G via an intermediate ungrounded representation U.When G is executed against 𝒦, it outputs denotation y.Our semantic parser scales from relational databases for restricted domains (e.g., US geography) to broad-coverage knowledge bases (e.g., Freebase).A knowledge base contains a set of entities denoted by ℰ (e.g., Barack Obama) and a set of relations denoted by ℛ (e.g., president, birth_place).[We treat types as unary relations.]Grounded Meaning RepresentationWe represent grounded meaning representations in FunQL <cit.> amongst many other alternatives such as lambda calculus <cit.>, λ-DCS <cit.> or graph queries <cit.>.FunQL is a variable-free query language, where each predicate is treated as a function symbol that modifies an argument list. For example, the FunQL representation for the utterance which states do not border texas is: 0.2in answer(exclude(state(all), next_to(texas)))where next_to is a domain-specific binary predicate that takes one argument (i.e., the entity texas) and returns a set of entities (e.g., the states bordering Texas) as its denotation.all is a special predicate that returns a collection of entities.exclude is a predicate that returns the difference between two input sets.An advantage of FunQL is that the resultingencodes semantic compositionality and derivation of the logical forms. This property makes FunQL logical forms natural to be generated with recurrent neural networks <cit.>.However, FunQL is less expressive than lambda calculus, partially due to the elimination of variables. A more compact logical formulation which our method also applies to is λ-DCS <cit.>. In the absence of anaphora and composite binary predicates, conversion algorithms exist between FunQL and λ-DCS. However, we leave this to future work. Ungrounded Meaning Representation We also use FunQL to express ungrounded meaning representations. The latter consist primarily of natural language predicates and domain-general predicates. Assuming for simplicity that domain-general predicates share the same vocabulary in ungrounded and grounded representations, the ungrounded representation for the example utterance is: 0.2in answer(exclude(states(all), border(texas))) where states and border are natural language predicates.In this work we consider five types of domain-general predicates illustrated in Table <ref>. Notice that domain-general predicates are often implicit, or represent extra-sentential knowledge.For example, the predicate all in the above utterance represents all states in the domain which are not mentioned in the utterance but are critical for working out the utterance denotation. Finally, note that for certain domain-general predicates, it also makes sense to extract natural language rationales (e.g., not is indicative for exclude). But we do not find this helpful in experiments.In this work we constrain ungrounded representations to be structurally isomorphic to grounded ones.In order to derive the target logical forms, all we have to do is replacing predicates in the ungrounded representations with symbols in the knowledge base.[As a more general definition, we consider two semantic graphs isomorphic if the graph structures governed by domain-general predicates,ignoring local structures containing only natural language predicates, are the same (Section <ref>). ] § MODELINGIn this section, we discuss our neural model which maps utterances to target logical forms.The semantic parsing task is decomposed in two stages: we first explain how an utterance is converted to an intermediate representation (Section <ref>), and then describe how it is grounded to a knowledge base (Section <ref>). §.§ Generating Ungrounded RepresentationsAt this stage, utterances are mapped to intermediate representations with a transition-based algorithm. In general, the transition system generates the representation by following a derivation tree (which contains a set of applied rules) andsome canonical generation order (e.g., pre-order). For FunQL, a simple solution exists since the representation itself encodes the derivation. Consider again answer(exclude(states(all), border(texas))) which is tree structured. Each predicate (e.g., border) can be visualized as a non-terminal node of the tree and each entity (e.g., texas) as a terminal. The predicate all is a special case which acts as a terminal directly. We can generate the tree top-down with a transition system reminiscent of recurrent neural network grammars (RNNGs; ).Similar to RNNG, our algorithm uses a buffer to store input tokens in the utterance and a stack to store partially completed trees.A major difference in our semantic parsing scenario is that tokens in the buffer are not fetched in a sequential order or removed from the buffer.This is because the lexical alignment between an utterance and its semantic representation is hidden.Moreover, some domain-general predicates cannot be clearly anchored to a token span.Therefore, we allow the generation algorithm to pick tokens and combine logical forms in arbitrary orders, conditioning on the entire set of sentential features.Alternative solutions in the traditional semantic parsing literature include a floating chart parser <cit.> which allows to construct logical predicates out of thin air. Our transition system defines three actions, namely nt, ter, and red, explained below. nt(x) generates a non-terminal predicate.This predicate is either a natural language expression such as border, or one of the domain-general predicates exemplified in Table <ref> (e.g., exclude). The type of predicate is determined by the placeholder x and once generated, it is pushed onto the stack and represented as a non-terminal followed by an open bracket (e.g., `border(').The open bracket will be closed by a reduce operation.ter(x) generates a terminal entity or the special predicate all.Note that the terminal choice does not include variable (e.g., $0, $1), since FunQL is a variable-free language which sufficiently captures the semantics of the datasets we work with.The framework could be extended to generate directed acyclic graphs by incorporating variables with additional transition actions for handling variable mentions and co-reference. red stands for reduce and is used for subtree completion.It recursively pops elements from the stack until an open non-terminal node is encountered.The non-terminal is popped as well, after which a composite term representing the entire subtree, e.g., border(texas), is pushed back to the stack.If a red action results in having no more open non-terminals left on the stack, the transition system terminates.Table <ref> shows the transition actions used to generate our running example.The model generates the ungrounded representation U conditioned on utterance x by recursively calling one of the above three actions. Note that U is defined by a sequence of actions (denoted by a) and a sequence of term choices (denoted by u) as shown in Table <ref>.The conditional probability p(U|x) is factorized over time steps as:p(U|x)= p(a, u | x) = ∏_t=1^T p(a_t | a_<t, x) p(u_t | a_<t, x)^𝕀(a_t ≠red) 3where 𝕀 is an indicator function. To predict the actions of the transition system, we encode the input buffer with a bidirectional LSTM <cit.> and the output stack with a stack-LSTM <cit.>.At each time step, the model uses the representation of the transition system e_t to predict an action:p(a_t | a_<t, x) ∝exp (W_a · e_t)where e_t is the concatenation of the buffer representation b_t and the stack representation s_t.While the stack representation s_t is easy to retrieve as the top state of the stack-LSTM, obtaining the buffer representation b_t is more involved.This is because we do not have an explicit buffer representation due to the non-projectivity of semantic parsing.We therefore compute at each time step an adaptively weighted representation of b_t <cit.> conditioned on the stack representation s_t.This buffer representation is then concatenated with the stack representation to form the system representation e_t.When the predicted action is either nt or ter, an ungrounded term u_t (either a predicate or an entity) needs to be chosen from the candidate list depending on the specific placeholder x.To select a domain-general term, we use the same representation of the transition system e_t to compute a probability distribution over candidate terms:p(u_t^GENERAL | a_<t, x) ∝exp (W_p · e_t)To choose a natural language term, we directly compute a probability distribution of all natural language terms (in the buffer) conditioned on the stack representation s_t and select the most relevant term <cit.>:p(u_t^NL | a_<t, x) ∝exp (W_s · s_t) When the predicted action is red, the completed subtree is composed into a single representation on the stack.For the choice of composition function, we use a single-layer neural network as in dyer2015transition, which takes as input the concatenated representation of the predicate and arguments of the subtree. In the following, we explain representation learning of the system and how predictions are made at each time step. We use lowercase boldface letters to denote vectors (e.g., 𝐡), uppercase boldface letters to denote matrices (e.g., 𝐖), and lowercase letters to denote variable names and scalars (e.g., x). The sentence, or buffer x consists of a sequence of tokens x = [x_1, ⋯, x_m] where m denotes the sentence length.We encode x with a bi-directional LSTM <cit.>. The contextual representation for each word is represented as 𝐱_t and the buffer representation 𝐱 is a list of contextual word representations.The stack is modeled with a stack-LSTM <cit.>, which is an LSTM augmented with a pointer to states.This architecture simulates state changes in the LSTM with respect to push and pop operations, which modify the content and move the pointer. The hidden state under the pointer, denoted by 𝐬_t, acts as the representation of the stack. When TER or NT is invoked, new content is pushed onto the stack and 𝐬_t is recurrently updated to 𝐬_t+1 as a normal LSTM. When RED is invoked, nodes of the subtree on top of the stack are popped and then composed into a single representation. During popping, the stack pointer moves backward to an intermediate state 𝐬_t: t+1. After that, the composite representation is pushed back to the stack and a recurrent update to 𝐬_t+1 is performed. For more explanations of the stack LSTM, we refer readers to the original work of dyer2015transition. For composition function of the RED operation, we use a feed-forward neural network as the composition function, which takesas input the parent representation of the subtree and the average of its children representations. Given the buffer representation 𝐱 and the stack representation 𝐬_t, we now proceedto discuss predictions of the transition system.At each time step, the representation of the transition system, denoted by 𝐞_t, is computed with a neural network that combines 𝐱 and 𝐬_t. 𝐞_t = MLP (𝐱̃_t, 𝐬_t) 𝐱̃_t = ∑_i=1 c^i_t 𝐱_i c^i_t = exp{𝐰^T_ctanh (𝐖_x𝐱_i+𝐖_s𝐬_t) }/∑_j=1^m exp{𝐰^T_ctanh (𝐖_x𝐱_j+𝐖_s𝐬_t)}where 𝐰_c, 𝐖_x, and 𝐖_s are weight parameters.We compute the probability of the next action, and also domain-general predicate choices, based on the transition system representation 𝐞_tp(a_t ) ∝exp𝐚_t𝐞_t^T + b Meanwhile, if an NT (or TER) action is invoked and a natural language predicate (or entity) needs to be selected, we use another neural network which interacts between 𝐱 and 𝐬_t, and outputs a probability distribution of the candidate non-terminal (terminal) symbols u_t: p(u_t) ∝exp𝐮_t 𝐬_t^T + b§.§ Generating Grounded RepresentationsSince we constrain the network to learn ungrounded structures that are isomorphic to the target meaning representation, converting ungrounded representations to grounded ones becomes a simple lexical mapping problem. For simplicity, hereafter we do not differentiate natural language and domain-general predicates.To map an ungrounded term u_t to a grounded term g_t, we compute the conditional probability of g_t given u_t with a bi-linear neural network:p(g_t | u_t) ∝expu⃗_⃗t⃗· W_ug·g⃗_⃗t⃗^⊤where u⃗_⃗t⃗ is the contextual representation of the ungrounded term given by the bidirectional LSTM, g⃗_⃗t⃗ is the grounded term embedding, and W_ug is the weight matrix.The above grounding step can be interpreted as learning a lexicon: the model exclusively relies on the intermediate representation U to predict the target meaning representation G without taking into account any additional features based on the utterance. In practice, U may provide sufficient contextual background for closed domain semantic parsing where an ungrounded predicate often maps to a single grounded predicate, but is a relatively impoverished representation for parsing large open-domain knowledge bases like Freebase. In this case, we additionally rely on a discriminative reranker which ranks the grounded representations derived from ungrounded representations (see Section <ref>).The reranker uses p(g_t | u_t) as feature along with other sentential features to select the final grounded representation.This way the ungrounded representation still imposes structural constrains and helps guide the target search space.§.§ Training ObjectiveWhen the target meaning representation is available, we directly compare it against our predictions and back-propagate.When only denotations are available, we compare surrogate meaning representations against our predictions <cit.>.Surrogate representations are those with the correct denotations, filtered with rules (see Section <ref>).When there exist multiple surrogate representations,[The average Freebase surrogate representations obtained with highest denotation match (F1) is 1.4.] we select one randomly and back-propagate. Consider utterance x with ungrounded meaning representation U, and grounded meaning representation G. Both U and G are defined with a sequence of transition actions (same for U and G) and a sequence of terms (different for U and G).Recall that a = [a_1, ⋯, a_n] denotes the transition action sequence defining U and G; let u = [u_1, ⋯, u_k] denote the ungrounded terms (e.g., predicates), and g = [g_1, ⋯, g_k] the grounded terms. We aim to maximize the likelihood of the grounded meaning representation p(G | x) over all training examples.This likelihood can be decomposed into the likelihood of the grounded action sequence p(a|x) and the grounded term sequence p(g|x), which we optimize separately.For the grounded action sequence (which by design is the same as the ungrounded action sequence and therefore the output of the transition system), we can directly maximize the log likelihood log p(a | x) for all examples:ℒ_a= ∑_x ∈𝒯log p(a | x)=∑_x ∈𝒯∑_t=1^n log p(a_t | x)where 𝒯 denotes examples in the training data. For the grounded term sequence g, since the intermediate ungrounded terms are latent, we maximize the expected log likelihood of the grounded terms ∑_u [ p(u | x) log p(g | u, x)] for all examples, which is a lower bound of the log likelihood log p(g | x) by Jensen's Inequality:ℒ_g =∑_x ∈𝒯∑_u [ p(u | x) log p(g | u, x)]=∑_x ∈𝒯∑_u [ p(u | x) ∑_t=1^klog p(g_t | u_t)] ≤∑_x ∈𝒯log p(g | x)The final objective is the combination of ℒ_a and ℒ_g, denoted as ℒ_G = ℒ_a + ℒ_g. We optimize this objective with the method described in lei2016rationalizing and xu2015show. Optimization of the neural parameters with respect to ℒ_a is straightforward; while the expected loss ℒ_g is fully differentiable, it requires marginalization over ungrounded meaning representations.An alternative would be to use a stochastic gradient method <cit.> that approximates the true gradient induced by ℒ_g with a set of sampled ungrounded representations.§.§ Reranker As discussed above, for open domain semantic parsing, solely relying on the ungrounded representation would result in an impoverished model lacking sentential context useful for disambiguation decisions.For all Freebase experiments, we followed previous work <cit.> in additionally training a discriminative ranker to re-rank grounded representations globally.The discriminative ranker is a maximum-entropy model <cit.>.The objective is to maximize the log likelihood of the correct answer y given x by summing over all grounded candidates G with denotation y (i.e.,[[ G ]]_𝒦 = y):ℒ_y=∑_(x, y) ∈𝒯log∑_[[ G ]]_𝒦 = y p (G | x) p (G | x) ∝exp{ f(G, x) }where f(G, x) is a feature function that maps pair into a feature vector. We give details on the features we used in Section <ref>.§ EXPERIMENTSIn this section, we verify empirically that our semantic parser derives useful meaning representations. We give details on the evaluation datasets and baselines used for comparison. We also describe implementation details and the features used in the discriminative ranker. §.§ Datasets We evaluated our model on the following datasets which cover different domains, and use different types of training data, i.e., pairs of natural language utterances and grounded meanings or question-answer pairs.GeoQuery <cit.> contains 880 questions and database queries about US geography. The utterances are compositional, but the language is simple and vocabulary size small. The majority of questions include at most one entity. Spades <cit.> contains 93,319 questions derived from clueweb09 <cit.> sentences. Specifically, the questions were created by randomly removing an entity, thus producing sentence-denotation pairs <cit.>.The sentences include two or more entities and although they are not very compositional, they constitute a large-scale dataset for neural network training. WebQuestions <cit.> contains 5,810 question-answer pairs. Similar to spades, it is based on Freebase and the questions are not very compositional. However, they are real questions asked by people on the Web. Finally, GraphQuestions <cit.> contains 5,166 question-answer pairs which were created by showing 500 Freebase graph queries to Amazon Mechanical Turk workers and asking them to paraphrase them into natural language.§.§ Implementation DetailsAmongst the four datasets described above, GeoQuery has annotated logical forms which we directly use for training.For the other three datasets, we treat surrogate meaning representations which lead to the correct answer as gold standard.The surrogates were selected from a subset of candidate Freebase graphs, which were obtained by entity linking.Entity mentions in Spades have been automatically annotated with Freebase entities <cit.>.For WebQuestions and GraphQuestions, we follow the procedure described in reddy2016transforming. We identify potential entity spans using seven handcrafted part-of-speech patterns and associate them with Freebase entities obtained from the Freebase/KG API.[<http://developers.google.com/freebase/>] We use a structured perceptron trained on the entities found in WebQuestions and GraphQuestions to select the top 10 non-overlapping entity disambiguation possibilities.We treat each possibility as a candidate input utterance, and use the perceptron score as a feature in the discriminative reranker, thus leaving the final disambiguation to the semantic parser.Apart from the entity score, the discriminative ranker uses the following basic features.The first feature is the likelihood score of a grounded representation aggregating all intermediate representations.The second set of features include the embedding similarity between the relation and the utterance, as well as the similarity between the relation and the question words.The last set of features includes the answer type as indicated by the last word in the Freebase relation <cit.>.We used the Adam optimizer for training with an initial learning rate of 0.001, two momentum parameters [0.99, 0.999], and batch size 1. The dimensions of the word embeddings, LSTM states, entity embeddings and relation embeddings are [50, 100, 100, 100].The word embeddings were initialized with Glove embeddings <cit.>.All other embeddings were randomly initialized. §.§ Results Experimental results on the four datasets are summarized in Tables <ref>–<ref>.We present comparisons of our system which we call ScanneR (as a shorthand for SymboliC meANiNg rEpResentation) against a variety of models previously described in the literature. results are shown in Table <ref>. The first block contains symbolic systems, whereas neural models are presented in the second block.We report accuracy which is defined as the proportion of the utterance that are correctly parsed to their gold standard logical forms. All previous neural systems <cit.> treat semantic parsing as a sequence transduction problem and use LSTMs to directly map utterances to logical forms. ScanneR yields performance improvements over these systemswhen using comparable datasources for training. jia2016data achieve better results with synthetic data that expands GeoQuery; we could adopt their approach to improve model performance, however, we leave this to future work. Table <ref> reports ScanneR's performance on Spades. For all Freebase related datasets we use average F1 <cit.> as our evaluation metric.Previous work on this dataset has used a semantic parsing framework similar to ours where natural language is converted to an intermediate syntactic representation and then grounded to Freebase. Specifically, bisk2016evaluating evaluate the effectiveness of four different CCG parsers on the semantic parsing task when varying the amount of supervision required. As can be seen, ScanneR outperforms all CCG variants (from unsupervised to fully supervised) without having access to any manually annotated derivations or lexicons.For fair comparison, we also built a neural baseline that encodes an utterance with a recurrent neural network and then predicts a grounded meaning representation directly <cit.>.Again, we observe that ScanneR outperforms this baseline. Results on WebQuestions are summarized in Table <ref>. ScanneR obtains performance on par with the best symbolic systems (see the first block in the table). It is important to note that bast2015more develop a question answering system, which contrary to ours cannot produce meaning representations whereas berant2015imitation propose a sophisticated agenda-based parser which is trained borrowing ideas from imitation learning.reddy2016transforming learns a semantic parser via intermediate representations which they generate based on the output of a dependency parser. ScanneR performs competitively despite not having access to any linguistically-informed syntactic structures. The second block in Table <ref> reports the results of several neural systems.xu2016question represent the state of the art on WebQuestions. Their system uses Wikipedia to prune out erroneous candidate answers extracted from Freebase. Our model would also benefit from a similar post-processing step.As in previous experiments, ScanneR outperforms the neural baseline, too. Finally, Table <ref> presents our results on GraphQuestions. We report F1 for ScanneR, the neural baseline model, and three symbolic systems presented in su2016generating. ScanneR achieves a new state of the art on this dataset with a gain of 4.23 F1 points over the best previously reported model.§.§ Analysis of Intermediate Representations Since a central feature of our parser is that it learns intermediate representations with natural language predicates, we conducted additional experiments in order to inspect their quality.For GeoQuery which contains only 280 test examples, we manually annotated intermediate representations for the test instances and evaluated the learned representations against them. The experimental setup aims to show how humans can participate in improving the semantic parser with feedback at the intermediate stage. In terms of evaluation, we use three metrics shown in Table <ref>.The first row shows the percentage of exact matches between the predicted representations and the human annotations.The second row refers to the percentage of structure matches, where the predicted representations have the same structure as the human annotations, but may not use the same lexical terms. Among structurally correct predictions, we additionally compute how many tokens are correct, as shown in the third row.As can be seen, the induced meaning representations overlap to a large extent with the human gold standard.We also evaluated the intermediate representations created by ScanneR on the other three (Freebase) datasets.Since creating a manual gold standard for these large datasets is time-consuming, we compared the induced representations against the output of a syntactic parser. Specifically, we converted the questions to event-argument structures with EasyCCG <cit.>, a high coverage and high accuracy CCG parser. EasyCCG extracts predicate-argument structures with a labeled F-score of 83.37%. For further comparison, we built a simple baseline which identifies predicates based on the output of the Stanford POS-tagger <cit.> following the ordering VBD ≫ VBN ≫ VB ≫ VBP ≫ VBZ ≫ MD.As shown in Table <ref>, on Spades and WebQuestions, the predicates learned by our model match the output of EasyCCG more closely than the heuristic baseline. But for which contains more compositional questions, the mismatch is higher.However, since the key idea of our model is to capture salient meaning for the task at hand rather than strictly obey syntax, we would not expect the predicates induced by our system to entirely agree with those produced by the syntactic parser.To further analyze how the learned predicates differ from syntax-based ones, we grouped utterances in Spades into four types of linguistic constructions: coordination (conj), control and raising (control), prepositional phrase attachment (pp), and subordinate clauses (subord).Table <ref> also shows the breakdown of matching scores per linguistic construction, with the number of utterances in each type.In Table <ref>, we provide examples of predicates identified by ScanneR, indicating whether they agree or not with the output of EasyCCG. As a reminder, the task in Spades is to predict the entity masked by a blank symbol (__). As can be seen in Table <ref>, the matching score is relatively high for utterances involving coordination and prepositional phrase attachments.The model will often identify informative predicates (e.g., nouns) which do not necessarily agree with linguistic intuition.For example, in the utterance wilhelm_maybach and his son __ started maybach in 1909 (see Table <ref>), ScanneR identifies the predicate-argument structure son(wilhelm_maybach) rather than started(wilhelm_maybach).We also observed that the model struggles with control and subordinate constructions. It has difficulty distinguishing control from raising predicates as exemplified in the utterance ceo john_thain agreed to leave __ from Table <ref>, where it identifies the control predicate agreed. For subordinate clauses, Scanner tends to take shortcuts identifying as predicates words closest to the blank symbol. § DISCUSSION We presented a neural semantic parser which converts natural language utterances to grounded meaning representations via intermediate predicate-argument structures. Our model essentially jointly learns how to parse natural language semantics and the lexicons that help grounding.Compared to previous neural semantic parsers, our model is more interpretable as the intermediate structures are useful for inspecting what the model has learned and whether it matches linguistic intuition.An assumption our model imposes is that ungrounded and grounded representations are structurally isomorphic. An advantage of this assumption is that tokens in the ungrounded and grounded representations are strictly aligned.This allows the neural network to focus on parsing and lexical mapping, sidestepping the challenging structure mapping problem which would result in a larger search space and higher variance. On the negative side, the structural isomorphism assumption restricts the expressiveness of the model, especially since one of the main benefits of adopting a two-stage parser is the potential of capturing domain-independent semantic information via the intermediate representation.While it would be challenging to handle drastically non-isomorphic structures in the current model, it is possible to perform local structure matching, i.e., when the mapping between natural language and domain-specific predicates is many-to-one or one-to-many.For instance, Freebase does not contain a relation representing daughter, using instead two relations representing female and child.Previous work <cit.> models such cases by introducing collapsing (for many-to-one mapping) and expansion (for one-to-many mapping) operators. Within our current framework, these two types of structural mismatches can be handled with semi-Markov assumptions <cit.> in the parsing (i.e., predicate selection) and the grounding steps, respectively. Aside from relaxing strict isomorphism, we would also like to perform cross-domain semantic parsing where the first stage of the semantic parser is shared across domains. Acknowledgments We would like to thank three anonymous reviewers, members of the Edinburgh ILCC and the IBM Watson, and Abulhair Saparov for feedback. The support of the European Research Council under award number 681760 “Translating Multiple Modalities into Text” is gratefully acknowledged. acl_natbib | http://arxiv.org/abs/1704.08387v3 | {
"authors": [
"Jianpeng Cheng",
"Siva Reddy",
"Vijay Saraswat",
"Mirella Lapata"
],
"categories": [
"cs.CL"
],
"primary_category": "cs.CL",
"published": "20170427002420",
"title": "Learning Structured Natural Language Representations for Semantic Parsing"
} |
ad1]Xi Dengad1]Satoshi Inabaad1]Bin Xiead2]Keh-Ming Shyuead1]Feng Xiao cor[ad1]Department of Mechanical Engineering, Tokyo Institute of Technology, 4259 Nagatsuta Midori-ku, Yokohama, 226-8502, Japan.[ad2]Department of Mathematics, National Taiwan University, Taipei 106, Taiwan [cor]Corresponding author: Dr. F. Xiao (Email: [email protected]) We present in this work a new reconstruction scheme, so-called MUSCL-THINC-BVD scheme, to solve the five-equation model for interfacial two phase flows. This scheme employs the traditional shock capturing MUSCL (Monotone Upstream-centered Schemes for Conservation Law) scheme as well as the interface sharpening THINC (Tangent of Hyperbola for INterface Capturing) scheme as two building-blocks of spatial reconstruction using the BVD (boundary variation diminishing) principle that minimizes the variations (jumps) of the reconstructed variables at cell boundaries, and thus effectively reduces the numerical dissipations in numerical solutions. The MUSCL-THINC-BVD scheme is implemented to all state variables and volume fraction, which realizes the consistency among volume fraction and other physical variables. Benchmark tests are carried out to verify the capability of the present method in capturing the material interface as a well-defined sharp jump in volume fraction, as well as significant improvement in solution quality. The proposed scheme is a simple and effective method of practical significance for simulating compressible interfacialmultiphase flows. Compressible multiphase flows five-equation model interface capturing THINC reconstruction BVD algorithm Implementation of BVD (boundary variation diminishing) algorithm in simulations of compressible multiphase flows [ December 30, 2023 ================================================================================================================ § INTRODUCTIONCompressible multiphase flow is one of active and challenging research areas of great importance in both theoretical studies and industrial applications. For example, shock/interface interactions are thought to be crucial to the instability and evolution of material interfaces that separate different fluids as can be observed in a wide spectral of phenomena<cit.>. The material interfaces greatly complicate the physics and make problems formidably difficult for analytical and experimental approaches in general. In many cases, numerical simulation turns out to be the most effective approach to provide quantitative information to elucidate the fundamental mechanisms behind the complex phenomena of multiphase flows. In comparison to computation of single phase flow, development of numerical methods for multiphase flow faces more challenging tasks. The major complexity comes from the moving interfaces between different fluids that usually associate with strong discontinuities, singular forces and phase changes. Given the numerical methods developed for multiphase incompressible flow with interfaces having been reached a relatively mature stage, the numerical solvers for compressible interfacial multiphase flow are apparently insufficient. For incompressible multiphase flows with moving interfaces where the density and other physical properties, e.g. viscosity and thermal conductivity, are constant in each fluid, the one-fluid model <cit.> can be implemented in a straightforward manner with an assumption that the physical fields change monotonically across the interface region. So, provided an indication function which identifies the moving interface, one can uniquely determine the physical property fields for the whole computational domain. Some indication functions, such as volume of fluid (VOF) function <cit.> and level set function <cit.>, have been proposed and proved to be able to well define the moving interface with compact thickness and geometrical faithfulness if solved by advanced numerical algorithms. However, substantial barrier exists when implementing the one-fluid model to compressible interfacial multiphase flow. The new difficulties we face when applying the one-fluid model [More precisely, it should be called single-state model or single-equivalent-fluid (SEF) model<cit.>.We call such model SEF in the present paper.]to compressible interfacial multiphase flow lie in two aspects: * (I) Density and energy in compressible flow have to be computed separately in addition to the indication function, hence special formulations are required to reach a balanced state among all variables for the interface cell where a well-defined interface falls in; * (II) The numerical dissipation in the so-called high-resolution schemes designed for solving single phase compressible flow involving shock waves tends to smear out discontinuities in numerical solutions including the material interfaces, which is fatal to simulations of interfacial multiphase flows even if the schemes can produce acceptable results in single phase cases.For issue (I) mentioned above, mixing or averaging models that consist of Euler or Navier-Stokes equations along with interface-indication function equations for each of fluid components have been derived and widely used as efficient approximations to the state of the interface cell where two or more species co-exist.A simple single-fluid model was reported in <cit.> for interfacial multiphase compressible flows using either explicit time marching or semi-implicit pressure-projection solution procedure. The latter results in a unified formulation for solving both compressible and incompressible multiphase flows. As the primitive variables are used in these models, the conservation properties are not guaranteed, and thus might not be suitable for high-Mach flows involving shock waves.Conservative formulations, which have been well-established for single phase compressible flows with shock waves, however may lead to spurious oscillations in pressure or other thermal fields <cit.>. It was found that special treatments are required in transporting the material interface and mixing/averaging the state variables to find the mixed state of fluids in the interfacial cell that satisfies pressure balance across material interface for multiple polytropic and stiff gases <cit.>, and for van der Waals and Mie-Grüneisen equations of state (EOS) <cit.>. A more general five-equation model <cit.> was developed for a wide range fluids. These models apply to multiphase compressible flows with either spread interfaces or sharp interfaces. The five-equation model will be used in the present work as the PDE (partial differential equation) set to solve.Provided the SEF models with some desired properties, such as hyperbolicity, conservation and well-balanced mixing closure without spurious oscillations in thermal variables, we can in principle implement numerical methods for single phase compressible flow (e.g. standard shock-capturing schemes) to solve multiphase ones. TVD (Total Variation Diminishing) schemes, such as the MUSCL (Monotone Upstream-centered Schemes for Conservation Law) scheme <cit.>, can resolve discontinuities without numerical oscillations, which is of paramount importance to ensure the physical fields to be bounded and monotonic in the transition region. However, TVD schemes suffer from excessive numerical dissipation, which brings the problem (II) listed above to us. The intrinsic numerical dissipation smears out the flow structures including the discontinuities in mass fraction or volume fraction that are used to represent the material interfaces. Consequently, material interfaces are continuously blurred and spread out, which is not acceptable in many applications, especially for the simulations that need long-term computation. Applying high order schemes like WENO (Weighted Essentially Non-Oscillatory) scheme <cit.> to solve compressible multiphase flow are found in the literatures <cit.>. However, implementing high order schemes might generate numerical oscillations for compressible multiphase flow with more complex EOS, such as the Mie-Grüneisen equation of state. In <cit.>, to reduce numerical oscillation introduced by high order schemes, the state variables have to be cast into characteristic fields. Although with this effort, stability cannot be guaranteed in the long-term computation even using a forbiddingly small of time step. In a recent work <cit.>, to further reduce the numerical oscillations and to deal with complicated EOS, an approximate intermediate state at each cell edges is obtained in a more careful way to conduct characteristic decomposition. Furthermore, high order monotonicity-preserving scheme <cit.> was used to ensure volume fraction remain bounded. In general, the implementation of high order shock capturing schemes in compressible multiphase flows will increase complexity of algorithm and may invoke computational instability.In order to keep material interfaces being a compact thickness during computation, special treatments are required to sharpen or steepen the interfaces. The existing methods for this purpose can be categorized into interface-tracking and interface-capturing. Interface-tracking methods like Arbitrary Lagrangian-Eulerian (ALE) <cit.>, free-Lagrange <cit.>, front-tracking <cit.> and level set/ghost fluid <cit.>have the distinct advantage of treating interface as a sharp discontinuity. For example, ALE and free-Lagrange methods treat interfaces as boundaries of distorted computational grids, which however makes them complicated and computationally expensive when there are large interface deformations and topological changes. Another drawback is that these methods are typically not numerically conservative at material interfaces, which may lead to a wrong prediction about the position of interfaces and shock waves. Interface-capturing methods resolve the interfaces on fixed Eulerian grids, usingspecial numerical techniques to sharpen the interface from spreading out. For example, in <cit.> the advection equation of the interface function is treated by artificial compression method. As a post-processing approach, anti-diffusion techniques have been introduced by <cit.> and <cit.>. Another approach is to reconstruct the volume fraction under the finite volume framework by THINC (Tangent of Hyperbola for INterface Capturing) function <cit.>. By virtue of the desirable characteristics of the hyperbolic tangent function in mimicking the jump-like profile of the volume fraction field, the sharp interface can be accurately captured in a simple way <cit.>. However, unlike incompressible multiphase flow, a common occurrence when applying various interface-sharping methods explicitly on the SEF models is that velocity and pressure oscillations may occur across the interface <cit.> due to the inconsistency between the physical variables and the sharpened or compressed volume fraction field. As stated in <cit.>, in contrast to incompressible flows where density of fluid is fixed, artificial interface sharpening scheme cannot be applied alone to volume fraction function in compressible multiphase cases. In compressible multiphase flows, neither fluid density nor volume fraction alone is sufficient to determine the interface location and the fluid density. As some remedies, density correction equations are formulated in <cit.>. In <cit.>, a homogeneous reconstruction has been proposed where the reconstructed volume fraction is used to extrapolate the remaining conservative variables across the interface to ensure the thermal and mechanical consistency across the isolated material interfaces. In <cit.>, a consistent compression method has been discussed to maintain equilibrium across interfaces. To alleviate the defect of shock capturing scheme when solving single-fluid model for compressible multiphase flows, a novel spatial reconstruction is presented in this work to resolve contact discontinuities including material interfaces with substantially reduced numerical dissipation, which then maintains the sharpness of the transition layer of material interfaces throughout even long term computations. The scheme, so-called MUSCL-THINC-BVD, implements the boundary variation diminishing (BVD) algorithm <cit.> with the traditional MUSCL scheme and the interface-sharpening THINC scheme as two building-blocks for reconstruction. The BVD algorithm choose a reconstruction function between MUSCL and THINC, so as to minimize the variations (jumps) of the reconstructed variables at cell boundaries, which in turn effectively removes the numerical dissipations in numerical solutions. More importantly, we apply MUSCL-THINC-BVD scheme to all state variables and volume fraction, so sound consistency is achieved among volume fraction and other physical variables. Resultantly, the manipulations to the physical variables according to the volume fraction in other existing methods are not needed in the present method. The numerical model is formulated under a standard finite volume framework with a Riemann solver in the wave propagation form <cit.>. The numerical results of benchmark tests verify the capability of the present method in capturing the material interface as a well-defined sharp jump in volume fraction, as well as significant improvement in solution quality. The format of this paper is outlined as follows. In Section <ref>, the governing equations of the five-equation model and closure strategies are stated. In Section <ref>,after a brief review of the finite volume method in wave-propagation form for solving the quasi-conservative five-equation model, the details of the new MUSCL-THINC-BVD scheme for spatial reconstruction are presented.In Section <ref>, numerical results of benchmark tests are presented in comparison with other high-order methods. Some concluding remarks end the paper in Section <ref>.§ COMPUTATIONAL MODELS§.§ Governing equationsIn this work, the inviscid compressible two-component flows are formulated by the five-equation model developed in <cit.>. By assuming that the material interface is in mechanical equilibrium of mixed pressure and velocity, the five-equation model consists of two continuity equations forphasic mass, a momentum equation, an energy equation and an advection equation of volume fraction as follows∂/∂ t ( α_1ρ_1 ) +∇· ( α_1ρ_1𝐮 ) = 0, ∂/∂ t ( α_2ρ_2 ) +∇· ( α_2ρ_2𝐮 ) = 0, ∂/∂ t ( ρ𝐮 ) +∇· ( ρ𝐮⊗𝐮 ) + ∇ p= 0, ∂ E/∂ t+∇· ( E 𝐮 +p 𝐮 )= 0, ∂α_1/∂ t +𝐮·∇α_1=0,where ρ_k and α_k∈ [0,1]denote in turn the kth phasic density and volume fraction for k=1,2, 𝐮 the vector of particle velocity, p the mixture pressure and E the total energy. When considering more than two-phases, the five-equation model can be extended by supplementing additional continuity equations and volume fraction advection equations for each new phase.§.§ Closures strategyTo close the system, the fluid of each phase is assumed to satisfy the Mie-Grüneisen equation of state, p_k (ρ_k,e_k ) = p_∞,k(ρ_k) + ρ_kΓ_k(ρ_k) ( e_k- e_∞,k(ρ_k)),where Γ_k = (1/ρ_k)(∂ p_k/∂ e_k)|_ρ_k is the Grüneisen coefficient, and p_∞,k, e_∞,k are the properly chosen states of the pressure and internal energy along some reference curves (e.g., along an isentrope or other empirically fitting curves) in order to match the experimental data of the material examined <cit.>. Usually, parametersΓ_k, p_∞,k and e_∞,k can be taken as functions only of the density. This equation of state can be employed to approximate a wide variety of materials including some gaesous or solid explosives and solid metals under high pressure.The further closure of the system is completed by defining the mixed volume fraction, density andinternal energy as α_1+α_2=1,α_1ρ_1+α_2ρ_2=ρ,α_1ρ_1 e_1+α_2ρ_2 e_2=ρ e, Derived in <cit.>, under the isobaric assumption the mixture Grüneisen coefficient and p_∞,k, e_∞,k can be expressed asα_1/Γ_1(ρ_1)+α_2/Γ_2(ρ_2)=1/Γ,α_1ρ_1 e_∞,1(ρ_1)+α_2ρ_2 e_∞,2(ρ_2)=ρ e_∞,α_1p_∞,1(ρ_1)/Γ_1(ρ_1)+α_2p_∞,2(ρ_2)/Γ_2(ρ_2)= p_∞(ρ)/Γ(ρ).The mixture pressure is then calculated by p =( ρ e -∑_k=1^2α_kρ_k e_∞,k(ρ_k)+ ∑_k=1^2α_kp_∞,k(ρ_k)/Γ_k(ρ_k) ) / ∑_k=1^2α_k/Γ_k(ρ_k).It should be noted that the mixing rule of Eq.(<ref>) and Eq.(<ref>) ensure that the mixed pressure is free of spurious osclillations, which is particularly important to prevent the spurious pressure oscillations across the material interfaces <cit.>.§ NUMERICAL METHODS For the sake of simplicity, we introduce the numerical method in one dimension. Our numerical method can be extended to the multidimensions on structured grids directly in dimension-wise reconstruction fashion. We will first review the finite volume method in the wave propagation form<cit.> used in this work and then give details about the new MUSCL-THINC-BVD reconstruction scheme.§.§ Wave propagation methodWe rewrite the one dimensional quasi-conservative five-equation model(<ref>) as∂q/∂ t + ∂ f(q)/∂ x+B(q) ∂q/∂ x = 0,where the vectors of physical variables q and flux functions f areq=( α_1ρ_1, α_2ρ_2,ρ u, E, α_1 )^T,f=( α_1ρ_1 u, α_2ρ_2 u,ρ uu + p,Eu + pu, 0)^T,respectively. The matrix B is defined asB =( 0, 0, 0, 0, u),where u denotes the velocity component in x direction. We divide the computational domain into N non-overlapping cell elements, 𝒞_i: x ∈ [x_i-1/2,x_i+1/2 ], i=1,2,…,N, with a uniform grid with the spacing Δ x=x_i+1/2-x_i-1/2. For a standard finite volume method, the volume-integrated average value q̅_i(t) in the cell C_i is defined asq̅_i(t)≈1/Δ x∫_x_i-1/2^x_i+1/2q(x,t) dx.Denoting all the spatial discretization terms in (<ref>) byℒ(q̅(t)),the semi-discrete version of the finite volume formulationcan be expressed as a system of ordinary differential equations (ODEs)∂q̅(t)/∂ t=ℒ (q̅(t)).In the wave-propagation method,the spatial discretization for cell C_i is computed by ℒ( q̅_i(t)) = -1/Δ x (𝒜^+Δq_i-1/2 +𝒜^-Δq_i+1/2 + 𝒜Δq_i ) where𝒜^+Δq_i-1/2 and 𝒜^-Δq_i+1/2, are the right- and left-moving fluctuations, respectively, which enter into the grid cell, and 𝒜Δq_i is the total fluctuation within C_i. We need to solve Riemann problems to determine these fluctuations. The right- and left-moving fluctuations can be calculated by𝒜^±Δq_i-1/2 = ∑_k = 1^3 [s^k( q_i-1/2^L,q_i-1/2^R ) ]^±𝒲^k( q_i-1/2^L,q_i-1/2^R ), where moving speeds s^k and the jumps 𝒲^k (k=1,2,3) of three propagating discontinuities can be solved by Riemann solver <cit.> with the reconstructed values q_i-1/2^L and q_i-1/2^R computed from the reconstruction functions q̃_i-1(x) and q̃_i(x) to the left and right sides of cell edge x_i-1/2, respectively. Similarly, the total fluctuation can be determined by 𝒜Δq_i = ∑_k = 1^3 [s^k( q_i-1/2^R,q_i+1/2^L ) ]^±𝒲^k( q_i-1/2^R,q_i+1/2^L )We will describe with details about the reconstructions to get these values, q_i-1/2^L and q_i-1/2^R, at cell boundaries in next subsection as the core part of this paper. In practice, given the reconstructed values q_i-1/2^L and q_i-1/2^R,the minimum and maximum moving speeds s^1( q_i-1/2^L,q_i-1/2^R) and s^3( q_i-1/2^L,q_i-1/2^R) can be estimated by HLLC Riemann solver <cit.>ass^1=min{u_i-1/2^L-c_i-1/2^L,u_i-1/2^R-c_i-1/2^R},s^3=max{u_i-1/2^L+c_i-1/2^L,u_i-1/2^R+c_i-1/2^R},where c_i-1/2^L and c_i-1/2^R are sound speeds calculated by reconstructed values q_i-1/2^L and q_i-1/2^R respectively. Thenthespeed of the middle wave is estimated by s^2=p_i-1/2^R-p_i-1/2^L+ρ_i-1/2^L u_i-1/2^L (s^1-u_i-1/2^L)-ρ_i-1/2^R u_i-1/2^R (s^3-u_i-1/2^R)ρ_i-1/2^L (s^1-u_i-1/2^L)-ρ_i-1/2^R (s^3-u_i-1/2^R).The left-side intermediate state variables q_i-1/2^*L is evaluated byq_i-1/2^*L=(u_i-1/2^L-s^1) q_i-1/2^L+(p_i-1/2^Ln_i-1/2^L-p_i-1/2^*n_i-1/2^* )s^2-s^1where the vector n_i-1/2^L=(0,0,1,u_i-1/2^L,0), n_i-1/2^*=(0,0,1,s^2,0) and the intermediate pressure may be estimated asp_i-1/2^*=ρ_i-1/2^L (u_i-1/2^L-s^1)(u_i-1/2^L-s^2)+p_i-1/2^L=ρ_i-1/2^R (u_i-1/2^R-s^1)(u_i-1/2^R-s^2)+p_i-1/2^R. Analogously, the right-side intermediate state variables q_i-1/2^*R is q_i-1/2^*R=(u_i-1/2^R-s^3) q_i-1/2^R+(p_i-1/2^Rn_i-1/2^R-p_i-1/2^*n_i-1/2^* )s^2-s^3 Then we calculate the jumps 𝒲^k( q_i-1/2^R,q_i+1/2^L) as𝒲^1=q_i-1/2^*L-q_i-1/2^L, 𝒲^2=q_i-1/2^*R-q_i-1/2^*L, 𝒲^3=q_i-1/2^R-q_i-1/2^*R. Given the spatial discretization, we employ three-stage third-order SSP (Strong Stability-Preserving) Runge-Kutta scheme <cit.>q̅^∗= q̅^n +Δ t ℒ( q̅^n ), q̅^∗∗= 3/4q̅^n +1/4q̅^∗ + 1/4Δ t ℒ ( q̅^∗ ), q̅^n+1= 1/3q̅^n + 2/3q̅^∗ + 2/3Δ t ℒ ( q̅^∗∗ ), to solve the time evolution ODEs, where q̅^∗ and q̅^∗∗ denote the intermediate values at the sub-steps. §.§ MUSCL-THINC-BVD reconstructionIn the previous subsection, we left the boundary values, q_i-1/2^L and q_i-1/2^R, to be determined, which are presented in this subsection.We denote any single variable for reconstruction by q , which can be primitive variable, conservative variable or characteristic variable. The values q_i-1/2^L and q_i+1/2^R at cell boundaries are computed from the piecewise reconstruction functions q̃_i(x) in cell C_i. In the present work, we designed theMUSCL-THINC-BVD reconstruction scheme to capture both smooth and nonsmooth solutions. The BVD algorithm makes use of the MUSCL scheme <cit.> and the THINC scheme <cit.> as the candidates for spatial reconstruction. In the MUSCL scheme,a piecewise linear function is constructed from the volume-integrated average values q̅_i, which readsq̃_i(x)^MUSCL=q̅_i+σ_i(x-x_i) where x ∈ [x_i-1/2,x_i+1/2] and σ_i is the slope defined at the cell center x_i=1/2(x_i-1/2+x_i+1/2). To prevent numerical oscillation, a slope limiter <cit.> is used to get numerical solutionssatisfying the TVD property. We denote the reconstructed value at cell boundaries from MUSCL reconstruction as q_i-1/2^R,MUSCL and q_i+1/2^L,MUSCL.The MUSCL scheme, in spite of popular use in various numerical models, has excessive numerical dissipation and tends to smear out flow structures, which might be a fatal drawback in simulating interfacial multiphase flows. Being another reconstruction candidate, the THINC <cit.> uses the hyperbolic tangent function, which is a differentiable and monotone function that fits well a step-like discontinuity. The THINC reconstruction function is written asq̃_i(x)^THINC=q̅_min+q̅_max2(1+θ tanh(β(x-x_i-1/2x_i+1/2-x_i-1/2-x̃_i))), where q̅_min=min(q̅_i-1,q̅_i+1), q̅_max=max(q̅_i-1,q̅_i+1)-q̅_min and θ=sgn(q̅_i+1-q̅_i-1). The jump thickness is controlled by parameter β. In our numerical tests shown later a constant value of β=1.6 is used. The unknown x̃_i, which represents the location of the jump center, is computed from q̅_i = 1/Δ x∫_x_i-1/2^x_i+1/2q̃_i(x)^THINCdx. Then the reconstructed values at cell boundaries by THINC function can be expressed by q_i+1/2^L,THINC=q̅_min+q̅_max2(1+θtanh(β)+A1+A tanh(β))q_i-1/2^R,THINC=q̅_min+q̅_max2(1+θ A)where A=B/cosh(β)-1/tanh(β) and B=exp(θ β(2 C-1)), where C=q̅_i-q̅_min+ϵq̅_max+ϵ and ϵ=10^-20 is a mapping factor to project the physical fields onto [0,1].The final effective reconstruction function is determined by the BVD algorithm <cit.>, which choose the reconstruction function between q̃_i(x)^MUSCL and q̃_i(x)^THINC so that the variations of the reconstructed values at cell boundaries are minimized. BVD algorithm prefers the THINC reconstruction q̃_i(x)^THINC within a cell where a discontinuity exists. It is sensible that the THINC reconstruction should only be employed when a discontinuity is detected. In practice, a cell where a discontinuity may exist can be identified by the following conditionsδ<C<1-δ,(q̅_i+1-q̅_i)(q̅_i-q̅_i-1)>0,where δ is a small positive (e.g.,10^-4). In summary, the effective reconstruction function of MUSCL-THINC-BVD scheme readsq̃_i(x)^BVDl={[ q̃_i(x)^THINC if δ<C<1-δ, and (q̅_i+1-q̅_i)(q̅_i-q̅_i-1)>0, and TBV_i,min^THINC<TBV_i,min^MUSCL;q̃_i(x)^MUSCL otherwise ].. where the minimum value of total boundary variation (TBV) TBV_i,min^P, for reconstruction function P =THINC or MUSCL, is defined as TBV_i,min^P=min( |q_i-1/2^L,MUSCL-q_i-1/2^R,P|+|q_i+1/2^L,P-q_i+1/2^R,MUSCL|,|q_i-1/2^L,THINC-q_i-1/2^R,P|+|q_i+1/2^L,P-q_i+1/2^R,THINC|,|q_i-1/2^L,MUSCL-q_i-1/2^R,P|+|q_i+1/2^L,P-q_i+1/2^R,THINC| ,|q_i-1/2^L,THINC-q_i-1/2^R,P|+|q_i+1/2^L,P-q_i+1/2^R,MUSCL|) . Thus, THINC reconstruction function will be employed in the targeted cell if the minimum TBV value of THINC is smaller than that of MUSCL. In Fig. <ref>, we illustrate one possible situation corresponding to |q_i-1/2^L,MUSCL-q_i-1/2^R,THINC|+|q_i+1/2^L,THINC-q_i+1/2^R,MUSCL| when evaluating TBV_i,min^THINC. As stated in <cit.>, the BVD algorithm will realize the polynomial interpolation for smooth solution while for discontinuous solution a step like function will be preferred. As shown in numerical tests in this paper, discontinuities including material interface can be resolved by the MUSCL-THINC-BVD scheme with substantially reduced numerical dissipation in comparison with other existing methods. The material interface can be captured sharply and any extra step, like anti-diffusion or other artificial interface sharpening techniques used in the existing works <cit.>, is not needed here.More importantly, theMUSCL-THINC-BVD scheme is applied not only the volume fraction but also to other all physical variables, which automatically leads to the consistent reconstructions among the physical fields. As observed in our numerical results, no suprious numerical oscillation is generated in vicinity of material interfaces.It is usualy not trival to other anti-diffusion or artificial compression methods aforementioned.For example, in <cit.> anti-diffusion post-processing steps are required to adjust other state variables across interfaces to get around the oscillations. As discussed in <cit.>, when high-order reconstructions, such as MUSCL or WENO shemes, are applied, special attention must be paid to decide which physical variables should be reconstructed. It is concluded that one should implement high-order reconstructions to primitive variables or characteristic variables to prevent numerical oscillations in velocity and pressure across material interfaces. However, reconstructing conservative variables or flux functions may cause numerical oscillations. We show in the rest part of this section that THINC reconstruction ensures the consistency among the reconstructed variables across material interface even if the reconstruction is conducted for the conservative variables.We consider one-dimensional interface only problem where initial condition consists of constant velocity u=u_0, uniform pressure p=p_0 and constant phasic densities ρ_1=ρ_10 and ρ_2=ρ_20. Across a material interface, other variables,such as mixture densities ρ, mass fraction α_1ρ_1 and volume fraction α_1 have jumps. Without loss of generality, a positive velocity u=u_0>0 is considered here. Then the fluctuations for cell C_i can be calculated as𝒜^-Δq_i+1/2= 0, 𝒜^+Δq_i-1/2=u_0 ( q_i-1/2^R - q_i-1/2^L ), 𝒜Δq_i=u_0 ( q_i+1/2^L - q_i-1/2^R ).By using Eq. (<ref>) and Euler one-step forward time scheme, the cell average q̅_i^n can be updated by q̅_i^n+1 = q̅_i^n -Δ t/Δ x u_0 ( q_i+1/2^L - q_i-1/2^L ),or a component form, [ α_1ρ_1 α_2ρ_2 ρ u Eα_1 ]_i^n+1 =[ α_1ρ_1 α_2ρ_2 ρ u Eα_1 ]_i^n - Δ t/Δ x u_0 [( α_1ρ_1 )_i+1/2^L-( α_1ρ_1 )_i-1/2^L( α_2ρ_2 )_i+1/2^L-( α_2ρ_2 )_i-1/2^Lu_0 ( ρ_i+1/2^L-ρ_i-1/2^L ) E_i+1/2^L-E_i-1/2^L(α_1 )_i+1/2^L-(α_1 )_i-1/2^L ].We denote the reconstruction operator to compute q_i+1/2^L by𝒟_i+1/2(q)^L. As shown below, when we implement the reconstructions to the conservative variables, spurious oscillations may be generated in velocity u. To facilitate discussions, we address theconsistency in velocity u byDefinition 1. The reconstruction is u-consistent if the numerical results from (<ref>) satisfy u^n+1=u^n for isolated material interface where velocity u and pressure p are uniform.In the present model, in order to calculate velocity u_i^n+1, we first compute density from the two conservation equations for phasic densities, ρ_i^n+1=ρ_i^n-Δ t/Δ xu_0((𝒟_i+1/2(α_1ρ_1)^L+𝒟_i+1/2(α_2ρ_2)^L)-(𝒟_i-1/2(α_1ρ_1)^L+𝒟_i-1/2(α_2ρ_2)^L)).From the momentum equation, we have (ρ u)_i^n+1=(ρ u)_i^n-Δ t/Δ xu_0(𝒟_i+1/2(ρ u)^L-𝒟_i-1/2(ρ u)^L)=ρ_i^n u_0-Δ t/Δ xu_0u_0(𝒟_i+1/2(ρ)^L-𝒟_i-1/2(ρ)^L),where we assume the operator should satisfy 𝒟_j(mX)=m𝒟_j(X) in which X represents the reconstructed variable, m is constant, then the velocity u^n+1 is retrieved by u_i^n+1=u_0ρ_i^n-Δ t/Δ xu_0(𝒟_i+1/2(ρ)^L-𝒟_i-1/2(ρ)^L)ρ_i^n-Δ t/Δ xu_0((𝒟_i+1/2(α_1ρ_1)^L+𝒟_i+1/2(α_2ρ_2)^L)-(𝒟_i-1/2(α_1ρ_1)^L+𝒟_i-1/2(α_2ρ_2)^L)).It is clear thatthe u-consistent condition requires 𝒟_i±1/2(ρ)^L=(𝒟_i±1/2(α_1ρ_1)^L+𝒟_i±1/2(α_2ρ_2)^L) to maintain u_i^n+1=u_0.I.e. the reconstruction operator 𝒟_i±1/2(ρ )^L should be consistent with 𝒟_i±1/2(α_1ρ_1)^L and 𝒟_i±1/2(α_2ρ_2)^L at cell faces i±1/2 to ensure ρ_i±1/2^L=(α_1ρ_1)_i±1/2^L+(α_2ρ_2)_i±1/2^L. Concerning u-consistent property, we have Proposition 1. All schemes satisfy u-consistent condition if reconstruction are conducted in terms of primitive variables.Proof. It is straightforward that u^n+1=u^n since the reconstruction is conducted with u, which has also been discussed in <cit.>. □Proposition 2. Piecewise constant reconstruction scheme for conservative variables satisfy u-consistent condition.Proof. For piecewise constant reconstruction, it is obvious that above condition (<ref>) can be satisfied, as ρ_i-1/2^L=ρ_i-1=(α_1ρ_1)_i-1+(α_2ρ_2)_i-1=(α_1ρ_1)_i-1/2^L+(α_2ρ_2)_i-1/2^L and ρ_i+1/2^L=ρ_i=(α_1ρ_1)_i+(α_2ρ_2)_i=(α_1ρ_1)_i+1/2^L+(α_2ρ_2)_i+1/2^L.□ Proposition 3. All linear reconstruction schemes for conservative variables satisfy u-consistent condition.Proof. For general linear schemes, the reconstructed values at cell boundaries can be expressed as linear combinations of the cell average values, q̅_l, on the stencils, 𝒞_l,l=i-l',⋯, i+l”. That is𝒟_i+1/2(q)^L=∑_lχ_lq̅_l,where the coefficients χ_l are the same for ρ, α_1ρ_1 and α_2ρ_2. With the conclusion in proposition 2given above for piecewise constant reconstruction, we know that 𝒟_i+1/2(ρ)^L=(𝒟_i+1/2(α_1ρ_1)^L+𝒟_i+1/2(α_2ρ_2)^L), which leads to u_i^n+1=u_0 from Eq. (<ref>). □Proposition 4. Non-linear schemes for conservative variables may not satisfy u-consistent condition.Proof. Taking 5th order WENO scheme as an example, the reconstructed values at cell faces are a combination of three third order linear schemes through nonlinear weights. The reconstructed value can be expressed asq_i+1/2^L=∑_k=1^3w_i+1/2^(k)(q)^L𝒟_i+1/2^(k)(q)^L,where 𝒟_i+1/2^(1)(q)^L is the third order approximation on stencil {𝒞_i-2, 𝒞_i-1, 𝒞_i}, 𝒟_i+1/2^(2)(q)^L on {𝒞_i-1, 𝒞_i, 𝒞_i+1} and 𝒟_i+1/2^(3)(q)^L on {C_i, C_i+1, C_i+2}. w_i+1/2^(k)(q)^L is the nonlinear weight corresponding to the 𝒟_i+1/2^(k)(q)^L. To satisfy u-consistent condition, it is necessary that w_i+1/2^(k)(α_1ρ_1)^L=w_i+1/2^(k)(α_2ρ_2)^L=w_i+1/2^(k)(ρ u)^L.Because the nonlinear weights are separately determined according the smoothness of the reconstructed variables, there is no guarantee that Eq. (<ref>) always holds.Thus,WENO scheme is not u-consistent when applied to conservative variables, which has also been reported in <cit.>. This observation applies to all high-resolution schemes using nonlinear weights to suppress numerical oscillations. □Proposition 5. THINC reconstruction satisfies u-consistent condition.Proof. We consider a material interface in cell C_i with volume fraction (α_1)_i=λ(α_1)_i-1+(1-λ)(α_1)_i+1, 0<λ<1, which divides the cell into two regions sharing uniform physical properties with the neighboring cells respectively.Then, we have (ρ u)_i=λ(ρ u)_i-1+(1-λ)(ρ u)_i+1.Without loss of generality, we assume (α_1)_i-1>(α_1)_i+1 and ρ_1>ρ_2, the reconstructed values of phasic densities at cell face i+1/2 are( α_1ρ_1)_i+1/2^L=(α_1ρ_1)_i+1+(α_1ρ_1)_i-1-(α_1ρ_1)_i+12(1- tanh(β)+A_11+A_1 tanh(β)); ( α_2ρ_2)_i+1/2^L=(α_2ρ_2)_i-1+(α_2ρ_2)_i+1-(α_2ρ_2)_i-12(1+ tanh(β)+A_21+A_2 tanh(β)).Recall (<ref>), we getC_1=λ and C_2=(1-λ), which leads to B_1=exp(β(1-2λ))=B_2 andA_1=A_2=A. From the momentum equation, we haveρ_i+1/2^L=ρ_i+1+ρ_i-1-ρ_i+12(1- tanh(β)+A_31+A_3 tanh(β)),Again we can find A_3=A. Then, we finally getρ_i+1/2^L=(α_1ρ_1)_i+1/2^L+(α_2ρ_2)_i+1/2^L.Thus, u-consistent condition is satisfied. □We remark that the conclusion of proposition 4 applies to any reconstruction function if it is used in exactly the same form to different reconstructed fields. Being an extreme case, when β is large enough, the THINC reconstruction build up two piecewise-constant states in the cell where discontinuities exist. It might be of physical importance in applications. To verify that THINC scheme satisfies u-consistent condition even used to reconstruct the conservative variables, we present the numerical results to the isolated interface problem similar to <cit.>, where the initial condition is set as follows(α_1ρ_1, α_2ρ_2, u_0, p_0, α_1, γ)={[ (10, 0, 0.5, 1/1.4, 1, 1.6)for 0.3 ≤ x < 0.7; (0, 0.5, 0.5, 1/1.4, 0, 1.4)else ]..The computational domain is [0,1]. We computed the same test with MUSCL<cit.>, WENO<cit.> and THINC scheme respectively for comparison. Reconstruction is conducted in terms of the conservative variables. Figure 2 shows the numerical results of WENO and THINC at time t=0.1 using a 200-cell mesh with the Courant-Friedrichs-Lewy (CFL) condition CFL = 0.5. From the results, we can see WENO scheme produces numerical oscillations due to its non-linear property. We also compare the results computed from MUSCL and THINC in Figure <ref>. MUSCL scheme also produces numerical oscillations when reconstructing with conservative variables. The numerical oscillations generated from the MUSCL reconstruction are much smaller than that from the WENO scheme. Previous works suggested the use of primitive variables or characteristic variables for reconstruction, such as <cit.> for MUSCL and<cit.> for WENO. The THINC reconstruction provides consistent reconstruction for all types of variables, which indicates more possibility in applications.§ NUMERICAL RESULTS Comparative tests in one- and two- dimensions are conducted in this section with WENO scheme and the proposed MUSCL-THINC-BVD scheme. Here we use the WENO scheme in <cit.> which is one of representative high order shock-capturing schemes. We denote it as WENO-JS in our tests. In order to reduce the numerical oscillations, the WENO reconstruction should be implemented for characteristic fields as <cit.>.The one dimension tests were conducted with a single CPU (Intel(R) Xeon(R) CPU E5-2687W, 3.10GHZ), while two dimensional tests were conducted with a NVIDIA GTX980ti GPU.§.§ Passive advection of a square liquid column To evaluate the ability of the proposed scheme to capture interface as well as to maintain the equilibrium of velocity and pressure fields, a simple interface-only problem in one dimension is considered in this test. The problem consists of a square liquid columnin gas transported with a uniform velocity u = u_0 = 10^2 m/s under equilibrium pressure p= p_0 =10^5 Pa in a shock tube of one meter. For initial condition, liquid is set in the region of x ∈ [0.4, 0.6]mand gas is filled elsewhere.We set initially the volume fraction of liquidα_1 =1-ϵ for the liquid region and α_1 =ϵ in the gas region, and the volume fraction of gas is then α_2 =1- α_1. The small positiveϵ is set 10^-8 in numerical tests in this paper. The densities for the liquid and gas phases are ρ_1= 10^3kg/m^3 and ρ_2 = 1kg/m^3, respectively. To model the thermodynamic behavior of liquid and gas, we use the stiffened gas equation of state where the material-dependent functions appeared in (<ref>) areΓ_k = γ_k -1,p_∞,k = γ_kℬ_k,e_∞,k = 0,with the parameter values taken in turn to be γ_1 = 4.4, ℬ_1 = 6 × 10^8Pa, and γ_2 = 1.4, ℬ_2 = 0 for the liquid and gas phases.The computations using WENO-JS and MUSCL-THINC-BVD are carried out respectively. Periodic boundary condition is used on the left and right boundaries during the computations. Figure <ref> shows numerical results ofpartial density and pressure fieldsat time t=10ms using a 200-cell mesh with CFL = 0.5.It is obvious that MUSCl-BVD can solve the sharp interface within only two cells while WENO scheme, in spite ofhigh-order accuracy, excessively diffuses the interface due to the intrinsic numerical dissipation as other conventional shock capturing schemes. Meanwhile, MUSCL-THINC-BVD can retain the correct pressure equilibrium and particle velocity without introducing spurious oscillations across the interfaces. We have not conducted any extra procedures to sharpen the interface, which are used in other existing works to keep the steepness of the jump in volume fraction field to identify the interface. TheMUSCL-THINC-BVD reconstruction is implemented to all state variables, which remains the thermo-dynamical consistency among the physical fields. We also compare the computational cost in Table <ref>. Since it is not necessary to cast state variables to characteristic fields, the computation cost of MUSCL-THINC-BVD is about half of the WENO scheme. §.§ Two-material impact problem Following<cit.>, we computed the two-phase impact benchmark problem. At the beginning, there is a right-moving copper (phase 1) plate with the speed u_1=1500 m/s interacting with a solid explosive (phase 2)at rest on the right of the plate under the uniform atmospheric condition which has pressure p_0 = 10^5 Pa and temperature T_0 = 300K throughout the domain.The material properties of the copper and (solid) explosive are modeled bythe Cochran-Chan equation of state where in (<ref>)we set the same Γ_k as in the stiffened gas case, but with p_∞,k, e_∞,k defined byp_∞,k(ρ_k) =ℬ_1k ( ρ_0k/ρ_k )^-ℰ_1k - ℬ_2k ( ρ_0k/ρ_k )^-ℰ_2k, e_∞,k(ρ_k) = -ℬ_1k/ρ_0k ( 1-ℰ_1k )[( ρ_0k/ρ_k )^1-ℰ_1k -0.2in - 1] +0.6in ℬ_2k/ρ_0k ( 1-ℰ_2k ) [( ρ_0k/ρ_k )^1-ℰ_2k -0.2in - 1]- C_vkT_0.Here γ_k, ℬ_1k, ℬ_2k, ℰ_1k, ℰ_2k, C_vk, and ρ_0k are material-dependent quantities, see Table <ref> for a typical set of numerical values for copper and explosive considered.The solution of this test is characterized by a left-moving shock wave to the copper, a right-moving shock waves to the inert explosive, and a material interface lying in between that separates these two different materials. We run this problem with a 200-cell grid and CFL=0.5 up tot= 85 μs.Figure <ref> shows the resultsfor the partial densities, velocity, and the copper volume fraction of bothWENO and MUSCL-THINC-BVD for comparison. Again, MUSCL-THINC-BVD can keep sharp interface without spurious numerical oscillation in velocity fields. It should be noted that due to complicated state equations, characteristic decomposition is conducted as in <cit.> when implementing WENO scheme. In previous work <cit.>, there is a slight overshoot on the partial densityα_1ρ_1 on the left of the interface when using THINC method for the volume fraction. This oscillation is not observed in present study due to the global consistency in MUSCL-THINC-BVD reconstructions for all physical fields. §.§ Shock interface interaction problem The interaction between a strong shock wave in helium and an air/helium interface has been studied. Typically, such problem is very challenging for some interface tracking methods. For example, the schemes which are not conservative on discrete level may miscalculate the position and speed of the waves resulted from the interaction <cit.>. The initial problem is set the same as <cit.>, where a Mach 8.96 shock wave is traveling in helium toward a material interface with air which is moving toward the shock wave simultaneously. The detail initial configuration is given by(α_1ρ_1, α_2ρ_2, u_0, p_0, α_1)={[ (0.386, 0, 26.59, 100, 1)for -1 ≤ x < -0.8; (0.1, 0, -0.5, 1, 1) for -0.8 ≤ x < -0.2; (0, 1, -0.5, 1, 0)for -0.2 ≤ x < 1 ]..The calculation domain is [-1,1] which is divided by 200 uniform mesh cells. The solutions at t=0.07 were computed with the CFL number of 0.1. The comparisons of numerical results between MUSCL-THINC-BVD and WENO schemes are presented in Figure <ref>. The results from MUSCL-THINC-BVD show much superior solution quality in resolving material interface without obvious numerical oscillations, while some oscillations are observed in the pressure and velocity fields by in the results of WENO scheme in the region of the reflected shock wave even althoughefforts have been made to implement reconstructions to characteristic variables <cit.>.§.§ Shock-bubble interactionIn this widely used benchmark test <cit.>, we investigate the interactions between ashock and a bubble which involves a shock wave of Mach 1.22 in air impacting a cylindrical bubble of refrigerant-22 (R22) gas. The experimental results can be referred in <cit.>. A planar rightward-moving Mach 1.22 shock wave in air impacts a stationary R22 gas bubble with radius r_0 = 25mm.The numerical test, both the air and R22 are modeled as perfect gases. Inside the R22 gas bubble, the state variables are (ρ_1,ρ_2, u,v, p,α_1 )= ( 3.863 ^3,1.225 ^3, 0, 0, 1.01325× 10^5 ,1-ε),while outside the bubble the corresponding parameters are (ρ_1, ρ_2, u, v, p, α_1 ) =( 3.863 ^3, 1.225 ^3, 0,0,1.01325× 10^5 , ε)and(ρ_1, ρ_2, u, v, p, α_1 )= (3.863 ^3, 1.686 ^3, 113.5 , 0, 1.59 × 10^5 , ε)in the pre- and post- shock regions, respectively, where ε=10^-8. The mesh size is Δ x = Δ y =1/8mm which corresponds to a grid-resolution of 400 cells across the bubble diameter.Zero-gradient boundary conditions are imposed at the left and right boundaries while symmetric boundaries are imposed at the top and bottom boundaries. Schlieren-type images of density gradient, |∇ρ|, at different time instants are presented in Figs.<ref>-<ref>, in which comparisons are made among WENO, MUSCL and MUSCL-THINC-BVD schemes. The MUSCL-THINC-BVD scheme maintains much better the compact thickness of the material interfaces and gives large-scale flow structures similar to the results computed from WENO and MUSCL schemes. Moreover, MUSCL-THINC-BVD scheme is able to reproduce finer flow structures due to largely reduced numerical dissipation.For example, the instability develops along the interface, which thenrolls up and produces small filamentsas shown in Figure <ref>. These fine structures in flow and interface tend to be smeared out by numerical schemes with excessive numerical dissipation <cit.> unless high-resolution meshes are used.Not only the well-resolved material interface, we can also observe that the reflected shock waves and transmitted shock waves can be captured more clearly compared with the original MUSCL schemes and competitive toWENO shock-capturing scheme. The resolution quality has been improved remarkably by MUSCL-THINC-BVD scheme to reproduce the complexflow features which are easily diffused out by conventional shock capturing schemes. We make comparisons further with published works which were computed on much finer grid. Shown in Fig. <ref> and <ref> we plot our results on a coarse mesh where the diameter uses 400 cells to compare with the results computed by anti-diffusion interface sharpening technique <cit.> and multi-scale sharp interface <cit.> on a finer grid where 1150 cell were used for the bubble diameter. From Fig. <ref>, it can be observed that similar small-scale structures have been recovered by the MUSCl-THINC-BVD scheme with much fewer cells. § CONCLUSION REMARKS In this work, we implement MUSCL-THINC-BVD scheme to simulate compressible multiphase flows by solving the five-equation model. This scheme can resolve discontinuous solutions with much less numerical dissipation. By treating interface as another contact discontinuity rather than implementing interface-sharping techniques explicitly, the new scheme can realize thermodynamical-consistent reconstruction straightforwardly. The results of test cases show a remarkable improvement in the solution quality to the problems of interest. Compared with the high-order shock-capturing schemes, the new scheme shows competitive or even better numerical results but with less computational cost. This work provides an effective but simple approach to simulate compressible interfacial multiphase flows.§ ACKNOWLEDGMENT This work was supported in part by JSPS KAKENHI Grant Numbers 15H03916 and 15J09915. 10trg-bookTryggvason, Grétar, Ruben Scardovelli, and Stéphane Zaleski. Direct numerical simulations of gas-liquid multiphase flows. 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Journal of Computational Physics 185.1 (2003): 120-138.shyue:jcp06 Shyue, Keh-Ming. “A wave-propagation based volume tracking method for compressible multicomponent flow in two space dimensions." Journal of Computational Physics 215.1 (2006): 219-244.hanHan, L. H., X. Y. Hu, and N. A. Adams. “Adaptive multi-resolution method for compressible multi-phase flows with sharp interface model and pyramid data structure." Journal of Computational Physics 262 (2014): 131-152. LuoLuo, Jun, X. Y. Hu, and N. A. Adams. “Efficient formulation of scale separation for multi-scale modeling of interfacial flows." Journal of Computational Physics 308 (2016): 411-420. HaasHaas, J-F., and Bradford Sturtevant. “Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities." Journal of Fluid Mechanics 181 (1987): 41-76. | http://arxiv.org/abs/1704.08041v2 | {
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APS/123-QED [email protected] Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, DenmarkNiels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark[Also at ]SPOC, Technical University of Denmark,DTU Fotonik, Ørsteds Plads, building 343, 2800 Kgs. Lyngby, Denmark Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, DenmarkNiels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, DenmarkHybrid systems of cold atoms and optical cavities are promising systems for increasing the stability of laser oscillators used in quantum metrology and atomic clocks. In this paper we map out the atom-cavity dynamics in such a system and demonstrate limitations as well as robustness of the approach. We investigate the phase response of an ensemble of cold strontium-88 atoms inside an optical cavity for use as an error signal in laser frequency stabilization. With this system we realize a regime where the high atomic phase-shift limits the dynamical locking range. The limitation is caused by the cavity transfer function relating input field to output field. However, the cavity dynamics is shown to have only little influence on the prospects for laser stabilization making the system robust towards cavity fluctuations and ideal for the improvement of future narrow linewidth lasers. 37.30.+i,06.30.Ft,42.50.Ct,42.62.Fi Dynamics of bad-cavity enhanced interaction with cold Sr atoms for laser stabilization J. W. Thomsen====================================================================================== § INTRODUCTIONOptical atomic clocks have undergone an immense development, and are continuously improving, with increased stability and accuracy every year <cit.>. The ability to reach exceedingly high accuracies within a reasonable time is made possible by the correspondingly huge effort to bring down the frequency noise in ultra stable laser sources <cit.>.The full potential of the high Q factor atomic transitions used in many optical atomic clocks can be reached only through improvements in the stability of the interrogation laser. Traditionally such interrogation lasers are stabilized to highly isolated optical reference cavities. This stabilization method is mainly limited by thermal fluctuations in the optical coating, mirror substrate and cavity spacer <cit.> demanding considerable experimental effort in order to construct cryogenically cooled mono-crystalline cavities and crystalline mirror coatings <cit.>. Several new approaches are being pursued in the so-called bad cavity regime <cit.>, in order to significantly suppress thermally induced length fluctuations. They use a combination of narrow linewidth ν atoms and optical cavities. These atomic systems have transitions at optical frequencies ν, with strongly forbidden transitions resulting in high Q factors, Q=ν/ν. By exploiting the high Q factor of the atomic transitions and using cavities with comparatively low Q factors the systems are far less sensitive to thermal fluctuations of the cavity components, and the experimental requirements are simplified. In these approaches active as well as passive atomic systems have been suggested <cit.>. The active atomic systems are optical equivalents of the maser, relying on co-operative quantum phenomena such as superradiance or superfluorescence of atoms inside the cavity mode. Several pioneering experiments have already demonstrated lasing under such conditions <cit.>. In the passive approach the atom-cavity system is used as a reference for laser stabilization where the narrow linewidth atomic transitions are interrogated inside an optical cavity. One proof-of-principle approach to this is based on using the NICE-OHMS technique <cit.> for generating sub-Doppler dispersion signals <cit.>. This has shown promising results for laser stabilization that could be able to compete with traditional cavity stabilization techniques <cit.>. By employing an optical cavity the coupling between atoms and optical field is improved by a factor of the cavity finesse, which significantly increases the total phase-shift experienced by the optical field. As the total phase-shift is increased, however, this limits the frequency range of linear behavior and thus the dynamical range of a servo locking the laser frequency. Additionally, the cavity servo response time might limit the signal quality if the condition of constant laser-to-cavity resonance must be strictly met.In this paper we show experimentally that the large total phase shift of the system not only improves the resonance slope, but also distorts the dispersion signal off atomic resonance. This becomes relevant for the interest of servo optimization in such a system <cit.> as it can limit the dynamical range of a servo lock. We show that this distortion originates from the transfer function of the cavity itself, and thus cannot be circumvented. We have realized a system with a theoretically attainable shot noise limited laser linewidth of Δν≈40 mHz, possibly allowing laser performance at the level of the state-of-the-art reported values <cit.>. We use the system to map out the dynamical range and investigate the consequences of an imperfect cavity servo, which causes a mismatch of the cavity resonance with respect to the laser frequency. Due to the bad cavity regime much looser bounds on the cavity resonance are allowed, as expected. This opens the possibility of using cavities with quasi-stationary lengths, and simultaneously underlines the insensitivity to cavity fluctuations.§ EXPERIMENTAL SYSTEMThe experimental system investigated here consists of an ensemble of cold strontium-88 atoms cooled to a temperature of T ≈ 5 mK. The atoms are trapped in a Magneto-Optical Trap (MOT) at the center of a TEM_00 Gaussian mode of an optical cavity, see FIG. <ref>. The cavity has a finesse of F=1240 and a linewidth of κ=2π· 630 kHz at λ=689 nm. A laser beam probing the narrow (5s^2) ^1S_0 → (5s5p)^3P_1 transition of ^88Sr is coupled into the cavity mode, and the cavity resonance is locked to the probe laser frequency at all times.Before entering the cavity the probe light is phase-modulated using a fiber-coupled Electro-Optical Modulator (EOM) in order to perform heterodyne detection of the transmitted signal. The modulation frequency is equal to the free spectral range (FSR) of the cavity resulting in sidebands at ω_0 ± jΩ for integer j andΩ=2π· 781.14 MHz. The sidebands are far detuned with respect to the (5s^2) ^1S_0 → (5s5p)^3P_1 transition having a linewidth of γ_nat=2π·7.5 kHz, and the interaction between the sidebands and the atoms can thus be neglected. This system is interrogated using a heterodyne measurement (the NICE-OHMS technique <cit.>) in order to extract the dispersion signal of the atom-cavity system, which can be used as an error signal to lock the probe laser frequency to resonance with the atoms. We operate in the bad cavity regime where any cavity fluctuations are suppressed in the atom-cavity signal by a factor of κ/γ_nat, here about 100.The field transmitted through the cavity is split and simultaneously recorded on a low bandwidth (50 MHz) photodiode and a high bandwidth (1 GHz) avalanche photodetector (APD). The low bandwidth signal records the total transmission intensity of the cavity. The high bandwidth signal is filtered around the modulation frequency Ω and demodulated in order to record the atom induced phase shift of the sideband relative to the carrier frequencies.The measurements are performed in a cyclic operation as the intense cooling light of the MOT results in an AC Stark shift of the ^3P_1 level and washes out coherence of the probing transition. The cooling light is thus shut off before each measurement, and the probing light then recorded for an interrogation period of 100s. At this timescale the probing laser has a linewidth of Γ_l=2π·800 Hz which is much narrower than the natural linewidth of the probing transition γ_nat=2π·7.5 kHz. This transition linewidth places us deep in the bad cavity regime, where the cavity linewidth is much broader than the atomic linewidth κ≫γ. This means that the system is much less sensitive to variations in the cavity resonance frequency which can originate from, e.g., temperature fluctuations in the cavity componentsOnly a single measurement is performed before reloading the trap with new atoms, since atom loss due to the finite temperature of the atoms becomes measurable after 500s. This results in a cyclic operation where the dispersion is measured only for a single frequency detuning of the interrogation laser at a time. Varying the loading time of the MOT allows control over the atom number and typically ranges from 50 ms to 800 ms for intra-cavity numbers of N=2·10^6 to N=4·10^7.§ THEORY OF MEASUREMENTWe investigate theoretically a system consisting of an ensemble of N atoms coupled to a single mode of an optical cavity in order to describe the experimental system presented in this work. The NICE-OHMS technique as it is used here relies on the transmitted signal of the atom-cavity system, and is a heterodyne measurement between the carrier laser frequency and its sidebands. The input laser field before the cavity can then be described byE_in=E_0∑_j=-∞^∞J_j(y)e^i(ω_l+jΩ)t,where E_0 is the amplitude of the electric field, J_j(y) is the j'th order Bessel function of the first kind, with the modulation index y. The laser carrier frequency is ω_l, whereas Ω is the modulation frequency applied in the EOM.The interaction of the light with the atom-cavity system may be described by using a Born-Markov master equation as described in appendix <ref> following <cit.>. The approach is based on a many-particle Hamiltonian Ĥ and a derived set of complex Langevin equations that includes the Doppler effect from the finite velocity of the atoms.Classically we may relate the input and output fields with a complex transfer function χ[θ(E_in)]. The field-dependent complex atomic phase experienced by the light when interacting with the atom-cavity system θ(E_in) is found by means of the full quantum mechanical theory of appendix <ref>. In order to cast the behavior of our system in terms of measurable quantities, we assume that the relation between the quantum mechanical phase θ(E_in) and the measured output power can be described by a linear model such that E_out=χ(θ) E_in. We then insert the theoretical value of θ(E_in) into the transfer function.Here we are mainly interested in the properties of such a transfer function. By increasing the finesse of the cavity with respect to the numbers reported in <cit.> we enable the system to move between a low-phase-shift regime and a high-phase-shift regime.FIG. <ref> shows typical dispersion-scans where the theoretical model incorporating a cavity transfer function, see <cit.>, has been plotted using the known experimental parameters. The dispersion signal serves as an error signal for all values of the atom number N, but is distorted when detuned from resonance at higher values of N. This distortion is not caused by the atomic phase-response itself, but rather by the classical conditions of the transfer function imposed by the cavity. §.§ Dispersion signal Only a single frequency component of the modulated light, namely the carrier component j=0,interacts with the atoms. This means that we can simplify the description of our system by defining a transfer function for each frequency component j of the light as it passes through the cavity <cit.>χ_j = Te^iϕ_j/1-Re^2iϕ_j,where T (R) is the power transmission (reflectivity) of a single cavity mirror, and ϕ_j is the complex phase experienced by the j'th component of the interrogation laser. We assume identical mirrors with no losses. The real part of the transfer function corresponds to the transmitted amplitude of the E-field in the system, while the imaginary part corresponds to the dispersion. Due to energy conservation the absolute-squared value of the complex transfer function cannot exceed one, |χ|^2≤1, for a system with no gain or frequency conversions. This classical condition thus imposes a maximal value on the dispersion signal which is independent on the nature of the phase-delay inside the cavity. We can describe the complex phase for any sideband component as simply the phase-shift experienced by a single-passage interaction with the cavity ϕ_j=ϕ_cav^j for j≠0, while the carrier component of the light experiences the atomic phase as wellϕ_0=ϕ_cav^0 + ϕ_D + iϕ_A,where ϕ_D and ϕ_A are the phase components caused by atomic dispersion and absorption from a single passage of the cavity. In the case of a medium with no gain, we have ϕ_A ≥0. The cavity phase-shift is given by ϕ_cav^j=ϕ_cav + jπ, and the cavity locking conditions of the experiment defines ϕ_cav. The output field can now be expressed by a superposition of frequency components and corresponding transfer functionsE_out=E_0 ∑_j=-∞^∞ J_j(y)χ_j e^i(ω_l+jΩ)t,where E_0 contains any overall phase. By recording the intensity on a photodetector we can filter out the beat signal between sideband and carrier by demodulating at the modulation frequency Ω. By optimizing the phase of the demodulation signal to record the imaginary part of the transfer function and subsequently pass the signal through a 2 MHz low-pass filter we obtain a DC signalS_Ω∝ 2i|E_0|^2 J_0(y)J_1(y)(χ_0χ_1^*-χ_0^*χ_1),which is a purely real number. We have only included up to second order sidebands, and used χ_j=(-1)^|j|-1χ_1 for j≠0. Higher order sidebands are negligible for modulation indices up to y≃1.If we assume that the system is in a steady state the cavity locking condition dictates that the cavity is on resonance with the carrier frequency at all times, corresponding to that used in <cit.>. This gives us ϕ_cav + ϕ_D = nπfor integer n. The complex transfer function of the carrier then becomes solely dependent on the absorption χ_0 = Te^-ϕ_A/1-Re^-2ϕ_A,whereas the sideband transfer functions have the phase information of the atomic interaction written onto them by the cavity lockϕ_j = ϕ_cav^j =ϕ_cav + jπfor j≠0=nπ - ϕ_D + jπ.Ignoring an overall sign from e^i nπ we getχ_j = Te^i(jπ-ϕ_D)/1-Re^2i(jπ-ϕ_D)for j≠0.Since χ_0 is purely real we can write the signal asS_Ω∝ J_0(y)J_1(y)χ_0Im(χ_1).We are thus particularly interested in the properties of the imaginary part of the transfer function if we wish to understand the behavior of our error signal. §.§ Transfer function propertiesHaving understood the behavior of our system we can now investigate why we see the folding behavior depicted in FIG. <ref> b) and c). If we ignore the origin of the phases it is clear that a cavity transfer function such as the one in Eq. <ref> must have a periodicity of 2π as a function of the phase shift experienced by the light inside the cavity. In connection with locking of the laser to an atom-cavity system we are mainly interested in the phase slope around atomic resonance where the absolute phase is zero, but the phase slope can be very large. Very close to atomic resonance, the transfer function is proportional to sin(ϕ)≈ϕ <cit.> and we can treat the transfer function as linear in phase. For a slightly larger frequency detuning, however, the existence of a maximal value for the transfer function results in some interesting behavior for a system with large total phase shift. In FIG. <ref> the imaginary part of a phase-dependent transfer function χ_j is shown with varying single-passage phase-shift and mirror reflectivity R. We see that the imaginary part of the transfer function itself behaves dispersion-like for a linearly varying phase. In this figure we have assumed that there are no losses in the cavity mirrors (T+R=1) and that there is no absorption in the cavity Im[ϕ] = 0 which would not be the case close to an atomic resonance. If the effects of absorption in the medium is taken into account, this reduces the maximal value of transfer function |χ|_max further. For Im[ϕ] = ϕ_A > 0 we will thus have |χ|_max< 1 asymptotically decreasing towards zero as a function of ϕ_A. As an aside, including absorption also decreases the phase slope at resonance. This slope will nevertheless still increase linearly with atom number when the saturation condition is fulfilled.As the reflectivity of the mirrors (R) is increased the light is stored in the cavity for longer and thus experiences a larger total phase shift. This increases the phase slope on resonance proportionally to the finesse F of the cavity and in turn leads to a decrease in the phase range where the transfer function χ is linear, see FIG. <ref>. This insight tells us that the dispersion signal observed from the atom-cavity transfer function will be distorted and even change the sign of the slope for detunings at which the values of the total phase shift is large.This sets a limit to the maximal dynamical range that we can expect of a locking mechanism based on this dispersion signal S_Ω. It results in an inversion of the dispersion slope for large absolute phase-shifts. Here the boundaries on the transfer function act to fold down the signal in a non-linear manner. While the sign of the slope is thus inverted the sign of the signal itself never changes with respect to that of the phase. The linear-phase regime decreases in size linearly towards zero as a function of the mirror reflectivity R in the regime where the cavity linewidth κ≪FSR (F ≫ 1). A maximal dynamical range of ϕ=π is reached for R≲0.17. For systems with much larger atom number (and thus larger phase-shift) it could thus be an advantage to go towards lower mirror reflectivity, and thus deeper into the bad-cavity regime. This would further reduce the sensitivity to cavity perturbations. For systems using much broader atomic transitions where the cavity might naturally have lower finesse <cit.>, these effects would only be visible for very large samples.The absolute phase value at which such mirroring occurs typically increases with larger detuning from the resonance. This effect is caused by the decrease in atomic absorption for increased detuning. This causes the phase value necessary for the slope inversion of the transfer function to increase. Away from resonance the dispersion is thus highly distorted, with respect to the atomic phase, due to the functional form of the cavity transfer function. § RESULTS AND DISCUSSIONHere we report on the phase response of the system when operating in a regime of high phase shift due to a combination of large atom numbers N and high reflectance of the cavity mirrors. At small frequency detuning we see a linear scaling of the dispersion slope with respect to the phase slope, which gives us a limit on the ultimate frequency linewidth of a laser locked to such a system <cit.>. The dynamical range of a laser frequency lock to the atom-cavity system becomes limited at high absolute phase-shifts. This is caused by the characteristics of the transfer function whose behavior will then dominate over the power broadened transition linewidth Γ_ p. We quantify this limitation and its implications for laser frequency locking. We have also investigated the effects of having a cavity resonance lock with non-optimal conditions. The modification of such locking conditions is of interest to any experimental realization of the frequency lock. §.§ Phase-slope and projected shot-noise limited linewidthIn the context of locking the frequency of a laser to the atom-cavity system, we are interested in obtaining an error signal that we can use as a feed-back signal, which must have a large slope and a large signal-to-noise ratio (SNR). The first condition is limited by the physical system, and is given by the phase-slope present at resonance. The second condition is limited by the noise present in the experimental system, and is to a high degree limited by technical circumstances that may be significantly reduced. These technical contributions to the noise include residual amplitude modulation (RAM) of the laser sideband components, atom number fluctuations and noise in the detectors. Because of this fundamental difference in the two conditions, we wish to focus on the limitations set by the physical system initially - namely the phase-slope at resonance.In FIG. <ref> a) the slope of the atomic induced phase shift at resonance is plotted as a function of the input power on a logarithmic scale for N=2.7·10^7. It was shown in <cit.> that the slope at resonance scales linearly with the number of atoms N in the cavity mode. This is still the case in our regime of N≈1-5·10^7 and P_in≃100 nW <cit.>, and we will thus focus on the strongly non-linear scaling with laser power here. This scaling was shown for a cavity finesse of ℱ=75 in <cit.>. Here we show results for a system with finesse of ℱ=1240, and confirm that the theory scales well with cavity finesse.The very nonlinear behavior of the phase-slope shows a clear optimum in absolute phase slope for input powers of about 8 nW and a subsequent decrease in the absolute slope towards zero. While the phase slope is small for low powers due to the reduced saturation of the atoms, the saturation feature becomes power broadened for higher powers, once more leading to a reduction in the slope. The optimal phase slope is thus obtained for very low input powers, however, as we shall see below, this is not the optimal value for laser stabilization.The full curve in FIG. <ref> a) is a theoretical plot and we indicate a number of different input powers. At these powers we have performed scans of the atom-cavity spectrum and compared them to the theoretical model, in order to obtain a noise-free value for the phase slope at resonance. The fact that we see fluctuations of power-, atom number-, and technical noise or drift in the experiment is reflected by the misalignment between the dots and the theoretical behavior. Using the phase-slope it is possible to calculate the theoretically obtainable shot-noise limited (SNL) linewidth of a laser locked to the system. Here we find the minimal achievable linewidth by assuming that the detector efficiency is unity, and the lock is perfect. This can be found theoretically by using the expression <cit.>:Δν=π h ν/2η_qeP_sig(dϕ/dν)^2(1+P_sig/2P_ref)Where dϕ/dν is the phase slope at resonance, P_sig is the carrier power and P_ref is the reference power, which in our case is the power in the first order sidebands. η_qe is the quantum efficiency of the detector which we assume here is one.In FIG. <ref> b) we calculate this SNL linewidth Δν and plot the curve corresponding to the slope of FIG. <ref> a). We see that the optimum value of input power changes when we consider the SNL linewidth. For low powers the SNL linewidth increases dramatically as the shot noise starts to dominate the signal. This results in a relatively flat region around the optimum power spanning about an order of magnitude from P_in≃10-100 nW. The minimal value of Δν is highly dependent on the ratio between sideband and carrier power. The optimal ratio of P_ carrier/2P_sideband=1 was used in these experiments. For these parameters we predict a minimal value of Δν≈40 mHz which is comparable to the smallest laser linewidths ever reported <cit.>. By increasing the atom number it is possible to simultaneously decrease the projected linewidth of the locked laser, and increase the optimal operation power P_in. §.§ Dynamical rangeIn FIG. <ref> the recorded signal S_Ω is shown for three different regimes where the maximal atomic phase shift is below, at, or above that corresponding to the maximal value of the transfer function. This shows the transition from a regime where the dispersion is largely unperturbed and represents the phase-response of the atoms well, to a regime where the response is significantly modified by the transfer function. At small phase-shifts we see a linear increase of the size of the signal proportional to the phase. At larger phase-shifts, however, the functional form of the cavity transfer function results in a mirroring effect of the dispersion signal for detunings above γ_power where the phase shift is maximal. This has no influence on the slope around resonance, and will thus not affect the performance of an ideal frequency lock. It could, however, still limit the performance of a real servo system where the response time is not infinitely fast. We define the dynamical range of a lock to the dispersion signal as the range around resonance within which the dispersion slope has constant sign. This range is dictated by the full width at half maximum (FWHM) of the power broadened transition linewidth. This corresponds to the width of the Lamb dip in the case of simple saturated absorption spectroscopy. The width, however, is modified by the slope of the Doppler broadened Gaussian dispersion feature. This dispersion causes line-pulling and thus decreases the dynamical range further. Lower temperatures will cause more pronounced line-pulling than higher temperature as the Doppler-broadened dispersion slope increases. While this effect actually causes a decrease in resonance slope it turns out that the fractional increase in the number of saturated (cold) atoms N_sat outweighs this effect and the resonance slope is thus effectively increased for decreasing temperatures T. Finally the signal is modified by the cavity transfer function. Below the threshold in maximal phase-deviation set by this transfer function this is simply a phase-dependent scaling of constant sign and will thus not modify the dynamical range. Above this threshold, which becomes relevant in high N systems such as the one reported here, we see a decrease of the dynamical range due to the slope-sign inversion dictated by the transfer function. A higher atom number N increases the total phase, and thus pushes the system further beyond the threshold set by the transfer function boundaries. FIG. <ref> shows the dependency on cavity atom number of the dynamical range for an in-coupling power of P_in=100 nW and a temperature of T=2.5 mK. This shows the initial dynamical range of Δ_dyn≃180 kHz below threshold and a drop to few tens of kHz above the threshold. For typical atom numbers in our system we rarely exceed this threshold. For very high atom numbers, however, the range decreases asymptotically towards zero.The dynamical range of a frequency locking scheme will be limited by the power-broadened transition linewidth γ_power in all cases of FIG. <ref>. For higher atom numbers N, then, we will see another inversion within the narrow saturation dispersion, see FIG. <ref> b). Such an inversion will bring us into a regime where the dynamical range is limited by the properties of the transfer function χ rather than the power-broadened transition linewidth γ_power. Notice that this is only true if we require the sign on the slope to be constant. The sign of the signal itself will never change, and thus some degree of locking might still be possible for a flexible servo-system.The dynamical range is of interest in particular regarding stability requirements for the interrogation laser. A standard requirement for the interrogation laser is that the interrogation laser linewidth should be smaller than the transition linewidth of the sample in order to resolve the line. If our initial interrogation laser linewidth is of the order of the natural linewidth (γ=7.5 kHz) this is well within the dynamical range below threshold. For very high atom numbers N≳ 1.1×10^8, however, the dynamical range decreases below the natural transition linewidth of the atoms. It is thus important that the interrogation laser is prestabilized to well within this dynamical range, before the atom-cavity error signal can be optimally utilized.The aspects of the dynamical range considered here indicates that there is some optimal atom number depending on how efficient the servo can be made. While the slope around resonance increases linearly with the number of atoms N, and the dynamical range decreases severely above N≈2.5·10^7, an intermediate error signal could be preferable. Such a signal, like the intermediate (medium green) signal of FIG. <ref> b), provides the largest area under the error curve of the three shown. The preferred signal will depend on the particular experimental servo parameters.§.§ Locking condition effectsSince our experimental realization is based on a cyclic operation, the cavity lock causes the length of the cavity to change dynamically throughout the experimental cycle. If the cavity dynamics is slower than required to obtain perfect locking, we see a small correction compared to the ideal locking signal of Eq. <ref>. This causes large deviations in the DC transmission signal but has a relatively small effect on the phase response. When the cavity lock responsiveness is too slow the condition of constant resonance between the cavity and the laser carrier frequency will no longer be fulfilled. The atomic dispersion information will no longer be written onto the sideband frequencies but remains, in part or fully, on the carrier frequency. This means that χ_0 is no longer purely real, and the dispersion term of the atomic phase shift affects the transmission. For high atomic phase-shifts, then, the transmission of the carrier component will be significantly reduced as the resonance condition is no longer necessarily fulfilled.The locking condition determines some initial phase ϕ_init written onto the cavity phaseϕ_cav=nπ-ϕ_init.Here we investigate three different cases. For the case of a fast cavity lock that can follow the system dynamics we have ϕ_init=ϕ_D as shown in Eq. <ref>. A second idealized case is where the cavity lock is independent of the atoms inside the cavity ϕ_init=0. This means that the length of the cavity simply follows the vacuum wavelength of the interrogation laser L=nλ_vac/2. The third, and the more realistic, case is where we have some perturbed phase due to the experimental conditions. In our case, the fact that the locking dynamics are relatively slow results in an initial phase given by the atoms under the influence of the cooling light ϕ_init=ϕ_MOT. The phase-shifts of the field components then becomesϕ_0=nπ +ϕ_D + iϕ_A - ϕ_init ϕ_j =(n+j)π - ϕ_initfor j≠0,for some integer n.First we look at the case of an atom-independent cavity lock. Here the cavity length ensures resonance with the laser beam assuming that there is only vacuum in the cavity. In this case the sidebands (j≠0) are always resonant, but the carrier frequency (j=0) will be affected only by the atomic phase. In the limit of a very broad cavity linewidth κ this situation is equivalent to having no active lock on the cavity length. The behavior under this condition thus gives us some insight into the case of a system operating in the deep bad cavity limit with stationary mirrors but resonant with the atomic transition. In the second case, relevant to our current system, a slow lock means that we lock to the atoms in the MOT while the cooling light is still on. The carrier frequency thus experiences some phase shift from the AC Stark shifted atoms (ϕ_MOT), and this phase is written on the cavity length. Since the cavity cannot respond sufficiently fast to the subsequent conditions where MOT light is turned off, this modifies the phase of all χ_j with ϕ_MOT. The phase-information from the non-perturbed atoms is now only on the carrier component. This heavily modifies the DC transmission, and also causes the antisymmetric behavior of the signal to be lifted as ϕ_MOT is not symmetric with respect to ϕ_D. The carrier phase becomesϕ_0 = nπ - ϕ_MOT +ϕ_D + iϕ_Afor integer n, and the sideband phases retain the phase written on the lock ϕ_j≠0=(n+j)π - ϕ_MOT. In this case χ_0 is no longer a purely real quantity, which modifies the signal. We have implemented this to first order by manually adding the measured phase-shift ϕ_MOT of the system to the transfer functions of the carrier and sideband frequencies. A full description must include the modified atom-light interaction in the cavity caused by this effective cavity detuning during the probing time.In FIG. <ref> we show an example of a NICE-OHMS signal giving the dispersive response of the system. The NICE-OHMS signal has the expected features for a system with a large number of atoms in the cavity N=2.5·10^7 where sharp features occur due to the limitations set by the transfer function. Three theoretical curves are plotted, which shows the theoretical behavior of the system assuming a fast cavity lock (black), a cavity locked independently of atoms (dashed blue), and a cavity locked to the AC Stark shifted atoms in the MOT (light red). While these different approaches only cause slight variations close to resonance, they are necessary to include in order to explain the signal for larger detunings. As expected, the features are slightly sharper in the case of a fast cavity lock.The consequences of a non-optimal cavity locking condition on a laser lock is also considered here. FIG. <ref> shows two theoretical curves corresponding to optimal (fast) locking conditions (black), and atom-independent locking (dashed blue). In the case of optimal locking the system is close to a steady state. This can be realized either because the cavity lock is fast enough to follow the shift caused by turning off the trapping light, or by using a system operating in a continuous fashion. For the parameters used here (T=2.5 mK and N=2.7·10^7) we see an optimal phase slope with the fast lock, for powers of about P_in^opt=8 nW. The phase slope is reduced for all values of the input power in the case of an atom-independent locking. The functional shape also changes, and the optimal input power is increased to about P_in^opt=25 nW. Notice that while the slope is definitely reduced, it is below a factor of two for powers relevant to laser locking. The optimal power also becomes more experimentally accessible, and the two cases are seen to give approximately identical slopes for powers larger than P_in=400 nW. This indicates that the performance of the cavity lock might not be of detrimental importance to the ultimate performance of the system within technically relevant parameter regimes.§ CONCLUSIONWe have experimentally investigated an atomic ensemble of cold ^88Sr atoms in an optical cavity in the regime of high atomic phase-shift. The phase response of the system is recorded using the NICE-OHMS technique, and has promising features for frequency stabilization. The system operates in the bad cavity regime which suppresses the fluctuations caused by the finite temperature of the cavity. For the case of a narrow atomic transition, the bad cavity regime can still permit a high cavity finesse which yields a large number of photon round-trips. This causes the accumulated phase to grow beyond the approximately linear regime of the cavity transfer function, and mirroring effects of the phase-response can occur. These mirroring effects nonlinearly flips the slope of the dispersion signal around some maximal value. We experimentally mapped out the transition from the regime where the dispersion signal is an approximately linear representation of the atomic phase shift, to the regime where this representation is highly distorted by the cavity transfer function properties. We investigated the limitations this might have on an error signal for frequency locking of a laser. The mirroring effects causes a limitation of the dynamical range of a servo lock which must be included in the optimization of future servo systems operating using this technique.We also investigated the ultimate performance of a laser stabilized to such a system and saw predictions consistent with earlier work <cit.>. These predictions rely on investigations of the phase slope achievable at resonance and do not take into account the limitations on a servo loop such as the dynamical range limitations that occur. We saw that the degradation of the signal slope caused by non-optimal cavity locking was not detrimental to the system and amounted to a factor of two for realistic experimental parameters. This means that even a slow cavity lock could produce promising results for laser stabilization, and opens for the possibility of leaving out the cavity lock entirely as long as the system is deep in the bad cavity regime.We acknowledge support by ESA Contract No. 4000108303/13/NL/PA-NPI272-2012 and by the European Union through the project EMPIR 15SIB03 OC18. § THEORYHere we give a very brief overview of the theory used to model the interaction of the light with the atom-cavity system. We follow <cit.> and model the system by using a Born-Markov master equation to describe the evolution of the system's density matrix ρ̂. This evolution can be written asd/dtρ̂=1/iħ[Ĥ,ρ̂]+ℒ̂[ρ̂].The many-particle Hamiltonian describing the coherent evolution in a rotating interaction picture is given by Ĥ=ħΔ/2∑_l=1^Nσ̂_l^z+ħη(â^†+â) +ħ∑_l=1^Ng_l(t)(â^†σ̂_l^-+σ̂_l^+â)where Δ=ω_a-ω_c is the atom-cavity detuning, σ̂^+,-,z are the Pauli spin matrices and η=√(2πκ P_in/ħω_c) is the classical drive amplitude. â and â^† denote the annihilation and creation operators of the cavity mode respectively. The coupling rate between atoms and cavity is given by g_l(t)=g_0cos(kz_l-δ_lt)e^-r_j^2/w_0^2,where g_0 is the vacuum Rabi frequency, k is the wave number of the cavity mode, z_l and r_l denote the longitudinal and axial positions of the l'th atom, δ_l=kv_l is the Doppler shift contingent on the atom velocity v_l, and finally w_0 is the radial waist size of the cavity mode. Here the probing laser is assumed on resonance with the cavity at all times, ω_l=ω_c.The incoherent evolution is described by the Liuvillian ℒ̂[ρ̂] and is given byℒ̂[ρ̂]= - κ/2{â^†âρ̂+ρ̂â^†â-2âρ̂â^†}- γ_nat/2∑_l=1^N{σ̂_l^+σ̂_l^-ρ̂+ρ̂σ̂_l^+σ̂_l^–2σ̂_l^-ρ̂σ̂_l^+}+ 1/2T_2∑_l=1^N{σ̂_l^zρ̂σ̂_l^z -ρ̂},where κ is the cavity decay rate, γ_nat is the atomic transition linewidth and 1/2T_2 is the inhomogeneous dephasing of the atomic dipole. The approach is thus based on a many-particle Hamiltonian Ĥ and a derived set of complex Langevin equations that includes the Doppler effect from the finite velocity of the atoms. The evolution is found by means of a Floquet analysis and solved for the steady state case. This will not be investigated further here, the interested reader is referred to <cit.>. 14BloomB. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, Nature 506, 71-75 (2014) HinkleyN. Hinkley, J. A. Sherman, N. B. 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Kramer, L. Pelzer, M. Stepanova, and P. O. Schmidt, Metrologia 54, 307-321 (2017) ChristensenPHDB. T. R. Christensen, Ph.D. thesis, University of Copenhagen, Niels Bohr Institute, 2016RiehleF. Riehle, Frequency Standards: Basics and Applications, (Wiley-VCH 2004) | http://arxiv.org/abs/1704.08245v2 | {
"authors": [
"Stefan Alaric Schäffer",
"Bjarke Takashi Røjle Christensen",
"Martin Romme Henriksen",
"Jan Westenkær Thomsen"
],
"categories": [
"physics.atom-ph",
"physics.optics"
],
"primary_category": "physics.atom-ph",
"published": "20170426175901",
"title": "Dynamics of bad-cavity enhanced interaction with cold Sr atoms for laser stabilization"
} |
Department of Astronomy, University of Michigan, 1085 S. University Ave., Ann Arbor, MI 48109, USA; [email protected] of Astronomy, University of Michigan, 1085 S. University Ave., Ann Arbor, MI 48109, USAMcWilliams Center for Cosmology, Department of Physics, Carnegie Mellon University, 5000 Forbes Ave., Pittsburgh, PA 15213, USADepartment of Astronomy, University of Michigan, 1085 S. University Ave., Ann Arbor, MI 48109, USA Joint Institute for Nuclear Astrophysics and Center for the Evolution of the Elements (JINA-CEE), USAWe report new spectroscopic observations obtained with the Michigan/Magellan Fiber System of 308 red giants (RGs) located in two fields near the photometric center of the bar of the Large Magellanic Cloud.This sample consists of 131 stars observed in previous studies (in one field) and 177 newly-observed stars (in the second field) selected specifically to more reliably establish the metallicity and age distributions of the bar.For each star, we measure its heliocentric line-of-sight velocity, surface gravity and metallicity from its high-resolution spectrum (effective temperatures come from photometric colors).The spectroscopic Hertzsprung-Russell diagrams—modulo small offsets in surface gravities—reveal good agreement with model isochrones.The mean metallicity of the 177-RG sample is [Fe/H]=-0.76±0.02 with a metallicity dispersion σ=0.28±0.03.The corresponding metallicity distribution—corrected for selection effects—is well fitted by two Gaussian components: one metal-rich with a mean -0.66±0.02 and a standard deviation 0.17±0.01, and the other metal-poor with -1.20±0.24 and 0.41±0.06. The metal-rich and metal-poor populations contain approximately 85% and 15% of stars, respectively.We also confirm the velocity dispersion in the bar center decreases significantly from 31.2±4.3 to 18.7±1.9 km s^-1 with increasing metallicity over the range -2.09 to -0.38.Individual stellar masses are estimated using the spectroscopic surface gravities and the known luminosities.We find that lower mass hence older RGs have larger metallicity dispersion and lower mean metallicity than the higher-mass, younger RGs.The estimated masses, however, extend to implausibly low values (∼ 0.1 M_⊙) making it impossible to obtain an absolute age-metallicity or age distribution of the bar. § INTRODUCTION The Large Magellanic Cloud (LMC) is the nearest gas-rich satellite of the Milky Way that has ongoing star formation. The visual structure of the LMC is dominated by a prominent central bar, and hence we define the galaxies of this kind as Barred Magellanic Sprials <cit.> or SB(s)m <cit.>. Magellanic systems are common in the local universe, but they are rarely found as close to a massive parent system as the LMC to the Milky Way <cit.>.Because of its relative proximity and prominence, the central bar of the LMC represents a particularly useful test case to explore how such structures develop and evolve in galactic systems.It is well known that the LMC has a stellar bar that has no counterpart in the distribution of neutral or ionized gas <cit.>.More interestingly, the stellar bar is found to have multiple centers.The photometric center of the LMC bar, i.e. the densest point on a star count map, lies about 0.4 kpc away from the center of its stellar disk, and even more than 1 kpc away from the dynamical center of the neutral gas disk <cit.>. The location of the stellar dynamical center is still under debate. According to the line-of-sight (LOS) kinematics, the dynamical center of the carbon stars is consistent with the photometric bar center <cit.>.In contrast, the average proper motion (PM) data observed by Hubble Space Telescope (HST) imply the stellar dynamical center coincides with the H i dynamical center <cit.>.When combining the LOS velocities and the PM data, <cit.> even found that the stellar dynamical center is intermediate between the photometric bar center and the H i dynamical center.<cit.> distinguish `true' bars, which are formed in a quasi-independent manner early in the history of a disk galaxy, from `secular' bars that develop over time from instabilities in disk galaxies.The off-center feature of the LMC bar strongly supports the secular evolution scenario resulting from tidal interactions.Many numerical simulations have been employed to reproduce the morphology and internal dynamics of the LMC including its off-center bar, but the detailed evolution remains uncertain.For example, <cit.> showed that the off-center bar can be formed if the LMC with an already-existing bar can collide with a low-mass Galactic subhalo. Alternatively, more recent works prefer a dwarf-dwarf galaxy interaction <cit.>, presumably, in the case of the LMC, involving the Small Magellanic Cloud (SMC).The star clusters in the LMC are rarely found to have ages between approximately 3 and 12 Gyr ago <cit.>, though this age interval appears to be filled by field stars <cit.>.This suggests either that the star formation associated with star clusters was suppressed during this `age gap', or that clusters older than about 4 Gyr are preferentially destroyed (though not all; the LMC does contain a set of about ten `ancient' star clusters comparable in age to globular clusters in the Galaxy).Whatever the cause for the age gap in the cluster population, the fact remains that if we aim to probe the history of the LMC for ages greater than about 4 Gyr, we must rely on field stars.Although many photometric studies have been carried out to do this <cit.>, interpretation is complicated by the increased insensitivity of the main-sequence turnoff with age, and the fact that all evolved stars older than about 1 Gyr essentially funnel into a single red-giant branch for a given metallicity <cit.>.In the LMC bar, crowding introduces an additional complication for ground-based studies, while HST observations tend to have limited field coverage.Spectroscopic studies of the LMC bar can be used as as a tool to determine the age-metallicity relation of the bar and to explore the star formation history of this enigmatic component of the LMC.Many spectroscopic studies in the LMC bar and/or inner disk has been carried out during the past two decades.<cit.> analyzed low resolution spectroscopy of 98 RR Lyrae stars in the LMC bar, and reported an average metallicity of -1.48 dex for this old population.<cit.> carried out the first spectroscopic study of intermediate-age and old field stars in the LMC bar.The spectra of 373 red giants (RGs) were obtained at the near-infrared Ca ii triplet (CaT) and used to derive radial velocities and metallicities.They found a metallicity distribution function (MDF) peaked at [Fe/H]=-0.40 dex with a tail of metal-poor stars down to [Fe/H]≤-2.1 dex and a systemic change in velocity dispersion with mean chemical abundance. <cit.> reanalyzed C05's spectra and also obtained an average metallicity of -0.40 dex with a new calibration. On the other hand, studies on the inner disk reported lower mean metallicities than that of C05.<cit.> derived [Fe/H]≈-0.5 dex for a field about 3^∘ north of the LMC bar at first, and then <cit.> recalibrated a new mean of -0.58 dex for the same field.<cit.> derived [Fe/H]=-0.48 dex from 89 stars located about 2^∘ NW from the center of the LMC.More recently, <cit.> performed a detailed chemical analysis of 106 RGs in the sample of C05, using spectra obtained with the FLAMES/GIRAFFE multifibre spectrograph.Their measurements confirmed that C05 had overestimated the metallicities of metal-rich stars (by 0.25±0.03 dex from our calculation).Unfortunately, a reliable MDF of the LMC bar cannot be derived from their results due to inherent metallicity-dependent biases in their sample.Since the nature of LMC bar remains poorly constrained in terms of the galaxy's interaction and chemical evolution history, we aim to expand previous spectroscopic studies by observing a new sample of stars chosen using well-defined selection criteria.The ultimate aim is to produce a spectroscopic survey of evolved stars that can be used to map the chemo-dynamical properties as a function of position and age over the entire LMC bar region.In addition to 131 RGs observed by C05, our sample contains 177 more RGs selected from the OGLE-II photometry database <cit.> for fully providing the bar MDF.The spectra were obtained with the Michigan/Magellan Fiber System <cit.> over two separate fields that both are near the photometric center of the LMC bar.This paper represents a first look at the results of this survey and provides a description of the techniques we employ.The structure of the paper is as follows.Section <ref> introduces the sample selection, the observations and the data reduction processes.Section <ref> explains the measurements of velocities and stellar parameters that we derive from our spectra. Section <ref> reports the main results of this work and describes some of their implications.In Section <ref>, we summarize and further discuss the key results of this study.We close with a cautionary tale regarding the inherent and still significant complications in using field stars to independently probe the age distribution/star-formation history in an intermediate-age/old population such as the LMC bar.§ OBSERVATIONS AND DATA REDUCTION§.§ Fields and Target Selection In standard operation, M2FS fields must be centered on a relatively bright `Shack-Hartmann' (SH) star, which provides low-frequency wavefront data to the active optics system of the Magellan/Clay Telescope.Two additional spatially-coherent fiber bundles are used to image a pair of guide stars during exposures.Finally, a set of bright acquisition stars imaged through science fibers and visible on the guide camera are used for each field to held centroid the fibers in the field in both translation and rotation.These requirements ancillary stars impose mild restrictions on any M2FS field locations, but especially in crowded regions like the LMC bar.The data obtained for this study were collected in two bar fields labelled as `LMCC' and `LMC1', respectively.The trailing letter of `LMCC' stands for `Cole' because this field was chosen to include as many stars as practical from the sample of C05.The centers of these fields are listed in Table <ref>, and their locations and coverage areas are shown in Figure <ref> along with the photometric bar center and the dynamical center constrained by stellar proper motions <cit.>.We used two different methods to select the stellar candidates in the LMCC and LMC1 fields. In LMCC, the candidate RGs were directly selected from the sample of C05 by first ranking their stars by metallicity and then selecting every other one to produce a suitably-sized subsample for M2FS followup.In LMC1, the candidate RGs were randomly selected from the OGLE-II BVI maps of the LMC <cit.>, according to the following photometric criteria:16.00< I < 17.00, 22.00 < I + 5×(V-I) < 24.25.These limits were chosen to sample fully the color range–and hence the evidently wide metallicity range–of the population.As shown in Figure <ref>, we further divided the selection region into 32 rectangles and selected 6 candidates from each.This approach ensures that we sampled stars over a full color/metallicity/age range as populated within the red giant branch and allows us to account for selection when we generate the underlying metallicity distribution in the bar (see Section <ref>).As already intimated, the operational characteristics of M2FS affect the selection process.M2FS employs aluminum fiber plugplates to position up to 256 optical fibers at the Nasmyth-East focal surface of the Magellan/Clay Telescope.Each fiber has an entrance aperture of 1.2 arcsec and is fitted in a ferrule 13 arcsec in diameter; the latter defines the minimum separation between deployed fibers.Otherwise, any fiber can be positioned anywhere within a field of 29 arcmin in diameter except at locations for various ancillary stars used for guiding, field alignment/acquisition, and active optical control (the central SH star noted above).In the end, we were able to assign 147 science fibers in the LMCC field, and 184 science fibers in LMC1. Table <ref> lists the positions and photometric information of the observed stars. We also plot the locations of both LMCC and LMC1 samples on the color-magnitude diagram (CMD) in Figure <ref>.§.§ Spectroscopic Observations The summary of our LMC bar observations is listed in Table <ref> including the total exposure time for each field.M2FS employs twin spectrographs (which are referred to as `blue' and `red' channels, respectively) that have identical optical properties and wavelength coverage but can be operated independently in a variety of spectral configurations.In our LMCC and LMC1 observations, both spectrographs were configured to sample a wavelength range of 5130–5189 Åat an effective resolution ℛ∼ 20,000.Each spectrograph images the raw spectra onto a four-channel 4096× 4112 E2V CCD with a pixel size of 15 μ m.During the readout process, we binned the data by 2×2, which still over-samples the data in both spectral and spatial directions. The CCDs were readout in `slow' mode for a typical gain of 0.75 e^-/ADU and a readout noise of 2.7 e^-.Immediately before or after science exposures, we acquired calibration frames from a Th-Ar arc lamp (for wavelength calibration) and a quartz lamp (for spectral tracing).In addition, several twilight frames (which also included arc and quartz exposures) were obtained at either the beginning or the end of the same observing night.These twilight data are used to check the wavelength/velocity calibration and correct fiber-to-fiber throughput variations (Section <ref>), and also to estimate systematic offsets in the best-fit physical parameters (Section <ref>).We also obtained a master `fiber map' with all the fibers plugged in while the telescope focal surface is illuminated by ambient daylight in the mostly-closed dome.The map produces a high signal-to-noise template for tracing spectra, and is useful as a parallel check on the relative fiber throughputs estimated from the twilights.Groups of bias (zero) and dark frames (also by 2×2 binning) were also obtained during each observing run and combined to produce master bias and dark frames that are used in the data reduction.Background subtraction is very important in this work because of the crowded LMC bar region.The central surface brightness of the LMC bar (see the red filled square in Figure <ref>) is 20.65mag/arcsec^2 in V-band <cit.>.In contrast, the V-band telluric sky background would have ranged from about 22.0 (Dec 2014) to 21.6 (Nov 2015) mag/arcsec^2 at the location of the LMC bar—about 3.5–2.4 times fainter than the LMC contribution—at the time of our observations.We have therefore adopted two distinct approaches to obtain the background spectra in the LMCC and LMC1 fields, respectively.In LMCC, 95 sky-background fibers were assigned in addition to the 147 science fibers, and their positions were selected on the image obtained by STScI DSS.For selecting good background positions, we first randomly picked a position within the LMCC field and kept it if meeting two conditions: (a) the chosen position is well separated (> 13 arcsec) from all other assigned fiber positions, and (b) the mean count of all pixels within a 55 square centered on that position is within 10% of the modal background value of the image.In LMC1, no sky fibers were assigned in advance due to the initial use of these data for more limited M2FS commissioning purposes. To estimate the background contribution, we pointed the telescope 15 arcsec away from the original position—known as `off-target' exposures—along three principal directions (North, South and East) during the Nov 2015 observation run.As a result, the background subtraction approaches differ slightly for the LMCC and LMC1 samples; we describe these in the following section.§.§ Data Reduction Condensed descriptions of standard M2FS data reduction processes can be found in <cit.>.All data processing was carried out using IRAF scripts and pipelines designed for M2FS data, and the typical final products are background-subtracted stellar spectra.Briefly, all data were first processed through overscan subtraction, bias correction and dark correction.We removed cosmic rays from almost all exposures (excluding arcs or other short exposures) using the Laplacian-filtering algorithm from <cit.>.We subtracted diffuse scattered light from the two-dimensional images by fitting a polynomial surface to the regions of the images not illuminated by the fibers.The spectral traces, defined by combining twilights/quartzes/fiber maps, were shifted to match the locations of the science spectra (named because they are extracted from the science exposures) that produced final science traces.The final science traces were used to extract the calibration arcs to ensure no offset or interpolation shift existing between the arc exposures and the science exposures.The extracted arcs were then fit to a moderate-order polynomial to determine the transformation from extracted-pixel to wavelength for every science fiber/target.The typical root-mean-square of these fits for the data used in this study was 0.3 km s^-1.The wavelength-calibrated data were then normalized using relative fiber throughputs derived from the twilights or fiber maps.For all the above steps, we processed the sky-background data—both from the sky-background fibers in LMCC and from the off-target exposures in LMC1—the same as the science data. Throughout the entire reduction process, we also calculated `variance spectra' in order to track the signal-to-noise ratio (SNR) for every pixel of every science spectrum.These data are used to properly weight individual pixels when fitting spectra to model atmospheres in order to derive stellar parameters (see Section <ref>). The last but important step is the background subtraction.In LMCC, we found no significant flux or spectral variations among sky-background fibers.We therefore averaged all these background spectra to make a master background spectrum and subtracted it directly from all science spectra.In LMC1, the background spectra were obtained from the off-target exposures one year later.After removing some anomalously high-flux spectra (about 10% of the total), we averaged all other spectra to made a second-year master background spectrum. Then this spectrum was scaled by a factor 1.8 before being subtracted from the deeper first-year science spectra.This factor accounts for differences in exposure times, atmospheric extinction, telluric background and seeing between the two sets of observations. The systematic reliability of this approach is confirmed in Section <ref> where we demonstrate good agreement in the metallicity scales obtained independently from the M2FS data in the LMCC and LMC1 fields.Figure <ref> shows examples of background-subtracted spectra in the LMCC and LMC1 fields.The spectra span a range from the highest to lowest SNRs for our targets, and also a range in metallicities (measured through the method introduced in the following section). § ANALYSIS§.§ Modeling of M2FS SpectraTo measure the LOS velocity and the stellar parameters, we employ a Bayesian method to fit the background-subtracted M2FS spectrum (see <cit.>).For each star, this approach generates a model spectrum M(λ) by combining a continuum-normalized template spectrum T(λ) and an assumed continuum spectrum P_l(λ),M(λ)=P_l(λ)T(λ),where P_l(λ) is an l-order polynomial.This model spectrum M(λ) is used to compare with the observed spectrum using a maximum-likelihood technique.The template spectrum, T(λ), is generated using a library of synthetic spectra, which is used in the Sloan Extension for Galactic Exploration and Understanding (SEGUE) stellar Parameter Pipeline <cit.>.This library contains a set of rest-frame, continuum-normalized, stellar spectra, which are computed over a grid containing three atmospheric parameters, i.e. the effective temperature (T_ eff), the surface gravity (logg, where g is in cm s^-2) and the metallicity ([Fe/H]). These parameters vary in the following ranges (with the grid steps):4000 ≤T_ eff ≤ 10000K,with Δ T_ eff=250K, 0 ≤ logg ≤ 5 dex, with Δlogg=0.25dex, -5 ≤[Fe/H] ≤ 1 dex, with Δ [Fe/H]=0.25dex,For the α elements, their total abundance is typically fixed based on the iron abundance via the ratio [α/Fe]. This library assumes a hard-wired relation between [α/Fe] and [Fe/H]:[α/Fe] =0.4for[Fe/H] < -1, -0.4× [Fe/H] for[Fe/H]∈[-1, 0], 0for[Fe/H]≥ 0. Another consideration is the wavelength shift between the template spectrum and the observed spectrum.This shift has two potential origins: the velocity shift due to the LOS velocity and the uncertainty of the wavelength solution in the observed spectrum. The first offset is accounted for by a parameter that shifts the wavelength as λ' = λ v_ los/c, where v_ los is the LOS velocity and c is the speed of light.The wavelength solution may have residual systematic deviations, which could be modeled as a polynomial Q_m(λ) of order m.Combining these two effects, the final shifted wavelength isλ' = λ[ 1+Q_m(λ)+v_ los/c],and so the final template spectrum is now represented as T(λ').To carry out the maximum-likelihood technique, we adopt the likelihood function used by <cit.>,ℒ (S(λ)|θ⃗, s_1, s_2) =∏_i=1^N_λ1/√(2π(s_1Var [S(λ_i)]+s_2^2))exp[ -1/2( S(λ_i) - M(λ_i) )^2/s_1 Var[S(λ_i)]+s_2^2],where S(λ_i) and M(λ_i) are the observed spectrum and the model spectrum, respectively, and θ⃗ is a vector of all free parameters in M(λ_i) (summarized in the following paragraph).There are also two nuisance parameters s_1 and s_2 that, respectively, rescale and add an offset to the observational variances to account for systematically misestimated noises.In practice, P_l(λ) is treated as a fifth order polynomial (i.e., l=5) incorporating six parameters.The wavelength-shifted T(λ') inherits three parameters from the library spectra and three coefficients related to the modification of Q_m(λ) (i.e., m=2).An additional parameter is varied to convolve the template spectra with the instrumental line-spread function (LSF), and thus θ⃗ ends up with 13 free parameters.Considering the two nuisance parameters s_1 and s_2 in Eq. <ref>, the full fitting method contains 15 free parameters in the end, among which four of them are the physical parameters we aim to measure: v_ los, T_ eff, logg and [Fe/H].In each fitting, we truncate the spectrum in the region 5130 ≤λ /Å≤ 5180, while for the template spectra, the rest-frame wavelength region 5120 ≤λ /Å≤ 5190 are adopted to account for the LOS velocities up to about ± 550 km s^-1. §.§ Priors of the Physical Parameters We use the MultiNest package <cit.> to scan the parameter space.MultiNest implements a nested-sampling Monte Carlo algorithm, and returns random samplings from the posterior probability distribution functions (PDFs) for all input parameters. We record the first four moments of each physical parameter's posterior PDF: mean, variance, skewness and kurtosis.MultiNest requests a set of prior distributions for all parameters to initiate the calculation. We adopted uniform priors over a specified range of values as listed in Table 2 of <cit.>.For the physical parameters v_ los, logg and [Fe/H], the priors were set within the ranges:-500 ≤ v_ los/( km s^-1) ≤ 500, 0 ≤log[g/( cm s^-2)]≤ 5, -5 ≤ [Fe/H]/ dex≤ 1. Different from <cit.>, T_ eff was fixed during the fitting process in this study, because it proved difficult to constrain the temperatures adequately given the narrow wavelength range of our spectra and the typically modest median SNR per pixel (ranging from 5 to around 25). In addition, we found that the best-fit values for logg and [Fe/H] were strongly correlated with T_ eff when using uniform temperature priors.To break this degeneracy, we calculated effective temperatures for our targets from the OGLE-II V-I color index using color-temperature relation for giants <cit.>. We adopted a single-extinction model of E(V-I)=0.15±0.07 mag, which is equivalent to E(B-V)≈0.11 mag, A_V≈0.34 mag and A_I≈0.20 mag in the UBVRI photometric system.Since the color-temperature relation reported by <cit.> is a weak function of metallicity, we simply assumed [Fe/H]=-0.8 dex, the mean value for our targets when we used the Bayesian method with T_ eff as a free parameter. §.§ Twilight Offsets and Errors We also applied the Bayesian method described above to the twilight spectra obtained on the same night and in the same spectrograph configuration.Using the solar effective temperature T_ eff, ⊙=5778K as the fixed temperature prior, we found that the parameters v_ los, logg and [Fe/H] fitted from those spectra deviate from the known solar values, v_ los, ⊙=0km s^-1, log_10[g_⊙/( cm s^-2)]=4.44 dex and [Fe/H]_⊙=0 dex.As discussed in <cit.>, we attribute this to systematic mismatches between the model spectrum and the observed spectra.Without independent information regarding how such mismatches may vary with spectral type, we defined the offsets between the fitting and the true twilight parameters to be zero-point shifts. Table <ref> lists the mean offsets and the standard deviations of all three fitting physical parameters in each field and each channel.According to Table <ref>, we apply the twilight offsets in [Fe/H] to all results obtained from the Bayesian analysis of our science spectra.For v_ los and logg, we chose not to apply offsets to the data as the required offsets are either not statistically significant, or they have no significant implications for our final results. The total error budget for our derived stellar parameters include contributions from the variances of the twilight offsets, the variances of the posterior PDFs of v_ los, logg and [Fe/H], and—for T_ eff—the uncertainty of the color excess σ[E(V-I)]=0.07 mag. A full table of the final results is available in machine-readable format online.The initial few lines of this table is provided in Table <ref>.We list only results for spectra with median SNR per pixel greater than 5 (SNR>5), which results in 133 (out of 147 observed) targets in LMCC and 179 (out of 184) in LMC1.Excluding double/blended stars (see Section <ref>), the final LMCC and LMC1 sample sizes are 131 and 177, respectively.Heliocentric corrections has been applied to all velocity results and were calculated using appropriate PyAstronomy[https://github.com/sczesla/PyAstronomy] routines. §.§ Metallicity from the Equivalent Widths In order to examine the accuracy of the Bayesian analysis (BA), we compare our metallicities with those derived using the traditional spectroscopic analysis.For this purpose, we selected 11 LMCC stars (see Table <ref>) that not only have high-SNR spectra (SNR>10) but also show significant differences in the measured metallicities between our work and VdS13 (|[FeI/H]_VdS13-[Fe/H]_Bayesian|>0.2 dex).We restrict the analysis to Fe i lines with accuratevalues (grade “D+” or better, σ < 0.22 dex, according to the NIST Atomic Spectra Database; ), which leaves us with three lines that can be measured in our spectra (5150.84, 5166.28 and 5171.60 Å).Our approach is to measure equivalent widths (EWs) of these lines through a semi-automated routine that fits Voigt line profiles to continuum-normalized spectra. We inspect these fits and then modify them if the automated routine clearly fails to identify the continuum, which often occurs when the routine incorrectly includes neighboring absorption lines in the calculation of the continuum.The uncertainties in the EW values are ≈ 10%.Before applying this EW method to our M2FS spectra, we first measure EWs of these lines in high-quality spectra of two metal-poor RG standard stars, Arcturus <cit.> and<cit.>.The metallicities derived for the two standard stars are in good agreement with previous work. For Arcturus, we derive [Fe/H] = -0.41 ± 0.11 dex, which agrees with the value [Fe/H] = -0.52 ± 0.04 dex derived by <cit.>. For , we derive [Fe/H] = -1.61 ± 0.34 dex, which agrees with the value [Fe/H] = -1.56 ± 0.15 dex derived by <cit.>. This indicates that for high resolution and high SNR stellar spectra, our EW method is reliable, and the choice of lines has little effect on the derived metallicities.As an additional test, we have also degraded the resolution and SNR of these spectra to match our typical M2FS spectra. After remeasuring the EWs, the derived [Fe/H] values for these two standard stars are found to be approximately 0.2 dex lower than when the high-quality spectra are used.The statistical uncertainties of the derived abundances, however, are quite large (about 0.4 dex), so we do not consider this to represent strong evidence of a systematic offset in [Fe/H] induced by the modest SNR of our spectra. Finally, we also applied the BA to degraded spectra of Arcturus and HD 175305.In each case, we obtain [Fe/H] results to within 0.2 dex of the `true' values.For these 11 selected stars, we use the newly measured EW values with the formerly measuredand (listed in Table <ref>) to derive the metallicity for each star.We adopt a constant microturbulence velocity parameter, v_t = 2.0 ± 0.4 , for each star because we lack any constraints on this parameter. We derive the metallicities using a recent version of the spectrum analysis code MOOG <cit.>. The calculations are repeated 250 times for each star, resampling the stellar parameters and EWs each time from normal distributions. Table <ref> lists the metallicities derived by our BA and EW methods, as well as those measured by VdS13 and C05. We calculated the weighted mean differences in the metallicities among those measurements as ⟨ [Fe/H]_ BA- [Fe/H]_ EW⟩ = 0.07 ± 0.12 dex,⟨ [Fe/H]_ V13- [Fe/H]_ EW⟩= 0.22 ± 0.14 dex,⟨ [Fe/H]_ C05- [Fe/H]_ EW⟩= 0.25 ± 0.15 dex.At face value, this analysis favors the BA results, but there are important caveats. For example, the limited wavelength coverage provides few reliable Fe i lines for the EW analysis, so the statistical uncertainties are large (see Table <ref>).The relatively low SNR of the spectra may also mask the presence of other weak lines in the spectra leading to a systematically low misplacement of the continuum level. This interpretation is supported by the results of our test to rederive [Fe/H] from the degraded standard star spectra described above. Nonetheless, lacking a set of standard stars observed in the same conditions for calibration, we find no need for any additional corrections to the BA metallicities beyond the twilight offset described in Section <ref>. We shall therefore adopt the BA metallicities as listed in Table <ref> throughout the rest of this paper.§.§ Comparison with Previous StudiesAs stated in Section <ref>, we have 133 RGs left in the LMCC sample that were selected from C05's sample.Among them, 39 were also reobserved by VdS13.These stars provide an opportunity to compare the physical parameter measurements on a star-by-star basis. In Figure <ref>, we compare the LOS velocities and the stellar parameters measured from our M2FS spectra to those reported by C05 (top panels) and VdS13 (middle and bottom panels). For the subsample of 133 stars, we measured a mean heliocentric LOS velocity of 262.9 ± 2.1 km s^-1 and a corresponding velocity dispersion of 24.4±1.6 km s^-1, compared to 258.7 ± 2.4 km s^-1 and 27.0±1.7 km s^-1 measured by C05 (Figure <ref>a).The median errors of the individual velocities are 0.4 km s^-1 (this work) and 7.5 km s^-1 (C05). There is a systematic offset of 4.2±0.8 km s^-1 in the velocity measurements between these two studies (in the sense M2FS-C05).Figure <ref>b illustrates that we typically measure lower metallicities than C05 did for stars in common.Excluding the two anomalous sources in the upper-left corner of Figure <ref>b (see Section <ref>), the remaining 131 stars reveal weighted mean metallicities of -0.76 ± 0.03 dex (this work) and -0.50 ± 0.03 dex (C05).The corresponding metallicity dispersions are 0.30±0.03 dex and 0.37±0.04 dex, respectively. Although the dispersions are statistically equivalent, the systematic metallicity offset of 0.27±0.04 dex between our work and C05 appears to be significant.For the subsample of 39 stars, we measured a mean heliocentric LOS velocity of 263.9±3.6 km s^-1, which is slightly greater than 262.4±3.6 km s^-1 by VdS13 and 258.9±3.9 km s^-1 by C05 (Figure <ref>c).The velocity dispersion is 22.0±2.4 km s^-1 from our measurements, compared to 22.1±2.5 km s^-1 (VdS13) and 24.3±2.7 km s^-1 (C05).The median errors in the single-star velocities are 0.3 km s^-1 (our work), 0.2 km s^-1 (VdS13), and 7.5 km s^-1 (C05).The systematic offset between our velocities and VdS13 is 1.5±0.2 km s^-1 (in the sense M2FS-VdS13).For the same 39 stars, we measured a mean metallicity value of [Fe/H] = -0.80±0.05 dex, compared to -0.71±0.06 dex from VdS13, and -0.56±0.06 dex from C05.Given the median errors on metallicity in three studies are 0.06 dex (this work), 0.12 dex (VdS13) and 0.13 dex (C05), we confirm that C05 seems to have overestimated the metallicity for metal-rich stars in the LMC bar by around 0.25 dex.There is less evidence that such an offset exists among more metal-poor stars ([Fe/H] < -1.0 dex) (; VdS13), though the number of such stars we can directly compare from the various studies is not large.In contrast,our metallicity measurements are in statistically good agreement with those of VdS13.Finally, we also comparedandbetween our work and VdS13 in Figures 5e and 5f, respectively. Although VdS13 and we used different color- relations to calculatefrom photometry, the results ofare in reasonably good agreement. The comparison ofshows more scatter with no clear pattern. Note that VdS13 calculated theirfrom photometry (see Section 5.1), while ourare measured by comparing the scientific spectra to a library of templates. Our results imply poor agreement between these methods, which is an important conclusion for our aim to measure the age distribution function of LMC bar stars directly (Section 5.1).§ RESULTS §.§ Spectroscopic Hertzsprung-Russell Diagram It is well known from stellar evolutionary models <cit.> and observations of star clusters <cit.> that the classical red giant branch (RGB) in the Hertzsprung-Russell (HR) diagramtypically consists of stars older than approximately 1 Gyr over a wide range of metallicities. The resulting age-metallicity degeneracy along the RGBstems from this evolutionary funneling effect <cit.>, as well as the competing effects of age and metallicity on the photometric colors of such stars. From an astrophysical standpoint, this degeneracy makes it difficult to disentangle the age or metallicity distribution of an intermediate-age or old RG population from photometric observations. A key aim of this study is to use spectroscopy to try to break this degeneracy as far as possible by providing independent information on RGBmetallicities and ages (viameasurements).To illustrate the potential of this approach, we plot four spectroscopic HR diagrams showing the relationship between the stars' surface gravities and their effective temperatures in Figure <ref>. On each diagram, the stars are color-coded by the same metallicity bins shown in the legend of Figure <ref>a. The first three diagrams (Figures 6a, 6b and 6c) are made to illustrate the complications in interpreting the LMCC sample. We reduce the LMCC sample size from 133 (in Figure <ref>a) to 62 (in Figure <ref>c) for the purpose of investigating the `peculiar' metal-rich stars shown in the upper-left corner of Figure <ref>a. No such star has been found in the LMC1 sample and Figure <ref>d shows a clear segregation by metallicity.At a first glance, the metal-dependent pattern in Figure <ref>d is very similar to that shown on the CMD of C05 (see their Figure 8).However, there is a significant difference between those two diagrams.In Figure <ref>d, the color-coded populations can be well separated by metallicity, with the extremely metal-poor stars located on the upper-left corner and the extremely metal-rich ones on the lower-right.On the other hand, the extremely metal-poor stars on C05's CMD were overlapped by many relatively metal-rich counterparts, since the bluest stars in their sample are mostly metal-rich and a board color range is covered by the stars with metallicity around the peak of their MDF (-0.6<[Fe/H]<-0.3 dex).As described in Section <ref>, the stars in our LMCC sample were selected exclusively from C05's sample.Their selection criteria on the CMD (also reflected by the large black dots in Figure <ref>) were chosen to include stars over a wide range of metallicity expected in the LMC bar.However, the blue edge of their selection region—chosen to include metal-poor RGs—is likely contaminated by metal-rich stars younger than about 1 Gyr, which arise either from the bar itself or from a superimposed disk population in the central region of the LMC (see the isochrones in Figure <ref> as an example).Such stars show up clearly in Figure <ref>a, populating in the region of high T_ eff and low logg (i.e. upper-left corner of the plot).When imposing the same blue color cut as we did for LMC1 (i.e. I + 5×(V-I)>22.00 according to Eq. <ref> and also see the tilt red dashed line in Figure <ref>), we exclusively remove these more massive giants, resulting in a sample of 109 stars shown in Figure <ref>b.Though a few old metal-poor stars are removed in this selection, the resulting HRD cannot be distinguished from that of the LMC1 sample (Figure <ref>d). C05's sample also extends to a higher luminosity than our LMC1 sample (see the horizontal red dashed line in Figure <ref>).So we select another subsample from the LMCC sample using the same limits for the LMC1 sample (Eq. <ref>).The remaining 62 stars in Figure <ref>c produce an HRD in good agreement with the HRD of the full LMC1 sample shown in Figure <ref>d.We note that a few apparently very metal-poor stars are located in the lower-right corner of both Figures 6b and 6d, where we would expect the most metal-rich giants. In the nomenclature of Table <ref>, their IDs are LMCC-b086, LMCC-r001, LMC1-b017 and LMC1-b080. On close inspection, we found that the spectra of LMC1-b017 and LMC1-b080 can be well fitted with a two-star rather than a single-star model. In Figure <ref>, their observed spectra are plotted on the left with their best-fit single-star model spectra. On the right, the best-fit spectra employing a two-star model is shown for comparison. For LMC1-b017, the reduced chi-square value decreases from 1.7 (single-star) to 1.0 (two-star); and for LMC1-b080, the reduced chi-square value decreases from 5.2 (single-star) to 2.0 (two-star). In both cases, the two-star fits are significantly better.These two stars appear to be either spectroscopic binaries or physically unrelated stars that happen to be blended photometrically.We favor the second interpretation because double-RG binaries should be quite rare on stellar evolutionary timing grounds.For the two cases shown in Figure <ref>, the relative velocity shifts of the individual stars are in 40.0±0.9 km s^-1 for LMC1-b017 and 81.5±0.4 km s^-1 for LMC1-b080. Such velocities would be very difficult to explain in binary systems consisting of RGs with masses around 1 M_⊙. Spectral blending—whether due to a physical or photometric binary—can also explain the inferred low metallicities of the stars.Since the stars have different systemic velocities, their lines will appear weaker relative to the approximately doubled continuum, resulting in a low inferred metallicity when fit as a single star. The other two anomalous stars in Figure <ref>b, i.e. LMCC-b086 and LMCC-r001, do not exhibit obviously composite spectra and it is unclear why they exhibit relatively high surface gravities for their low metallicities.Alternatively, these stars may be metal-poor subgiants located foreground to the LMC in the Galactic halo. Whatever the origin, we exclude these four stars from our RG sample and subsequent analysis. This resulting LMCC and LMC1 samples consist of 131 and 177 stars, respectively. Finally, we compare the results of our reduced LMC1 sample to the predictions of stellar models from <cit.> in Figure <ref>. We applied a systematic offset of -0.3 dex to thescale of the stellar models to better match the data in a systematic sense. Given this shift, the overall trends and ranges ofand T_ eff expected in the models appears to fit the data reasonably well. Matching these isochrones represents a practical way to constrain the ages of individual RGs if their masses are well known.We discuss the limitations of such mass calculations in Section <ref>.§.§ Metallicity Distribution Function Figure <ref> shows a set of MDFs derived from our reduced LMCC and LMC1 samples.In all cases, we have fit the observed MDFs (the histograms) with two Gaussian distributions (the curves; also known as the two Gaussian mixture model), each of which represents a population parameterized by a Gaussian mixture weight w_i, a mean value μ_i and a standard deviation σ_i.The Gaussian mixture weight is the fraction of a single Gaussian distribution relative to the sum of the two Gaussians, and all weights should sum to unity.The best-fit Gaussian parameters can be found in the legend of each panel in Figure <ref>.C05 first fitted the MDF of the LMC bar by two populations. Their original MDF contains 373 RGs and can be best fitted by two Gaussian distributions as (w_1, μ_1, σ_1)=(0.89, -0.37, 0.15) and (w_2, μ_2, σ_2)=(0.11, -1.08, 0.46).One goal of our LMCC selection was to acquire a subsample from C05 without changing their original MDF.It can be confirmed—by the dashed histogram in Figure <ref>a with its best-fitting curve—that there is no significant difference between the MDF of the original C05 sample and the MDF of 131 stars. With this subsample of 131 stars, we also confirm that the fitting results of two Gaussian distributions are the best among one to ten Gaussian mixture models, and so is the case for all the following MDFs. Figure <ref>a also provides a direct comparison between the MDFs of the same 131 RGs from C05's (dashed) and our (solid) metallicity measurements.Our MDF peaks at a lower metallicity for both the metal-rich and metal-poor populations; the differences relative to C05 are 0.27 dex and 0.14 dex (in the sense C05-M2FS), respectively.In Figures 9b and 9c, we compare the MDFs between the LMCC and LMC1 samples.The LMC1 MDF remains the same in both figures, with the best-fit parameters as (w_1, μ_1, σ_1)=(0.85±0.06, -0.69±0.02, 0.16±0.01) and (w_2, μ_2, σ_2)=(0.15±0.06, -1.23±0.20, 0.40±0.05). The LMCC MDFs, on the other hand, are changed due to the different sample sizes: all 131 stars in the reduced LMCC sample are included in Figure <ref>b; while in Figure <ref>c, only the 62 stars are considered for being located in the same CMD region determined by the LMC1 sample (see Section <ref> and Figure <ref>c).Both LMCC MDFs agree well with the LMC1 MDF.One problem with all the MDFs mentioned above is that they may reflect biases due to the sample selection effects.Since the LMC1 sample was selected in a very specific manner (see Section <ref> and Figure <ref>), it is feasible for us to correct the LMC1 MDF for its underlying selection bias.Our correction starts by counting stars within every rectangle shown in Figure <ref>.Within a given rectangle, the sub-MDF is corrected by multiplying a weight factor, which is a ratio of the total number of stars divided by the observed number of stars.This step is then carried out for all the rectangles, and all sub-MDFs are summed up before being renormalized to a final corrected MDF.The comparison between the corrected MDF and the `raw' MDF from the reduced LMC1 sample is shown in Figure <ref>d.The corrected LMC1 MDF has a slightly higher mean for the metal-rich component but with the same fraction, and can now be fitted as (w_1, μ_1, σ_1)=(0.85±0.08, -0.66±0.02, 0.17±0.01) and (w_2, μ_2, σ_2)=(0.15±0.08, -1.20±0.24, 0.41±0.06). This correction can be applied in a reproducible manner to the LMC1 sample, but not to the LMCC sample due to the different and ill-defined selection method adopted by C05.As a result, the corrected MDF based on 177 stars in the LMC1 sample represents our best estimate of the MDF of the central LMC bar.§.§ Kinematics The heliocentric LOS velocities range between 175.2 km s^-1 and 343.1 km s^-1 for our full RG sample (including all the LMCC and LMC1 RGs), implying that all stars in our sample should be considered as LMC members according to <cit.>.The mean heliocentric LOS velocity of 131 LMCC stars is 262.9±2.1 km s^-1 with a standard deviation of 24.4±1.6 km s^-1, compared to 258.1±2.1 km s^-1 and 28.3±1.5 km s^-1 for 177 LMC1 stars.For a combined sample of 239 RGs (i.e. all stars in Figures 6c and 6d), we measured a mean heliocentric LOS velocity of 258.5±1.8 km s^-1 with a velocity dispersion of 27.8±1.3 km s^-1, in good agreement with previous measurement of the LMC bar by C05.In Figure <ref>, we plot the heliocentric LOS velocities (left panels) and the corresponding velocity dispersions (right panels) both as a function of metallicity.The top panels show the kinematics of LMCC and LMC1 separately, while the bottom panels show the combined results from the two fields.In all cases, the whole sample is divided into four subsamples according to the metallicity bins in Table <ref>, in which we also report the mean heliocentric velocity and the corresponding velocity dispersion in each bin based on the sample of 239 RGs. It can be seen both in Figure <ref> and Table <ref> that the velocity dispersion increases significantly from 18.7±1.9 km s^-1 for the most metal-rich stars to 31.2±4.3 km s^-1 for the most metal-poor stars. § SUMMARY AND DISCUSSION In this paper, we have measured physical parameters for 312 stellar targets in the LMC bar from high-resolution spectra obtained with the multi-fiber facility M2FS on the Magellan/Clay Telescope.Assuming E(V-I)=0.15±0.07 mag constantly, we initially estimated an effective temperature of each star from the OGLE-II V-I color <cit.> according to the color-temperature relation for giants <cit.>.Then, each star's LOS velocity, surface gravity and metallicity were fitted simultaneously by comparing each background-subtracted spectrum statistically with a library of template spectra made by synthetic modeling <cit.>.All of our stellar spectra are obtained from two LMC bar fields, labelled as `LMCC' and `LMC1', respectively.Four of the 312 stellar targets are confirmed to be possible double-star sources. Among the remaining 308 stars, 177 LMC1 RGs are observed spectroscopically for the first time. The remaining 131 LMCC RGs were first observed by C05 and 39 were also subsequently observed by VdS13.The reobserved stars are used to directly compare the measurements of the physical parameters among studies. As a result, we found that our metallicity measurements gave a MDF with lower peaks of both metal-rich and metal-poor populations, comparing to C05.The differences in these populations are Δ[Fe/H]=0.29±0.02 dex and 0.12±0.29 dex, respectively, given the typical measurement errors in metallicity are 0.06 dex and 0.12 dex for us and C05.Based on 177 RGs in the LMC1 sample, we measured a new mean metallicity in the LMC bar of [Fe/H]=-0.76±0.02 dex (σ=0.28±0.03 dex) and generated a new sample-selection-effect corrected MDF for the LMC bar (Figure <ref>d).The MDF can be best fitted by two Gaussian distributions with a portion of 85% and 15%, respectively.The majority population is made of the more metal-rich stars and has a mean value of [Fe/H]=-0.66±0.02 dex (σ=0.17±0.01 dex), while the minority one is relatively metal-poor and has a mean [Fe/H]=-1.20±0.24 dex (σ=0.41±0.06 dex).Our newly-observed MDF is different from that reported by C05 in two aspects: first, our metal-rich population peaks of about 0.29±0.02 dex lower than that found by C05; second, our metal-poor population fraction (≈ 15 %) is slightly larger than that (≈ 11 %) found by C05.In both LMCC and LMC1 samples, the kinematics as a function of metallicity provides a good way to trace the evolution of the LMC bar (Figure <ref>).The kinematics also show a clear trend of the decreased velocity dispersion with increasing metallicity.This trend not only confirms that more metal-poor stellar populations are distinguished dynamically by metallicity, but also implies a true chemo-dynamical evolution of the bar that covers at least a few crossing times of the system.C05 found that there may exist an old, thicker disk or halo population in the LMC bar with a velocity dispersion 40.8± 1.7 km s^-1 among the most metal-poor stars in their sample (5 %).Our results indicate the existence of the thicker component in the LMC bar, though a smaller value 31.2±4.3 km s^-1 is measured from our most metal-poor (≈10 %) stars.This reflects the kinematics of the old populations in the LMC bar, and is in good agreement with the velocity dispersion of 30 km s^-1 for RR Lyrae stars <cit.>. §.§ Stellar Masses:A Cautionary Tale The ultimate aim of this study is to map the chemo-dynamical properties as a function of position and age, such as the age-metallicity relation and the star formation history of the LMC bar.To take the first step, we compared our spectroscopic results with PARSEC isochrones <cit.> in Figure <ref>, which provides a way to measure the age of a RG if its mass is well constrained.To calculate the masses of the LMC1 RGs, we followed the method introduced by <cit.> who used the well-known relations L=4π R^2 σ T^4_ eff and g=GM/R^2 (where σ is the Stefan-Boltzmann constant and G is the gravitational constant). In our calculation, the luminosities were calculated from apparent OGLE-II V-band magnitudes, the distance modulus to the LMC <cit.>, a constant extinction (A_V=0.34 mag), and the bolometric corrections (BCs) interpolated from the model-based BC tables .We also used the solar bolometric magnitude M_ Bol, ⊙=4.77 mag and luminosity L_⊙=3.844× 10^33 erg/s <cit.>.Shown in Figure <ref> are the calculated masses as a function of metallicity.To elucidate the relation between mass and metallicity, we binned our sample by mass and calculated a mean metallicity for each bin (red squares).It can be seen that the mean metallicity increases with mass increasing, or equivalently with age decreasing. This phenomenon is similar to the one found by C05 on their age-metallicity relation.But an important problem with our mass calculation is evident in Figure <ref>: The expected mass of RGs in this portion of the CMD would be expected to be in a range spanning approximately from 0.6 to 5 M_⊙ (see the two dashed vertical lines in Figure <ref>). However, a significant number of our RGs have calculated masses below the lower limit.Due to this anomalously large range of masses from our simple calculation, it seems impossible to derive reliable ages from any RG mass-age relation (e.g. from the PARSEC isochrones).The origin of the discrepancy in our mass/age scale remains unclear and would not be solved by a simple offset in logg or in our T_ eff scale.We note that C05 and <cit.> encountered similar problems, though to a different quantitative extent.Therefore, we limit ourselves in concluding that there is an age-metallicity relation within the bar with a positive slope over most of the age sampled by normal RGs.Although the potential for extracting the age-metallicity distribution from multiplexed spectroscopy of field stars in the LMC bar is evident from this discussion, improved stellar parameters—particularlyvalues—will be needed to fully exploit this approach. We thank the anonymous referee for helpful suggestions.We thank Jeff Crane, Steve Shectman and Ian Thompson for invaluable contributions to the design, construction and support of M2FS.M.M. and Y.-Y.S. are supported by National Science Foundation (NSF) grant AST-1312997.M.G.W. is supported by NSF grants AST-1313045 and AST-1412999.I.U.R. acknowledges support from the NSF under Grant No. PHY-1430152 (JINA Center for the Evolution of the Elements).apj | http://arxiv.org/abs/1704.08363v1 | {
"authors": [
"Ying-Yi Song",
"Mario Mateo",
"Matthew G. Walker",
"Ian U. Roederer"
],
"categories": [
"astro-ph.GA",
"astro-ph.SR"
],
"primary_category": "astro-ph.GA",
"published": "20170426221622",
"title": "An Expanded Chemo-dynamical Sample of Red Giants in the Bar of the Large Magellanic Cloud"
} |
[e-mail:][email protected] Department of Physics, Royal Holloway, University of London, Egham TW20 0EX, U.K. Department of Physics, Princeton University, Princeton, NJ08544, U.S.A. Department of Physics, Royal Holloway, University of London, Egham TW20 0EX, U.K. Charles University, Faculty of Mathematics and Physics, Department of Condensed Matter Physics, Ke Karlovu 5, 121 16, Praha, Czech Republic Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), Forschungszentrum Jülich GmbH, Lichtenbergstr. 1, 85748 Garching, Germany Heinz Maier-Leibnitz Zentrum (MLZ), Technische Universität München, Lichtenbergstr. 1, 85748 Garching, Germany Laboratoire Léon Brillouin (CEA-CNRS), CEA Saclay, F-91911 Gif-sur-Yvette, France Department of Physics, Royal Holloway, University of London, Egham TW20 0EX, U.K. Physik Department E21, Technische Universität München, 85748 Garching, Germany Physik Department E21, Technische Universität München, 85748 Garching, Germany Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, U.K. In the metallic magnet Nb_1-yFe_2+y, the low temperature threshold of ferromagnetism can be investigated by varying the Fe excess y within a narrow homogeneity range. We use elastic neutron scattering to track the evolution of magnetic order from Fe-rich, ferromagnetic Nb_0.981Fe_2.019 to approximately stoichiometric NbFe_2, in which we can, for the first time, characterise a long-wavelength spin density wave state burying a ferromagnetic quantum critical point. The associated ordering wavevector _ SDW=(0,0,l_ SDW) is found to depend significantly on y and T, staying finite but decreasing as the ferromagnetic state is approached. The phase diagram follows a two order-parameter Landau theory, for which all the coefficients can now be determined. Our findings suggest that the emergence of SDW order cannot be attributed to band structure effects alone. They indicate a common microscopic origin of both types of magnetic order and provide strong constraints on related theoretical scenarios based on, e.g., quantum order by disorder.[75.25.-j, 75.40.-s, 75.40.Cx, 75.50.Bb] Ultra-small-moment incommensurate spin density wave order masking a ferromagnetic quantum critical point in NbFe_2 F. M. Grosche==================================================================================================================The exploration of ferromagnetic quantum phase transitions in metals has motivated numerous theoretical and experimental studies <cit.>, which have led to the discovery of non-Fermi liquid states <cit.> and of unconventional superconductivity (e.g. <cit.>). The underlying question, however, whether a ferromagnetic quantum critical point (QCP) can exist in clean band magnets, remains controversial. Fundamental considerations <cit.> suggest that the ferromagnetic QCP is avoided in clean systems by one of two scenarios: either the transition into the ferromagnetic state becomes discontinuous (first order), or the nature of the low temperature ordered state changes altogether, for instance into nematic or long-wavelength spin density wave (SDW) order <cit.>. Whereas there are many examples for the first scenario, including ZrZn_2 <cit.>, Ni_3Al <cit.> and UGe_2 <cit.>, the transition into a modulated state on the border of band ferromagnetism has proven to be more challenging to investigate. Recent reports show that this scenario may apply more widely beyond the comparatively simple band ferromagnets for which it was first discussed: (i) the masking of the field-tuned quantum-critical end point of the continuous metamagnetic transition of Sr_3Ru_2O_7 by two SDW phases <cit.>, (ii) the evolution of FM into long-wavelength SDW fluctuations in the heavy-fermion system YbRh_2Si_2 <cit.>, which displays a high Wilson ratio <cit.> andbecomes FM under Co-doping <cit.>, (iii) the emergence at finite temperature of SDW order in the ferromagnetic local moment system PrPtAl <cit.>, and the appearance of modulated magnetic order at the border of pressure-tuned FM systems CeRuPO <cit.>, MnP <cit.>, or LaCrGe_3 <cit.>.The band magnet NbFe_2 is a particularly promising candidate for investigating the SDW scenario in a clean itinerant system, because it is located near the border of ferromagnetism at ambient pressure <cit.>, enabling multi-probe studies and, in particular, neutron scattering. Ferromagnetic order can be induced at low temperature by growing Fe-rich Nb_1-yFe_2+y with y as small as 1% (Figure <ref> <cit.>). Compton scattering results on the Fe-rich side of the phase diagram have been analyzed by assuming ferrimagnetism as the ground state <cit.>, but more direct probes of the local fields by Mößbauer spectroscopy point to ferromagnetism as the ground state <cit.>. The precise low temperature state for y=0 has remained unidentified since early NMR studies first suggested that stoichiometric NbFe_2 may display low-moment SDW order <cit.>. Repeated attempts to detect the SDW order in neutron scattering were unsuccessful, but recent results from ESR, μSR, and Mößbauer spectroscopy also point strongly towards SDW order <cit.>. Non-Fermi liquid forms of resistivity and low temperature heat capacity have been observed in slightly Nb-rich NbFe_2. <cit.>figure_1Phase diagram of Nb_1-yFe_2+y with results for bulk T_ C (squares) and T_ N (diamonds) from single-crystal neutron diffraction (filled symbols) embedded into previous results from polycrystals (empty symbols)<cit.>. Vertical solid lines indicate the T range of neutron diffraction measurements. Of the two ferromagnetic (FM) phases, the one on the more Fe-rich side is separated from the paramagnetic (PM) state by a spin-density wave (SDW) at low temperatures, where non-Fermi liquid (NFL) behaviour is found as well. T_ 0, the FM phase boundary buried by the SDW phase (dashed line) is an extrapolation of T_ 0 values (circles) measured or calculated for the single crystals <cit.>. The inset shows the relevant reciprocal-space region, which was accessible during the neutron diffraction experiments. Circles show the presence and crosses the absence of SDW peaks. The SDW peak pattern is consistent with moments pointing along the c-axis.figure_3T dependences of normalised FM intensities and FM ordered moments μ_ FM, SDW ordered moments μ_ s, and of SDW ordering wavevector values _ SDW of Nb_1-yFe_2+y obtained from measurement sequences going down (downward triangles) and up (upward triangles) in T for Samples A (y=+0.003, black, grey), B (y=+0.015, dark blue, light blue), and C (y=+0.020, red, orange). Each sample has been investigated in two separate experiments (empty and filled symbols, respectively) demonstrating the reproducibility of the results. Lines are guides to the eye. (a) FM (1 0 2) intensities have been normalised by the nuclear (1 0 2) intensities at T_ C. The nuclear (1 0 2) intensities have been subtracted. Conversion to FM ordered moments μ_ FM has been obtained as described in the Supplement <cit.>. (b) SDW ordered moments μ_ s have been obtained from integrated intensities at =(1,0,1+l_ SDW) and comparison with nuclear and FM (1 0 2) intensities <cit.>. Lines to the right of the maxima are fits described in <cit.>. Lines to the left of the maxima are guides to the eye. (c) For all studied samples _ SDW has the form (0,0,l_ SDW) in the whole T range. A thermal hysteresis is observed in l_ SDW of Samples B and C, which contain a SDW-FM phase transition.Here, we present the outcomes of a neutron diffraction study, which for the first time demonstrates unambiguously the existence of a long-wavelength modulated magnetic state (SDW) forming on the border of ferromagnetic order at low temperature in stoichiometric single crystals of NbFe_2. We track its evolution with temperature and composition and probe the underlying ferromagnetic order. We find that the SDW state indeed displays a very small ordered moment μ_ s<0.1 μ_ B/(), which explains why previous neutron scattering experiments failed to detect it. Our data confirms the second order nature of the PM-SDW phase transition and a temperature hysteresis in the SDW ordering wavevector suggests that the SDW-FM phase transition is first order. The observed characteristics of the SDW order including the evolution of its ordering wavevector, which we find decreases on approaching the FM transition, as well as our theoretical analysis of the resulting phase diagram suggest that the occurrence of long wavelength SDW order on the border of FM in NbFe_2 is not coincidental, but rather emerges from the proximity to a FM quantum critical point, which is buried within the SDW dome.Experimental.— Large single crystals of C14 Laves phase NbFe_2 (lattice constants a=4.84 Å and c=7.89 Å) with compositions chosen across the iron-rich side of the homogeneity range have been grown in a UHV-compatible optical floating zone furnace from polycrystals prepared by induction melting <cit.>. The single crystals have been characterised extensively by resistivity, susceptibility, and magnetisation measurements, as well as by x-ray diffraction and neutron depolarization <cit.>, the latter showing homogeneity in structure and chemical composition. In this study, three samples have been measured: (i) sample A, whichis almost stoichiometric (y = +0.003); (ii) sample B, which is slightly Fe-rich (y = +0.015); and (iii) sample C, which is more Fe-rich still (y = +0.020) <cit.>.For the neutron scattering experiments the samples were mounted on Al holders and oriented with (h0l) as the horizontal scattering plane. In order to enhance the signal to background ratio neutron diffraction was carried out at two cold triple-axis spectrometers in elastic mode: Panda at the Heinz Maier-Leibnitz Zentrum (MLZ) <cit.> and 4F2 at the Laboratoire Léon Brillouin (LLB). Panda was run with neutron wavevectors k_ i=k_ f=1.57 Å^-1 and 4F2 was used with k_ i=k_ f=1.30 Å^-1 <cit.>.Results.— The principal discovery of SDW Bragg reflections in the Nb_1-yFe_2+y system is presented in Fig. <ref>, which focuses on the reflection (1 0 1)^+ at =(1,0,1)+_ SDW with _ SDW=(0,0,l_ SDW). SDW Bragg reflections corresponding to an ordering wavevector _ SDW have been confirmed for all samples of this study. l_ SDW shows a significant y and T dependence, which will be discussed further below. We have also established the SDW's long-range character <cit.>. In addition to the data discussed above of the (1 0 1)^+ reflection, the vicinity of nuclear Bragg reflections has been scanned for further SDW satellite peaks. SDW satellite peaks are present at =(1,0,1)±_ SDW and =(1,0,2)±_ SDW but they are absent at =(1,0,0)±_ SDW and at =(0,0,l)±_ SDW with l = 1,2,3 (see inset of Fig. <ref>). This distribution of allowed and forbidden satellite Bragg peaks is consistent with moment orientation along the c-axis, suggesting a lineary polarized SDW state rather than spiral order. Based on AC susceptibility and torque magnetometry data, the c-direction has been determined to be the magnetic easy-axis independent of the chemical composition <cit.>.Magnetic neutron scattering from the FM order has been observed at the position of the weak nuclear Bragg point (1 0 2). Fig. <ref>a shows the T dependence of the intensities of the FM (1 0 2) Bragg reflections normalised by the intensities of the nuclear (1 0 2) reflections at T_ C for all three compositions. Conversion to FM ordered moments μ_ FM has been obtained as described in <cit.>. In the Fe rich Samples B and C, FM order is observed at low temperatures. The measurements of the FM (1 0 2) signals (Fig. <ref>a) and peaks of the SDW signals (Fig. <ref>b, see below) reveal onset temperatures at T=24.5 K in sample B and T=34 K in sample C.The T dependences of the SDW ordered moments μ_ s for all samples are shown in Fig. <ref>b. μ_ s has been obtained from integrated intensities at =(1,0,1+l_ SDW) and comparison with nuclear and FM (1 0 2) intensities as described in <cit.>. In the almost stoichiometric Sample A, SDW order emerges below T_ N=14.5 K and is present down to the lowest measured temperature of 1.4 K. In the slightly Fe-rich Sample B SDW order appears below T=32.3 K and is fully suppressed below T=18.5 K. Finally, in the most Fe-rich Sample C SDW order appears below T=38.3 K and is fully suppressed below T=30.5 K. The small size of the SDW moments (μ_ s<0.1 μ_ B/()) explains the difficulties in observing the SDW order in previous neutron diffraction experiments.In all three samples, the SDW intensity rises continuously below the onset temperature T_ N, suggesting a second-order PM-SDW transition. The peak SDW intensity coincides with the FM onset temperature T_ C in samples B and C, and there is a T-range below T_ C, in which SDW and FM order appear to coexist. This overlap can be attributed to a distribution of transition temperatures within the sample, giving bulk T_ N=30.1 K, T_ C=21.5 K, (Sample B) and T_ N=37.1 K, T_ C=32.2 K (Sample C) in good agreement with bulk magnetic response <cit.>.The SDW ordering wavevector _ SDW=(0,0,l_ SDW) is found to depend significantly on composition and temperature (Fig. <ref>c): (i) the c-axis pitch number shifts from l_ SDW(T_ N)=0.157(1) in Sample A to l_ SDW(T_ N)=0.095(1) in Sample C (corresponding to an incommensurate modulation along the c-axis with a pitch in the range λ_ SDW≈ 50-100 Å); (ii) in Samples B and C l_ SDW shows a significant T dependence, decreasing by about 20% with decreasing T; (iii) l_ SDW(T) stays finite at the SDW-FM transition, so there is a discontinuous change in the magnitude of the magnetic ordering wavevectors there from finite (q_ SDW) to zero (q_ FM); (iv) in Samples B and C l_ SDW(T) reproducibly displays significant thermal hysteresis with lower l_ SDW values when warming into the SDW phase from the FM state. (i) and (ii) means that l_ SDW is being reduced on approaching FM and (iii) and (iv) point to the first-order nature of the SDW-FM transition.Discussion.— The delicate SDW state of the Nb_1-yFe_2+y system occurs in a narrow composition and temperature range attached to the threshold of FM order (Fig. <ref>). Because the SDW-FM transition is first order, the emergence of the SDW state masks the FM QCP and buries it inside a SDW dome. Our neutron diffraction results have revealed the following main characteristics of the SDW state: (i) an incommensurate ordering wavevector _ SDW with its striking dependence on y and T, (ii) the long range nature in contrast to what would be expected for a spin glass <cit.>, and (iii) a small linearly polarised ordered moment. The small ordered moments, which contrast with the large fluctuating moments μ_ eff≈1 μ_ B derived from the temperature dependence of the magnetic susceptibility, and the linearly polarized rather than helical order suggest that the SDW state should be understood within a band picture, not a local moment picture. However, the y and T dependent values of _ SDW cannot be explained based on the electronic band structure alone.Within density functional theory (DFT), the implications of the electronic band structure for magnetic order have been examined in detail. Direct total energy calculations for different ordering patterns <cit.> suggest that energy differences between a number of magnetic ground states are very small. Moreover, the wavevector dependence of the bare band structure derived susceptibility χ_^(0) or Lindhard function <cit.> is not consistent with the long-wavelength SDW order reported here as it does not feature a significant enhancement near the measured _ SDW range, suggesting that SDW order in NbFe_2 has a more subtle origin. The observed dependence of q_ SDW (Fig. <ref>c), which decreases as the FM state is approached, indicates that χ_q is strongly modified by order parameter fluctuations on the threshold of FM.The close connection between modulated and uniform magnetic order in NbFe_2 is striking. It can be modelled effectively with the help of a Landau expansion of the free energy in terms of the two order parameters M (for FM) and P (for SDW order) <cit.>, which in zero field reduces toF/μ_0 = a/2M^2 +b/4M^4 +α/2P^2 + β/4P^4 + η/2P^2 M^2The use of this model is corroborated in this study by the observation that q_ SDW does not go to zero but stays finite at the SDW-FM transition. While the parameters a and b can be obtained from magnetization measurements, this is not possible for the parameters α, β and η. They can, however, be determined from the SDW ordered moment P observed in our neutron scattering study.We make use of the Landau theory result that P^2 = -α/β and that within the SDW phase the intercept a^* and slope b^* of the Arrott plot H/M vs. M^2, which can be determined in bulk magnetization measurements, become a^* = a-αη/β and b^* = b-η^2/β <cit.>. Writing a = a_1(T-T_0) and α = α_1 (T-T_N), we find that the parameters characterising the SDW order vary slowly with composition and are of similar magnitude as those characterising the FM order: α_1/α≃β / b ≃ 2± 1 and that the coupling parameter η≃β <cit.>. This implies that the ratio η/√(β b)>1 throughout, which rules out phase coexistence within the two-parameter Landau theory, in agreement with experimental observations. Most importantly, the parameter values point towards a common microscopic origin of both types of magnetic order and puts strong constraints on a microscopic theory of magnetism in NbFe_2.A number of theoretical studies <cit.> have noted that the FM quantum critical point is unlikely to be observed in clean, 3D ferromagnets, and will instead either be avoided by a change of the magnetic transition from second order to first order or masked by the emergence of long-wavelength SDW. Both scenarios can be attributed to nonanalytic terms in the free energy associated with soft modes. These contribute a singular q-dependence of the form q^2ln q to χ_q^-1 and thereby produce an intrinsic tendency towards long-wavelength modulated order as the FM QCP is approached. An alternative approach <cit.> arrived at similar conclusions by considering the contribution of order parameter fluctuations to the free energy, when different ordered states were imposed, causing differences in the phase space available to critical fluctuations of FM and modulated magnetic order. Helical order in the local moment system PrPtAl <cit.> has recently been presented as a likely manifestation of this scenario, but a demonstration in a band magnet is still outstanding. New neutron data demonstrate that SDW order emerges near the border of ferromagnetism in the C14 Laves phase system Nb_1-yFe_2+y, burying an underlying FM QCP. This suggests that the SDW order in Nb_1-yFe_2+y is caused by an intrinsic instability of a ferromagnetic quantum critical point to modulated magnetic order, which has been postulated on the basis of fundamental considerations but has not before been detected in a band magnet.This work is based upon experiments performed at the PANDA instrument operated by JCNS at the Heinz Maier-Leibnitz Zentrum (MLZ), Garching, Germany, and at the 4F2 instrument at LLB, CEA-Saclay, France. We acknowledge support by the EPSRC through grant EP/K012894/1 and by the German Science Foundation (DFG) through FOR 960 (CP) and SFB/TR 80 (CP). This research project has also been supported by the European Comission under the 7th Framework Programme through the 'Research Infrastructures' action of the 'Capacities' Programme, Contract No: CP-CSA_INFRA-2008-1.1.1 Number 226507-NMI3. We thank S. Friedemann, M. Brando, and S. 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"authors": [
"P. G. Niklowitz",
"M. Hirschberger",
"M. Lucas",
"P. Cermak",
"A. Schneidewind",
"E. Faulhaber",
"J. -M. Mignot",
"W. J. Duncan",
"A. Neubauer",
"C. Pfleiderer",
"F. M. Grosche"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170426231729",
"title": "Ultra-small moment incommensurate spin density wave order masking a ferromagnetic quantum critical point in NbFe$_2$"
} |
Overcoming the ill-posedness through discretization in vector tomography:Reconstruction of irrotational vector fields Technical (PhD Transfer) Report January 2011Student: Alexandra KoulouriSupervisor: Prof. M. PetrouCommunications & Signal Processing GroupDepartment of Electrical & Electronics EngineeringImperial College London romanarabic CHAPTER: INTRODUCTION Vector field tomographic methods intend to reconstruct and visualize a vector field in a bounded domain by measuring line integrals of projections of this vector field.In particular, we have to deal with an inverse problem of recovering a vector function from boundary measurements. As the majority of inverse problems, vector field method is ill posed in the continuous domain and therefore further assumptions, measurements and constraints should be imposed for the full vector field recovery. The reconstruction idea in the discrete domain relies on solving a numerical system of linear equations which derives from the approximation of the line integrals along lines which trace the bounded domain <cit.>.This report presents an extensive description of a vector field recovery method inspired by <cit.>, elaborating on fundamental assumptions and the ill conditioning of the problem and defines the error bounds of the recovered solution. Such aspects are critical for future implementations of the method in practical applications like the inverse bioelectric field problem.Moreover, the most interesting results from previous work on the tomographic methods related to ray and Radon transform are presented, including the basic theoretical foundation of the problem and various practical considerations.§ MOTIVATIONIn the present project, the final goal is the implementation of a different approach for the EEG (Electroencephalography) analysis employing the proposed vector field method. Rather than estimating strengths or locations of the electric sources inside the brain, which is a very complicated task, a reconstruction of the corresponding static bioelectric field will be performed based on the line integral measurements. This static bioelectric field can be treated as an “effective” equivalent state of the brain at any given instant. Thus for instance, health conditions and specific pathologies (e.g. seizure disorder) may be recognized.For this purpose, in the current report a robust mathematical and physical model and subsequently an efficient numerical implementation of the problem will be formulated. As a future stage, this theoretical and numerical formulations will be adapted to the real EEG problem with the help of experts and neuroscientists.§ REPORT STRUCTUREThe rest of the report consists of four sections. The 2^st chapter is introductory and gives a brief description of the previously related work as well as the mathematical definition and theorems used by vector field tomographic methods. The 3^rd chapter gives an overview of the numerical vector field recovery method. Mathematical definitions, physical assumptions and conditions for the recovery of an irrotational vector field from line integral measurements without considering any boundary condition are described. Also, the approximation errors derived from the discretization of the line integrals and the a-prior error bounds are estimated. In the 4^th chapter, verification of the theoretical model by performing simulations as well as the practical adaptation of this model to the inverse bioelectric field problem using EEGmeasurements are described. Finally in the 5^th chapter, future work is discussed.CHAPTER: PREVIOUS WORKIn this chapter, we review several techniques that are important mathematical prerequisites for a better understanding of the vector field reconstruction problem. Moreover, an overview of the most prominent potential applications is presented.§ MATHEMATICAL PRELIMINARIES The basic mathematical tools for the vector field problem formulation are presented in the following sections. §.§ Formulation of Vector Field Tomography problemThe reconstruction of a scalar function f(n) from its line integrals or projections in a bounded domain is a well known problem. Today there are many practical and research applications in different fields such as biomedicine (e.g. MRI, CT), acoustic and seismic tomography which employ this method with great success and accuracy. However, there is a variety of other applications, like blood flow (velocity) in vessels or the diffusion tensor MRI problem where the estimation and the imaging of a vector field can be essential for the extraction of useful information. In these cases tomographic vector methods intend to reconstruct these fields from scalar measurements (projections) in a similar way to the scalar tomographic methods.The mathematical formulation of the vector tomographic problem is given by the line integrals (projection measurements I_L)I_L=∫_LF· dr=∫F·r̂drwhere r̂ is the unit vector in the direction of line L and F is the vector field to be recovered.A different formulation for the projection measurement in 2D domain commonly used in bibliography involves the 1-D Dirac delta function such asI_L=∫∫_D(F_x(x,y)cosϕ+F_y(x,y)sinϕ)δ(xsinϕ-ycosϕ-ρ)dxdywhere xsinϕ-ycosϕ=ρ is the line function with parameters (ρ,ϕ) defined as shown in figure <ref> and D is the bounded domain where F(x,y)≠0.In general the vector tomographic problem is considered to be ill posed since F is defined by two or three components. However, with the application of certain constraints, restrictions and further assumptions there are ways to solve the problem. §.§ Helmholtz decompositionThe Helmholtz decomposition <cit.> is a fundamental theorem of the vector calculus analysis as we shall see later.It states that any vector F which is twice continuously differentiable and which, with its divergence and curl, vanishes faster than 1/r^2 at infinity, can be expressed uniquely as the sum of a gradient and a curl as follows:F=F_I+F_S ⇒ F=-∇Φ+ ∇×AThe scalar function Φ is called the scalar potential and A is the vector potential which should satisfy ∇·A=0.Since, ∇×F_I=∇×(∇Φ)=0, component F_I is called irrotational or curl-free while F_S is the solenoidal or divergence-free component as it satisfies ∇F_S=∇·(∇×A)=0.In the case of a 2D vector field F(x,y), the decomposition equation becomes F=-∇Φ+∇× A_z(x,y)ẑ . §.§ Vectorial Ray TransformIn tomographic theory, the line integral <ref> is called ray transform. This transform is closely related to the Radon transform <cit.> and coincides with it in two dimensions. In higher dimensions, the ray transform of a function is defined by integrating over lines rather than hyper-planes as the Radon transform.In particular, for a bounded volume V (F=0 outside V) and according to equation <ref>, the vectorial ray transform can be expressed asI(ϕ,θ,p)= ∫_L(ϕ,θ,p)F·r̂_ϕ,θ dr== ∫_L(ϕ,θ,p)F_x(x,y,z)cosϕsinθ dr+ ∫_L(ϕ,θ,p)F_y(x,y,z)sinϕsinθ dr+ ∫_L(ϕ,θ,p)F_z(x,y,z)cosθ drwhere F_x, F_y and F_z are the components of vector F, ϕ and θ define the direction of the r̂_ϕ,θ unit vector along line L(θ,ϕ,p) as shown in figure <ref> and p=(x_p,y_p). Point p gives the coordinates of the line in the plane which passes through the origin of the axes and it is vertical to r̂_ϕ,θ (see fig. <ref>).Consequently, the line integral <ref> can be written as a volume integral using the appropriate Dirac delta functions. Thus we haveI(ϕ,θ,p) =∫∫∫_VF_x(x,y,z)cosϕsinθδ_x_pδ_y_pdxdydz+∫∫∫_VF_y(x,y,z)sinϕsinθδ_x_pδ_y_pdxdydz+∫∫∫_VF_z(x,y,z)cosθδ_x_pδ_y_pdxdydz whereδ_y_p=δ(xsinϕ-ycosϕ-y_p)andδ_x_p=δ(-xcosϕcosθ-ysinϕcosθ+zsinθ-x_p) In two dimensions, θ=π/2 and p is the signed distance of the line from the origin of the axes. Thus, equation <ref> becomesI(ϕ,p)=∫∫_D(F_x(x,y)cosϕ+F_y(x,y)sinϕ)δ(xsinϕ-ycosϕ-p)dxdyand F(x,y)=0 outside D. §.§ Central Slice TheoremThe solution to the inverse scalar ray transform is based on the central slice theorem (CST). CST states that the values of the 2D FT of scalar function f(x,y) along a line with inclination angle ϕ are given by the 1D FT of its projection I(ϕ,p). This fact combined with the implementation of many practical algorithms (e.g. back-projection) gave rise to the development of accurate and robust reconstruction methods.In the vectorial ray transform, the CST does not help us solve the problem. However, the formulation of the problem based on the CST is important for better understanding the theoretical approaches which will be described in the next section.Let the Fourier Transform of I(ϕ,θ,p={x_p,y_p}) beI_ϕ,θ(k_1,k_2)=∫∫ I(ϕ,θ,x_p,y_p)e^-i(k_1x_p+k_2y_p)dx_pdy_p Then according to equations <ref> and <ref> we obtainI_ϕ,θ(k_1,k_2)=cosϕsinθF_x(u,v,w)+sinϕsinθF_y(u,v,w)+cosθF_z(u,v,w)where F_x, F_y and F_z are the Fourier transforms of F_x, F_y and F_z respectively and u=k_1sinϕ-k_2cosϕcosθ, v=-k_1cosϕ-k_2sinϕcosθ andw=k_2sinθ.Applying the Helmholtz decomposition (eq. <ref>) we haveF_x(x,y,z)=∂A_z/∂ y -∂ A_y/∂ z-∂Φ/∂ x F_y(x,y,z)=-∂A_z/∂ x +∂ A_x/∂ z-∂Φ/∂ y F_z(x,y,z)=∂A_y/∂ x -∂ A_x/∂ y-∂Φ/∂ z Their Fourier transform leads toF_x(u,v,w)=ivA_z(u,v,w)-iwA_y(u,v,w)-iuΦ(u,v,w) F_y(u,v,w)=iwA_x(u,v,w)-iuA_z(u,v,w)-ivΦ(u,v,w) F_z(u,v,w)=iuA_y(u,v,w)-ivA_x(u,v,w)-iwΦ(u,v,w)when A and Φ tend to zero on the volume boundaries.Therefore the Fourier transform of the projection can be written asI_ϕ,θ(k_1,k_2)= i(k_1cosϕcosθ+k_2sinϕ)A_x + i( k_1sinϕcosθ-k_2cosϕ)A_y-ik_1sinθA_zFor the two dimensional case, θ=π/2 and k_2=0 (as we have only one variable)I_ϕ(k)=ikA_z(kcosϕ,ksinϕ)It is important to be mentioned that irrotational component Φ vanishes in equations <ref> and <ref>.§ THEORETICAL APPROACHESThe most important theoretical and mathematical studies of the vector field tomographic reconstruction from boundary measurements, as well as the feasibility of this formulation to yield unique solutions under certain constraints, were investigated only by a small group of the research community working in this field. Norton <cit.>, Baun and Haucks <cit.> and Prince <cit.>, <cit.> gave a step by step mathematical solution to the problem on bounded domains employing the Radon transform theory. In the following subsection, a description of their ideas and their methods is presented. §.§ Tomographic Vector Field MethodsNorton in <cit.> and <cit.> was the first who defined the full mathematical formulation of the two dimensional problem. Norton proved that only the solenoidal component of a vector field F on a D bounded domain can be uniquely reconstructed from its line integrals. Moreover, he showed that when the field F is divergenceless i.e. there are no sources or sinks in D, then both components can be recovered.In particular, assuming a bounded vector fieldF, i.e F=0 outside a region D which satisfies the homogeneous Neumann conditions on ∂D (on the field's boundaries), he produced equation <ref> applying Helmzoltz decomposition (eq. <ref>) and the Central Slice Theorem (eq.<ref>). Therefore, he proved that only the solenoidal component ∇×A can be determined.Furthermore, in <cit.> he demonstrated that when vector field F is divergenceless (∇F=0), then irrotational component Φ can be recovered.From the divergence of the decomposition (eq. <ref>) we obtain∇F=-∇(∇Φ)+∇(∇×A)⇒∇^2Φ=0 Thus, Norton was led to the Laplacian equation ∇^2Φ=0. The solution of the Laplacian equation gives the irrotational component and a full reconstruction of the field is possible. Norton employed Green's theorem andF's boundary values for the estimation of Φ on D.Later Braun and Hauck <cit.> showed that the projection of the orthogonal components of the vector function (transverse projection measurement) leads to the reconstruction of the irrotational component ∇Φ. So, they proposed that for the full 2D field reconstruction, a longitudinal and a transverse measurement are neededI_∥=∫F·r̂dr I_⊥=∫F·r̂_⊥drwhere r̂ is the unit vector along the line and r̂_⊥ is the unit vector orthogonal to the line.Moreover, they examined the problem for non-homogeneous boundary conditions. In that case, the irrotational and solenoidal decomposition is not unique and they proposed to decompose the vector into three components: the homogeneous irrotational, the homogeneous solenoidal and the harmonic with its curl and divergence being zero. They verified their method carrying out fluid flow estimation experiments. With this method there is no need to assume that there are no sources inside the domain. However, the difficulty of taking transversal measurements as it was mention in <cit.> makes the method quite impractical, especially for Doppler back scattering methods.Prince <cit.> and Prince and Osman <cit.> extended the previous method to 3 dimensions, reconstructing both the solenoidal and the irrotational components of F from the inverse 3D Radon transform. Actually they evolved the Braun-Hauck's method by defining a more general inner product measurement which was called probe transform and it was expressed asG^p(a,ρ)=∫_^3p(a,ρ)·F(x,y,z)δ(xa_x+ya_y+za_z-ρ)dxdydzwhere p is the so-calledvector probe, ρthe distance of the projection plane from the origin and a=(a_x,a_y,a_z) the normal vector to the plane.With the application of Helmholtz decomposition for homogeneous field's boundaries (eq. <ref>) and the Cental Slice Theorem, equation <ref> becomes G^p(a,k)=(j2π k)p(a)·[Φ(ka)a+a× A(ka)]Therefore, if p is orthogonal to a then the irrotational component is eliminated, while when p is parallel to a then the solenoidal component vanishes. On this basic principle Prince and Osman based their model for the reconstruction of a 3D vector field. § PROPOSED APPLICATIONSA plethora of different applications have been proposed in the area of vector field tomography. Some of the earlier studies by Johnson et al.<cit.> and Johnson <cit.> were concerned with the reconstruction of the flow of a fluid by applying numerical techniques (iterative algebraic reconstruction techniques). Johnson et al. <cit.> used ultrasound measurements (acoustic rays) to reconstruct the velocity field of blood vessels. Later, other applications like optical polarization tomography <cit.> for the estimation of electric field in a Kerr material and oceanographic tomography <cit.> were also reported. In Kramar's thesis <cit.> a vector field method for the estimation of the magnetic field of the sun's coronal is presented, giving interesting results.Moreover, in vector field literature there are many other proposed applications in the area of Doppler back scattering, Optical tomography, Photoelasticity and Nuclear Magnetic Resonance Plasma physics <cit.>. However, there are only a few practical or commercial applications in this field. §.§ Vector Field Reconstruction and Biomedical ImagingVector field tomography has not received much attention in the area of medical applications. There are only a few papers <cit.>and one PhD thesis<cit.> which present relevant methods. The main area of research according to these papers are Doppler back scattering for blood flow estimation, although till now there are only simulations and theoretical formulations without performing any real experiment or employing real data.Moreover, the Lawrence Berkeley National Laboratory <cit.> has developed many tomographic mathematical tools and algorithms for medical imaging issues. As it is mention in <cit.>, their work has focused mainly on the implementation of algorithms for the reconstruction of the 3D diffusion tensor field from MRI tensor projections and iterative algorithms for solving the non-linear diffusion tensor MRI problem. § SUMMARYStudy of previous work indicates that the vector field tomography has practical potential. Till now much of the work was devoted to the theoretical development and formulation of the problem. Moreover it is clear that the acquisition of the measurements and the performance of real experiments are quite difficult tasks and a multidisciplinary collaboration is required. CHAPTER: VECTOR FIELD RECOVERY METHOD: A LINEAR INVERSE PROBLEM In the previous chapter, an extended description of the mathematical expression of the vector field tomographic problem in ray and Radon transform sense was presented. A different approach for vector field recovery method stemming from the numerical inverse problems theory will be considered here.The numerical solution of an inverse problem requires the definition of a set of equations mathematically adapted to the physical properties of the problem, subsequently, the design of the geometrical model where these equations operate and finally the discretization of the equations to form a numerical system such as the approximated solution to be as close as possible to the real solution of the model.So, the vector field recovery problem can be considered as an inverse problem which can be formulated as an operator equation of the formKx=ywith K being a linear operator between spaces X and Y over the fieldand where the geometrical and numerical models are designed according to the topological and error approximation requirements of the problem.The current method is based on the estimation of a vector field from the line integrals (projection measurements) in an unbounded domain. Thus, K operator is an integral operator and the vector field recovery method relies on solving a set of linear equations which derives from a set of numerically approximated line integrals which trace a bounded domain Ω, and are expressed as∫_LE· dr=Φ(a)-Φ(b)where E is the irrotational vector field, dr=r̂dr with r̂ the unit vector along line L and Φ gives the boundary measurements at starting point a and endpoint b.The initial idea was put forward in <cit.> where it was shown that there is potential for a vector field to be recovered in a finite number of points from boundary measurements. However, this initial idea followed an intuitive approach to the problem as it lacked the necessary conditions and assumptions about the recovered field and the formulation of the equations. Moreover, there was no clear and robust proof about the well or ill posedness of this inverse problem.Therefore, in the rest of this text:* the necessary preconditions and assumptions are defined such as the mathematical formulation of the problem fits the physical vector field properties as closely as possible;* an extended description of the vector field method employing the line integral formulation is given;* the ill posedness of the vector field method in the continuous domain is investigated and we show that the number of independent equations which stem from the problem's formulation give a final numerical system which is nearly rank deficient (ill conditioned);* the approximation and discretization errors resulted by the numerical implementation of the problem are formulated. Finally, the solution error of the numerical system is estimated and the conditions under which this error is bounded are presented, revealing that the discretization is a way of “self-regularization” of this ill conditioned inverse problem. § PRECONDITIONS AND ASSUMPTIONS Field E is assumed bounded and continuous in a domain Ω, bandlimited, irrotational and quasi-static.In particular, the quasi-static condition implies that the field behaves, at any instance, as if it is stationary. Moreover, the field is considered irrotational and satisfies ∇×E=0 and thus it can be represented by the gradient of a scalar function Φ. So, E=-∇Φ in a simply connected region (Poincare's Theorem <cit.>). Consequently, the gradient theorem gives∫_cE· dc=∫_c-∇Φ dc=Φ(a)-Φ(b)which will be the model equation of our problem and implies that the line integral of E along any curve c is path-independent.The irrotational property ∇×E=0 in integral form can be expressed by applying the well-known Stokes' theorem (curl-theorem) which relates the surface integral of the curl of a vector field over a surface S in Euclidean 3D space to the line integral of the vector field over its boundary ∂S such as∫_S∇×EdS=∫_∂SE· dcFor an irrotational field obviously∫_∂SE· dc=0So, the path integral of E over a closed curve (path) is equal to zero.Finally, vector field E is band limited, ∫∫_|Ω|Edxdy< ∞ and continuous in Ω thus is can be expanded in Fourier Series as E(x,y)=∑_ke_ke^ik⟨ x,y⟩+e̅_ke^-ik⟨ x,y⟩where k=(n_x,n_y) with n_x,n_y=0,±1,±2….As the physical and mathematical properties of the vector field have been defined, the mathematical and numerical formulation of the problem will be explained in the following section. § METHODOLOGY §.§ Mathematical ModelingThe formulation of the method is based on the idea in <cit.> and <cit.> to approximately reconstruct a vector field E at a finite number of points when a sufficiently large number of line integrals I_L_k along lines which trace the bounded domain, are known. The model equation for the recovery of an irrotational field inside a bounded convex domain Ωis given byI_L_k=∫_L_kE· drwhere line L_k traces the bounded domain and intersects it in two points. As the field is irrotational, the line integral is I_L_k=Φ(a_k)-Φ(b_k) where Φ are the boundary values at the intersection points a_k and b_k with domain Ω. So, a set of linear equations <ref> can be acquired for a finite number of known values Φ on the boundaries of Ω. §.§ Geometric ModelAs the model equations have been defined, the next step is the geometric model generation. Generally, the geometric model is a discrete domain of specific shape where the model equations are valid. For instance, if the line integral equations (model equations) were employed for field recovery from scalp potential recordings (e.g. EEG), then the geometric model would be a 3D mesh with scalp shape.In our initial approach for the evaluation of the method, a simple geometric model was designed as described in <cit.>. In particular, a discrete version of a 2D continuous square domain Ω=:{(x,y)∈[-U,U]^2} (fig.<ref>) was defined using elements of constant size [P× P] called cells or tiles and N=U/P.The goal was to recover the field in each cell solving a numerical system of a discretized version of model equations <ref>. The number and the positions of the tracing lines L_k in the bounded domain were defined by pairs of sensors (boundary measurements) which were placed in the middle of the boundary edges of all boundary cell (fig.<ref>). Each tracing line connected a pair of sensors which did not belong to the same side of the square domain and thus for a N=U/P number of edge cells or N sensors in each side of the domain, the connected pairs led to 6N^2 model equations (line integrals). The domain had N× N cells and the 2D vector field had two components. Thus the number of unknowns (value of each cell) was 2[N× N] and the number of equations was 6N^2 . §.§ Numerical ImplementationFor the numerical implementation of the method, the line integral <ref> was approximated using the Riemann's sum. Thus,I_L_k≈∑_(x_m,y_m)∈ L_kE(x_m,y_m)·r̂_ϕΔ rwhere E(x_m,y_m) are the unknown vector values at sampling points (x_m,y_m) along line L(ϕ,ρ)={(x,y)| xsinϕ-ycosϕ=ρ} with 0^0≤ϕ≤180^0 and ρ∈ and r̂_ϕΔ r=(cosϕx̂+sinϕŷ)Δ r (fig.<ref>).For the numerical approximation, the samples E(x_m,y_m) are assigned to values E[i,j]=(e_x[i,j],e_y[i,j]) based on an interpolation scheme. The simple case of the nearest neighbor approximation in a 2D square domain [-U,U]^2 leads to i=⌈x_m+U/P⌉ and j=⌈y_m+U/P⌉.Finally, all the approximated line integrals give a set of algebraic equations which can be represented byb=Ax where x contains the unknown vector field components in finite points with index u, b is the column vector with the measured values of the line integrals I_L_k. The elements of the transfer matrix A, a_k,u represent the weight of projection of the unknown element u on the I_L_k. For the case of a discrete square domain as it was described in subsection <ref>, the 4N sensors around the boundaries give 6N^2 equations and as the number of the unknowns is 2N^2, at first sight we conclude that we deal with an over-determined system.§ ILL POSEDNESS AND ILL CONDITIONING OF THE INVERSE PROBLEMIf we ignore the physical properties of the field, the reconstruction formulation described in the subsections <ref> and <ref> seems correct and robust as the system <ref> of algebraic equations is over-determined and thus with the least square method, the problem can be solved. However, the majority of the inverse problems are typically ill-posed. In the current problem, the irrotational assumption of the field implies that a line integral along a closed path is zero. More precisely, taking all pairs of sensors which do not belong to the same side we obtain tracing lines which create closed paths (fig. <ref>). Particulary, when the tracing lines L_1, L_2 … form a closed curve, according to equations <ref> and <ref> we obtainI_L_1+I_L_2+…+I_L_n=0or-I_L_1=I_L_2+…+I_L_nThis indicates that a line integral can be expressed as a linear combination of other line integrals and thus we have linearly dependent equations I_L_k. For instance, in figure <ref> the lines which connect sensors S_2, S_9 and S_23 create a closed loop and thus for equations (line integrals) I_L_2-9, I_L_9-23 and I_L_23-2 we haveI_L_2-9+I_L_9-23+I_L_23-2 ==∫_L_2-9E· dr+ ∫_L_9-23E· dr+ ∫_L_23-2E dr==∫_L_2-2E· dr =0Therefore, only two equations can be assumed independent since any one of the three can be expressed as a linear combination of the other two equations.The linear dependencies are quite significant in the continuous domain. On the other hand, in the discrete domain, where the integral is approximated by a summation, the dependencies are not so obvious as the accuracy of the continuous domain is lacking. Thus, taking Riemann's summation (eq.<ref>), leads to sums of equations close to zeroI_L_1+I_L_2+…+I_L_n≈ 0If the accuracy improves, i.e rather than applying nearest neighbor approximation, a different interpolation scheme like bilinear, cubic or more sophisticated techniques e.g. finite elements and a grid refinement of the bounded domain, can lead to more obvious equation dependencies.One important task is the examination of the stability of the solution of linear system <ref> i.e. to check whether the independent equations are enough to give a unique solution and whether matrix A has full rank. §.§.§ Ill Conditioning: Number of Independent EquationsThe number of independent equations is important for the solution of the problem since in the case that this number is less than the unknowns, the system is under-determined and different mathematical tools are needed.In order to define the number of independent equation, we have to exclude tracing lines which “close” paths such as making sure that any line starting from one sensor does not end up to the same sensor. The number of independent equations can be defined based on the fundamental properties of graph theory <cit.>.More specifically, we assume that the 4N sensors along the boundary of the square domain are the “vertices” of a graph G and that the lines which connect two sensors are the G graph's “edges”. The main property that this graph should satisfy is that any two “vertices” are linked by a unique path or in other words that the graph should be connected and without cycles. According to graph theory, a “tree” is a undirected simple graph G that satisfies the previous condition.Moreover, it is known that a connected, undirected, acyclic graph withN “vertices” has N-1 “edges”. Therefore in a 2D square domain (subsection <ref>) with 4N sensors along the boundaries, the maximum number of tracing lines in order to avoid loops is 4N-1 and so the number of independent equations (independent line integrals) is 4(N-1). In all, the system of equations may have at most 4N-1 independent equations. Obviously, for a system with 2N^2 unknowns, the 4(N-1) equations lead to an under-determined case.As a result, the transfer matrix A of system <ref> approximates a rank-deficient matrix i.e. there are nearly linearly dependent lines and the linear system may be inconsistent and severely ill conditioned. §.§ Ill Conditioning IndicatorsGenerally speaking, the ill posedness technically applies to continuous problems. The discrete version of an ill posed problem may or may not be severely ill conditioned.The discrete approximation will behave similarly to the continuous case as the accuracy of the approximation increases. With a “rough” approximation scheme and coarse discretization of the bounded domain, the linear dependencies of the equations are eliminated and transfer matrix A of system <ref> is not rank deficient.The condition number of transfer matrix A (system <ref>) and the magnitude of the singular values of A are reliable indicators of how close to rank deficiency and consequently to ill conditioning the numerical system is. For the case where a bounded domain Ω:{(x,y)→[-5.5,5.5]^2⊂^2} is discretized employing cells of size P× P=1× 1, and there are 4× U/P= 4× N=121 sensors along the boundaries (see subsection <ref>), 6N^2=726 equations and the field is created by a single charge in position (-19,19) on the z-plane as in <cit.>, the condition number is 84, which is not so large in order to deal with a severely ill-conditioned system. Moreover, the simulation results, which will be presented in the next chapter, show that the estimated solution is not far from the real one.So, under certain conditions, system <ref> which was derived from the numerical approximation of the line-integral can give acceptable results and the discretization process can be assumed as a kind of “self-regularization” (regularization by discretization) of the continuous ill posed problem.The aim of the work presented next is the mathematical definition of the numerical errors due to the line-integrals approximations and how these errors are related with the ill conditioning of the linear system <ref> defining an upper error bound of the system's numerical solution. § APPROXIMATION ERRORS For the numerical solution of a line integral system, one has to discretize the continuous problem and reduce it to a finite system of linear equations. The discretization process introduces approximation and rounding errors to the set of line integrals (model equations <ref>) which can be interpreted as perturbations. First, the mathematical formulation and nature of these errors will be described in the following analysis. Subsequently, the error of the numerical solution will be estimated. §.§Description and Derivation of the Approximation ErrorsThe estimation of the line integral (eq.<ref>) in a 2D domain is given by Riemmann's integralI_L_ρ,ϕ=lim_max{Δ r_k}→0∑_kE(x_k,y_k)·r̂_ϕΔ r_kwhere (x_k,y_k) are the coordinates of the samples E(x_k,y_k) of the field along line L_ρ,ϕ={(x,y)|xsinϕ-ycosϕ=ρ} ∈Ω with r̂_ϕ=(cosϕ,sinϕ) the unit vector in the direction of the line (fig.<ref>) and Δ r_k the sampling step.The coordinates of the samples are x_k=x_k-1+Δ r_k cosϕ and y_k=y_k-1+Δ r_k sinϕ such as (x_k,y_k)∈Ω.Substituting E with its vector components (e_x,e_y), equation <ref> is expressed asI_L_ρ,ϕ = lim_max{Δ r_k}→0∑_k{e_x(x_k,y_k)cosϕΔ r_k + e_y(x_k,y_k)sinϕΔ r_k}The numerical treatment of the problem is based on the assignment of each sample E(x_k,y_k) to a value E[i,j] of an element [i,j] of the discrete domain. This assignment is actually a quantization process or “mapping” of the vector field samples to a finite set of possible discrete values (fig.<ref>).So, each sample E(x_k,y_k) is the sum of E[i,j] plus an error vector δE(x_k,y_k). This can be expressed ase_x(x_k,y_k)=e_x[i,j]+ δ e_x(x_k,y_k)or briefly ase_x^k=e_x[i,j]+δ e_x^kSimilarly, the y-component ise_y^k=e_y[i,j]+δ e_y^kLet us assume that there is a line segment Δ L_ij⊆ L_ρ,ϕ with length Δ L_ijwhich lies on element(cell) [i,j] as in figure <ref>. Moreover, considering the sampling step to be constant Δ r_k=Δ r and N_ij the number of samples E^k=(e_x^k,e_y^k) in Δ L_ij, Riemann's sum along segment Δ L_ij isI_Δ L_ij,ρ,ϕ=∑_(e_x^k,e_y^k)∈Δ L_ij{e_x^kcosϕΔ r+e_y^ksinϕΔ r}= = ∑_(e_x^k,e_y^k)∈Δ L_ij{(e_x[i,j]+δ e_x^k)cosϕΔ r+(e_y[i,j]+δ e_y^k)sinϕΔ r}= = (N_ije_x[i,j]+∑_kδ e_x^k)cosϕΔ r +(N_ije_y[i,j] +∑_kδ e_y^k)sinϕΔ r Where N_ij∈ℵ is the number of samples of Δ L_ij⊆ L_ρ,ϕ and segment Δ L_ij lies inside element [i,j] of the discrete domain.The error vector caused by the discretization process in an element [i,j] is defined according to(δẽ_x[i,j],δẽ_y[i,j])= (∑_k=N_1^N_2δ e_x^k ,∑_k=N_1^N_2δ e_y^k)where N_2-N_1+1=N_ij is the number of samples along Δ L_ij.The line integral (eq.<ref>) can be represented as a sum of line integrals (eq.<ref>) along segments Δ L_ij⊆ L_ρ,ϕ.Thus,I_L_ρ,ϕ= lim_Δ r→0∑_Δ L_ijI_Δ L_ij,ρ,ϕ= = lim_Δ r→0∑_Δ L_ij∑_∈Δ L_i,j(e_x^k,e_y^k){e_x^kcosϕΔ r+e_y^ksinϕΔ r}= = lim_Δ r→0∑_Δ L_ij{(N_ije_x[i,j]+δẽ_x[i,j])cosϕΔ r + (N_ije_y[i,j]+δẽ_y[i,j])sinϕΔ r}For simplicity, in the following equations indices i and j are substituted by a single index m.So equation <ref> becomesI_L_ρ,ϕ=lim_Δ r→0∑_Δ L_m{(N_m e_x[m]+δẽ_x[m])cosϕΔ r + (N_me_y[m]+δẽ_y[m])sinϕΔ r}The quantization error term due to the discretization process is defined asδẼ=lim_Δ r→0∑_Δ L_m(δẽ_x[m]cosϕΔ r +δẽ_y[m]sinϕΔ r)Thus, equation <ref> becomesI_L_ρ,ϕ=lim_Δ r→0∑_Δ L_m{N_m Δ r(e_x[m]cosϕ+e_y[m]sinϕ)}+δẼSampling step Δ r is finite and min{Δ L_m}≫Δ r>0. Term N_mΔ r represents the length of a segment of line L_ρ,ϕ which “corresponds” to element [m], where N_m is the number of samples in element [m] (fig.<ref>). As the sampling step is Δ r≫0, N_mΔ r≠Δ L_m as shown in figure <ref>. Thus, N_mΔ r=Δ L_m-ε_mΔ r with |ε_m|<1 being a sampling error coefficient.The length of line L_ρ,ϕ isL_ρ,ϕ=∑_m N_mΔ r=∑_m(Δ L_m-ε_mΔ r)=∑_mΔ L_m∑_mε_mΔ r=0. Therefore, equation <ref> is equal toI_L_ρ,ϕ= lim_Δ r→0∑_Δ L_m{(Δ L_m-ε_mΔ r)(e_x[m]cosϕ+e_y[m]sinϕ)}+δẼ= = ∑_Δ L_m{Δ L_m(e_x[m]cosϕ+e_y[m]sinϕ)}+δẼFinally, the line integral I_L_ρ,ϕ can be expressed as the sum of two terms, one which is the discretization process term ∑_Δ L_m{Δ L_m(e_x[m]cosϕ+e_y[m]sinϕ)} and another which is error term δẼ resulted by the discretization.So,I_L_ρ,ϕ=∑_Δ L_m{Δ L_m(e_x[m]cosϕ+e_y[m]sinϕ)}+δẼFor the computational estimation of the line integrals, a finite number N of samples E(x_k,y_k) along line L_ρ,ϕ are assigned to the discrete values E[m] based on an interpolation scheme (e.g. “rough” nearest neighbor). Thus, the approximated line integral equation has the formĨ_L_ρ,ϕ= ∑_Δ L_m{N_mΔ r(e_x[m]cosϕ +e_y[m]sinϕ)}= = ∑_Δ L_m{(Δ L_m-ε_mΔ r) (e_x[m]cosϕ + e_y[m]sinϕ)}= = ∑_Δ L_m{Δ L_m (e_x[m]cosϕ + e_y[m]sinϕ)}-ε(Δ r)Whereε(Δ r)=∑_Δ L_mε_mΔ r{e_x[m]cosϕ + e_y[m]sinϕ}is the sampling error. Taking the difference between I_ρ,ϕ (eq.<ref>) and Ĩ_ρ,ϕ (eq.<ref>)I_ρ,ϕ-Ĩ_ρ,ϕ=δẼ+ε(Δ r)There are two different types of error, δẼ which is resulted by the discretization process and ε(Δ r) which is caused by sampling step Δ r>0 along the integral line. Obviously, as the sampling step Δ r→ 0, ε(Δ r) is eliminated. §.§ A Priori Error EstimateThe numerical estimation of the field is based on the solution of the linear system <ref> (subsection <ref>) in a Least Square (LS) sense employing the numerical approximation of the line integrals.For a grid N× N with 4N known boundary values, the linear system has 6N^2 equations and the unknown vector components are 2N^2. Hence, the LS system is expressed asb=A̅x̅_LS [ [ I_L_ρ_1,ϕ_1; I_L_ρ_2,ϕ_2; …; I_L_ρ_p,ϕ_p ]] = [ a̅_11 a̅_12 … a̅_1l/2+1 … a̅_1l; a̅_21 a̅_22 … … … …; … … … … … …; … … … … … a̅_pl ][ [ e_x_LS[1]; e_x_LS[2]; ⋮; e_y_LS[1]; ⋮ ]]with l=2N^2 and l≪ p=6N^2 i.e. it is an over-determined system and* b=[I_L_ρ_1,ϕ_1, I_L_ρ_2,ϕ_2, …, I_L_ρ_p,ϕ_p]^T are the observed measurements without any additional noise.* x= [e_x_LS[1],…,e_x_LS[l/2],e_y_LS[1],…, e_y_LS[l]]^T=[E_x|E_y]^T are the field values to be recovered.* A̅ has the coefficientsa̅_ku = {[ (Δ L_u_k-ε_kuΔ r)cosϕ_k; (Δ L_u-l/2_k-ε_ku-l/2Δ r)sinϕ_k; 0 ]. and1≤ k ≤ p. A̅: the perturbed transfer matrixMatrix A̅ of the linear system can be written as A̅=A-δA where A is the “unperturbed” transfer matrix with elements a_ku=a̅_ku-ε_kuΔ r cosϕ_k=Δ L_u_kΔ r cosϕ_k for 1≤ u≤ 1/2 and a_ku=a̅_ku-ε_kuΔ r sinϕ_k=Δ L_u_kΔ rsinϕ_k for l/2+1≤ u≤ l and δA is a perturbation of matrix A due to Δ r>0 (eq.<ref>) and has elements of the form ε_kuΔ r cosϕ_k and ε_kuΔ r cosϕ_k.The validation of the LS solution is performed by theoretically estimating the relative error (RE)x_exact-x̅_LS/x_exact using the . Euclidean norm <cit.> where x_exact is the column vector with the real values of the fieldwhile x̅_LS is the least square solution of system <ref>.The column vector x_exact derives from the set of line integrals I_L_ρ_1,ϕ_1,I_L_ρ_2,ϕ_2…,I_L_ρ_p,ϕ_p given by equation <ref> which form the systemb=Ax_exact+δẼwhere δẼ= [δẼ_1,… ,δẼ_p]^T is a column vector with the discretization error (eq.<ref>) of equations <ref>, b are the observed measurements and A the “unperturbed” transfer matrix.Next the following lemma is proven. The relative solution error (RE) of systems A̅x_LS=b and Ax_exact+δẼ=b is given byx_exact-x̅_LS/x_exact≤ e_AA̅^†_2A_2+e_Ak(A)+e_bA̅^†_2 b/x_exactwhere A and A̅ ∈^p× l with p>l, A̅=A-δA, δA_2=e_AA_2≠0 and δẼ=e_bb≠0 with Rank(A̅)≥ Rank(A)=v≤ p (see Appendix <ref>).Moreover, A_2=max_x=1Ax=√(σ_max) is the matrix 2-norm of A with σ_max the maximum singular value of A and A^†_2=1/√(σ_min) where σ_min the minimum singular value and k(A) the condition number defined as k(A)=A_2A^†_2 andwhere symbol † refers to the pseudo-inverse of the rectangular matrices A and A̅ such as A^†=(A^TA)^-1A^T For the estimation of the relative error (RE) we employ the decomposition theorem A^†-A̅^† discussed in <cit.> and <cit.>.According to the decomposition theorem <cit.>,A^†-A̅^†=-A̅^†(A-A̅)A^†- A̅^†(I-AA^†)+(I-A̅^†A̅)A^†Therefore, if b̅=b-δẼ, x_exact-x̅_LS = A^†b̅- A̅^†b=A^†b̅- A̅^† (b̅+δẼ)=(A^† - A̅^†)b̅-A̅^†e == [-A̅^†(A-A̅)A^†- A̅^†(I-AA^†)+ (I-A̅^†A̅)A^†]b̅-A̅^†δẼ(I-AA^†)b̅=0 as b̅∈ Range(A) (i.e b̅=Ax_exact).So, equation <ref> becomesx_exact-x̅_LS = [-A̅^†(A-A̅) A^†+(I-A̅^†A̅) A^†]b̅-A̅^†δẼ==-A̅^†δAx_exact-A̅^†δẼ+(I-A̅^†A̅)A^†b̅ Moore Penrose pseudo-inverse identities A^†=A^†AA^†= (A^†A)^TA^†=A^T(A^†)^TA^† <cit.><cit.> imply that A^†b̅=A^T(A^†) ^TA^†b̅=A^T(A^†)^Tx_exact.Hence, equation <ref> becomesx_exact-x̅_LS =-A̅^†δAx_exact-A̅^†δẼ+(I-A̅^ †A̅)A^T(A^†)^Tx_exact==-A̅^†δAx_exact-A̅^†e+(I-A̅ ^†A̅)(A̅+δA)^T(A^†)^Tx_exact(I-A̅^†A̅)A̅^T=0 i.e. A̅^T ∈ N (A̅^†A̅)as A̅^T=A̅^†A̅A̅^T<cit.>.Thus,x_exact-x̅_LS =-A̅^†δAx_exact-A̅^†δẼ+(I-A̅^†A̅)δA^T(A^†)^Tx_exactAccording to the matrix norm inequalities of 2-norm ._2 <cit.> for matrices B and C ∈^m× n BC_2≤B_2C_2 and B+C_2≤B_2+C_2 and Bx_2≤B_2x when x is a vector ∈^n× 1.So,x_exact-x̅_LS/x_exact = -A̅^†δAx_exact +(I-A̅^†A̅)δA^T (A^†)^Tx_exact-A̅^†δẼ_2/x_exact≤≤-A̅^†δAx_exact_2+(I- A̅^†A̅)δA^T(A^†)^Tx_exact_2+-A̅^†δẼ_2/x_exact≤≤δA_2 A̅^†_2+ (I-A̅^†A̅)_2δA^T_2A^†_2 +A̅^†_2 δẼ/x_exact≤≤δA_2 A̅^†_2+δA_2A^†_2+A̅^†_2 δẼ/x_exactif we set δA_2=e_AA_2, δẼ=e_bb and k(A)=A^†_2A_2 the condition number of A thenx_exact-x̅_LS/x_exact≤ e_AA̅^†_2A_2+e_Ak(A)+e_bA̅^†_2 b/x_exact§.§.§ Conditions for Bounded Relative ErrorThe relative error (RE) is bounded when the sampling error is small i.e. e_A→0 and the discretization error δẼ=e_bb>0 with 0<e_b≪1.If the sampling step Δ r→ 0 then δA≈0 and A̅≈A (the sampling error is eliminated ε(Δ r)≃ 0) and the relative error RE isx_exact-x̅_LS/x_exact≤e_bA^†_2 b/x_exact≤e_b/1-e_bA^†_2A_2=e_b/1-e_bk(A) When the discretization error δẼ_2>0 with δẼ=e_bb and 0<e_b≪1 the equations of the LS system <ref> have the form (eq.<ref>)∑_u=1^l{Δ L_u (e_x_LS[u]cosϕ + e_y_LS[u]sinϕ)}where Δ L_u>0. This formulation is an approximation of the line integral equation (eq.<ref>) and the summation of any set of these equations cannot be zero in any closed path. So, the rectangular transfer matrix of LS system <ref> is not rank deficient as there are no linearly dependent equations and the condition number k(A)<∞.Thus, the relative error (RE)x_exact-x̅_LS/x_exact≤e_b/1-e_bk(A)<∞is bounded.If the resolution of the discrete domain improves (grid refinement) such as the discretization error δẼ_2≃0, the linear equations areI_L_ρ,ϕ=lim_Δ L_u→ 0∑_uΔ L_u(e_x[u]cosϕ+e_y[u]sinϕ)}→∫_L_ρ,ϕe_xcosϕ dl+e_ysinϕ dlthen the system becomes severely ill conditioned. The condition number k(A)→∞ due to the linear dependencies between the equations and the RE bound tends to infinity.Particulary, for a grid N× N (N is the resolution), 4N boundary values and 2N^2 unknowns, when N→∞ (i.e. grid refinement), the number of independent equations F(N)→ 4(N-1) as the equations approximate the continues line integrals (section <ref>). Then, the number of independent equations cannot exceed the number of the unknowns as lim_N→∞F(N)/2N^2=lim_N→∞4(N-1)/2N^2=0 andsystem <ref> becomes rank deficient and consistent (i.e. infinity number of solutions).In other words rank deficiency of transfer matrix A (e.g Rank(A)=r) implies that there are singular valuesσ_r→ 0. So according to singular value decomposition (SVD) A=UΣV^† where matrix U and matrix V are unitary, and Σ is an matrix whose only non-zero elements are along the diagonal with σ_i>σ_i+1≥ 0(The columns of U andV are known as the left {u_i} and right {v_i} singular vectors) for σ_r=0⇒ Au_r=0 and thus A(x+au_r)=b which show that the uniqueness test fails(2^nd Hadamards criterion).So, a theoretical relationship between the discretization and the ill-posedness of the inverse problem has been presented. Particularly, the discretization process is a way to regularize the continuous ill posed problem since the discretization ensures a finite upper bound to the solution error. This is called self-regularization (regularization by projection) or regularization by discretization. The proper choice of the discretization parameters is important for the problem regularization. An extreme coarse discretization of the domain increases the ill conditioning of the problem in the sense that if e_b→1 (high discretization error), the RE is again unbounded, leading again to an unstable linear system. Graphically this is presented in figure <ref>So, a bounded discretization error δẼ is essential for an accurate solution to balance the solution approximation error and the ill conditioning of the linear system.§ SUMMARYThe proposed method intends to approximate a vector field from a set of line integrals. As we showed, from 4N boundary measurements, the maximum number of equations which can be obtained taking all possible pairs of boundary measurements is2N(4N-1) and the maximum number of independent line integrals cannot exceed the 4N-1 equations. In the case of a square domain with 2N^2 unknowns, this number of independent continuous line integrals is not sufficient for the recovery of a 2D irrotational vector field. However, we provided a theoretical proof showing that the discretization can be an efficient way of regularizing the continuous ill posed problem and therefore the problem is tractable by solving it in the discrete domain. CHAPTER: SIMULATIONS AND A REAL APPLICATION The current chapter is divided into two main parts: in the first part the validation and verification of the line integral method is presented performing some basic simulations while in the second part an introduction to the inverse bioelectric field problem is reported as a future real application of the proposed vector field reconstruction method.In particular, for the evaluation of the method we perform simulations for the approximately reconstruction of an electrostatic field produced by electric monopoles in a 2D square domain based on the vector field recovery method. According to the preconditions and assumptions of the mathematical model, the electrostatic field is a good example for the validation of the method as it satisfies the quasi static condition and the irrotational property.Moreover, the numerical implementation of the method is based on the modeling described in subsection <ref> where nearest neighbor approximation was employed. For the verification of the theoretical predictions, the magnitude of the singular values of transfer matrix A of system <ref>, will provide a measure of the invertibility and conditioning of the numerical system <ref>. The sampling step along the line integrals will be very small (much smaller than the cell size) in all simulations.Finally, the proposed vector field method will be presented as an equivalent mathematical counterpart of the partial differential formulation of the inverse bioelectric field problem <cit.>. Comparisons of the two approaches will be made.§ RESULTSThe qualitative and quantitative evaluation of the method is very important for examination of its robustness. The validation is concerned with how the mathematical and geometric formulations represent the real physical problem and the verification assesses the accuracy with which the numerical model approximates the real one. The mathematical and geometric properties of the experimental model are defined in the “Simulation Setup” subsection while in subsection “Simulations using Electric Monopoles” the evaluation of the previous theoretical findings is performed. §.§ Simulation Setup Geometrical ModelIn the current simulations we intend to recover a vector field in a 2D square bounded domain Ω. For the numerical estimation of the field, the discretization of Ω is essential. Discretization of the domain can be described asΩ:{(x,y)∈[-U,U]^2}→_Discretization{[i,j]∈ [1:N,1:N]}where i=⌊x+U/P⌋, j=⌊y+U/P⌋ (nearest neighbor approximation) with [P× P] the cell's size and N=2U/P the spatial resolution (sampling rate) of the discrete domain.Mathematical ModelThe numerical representation of the line integral equations is given byĨ_L_ρ,ϕ=∑N_ijE[i,j]·(cosϕ,sinϕ)Δ rwhere N_ij is the number of samples in cell [i,j] of the domain and Δ r the sampling step. The linear system of equations is designed according to subsection <ref>. Source ModelA set of monopoles is employed for the production of the real electric field. The field created by monopoles (point sources) is given by E=k∑_iQ_i/(r-r_i)^2r̂_i where (r-r_i)^2 is the distance between charge Q_i and the E-field evaluation point r and r̂_i the unit vector pointing from the particle with charge Q_i to the E-field point. Φ=k∑_iQ_i/|r-r_i| is the potential function for the estimation of the potential values at the boundaries of the domain.So, in the inverse vector field problem, one seeks to estimate field E[i,j] in each cell [i,j] when the potential values in the middle of the boundary edges of the boundary cells are known.The linear system <ref> is formulated from a set of approximated line integrals (<ref>), where value Ĩ_L_ρ,ϕ is the potential difference between two boundary points.In most simulations, the point sources of the field were positioned outside the bounded domain in order to avoid singularities, since in practical problems the value of the field cannot be infinite.In the next subsection we examine the following scenarios:* increase the resolution of the interior of the discrete domain and the observed boundary measurements and examine the relationship between the resolution and the ill conditioning of the system <ref>;* create a field using more point sources with arbitrary charges and positions but still outside the bounded domain for constant resolution;* place the point source inside the domain. The goal is to examine experimentally the ill conditioning of the system for different source distributions (close and far from the bounded domain) and the relationship between the spatial resolution of the domain and the ill conditioning. For the examination of the ill conditioning, the singular values of transfer matrix A of the linear system <ref> are estimated performing the singularvalue decomposition (SVD) of A. It is known that a slowly decreasing singular value spectrum with a broader range of nonzero singular values indicates a better conditioning while a rapidly decreasing to zero shows increase of ill conditioning. Moreover, a second indicator is the condition number k(A)=σ_max/σ_min where σ_max and σ_min are the maximum and minimum singular values of A, respectively. The condition number is a gauge of the transfer error from matrix A and vector b (eq. <ref>) to the solution vector x. When the condition number is close to 1, the conditioning of the system is good and the transfer error is low. So, small changes in A or b produce small errors in x. Finally, we estimate the relative error of the magnitude and the phase between the approximated and the real field usingRE=x_LS-x_exact_2/x_exact_2where x_2=√(∑_i x_i^2). §.§ Simulations using Electric Monopoles* In the first set of simulations the sources are selected to be far from the recovery domain in order the field to be smooth and thus to examine only the relationship between resolution and ill conditioning due to grid refinement. 1^st Example For a bounded domain Ω:{(x,y)∈[-U,U]^2⊂^2}=[-5.5,5.5]^2 and two point sources with the same charge Q placed at (-19,0) and (19,19) on the x,y coordinate system (far from the domain Ω) and small sampling step Δ r=10^-4U along the integral line, we obtain the following.When P× P=1× 1, the cell size and grid is 11×11 (resolution is 11), the recovered field is depicted in figure <ref>B.The condition number of the transfer matrix A of the linear system is k(A)=174 and the relative errors (eq. <ref>) of the magnitude and the phase between the real and the reconstructed field are RE_magnitude=0.11 and RE_phase=0.05 respectively.When the cell size is P=0.5, the condition number is k(A)=10^4 and the relative errors RE_magnitude=0.08 and RE_phase=0.045. The recovered field is showed in figure <ref>B. In both cases Δ r is much less than the cell size P. As we can observe the condition number in the second case in much higher than in the first case where the spatial resolution is “coarse”. This implies that the solution ofthe second system Ax_LS=b is more sensitive and unstable to A or b perturbations.System instability means that the continuous dependence of the solution upon the input data cannot be guaranteed and in the presence of input noise (perturbations) the system behavior is unpredictable. In the current simulations there is no additional noise or other external sources of error, so the ill conditioning of the system (intrinsic ill conditioning) due to the high condition number k(A) of order 10^k has an effect on the computed solution in the sense that a loss of k digit accuracy in the solution is applied (rule of thumb <cit.>). Thus, for a floating point arithmetic (16 digits floating point numbers are used generally in these simulations) only 16-k digit solution accuracy can be achieved. So, in the absence of sources of noise, there is mainly a loss in accuracy when the condition number is high while there no great effect on the system's stability.The approximation error RE is lower for the “refined” system (Res. 22) than that of the “coarse” system (Res. 11). In the “refined” system, there is a loss of 4 digits in the solution accuracy which is not so high for floating point measurements and due to grid refinement, more spatial frequencies of the field can be recovered. Thus, the solution of the “refined” system is more accurate. However, the “refined” system is more unstable and if the spatial resolution of the problem tends to the continuous case then obviously the linear system will tend to rank deficiency.Moreover, grid refinement results the faster descent of the singular values of the spectrum to zero and frequently a narrower range of non zero singular values, which also indicate that the conditioning of the transfer matrix A and the stability of the linear system deteriorates. This is clear according to figure <ref> where the continuous line depicts the singular values spectrum when the grid resolution is N=U/P=11 and the dashed line the spectrum for N=22. 2^nd Example Further results for 2 point sources with Q=10^-8 localized again at (-19,0) and (19,19) (far away from the bounded domain Ω:{(x,y)∈[-2.5,2.5]^2}) and 3 different cell sizes P=1,0.5,0.25are presented in table 4.1 and figure <ref>.* When a large number of point sources (160 sources with arbitrary charges) are placed far from domain Ω (fig.<ref>) then RE_magnitude = 0.1567 RE_phase = 0.1527 and the recovered field is depicted in figure <ref>.If the point sources are closer to the field of interest, like in figures <ref> and <ref>, and keeping constant resolution, the relative errors are RE_magnitude=0.63 and RE_phase=0.33. Clearly, the relative errors increase as the field sources are placed closer to the recovery domain. * In the case where the sources are localized inside the bounded domain, therelative error of the magnitude of the field istoo high while the phase error is low and the vector arrows point to the sources' positions.The field close to sources is not smooth, i.e. the real field has both high and low spatial frequency terms and thusfor poor spatial resolution of the bounded domain (fig. <ref>): only the lowspatial frequency terms can be recovered. The same actually happens in the previous case(fig.<ref>)when the point sources are placed closer to the grid. Infigure <ref> the relative error of the magnitude ishigh due to low resolution. However, the recovered field points at the source locations which indicates that the low spatialfrequency terms give some extremely useful information about the direction of the field.Condition number vs ResolutionFinally, figure <ref> depicts the grid resolution against the condition number of the transfer matrix of the linear system. By construction the transfer matrix of the linear system depends on the geometrical characteristics of the region where we intend to estimate the field and the change of the resolution. So, as our experiments are performed in a rectangular region then for constant resolution all the examples share the same transfer matrix and what it changes in our linear system basically is the boundary measurements b. Therefore, the following graph presents the relationship between the rectangular resolution and the conditioning of the transfer matrix.So, one more time we conclude that different discretization choices of the grid impact the formulation of matrix A and the increase in spatial resolution actually increases the ill-conditioning of this inverse problem.§.§.§ The effect of additive noise on the reconstructionIn this paragraph we examine the effect of noise in b measurements of the linear system Ax_LS=b for the 2^nd example of the previous subsection where the field was produced by 2 point sources with Q=10^-8 localized at (-19,0) and (19,19) far from the bounded domain Ω:{(x,y)∈[-2.5,2.5]^2}. According to figure <ref> the lower acceptable value of condition number K(A) is 72 such as to balance ill conditioning-resolution N× N=9× 9. So the majority of the rest simulations with noisy measurements will be performed for this resolution however, some further simulations with higher condition number will be estimated in order to examine the stability of the system in the presence of noise. Moreover, we employ the relative magnitude of the input and output error to provide a measure of stability factor S:e_out=x_LS-x_exact_2/x_exact_2e_in=b̅-b_2/b_2S=e_out/e_in Noisy measurements are estimated according to b̅=b+n· randn() where randn() is Matlab functionwhich produce random numbers whose elements are normally distributed with mean 0, variance σ^2=1 and n∈[5 ·10^-3,5· 10^-2] In figure <ref> for each n value, we estimate the mean value of e_outusing 100 different noise vectors. Stability factor was estimated between 8.5∼9.5 (we have to mention that S is meaningless for b̅→̅b̅.)Particulary, x-axis depicts the Signal to Noise ratios (SNR)of our inputs which are given by 20log_10(1/mean(e_in)_n)while y-axis the mean values of mean(e_out)_n. Obviously for higher values of the SNR (low additive noise) the output relative error is low. For the case where the condition number is a bit higher i.e. A=132.6 and N× N=16× 16 the curve between e_out and SNR is depicted in figure <ref>. Comparing figures <ref> and <ref> and stability factors S we can mention that the increase in the ill conditioning of the problems(even a small increase) affects the accuracy of the output solution in the presence of noise. Obviously when the level of noise is high and we need a better accuracy(grid refinement) then extra regularization such as Tikhonov should be considered and possible pre-processing of the measurements to reduce noise level. §.§ DiscussionDifferent discretization choices of the grid will impact the formulation of matrix A. The increase in discretization resolution on the bounded domain Ω actually increases the ill-conditioned nature of the inverse problem. The ill conditioning of the linear system results in the instability of the solution of the system especially in the presence of noise. In the current simulation, there was no additional noise. However, the majority of real problems suffer from additive or multiplicative noise and a future examination of the effect of noise to the inverse problem will be performed.In order to alleviate the ill conditioning of the linear system, one may perform a coarse discretization of the bounded domain. This leads to a “rough” approximation of the vector field (see fig. <ref>). For a non-smooth field (fig.<ref>) the approximation error is considerable. Particularly, let us consider vector field E(x,y) in bounded domain Ω as the sum of spatial frequency terms according to Fourier series expansionE(x,y)=∑_ke_ke^ik⟨ x,y⟩+e̅_ke^-ik⟨ x,y⟩The spatial frequency k is directly related to the spatial resolution of the domain and basically the resolution determines the spatial frequency band limit of the vector field to be recovered. If the resolution of the domain is high then more spatial frequency terms can be recovered. However, the “uncontrolled” increase in resolution worsens the problem conditioning. So, the mathematical and experimental choice of the optimal discretization of the bounded domain, such as to minimize the approximation error while maximizing the stability of the linear system needs to be assessed.§ PRACTICAL ASPECTS OF THE METHODA major objective of this section is to introduce the inverse bioelectric problem as one of the most prominent biomedical potential applications of the proposed vector field method. Moreover, a comparison of the line integral method with the current partial differential formulation will be presented.§.§ Inverse Bioelectric Field ProblemThe bioelectric field <cit.> is the manifestation of the current densities inside the human body as a result of the conversion of the energy from chemical to electric form (excitation) in the living nerves, muscles cells and tissues in general.Bioelectric inverse techniques intend to estimate the source distributions inside specific parts of the body, e.g brain and heart, employing the EEG or ECG recordings (passive methods). The ordinary mathematical modeling of this problem is based on the solution of a boundary value problem of an elliptic partial differential equation (PDE). As we will see later, this problem is ill-posed as it does not satisfy some of the three Hadamard's criteria: existence, uniqueness of the solution and continuous dependence of the solution upon the data.§.§ PDE MethodsThe bioelectric field problem can be formulated in terms of either the Poisson or the Laplace equation depending on the physical properties of the problem <cit.>.In particular, for the problem formulation, physical characteristics of the electrical sources of the human body have to be defined <cit.>. The primary source ρ(x,y) of electric activity is produced as a result of the transformation of the energy inside the cells from chemical to electric, which consequently induces an electric current of the form σE, where σ is the bulk conductivity of the volume. In general, the electric density is time-varying, however, the passive way of the data acquisition (not imposed external potentials like in electrical impedance tomography) using EEG or ECG to record bioelectric source behavior in low frequencies (frequencies below several kHz) <cit.>,<cit.> enables the quasi-static treatment of the problem. The static consideration of the field in a negligible short time implies that the capacitance component of the tissues (jωE) can be assumed negligible.So, the total current density is given by J=ρ(x,y,z)+σE and due to the quasi-static condition it obeys ∇·J=0. A last important point is that electromagnetic wave effects are also neglected <cit.> and therefore the electric field is given by E=-∇Φ.§.§.§ Poisson Equation FormulationThe commonly used inverse bioelectric methods try to calculate the internal current sources J given a subset of electrostatic potentials measured on the scalp or other part of the body, the geometry and electrical conductivity properties within this human part.The physical and mathematical model of the inverse bioelectric problem is derived from ∇·J=0 and the scalar potential representation of the field E=-∇Φ in a bounded domain Ω. Hence, the inverse problem is based on the solution of a Poisson-like equation:∇σ∇Φ=-ρΩwith the Cauchy boundary conditions:Φ=Φ_0 Σ⫅Γ_T σ∇Φn̂=0Γ_T=∂Ωwhere Φ is the electrostatic potential, σ is the conductivity tensor, ρ are the current sources per unit volume, and Γ_T and Ω represent the surface and the volume of the body part (e.g. head), respectively.The Dirichlet condition (Φ=Φ_0 Σ⫅Γ_T) is a mathematical abstraction of potential measurements on Σ, which are in reality obtained from only a finite number of electrodes, andthe Neumann condition (σ∇Φn̂=0Γ_T=∂Ω) is describing that no current flows out of the body.For the solution of the problem, the modeling of the sources is extremely essential and the difficulties of the design of an accurate model impose significant limitations to the current solution techniques.In particular, usually the sources are mathematically assumed to be dipoles with unknown magnitude, position, and orientation <cit.>,<cit.>. The simplest problem uses the assumption that ρ = Q(m, P, O) is a single dipole, characterized bythree parameters (six degrees of freedom), corresponding to magnitude m, position P (x, y, z), and orientation O. These parameters are adjusted such that the resulting electrostatic potential best matches the given measured data (trial and error method)<cit.>. More general models with several dipoles and consequently increased number of unknown parameters have been designed. Unfortunately, more general cases where there are no previous assumptions about the sources have no unique solution. Solutions can only be found with suitable regularization or a restriction of the solution space by assuming ρ of a special form.Generally, the inverse bioelectric field problem (<ref>) is ill posed as for an arbitrary source distribution, it does not have a unique solution and the solution does not depend continuously on the data i.e. small errors in measurements may cause large errors in the solution.More specifically, according to Helmholtz's theorem earlier, and Thevenin's (Norton's) theorem later <cit.>,<cit.>, it is always possible to replace a combination of sources and associated circuity with a single equivalent source and a series of impedances. Thus, circuits with different structure (impedances' sequence and source location) can give the same equivalent circuit (Thevenin circuit). Similarly, in the electric field which can be assumed as an equivalent representation of a circuit, a different source distribution can give the same equivalent source-generator (Superposition Principle) which gives rise to the observed weighted boundary potentials. In other words, the same observed potential measurements could be produced from different source distributions and thus the problem has not unique solution. So, for the estimation of the sources the common method <cit.> is to breaks up the solution domain into a finite number of subdomains. In each subdomain, a simplified model of the actual bioelectric source (such as a dipole) is assumed. The problem then is to find the magnitudeand direction of each of the simplified sources in the subdomains.In addition, theoretically the elliptic equations boundary value problem of Poisson PDE with Cauchy boundary conditions leads to an over-specified or too “sensitive” boundary value problem <cit.> resulting large errors in the solution for small changes in boundary conditions. So, regularization techniques should be performed for the stability of the solution <cit.>.§.§.§ Laplace Equation FormulationIf the inverse problem is formulated such as there are no sources on the bounded domain, then we obtain the homogeneous counterpart of the Poisson equation. These are the cases where there is interest on the estimation of the potential distribution on a surface (e.g. cortex or surface of the heart) and not in the whole volume from boundary measurements (like scalp (EEG) or torso (ECG) recordings) and hence the sources are out of the region of interest.So, the Cauchy boundary value problem is formed by Laplacian equation ∇σ∇Φ=0. This is also an ill posed problem in the sense that the solution does not depend continuously on the data. Numerically, the solution of the Laplace's equation can be formulated as AΦ=Φ_boundaries. As the continuous problem is ill posed due to its physical nature <cit.>,<cit.>, matrix A will be ill conditioned.There are different regularization methods and numerical techniques for the approximate solution of this system <cit.>,<cit.>. In the current section, we will not give further details about these techniques. However, a brief comparison between Laplace's equation and the line integrals methods for the estimation of a vector field will be presented. §.§ Vector Field MethodIn the previous description, the inverse bioelectric theory intends to estimate the positions and strengths of the sources within a volume (e.g brain) that could have given rise to the observed potential recordings under the assumptions that the field is quasi-static and irrotational. Obviously, the physical principles of the bioelectric field are fitted well with the preconditions and assumptions of the proposed vector field tomographic method. Thus, the proposed method can be applied to recover a bioelectric field in discrete points using the boundary information without needing any prior information about the sources' formulation while the boundary conditions are automatically satisfied as they are directly incorporated in the line integral formulation. Vector Field Recovery with Laplace and Vector Field MethodWe present the recovery of vector field in a bounded domain [-5.5,5.5]^2 with cell size P× P=1×1, grid N× N=11×11 and sampling step Δ r=0.0001P, solving two systems of linear equations where one was derived from the line integral method as it was described previously and another from the numerical solution of the Laplace equation∇^2Φ=∂^2ϕ/∂ x^2+∂^2ϕ/∂ y^2=0based on the finite difference approximation∂^2 ϕ/∂ x^2=ϕ_i+1,j-2ϕ_i,j+ϕ_i-1,j/h^2 ∂^2 ϕ/∂ y^2=ϕ_i,j+1-2ϕ_i,j+ϕ_i,j-1/h^2For the recovery of the field in figure <ref>A, the rows of system AΦ=Φ_boundaries were defined according to4ϕ_i,j-ϕ_i-1,j-ϕ_i+i,j-ϕ_i,j-1-ϕ_i,j+1=0with known boundary potential values ϕ_0,j,ϕ_i,0,ϕ_0,N and ϕ_N,j. After the estimation of ϕ the vector field was calculated byE=( ϕ_i+1,j-ϕ_i-1,j/h,ϕ_i,j+1-ϕ_i,j-1/h)with h=P. The coarse recovery of an electrostatic field employing the vector field method presented better and more accurate results than the finite difference Laplacian solution. Here, we have to consider that E is estimated from the potential values Φ and thus extra numerical errors were introduce to the Laplacian solution. CHAPTER: CONCLUSIONSOver the past two decades there has been an increasing interest in vector field tomographic methods and many mathematical approaches based on ray and Radon Transform theory have been reported. Nowadays, there is a lack of potential applications, although all the theoretical findings and the previous work may be extremely essential for the evolution and the implementation of robust techniques. Particulary, the theoretical and experimental study of the proposed vector field method for real problems with simple physical and geometric properties is the first step towards the deeper understanding and the evolution of the method for more complex real applications. Briefly, our future work will be focused on three basic tasks: * robust theoretical formulation of the vector field method according to inverse problems theory;* more sophisticated numerical approaches based on finite elements theory and further experiments and simulations;* practical aspects of the method especially in the field of inverse bioelectric field problem with comparisons between the vector field method and the current PDE methods.§ FUTURE WORK* Theoretical FoundationIn subsection <ref>, basic physics principles are used to describe the ill conditioning of the vector field method and a more intuitive consideration of the ill posedness of the problem was given. Next step is the robust mathematical formulation of this inverse problem in Hadamard's sense, examining the uniqueness of the solution and the stability of the solution to the noise in the input data.In the mathematical theory of the inverse problems, an inverse problem is described by the equationKx=ywhere given y and K we have to solvefor x.The vector field method is an inverse problem where operator K is an integral operator. This integral operator is compact in many natural topologies and in general it is not one-to-one <cit.>. Almost always compact operators K are ill posed. However, this basic knowledge is not enough for the mathematical interpretation of the ill posedness of the vector field problem, so further theorems of the inverse problem theory, and assumptions should be considered. A robust and complete mathematical formulation of the problem is essential for a better understanding of the problem's nature and the potential numerical adaptations of this theoretical problem to real applications.* Numerical Implementation and Further ExperimentsThe numerical solution of the vector field recovery method is based on the discretization of the problem. Ill posedness of the continuous problem is expressed as ill conditioning of the system of linear equations in the discrete version. The conditioning of the problem depends on the resolution of the discrete domain and the applied interpolations schemes. A “naive” discretization may lead to disastrous results as forvery “coarse” discretization, the approximation errors increase while when an extreme grid refinement is applied, the conditioning of the numerical system deteriorates.In particular,the grid refinement can be used to increase the resolution inbounded domains, but must be used with care because refined meshesworsen the conditioning of the inverse problem and simultaneouslyincrease the computational cost.So, there is an openquestion about the optimal discretization of the bounded domain,such as to minimize the approximation error while maximizing thestability of the linear system. For this purpose, discretization error bounds should beestimated mathematically and experimentally.Moreover, the discretization of the domain employing otherelements,not cells or tiles as in the square domain, embedding more sophisticated techniques should be considered.Another option to overcome the ill conditioning of the problem is by increasing the observed data available on the boundary (increase boundary resolution). For the case of a simple 2D square domain, there is an initial approach for the definition of the optimum number and positions of the boundary measurements reported in <cit.>. However, further investigation and generalization for arbitrary shapes of the bounded domain need to be considered. Moreover, it should be mentioned that the increase of boundary measurements is not a very efficient way to solve the problem as in practical applications (e.g. EEG) there are only a few measurements and from these few measurements we have to extract as much information as possible. Finally, the increased boundary resolution, not as a consequence of the increasing measured data but as a results of interpolation schemes <cit.>, does not improve the problem conditioning in real problems. * After the theoretical and the numerical schemes have been completed, the next steps are the EEG application.In ordinary EEG source analysis, the inverse problem estimates the positions and strengths of the sources within the brain that could have given rise to the observed scalp potential recordings. In the current project, the final stage is the implementation of a different approach for the EEG analysis employing the proposed method. Rather than performing strengths estimation or location of the sources inside the brain, which are very complicated tasks, a reconstruction of the corresponding static bioelectric field will be performed based on the line integral measurements. This bioelectric field can be treated as an “effective” equivalent state of the brain. Thus for instance, health conditions or a specific pathology (e.g. seizure disorder) may be recognized. This can be achieved by designing a slice-by-slice reconstruction algorithm for vector field recovery or the 3D extensionof the proposed theoretical model where new numerical problems concerning the design of huge linear systems (sparse matrices and different computational strategies) arise. CHAPTER: RANK OF PERTURBED MATRIX A̅As it is described in <cit.> p.216, the rank of a perturbed matrix A̅=A-E is equal to or greater than the rank(A) when the entries of E have small magnitude. According to the proof on p.217 <cit.> matrix A with Rank(A)=p can be transformed in an equivalent matrix (rank normal form) when elementary row and column operators are applied i.e.A∼([ I_p 0; 0 0 ])Equivalently for the A-E whereE=([ E_11^p× pE_12;E_21E_22 ]) we obtainA-E∼([ I_p-E_11^p× p 0; 0 S ])Thus,Rank(A-E) =Rank(I_p-E_11^p×p)+Rank(S)==Rank(A)+Rank(S)⩾ Rank(A) In practical problem, S=0 is very unlikely as the perturbation (discretization or roundoff errors) is almost “random”. Therefore Rank(A-E)>Rank(A). plain | http://arxiv.org/abs/1705.00708v1 | {
"authors": [
"Alexandra Koulouri"
],
"categories": [
"q-bio.QM"
],
"primary_category": "q-bio.QM",
"published": "20170427200036",
"title": "Overcoming the ill-posedness through discretization in vector tomography: Reconstruction of irrotational vector fields"
} |
Institute for Theoretical Physics and Astrophysics, University of Würzburg, Am Hubland, 97074 Würzburg, Germany Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany Department of Physics, University at Buffalo, State University of New York, Buffalo, New York 14260, USADepartment of Physics, University at Buffalo, State University of New York, Buffalo, New York 14260, USADepartment of Physics, University at Buffalo, State University of New York, Buffalo, New York 14260, USADepartment of Physics, University at Buffalo, State University of New York, Buffalo, New York 14260, USA The two-dimensional character and reduced screening in monolayer transition-metal dichalcogenides (TMDs) lead to the ubiquitous formation of robust excitons with binding energies orders of magnitude larger than in bulk semiconductors. Focusing on neutral excitons, bound electron-hole pairs, that dominate the optical response in TMDs, it is shown that they can provide fingerprints for magnetic proximity effects in magnetic heterostructures. These proximity effects cannot be described by the widely used single-particle description, but instead reveal the possibility of a conversion between optically inactive and active excitons by rotating the magnetization of the magnetic substrate. With recent breakthroughs in fabricating Mo- and W-based magnetic TMD heterostructures, this emergent optical response can be directly tested experimentally.Magnetic Proximity Effects in Transition-Metal Dichalcogenides: Converting Excitons Igor Žutić December 30, 2023 ===================================================================================Proximity effects can transform a given material through its adjacent regions to become superconducting, magnetic, or topologically nontrivial <cit.>. In bulk materials, the sample size often dwarfs the characteristic lengths of proximity effects allowing their neglect. However, in monolayer (ML) van der Waals materials such as graphene or transition-metal dichalcogenides (TMDs), the situation is drastically different, even short-range magnetic proximity effects exceed their thickness <cit.>.MX_2 (M = Mo,W, X = S, Se, Te) ML TMDs have unique optical properties that combine a direct band gap, very large excitonic binding energies (up to ∼0.5 eV), and efficient light emission <cit.>. A hallmark of TMDs is their valley-spin coupling which leads to a valley-dependent helicity of interband optical transitions as well as important implications for transport and qubits <cit.>. Lifting the degeneracy between the valleys K and K' was identified as the key step in manipulating valley degrees of freedom. However, a common approach was focused on very large magnetic fields required by a small Zeeman splitting of ∼0.1-0.2 meV/T <cit.>. Instead, recent experimental breakthroughs demonstrate a viable alternative by using optically detected magnetic proximity effects dominated by excitons <cit.>.While a single-particle description already suggests unusual implications of magnetic proximity effects <cit.>, strong many-body interactionsqualitatively alter the optical response in TMDs and yield a wealth of unexplored phenomena <cit.>. Here, we provide the missing description of Coulomb interaction in magnetic proximity effects and elucidate how they transform the observed excitons in TMDs on magnetic substrates. In the seemingly trivial case of an in-plane magnetization, M, where a single-particle description implies no lifting of the valley degeneracy <cit.>, we predict that dark neutral excitons X^0 become bright. The term dark (bright) represents optically forbidden (allowed) dipole transitions with an antiparallel (parallel) electron spin configuration.Figure <ref>(a) shows the band structure of ML TMDs reflecting strong spin-orbit coupling (SOC) due to the d orbitals of the heavy metal atoms and broken inversion symmetry and the considered geometry. For bands with a 2D representation, the SOC Hamiltonian can be written as H_SO=Ω(k)·s using the SOC field Ω(k) <cit.>, where k is the wave vector and s is the vector of spin Pauli matrices. In ML TMDs, this leads to Ω(k)=λ(k)ẑ, where λ(k) is odd in k and ẑ is the unit vector normal to the ML plane. At the K point, λ(k) reduces to the values λ_c(v) in the conduction (valence) band CB (VB), Fig. <ref>(a). The limiting case of this description with a magnetic substrate, see Fig. <ref>(b), neglecting many-body effects is given by the Hamiltonian <cit.> H_tot=H_0+H_ex+H_R, a sum of the “bare" ML TMD, a proximity-induced exchange term, and Rashba SOC withH_0 = ħ v_F(k_xσ_xτ_z+k_yσ_y)+(E_g/2)σ_z+ τ_z s_z[λ_c(1+σ_z)/2+λ_v(1-σ_z)/2],where σ_i and τ_i denote Pauli matrices for the CB/VB and the valley, v_F the Fermi velocity, and E_g the band gap in the absence of SOC. Writing M=Mn̂, we haveH_ex=-n̂· s[J_c(1+σ_z)/2+J_v(1-σ_z)/2],where J_c(v) is the exchange splitting induced in the CB (VB), while in the Rashba SOC, H_R=λ_R(s_yσ_xτ_z-s_xσ_y), λ_R is the Rashba SOC parameter. From the resulting single-particle description, H_totη^τ_nk=ϵ^τ_n(k)η^τ_nk with the energies ϵ^τ_n(k) and the corresponding eigenstates η^τ_nk, we develop a generalized Bethe-Salpeter equation (BSE) to elucidate many-body manifestations of magnetic proximity effects. The BSE can be conveniently written as <cit.>[Ω^τ_S -ϵ^τ_c(k)+ϵ^τ_v(k)] 𝒜^Sτ_vck =∑_v'c'k'𝒦^τ_vck,v'c'k'𝒜^Sτ_v'c'k',where in a given valley τ the band index n=c(v) denotes one of the two CBs (VBs), Ω^τ_S is the energy of the exciton state |Ψ^τ_S⟩=∑_vck𝒜^Sτ_vckĉ^†_τ ckĉ_τ vk|GS⟩ <cit.> with the coefficients 𝒜^Sτ_vck, the creation (annihilation) operator of an electron in a CB c (VB v) ĉ^†_τ ck (ĉ_τ vk) in this valley, and the ground state |GS⟩ with fully occupied VBs and unoccupied CBs. Here, the kernel, 𝒦^τ_vck,v'c'k', includes the Coulomb interaction between electrons in the layer, determined from the dielectric environment, geometry, and form factors calculated from η^τ_nk <cit.>. The influence of magnetic substrates therefore modifies not only the single-particle energies ϵ^τ_c/v(k), but also the many-body interactions through this M-dependent kernel <cit.>, which could be generalized to include other quasiparticle excitations beyond excitons.While experiments demonstrate the proximity-induced exchange splitting in ML TMDs using an adjacent ferromagnet <cit.>, the employed single-particle description poses large uncertainties and excludes detected excitons. Equation (<ref>) now allows us to compute excitons in TMDs as M is rotated. Generally, 𝒦^τ_vck,v'c'k' couples all CBs and VBs in a valley. Only if spin is a good quantum number, in the absence of Rashba SOC and Mẑ, do the CBs (VBs) decouple. Each exciton is then formed from only one CB and VB and can be labeled by the spin configuration of those bands. This is no longer exactly true for arbitrary M orientation, but our results <cit.> show that typically the coupling between different CBs (VBs) is small and excitons are still mainly formed from one specific CB and VB as depicted in Fig. <ref>(c).The proximity-modified optical response, including excitons, can be accurately studied through the absorption,α(ω)=4e^2π^2/cω1/A∑_Sτ|∑_vck𝒟_vck𝒜^Sτ_vck|^2δ(ħω-Ω^τ_S),where ω is the photon frequency of light propagating along the -ẑ direction, c the speed of light, A the 2D unit area, and the single-particle velocity matrix elements are given by 𝒟^σ^±_vck=[η^τ_ck]^†v̂_±η^τ_vk for circularly polarized light and by 𝒟^x/y_vck=[η^τ_ck]^†v̂_x/yη^τ_vk for linearly polarized light with v̂_±=(v̂_x±ıv̂_y)/√(2), v̂_x/y=∂ H_tot/∂(ħ k_x/y). Optically inactive excitons or excitations imply that ∑_vck𝒟_vck𝒜^Sτ_vck has to vanish—due to either orbital restrictions or the spin configuration. The δ function is modeled by a Lorentzian with broadening Γ.A common approach for robust magnetic proximity effects in 2D materials is to minimize the hybridization effects and employ a magnetic insulator or a semiconductor <cit.>. First-principles results suggest a giant proximity-induced exchange splitting in MoTe_2/EuO <cit.>, which has also guided our choice of parameters. We use a reduced exchange coupling of J_c=100 meV and J_v=85 meV to reflect the fact that the calculated (111) interface is polar <cit.> and will undergo interface reconstruction <cit.>. The use of the optical response can provide a cleaner detection of magnetic proximity effects than through transport measurements, which can be complicated by various artifacts and complex interfaces. Similar transport difficulties are already known from the case of spin injection and detection <cit.>. Using ML MoTe_2 parameters <cit.>, a background dielectric constant ε=12.45 and the ML polarizability parameter r_0=6.3 nm to model ML MoTe_2 on EuO <cit.>, we set λ_R=0 in the following for simplicity. However, we find that Rashba SOC does not significantly change our results <cit.>.Similar to many experiments on ML TMDs <cit.>, the calculated absorption for M=0 in Fig. <ref>(a) is polarization independent and dominated by the so-called A and B peaks of bright neutral excitons X^0, corresponding to dipole-allowed transitions from the upper (A) and lower (B) valence band, respectively [see also Fig. <ref>(a)]. In contrast, these peaks are completely absent in the single-particle picture [insets in Figs. <ref>(a)-(c)], which is insufficient to properly include the excitonic effects <cit.>.The polarization independence of the absorption is a consequence of the valley degeneracy, lifted by an out-of-plane M as seen in Fig. <ref>(b). To understand the removal of the valley degeneracy, we focus on circularly polarized light because σ_+ (σ_-) couples exclusively to the K (K') valley. Since the exchange splitting is different for the CBs and VBs (with J_c>J_v>0), the single-particle gap energy for spin-up (spin-down) transitions at K is decreased (increased), resulting in a redshift of the A peak and a blueshift of the B peak for σ_+. An opposite behavior for σ_- is seen at K', leading to a splitting between σ_+ and σ_- of 29 meV and 30 meV for the A and B peaks, respectively. The absorption for x-polarized light is the symmetric combination of σ_+ and σ_-.Despite the common valley degeneracy and polarization independence in Figs. <ref>(a) and (c), there is a striking difference in the position of the two main peaks and the emergence of a new low-energy peak for M⊥ẑ. A clue for this behavior comes from Fig. <ref>(c). While there are well-defined dark and bright excitons for an out-of-plane M, the situation changes when M is rotated in plane. As the spin projections of a CB and VB forming a dark exciton at Mẑ are no longer perfectly antiparallel if M acquires an in-plane component, the single-particle matrix elements 𝒟_vck between these two bands and hence also the exciton dipole matrix element become finite and the formerly dark excitons become bright. Following the rotation of M from ϕ=0 to π (recall Fig. <ref>) illustrates in Fig. <ref> a peculiar transfer of spectral weight. In addition to the two bright excitons at ϕ=0, with an increase in ϕ two dark excitons gradually become bright, consistent with four bright excitons at ϕ=π/2 in Fig. <ref>(c). Such a brightening of excitons due to spin flips, also predicted due to out-of-plane electric fields <cit.>, has recently been observed in very large in-plane magnetic fields, B ≈ 30 T <cit.>, while found negligible even at B>8 T <cit.> in ML TMDs. However, Fig. <ref> shows also a reverse process: As ϕ is increased, there is a darkening of the bright excitons. By reversing M, there is a complete conversion between the dark and bright excitons.Together with the spectral weight transfer, Fig. <ref> reveals that all exciton peaks are shifted in energy. From the full BSE calculation, it is possible to obtain a simplified description based on the ϕ evolution of the single-particle optical gap, neglecting the M-dependent changes in the exciton binding energy, which in other cases can be significant as shown in Fig. S4 in Ref. Note:SM. The corresponding peak positions, marked by dashed lines in Fig. <ref>, are given byħω^exc_cv(ϕ)=ϵ^τ_c(ϕ)-ϵ^τ_v(ϕ)-E^cv_b,where E^cv_b is the binding energy of the 1s exciton formed from CB c and VB v at ϕ=0, which does not significantly change for ϕ≠0 here, and ϵ^τ_c(ϕ) and ϵ^τ_v(ϕ) are their respective single-particle band edges at k=0. For λ_R=0, these can be computed analytically asϵ^τ_v,±(ϕ)=-(E_g/2)±√(J^2_vsin^2ϕ+(τλ_v-J_vcosϕ)^2), ϵ^τ_c,±(ϕ)=(E_g/2)±√(J^2_csin^2ϕ+(τλ_c-J_ccosϕ)^2)for the two VBs v=(v,±) and the two CBs c=(c,±). The peak positions calculated from Eq. (<ref>) agree well with those computed from the full BSE in Eq. (<ref>), demonstrating the usefulness of this simplified picture.Symmetry requires the energy spectra (and consequently the absorption) at the K and K' points to be connected via ϵ^-τ_n(ϕ)=ϵ^τ_n(π-ϕ), which is also reflected by Eq. (<ref>) and Figs. <ref>(a) and (b). With our parameters for MoTe_2 on EuO, the strong exchange splitting in the CB determines the spin ordering in both valleys, while the VB spin ordering is unaffected [see Fig. <ref>(c)]. Hence, the exciton with lowest energy in the K valley is bright for ϕ=0, whereas it is dark for ϕ=π. Conversely, the exciton with lowest energy in the K' valley is dark for ϕ=0 and bright for ϕ=π. While we have chosen MoTe_2/EuO with its predicted huge magnetic proximity effects, it is important to explore if the main trends will persist in other TMD-based heterostructures having weaker proximity effects. Motivated by recent optical measurements <cit.>, we repeat our analysis of the absorption spectra for WSe_2/EuS. In this system, observed valley splittings of 2.5 meV for out-of-plane B=1 T greatly exceed the Zeeman effect of 0.1-0.2 meV/T in TMDs on nonmagnetic substrates. Consistent with our analysis, it can be attributed to magnetic proximity effects induced by the out-of-plane M in EuS. To describe WSe_2, we choose standard parameters <cit.> and recall that for M=0 there is a reversed CB ordering between MoTe_2 and WSe_2 [Fig <ref>(a)]. EuS, a magnetic insulator extensively used to demonstrate robust spin-dependent effects, including the first demonstration of solid-state spin filtering <cit.>, has significantly smaller exchange splittings than EuO, chosen in Fig. <ref> to be J_c=12.5 meV and J_v=5 meV. We again neglect the Rashba SOC based on the studies of MoTe_2/EuO, where even at a large value, λ_R=50 meV, arising due to the structural inversion asymmetry, all the key features are preserved and just slightly blueshifted as compared to the absorption spectra in Fig. <ref> <cit.>.Focusing on the A peaks for σ_+ and σ_-, Figs. <ref>(a) and (b) exhibit a lifting of the valley degeneracy by an out-of-plane M with the peaks shifted oppositely for both polarizations. This behavior, experimentally observed also in Ref. Zhao2017:NN, is similar to Fig. <ref>, but with smaller splittings of ∼ 14 meV for the A and B excitonic peaks. Likewise, the dark excitons below the A peak become bright due to an in-plane M, albeit with much less spectral weight lost by the A peak to the formerly dark excitons. This smaller transfer of spectral weight reflects the much smaller values of J_c(v) than for EuO. Figure <ref>(c) shows the peak positions of the bright and dark excitons with the magnetization reversal from ẑ to -ẑ. Similar to Fig. <ref> and to experimental observations in Ref. Zhao2017:NN, the relative ordering of bright, as well as dark, excitonic peaks for both σ_+ and σ_- is reversed when M is rotated from ẑ to -ẑ (in our setup, light is propagating along -ẑ).Despite these similarities, there are also striking differences between the evolution of excitons in the two heterostructures. Unlike Fig. <ref>, the bright excitons do not become dark as M is reversed in Fig. <ref>(c) <cit.>. Instead, the oscillator strength of the formerly dark excitons has its maximum for in-plane M and then decays to zero as M is rotated to -ẑ [see the symbol sizes qualitatively representing the exciton oscillator strengths in Fig. <ref>(c)]. This is a consequence of J_c(v) being dominated by λ_c(v) and hence the relative spin ordering in both valleys is unchanged compared to the case of M=0. The smaller values of J_c(v) in Fig. <ref> compared to Figs. <ref> and <ref> also result in the shifts of the exciton peaks with ϕ which are noticeably smaller in Fig. <ref>(c) than in Fig. <ref>. Nevertheless, our calculations for WSe_2/EuS show that, even in TMD-based magnetic heterostructures with weaker ferromagnetic exchange, an in-plane M is expected to result in pronounced signatures of the dark excitons.So far, magnetic proximity effects in TMDs employing ferromagnetic insulators and semiconductors were measured at cryogenic temperatures. However, this is not a fundamental limitation: Common ferromagnetic metals could enable room temperature proximity effects, while the metal/ML TMD hybridization can be prevented by inserting a thin insulating layer. A similar approach for Co- or graphene-based heterostructures that predicted a gate-controlled sign change in the proximity-induced spin polarization <cit.> was recently confirmed experimentally at 300 K <cit.>, suggesting important opportunities to study unexplored phenomena in TMDs. Unlike B fields of ∼ 30 T <cit.> that exceed typical experimental capabilities, the removal of valley degeneracy using magnetic substrates is not complicated by orbital effects and yet could enable even larger valley splittings <cit.>. Magnetic proximity offers another way to control and study many-body interactions in the time-reversed valleys of ML TMDs. 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Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Layer-dependent Ferromagnetism in a van der Waals Crystal down to the Monolayer Limit, Nature (London) 546, 270 (2017).§ BETHE-SALPETER EQUATION In this work, we employ the procedure described in Refs. Rohlfing2000:PRB,Scharf2016:PRB to compute neutral excitons. If intervalley coupling is not taken into account, the excitons can be calculated for each valley separately. Then, neutral excitons with momentum q_exc=0 can be described by their quantum number S and their valley τ and calculated from the Bethe-Salpeter equation (BSE),[Ω^τ_S-ϵ^τ_c(k)+ϵ^τ_v(k)] 𝒜^Sτ_vck =∑_v'c'k'𝒦^τ_vck,v'c'k'𝒜^Sτ_v'c'k',given as Eq. (3) in the main text <cit.>. Here, the band index n=c(v) denotes one of the two conduction (valence) bands, ϵ^τ_n(k) the corresponding single-particle/quasiparticle eigenenergies in valley τ with momentum k, and Ω^τ_S the energy of the exciton state |Ψ^τ_S⟩. We have used the ansatz |Ψ^τ_S⟩=∑_vck𝒜^Sτ_vckĉ^†_τ ckĉ_τ vk|GS⟩ for the exciton state with the coefficients 𝒜^Sτ_vck, the creation (annihilation) operator of an electron in a conduction band c (valence band v) ĉ^†_τ ck (ĉ_τ vk) in valley τ, and the ground state |GS⟩ with fully occupied valence bands and unoccupied conduction bands.Equation (<ref>) describes an eigenvalue problem for the eigenvalue Ω^τ_S and the eigenvector 𝒜^Sτ_vck. The interaction kernel <cit.>, consists of the direct and exchange terms,𝒦^τ_vck,v'c'k'=𝒦^d,τ_vck,v'c'k'+𝒦^x,τ_vck,v'c'k', 𝒦^d,τ_vck,v'c'k'=-∫^̣2r^̣2r' W(r-r'){[ψ^τ_ck(r)]^†ψ^τ_c'k'(r)}×{[ψ^τ_v'k'(r')]^†ψ^τ_vk(r')}, 𝒦^x,τ_vck,v'c'k'=∫^̣2r^̣2r' V(r-r'){[ψ^τ_ck(r)]^†ψ^τ_vk(r)}×{[ψ^τ_v'k'(r')]^†ψ^τ_c'k'(r')},if only intravalley Coulomb interactions are taken into account. Here, ψ^τ_nk(r) denote the wave functions of the single-particle states with energies ϵ^τ_n(k) (see above), V(r) the bare Coulomb potential determined from the dielectric environment, and W(r) the screened Coulomb potential, all in real space.The single-particle eigenstates ψ^τ_nk(r)=exp(ık·r)η^τ_nk/√(A) (with unit area A) are determined from H_totη^τ_nk=ϵ^τ_n(k)η^τ_nk, where the single-particle Hamiltonian H_tot=H_0+H_ex+H_R is defined in the main text. Inserting ψ^τ_nk(r) into Eqs. (<ref>) and (<ref>), we obtain𝒦^d,τ_vck,v'c'k'=-W(k-k')f_cc'(k,k')f_v'v(k',k)/A, 𝒦^x,τ_vck,v'c'k'=-V(k-k')f_cv(k,k)f_v'c'(k',k')/A,where the form factors f^τ_nn'(k,k')=[η^τ_nk]^†η^τ_n'k' are calculated from the single-particle states and W(k) and V(k) are the Fourier transforms of W(r) and V(r). Due to orthogonality of the eigenspinors η^τ_nk, f^τ_cv(k,k)=0 and 𝒦^x,τ_vck,v'c'k' vanishes. Hence, we are left with 𝒦^τ_vck,v'c'k'=𝒦^d,τ_vck,v'c'k' in our model. In addition to the single-particle states, W(k) is needed to compute 𝒦^τ_vck,v'c'k', with expressions for the Coulomb potentials W(k) and V(k) discussed in Sec. <ref> below.§ BARE COULOMB POTENTIAL AND GEOMETRIC CORRECTIONS If the conduction bands are completely empty and all valence states are occupied, the screened potential and the bare Coulomb potential, that is, the potential unscreened by free charge carriers, coincide, W(q)=V(q). The bare Coulomb potential is determined only from the dielectric environment (see Fig. <ref>) and can be obtained from the Poisson equation. Then, the bare Coulomb interaction between two electrons in the xy-plane (z=z'=0) can be calculated as <cit.> V(q)=2π e^2/q(ε̃^2-ε_tε_b)+(ε̃^2+ε_tε_b)cosh(qd)+ε̃(ε_t+ε_b)sinh(qd)/ε̃[(ε̃^2+ε_tε_b)sinh(qd)+ε̃(ε_t+ε_b)cosh(qd)]. For thin layers qd≪1, 1/V(q) can be expanded in powers of qd, which yieldsV(q)≈2π e^2/ε q+r_0q^2,whereε=(ε_t+ε_b)/2is the average dielectric constant of the bottom and top materials surrounding the monolayer transition-metal dichalcogenide (ML TMD), andr_0=ε̃d/2(1-ε_t^2+ε_b^2/2ε̃^2)can be interpreted as the polarizability of the monolayer. In the limit of ε_t/b≪ε̃, r_0=ε̃d/2 and we recover the result derived in Refs. Keldysh1979:JETP,Cudazzo2011:PRB.The interaction given by Eqs. (<ref>)-(<ref>) has proven to be highly successful in capturing the excitonic properties of ML TMDs <cit.> and is also used in our calculations. With ε_t=1 for air, ε_b=23.9 for EuO and ε_b=11.1 for EuS, we obtain average ε=12.45 for air/MoTe_2/EuO and ε=6.05 for air/WSe_2/EuS from and model the geometric correction by r_0=6.3 nm for air/MoTe_2/EuO and by r_0=4.0 nm for air/WSe_2/EuS.§ EFFECT OF RASHBA SPIN-ORBIT COUPLING In the main text, we have presented results for λ_R=0. The reason for this is twofold: First, even for a large value such as λ_R=50 meV, all key features discussed in the absence of Rashba spin-orbit coupling (SOC) are preserved. The second reason is that it allows for a simpler description of the effect the magnetic substrate has on the absorption spectrum of a ML TMD and that it is possible to provide simple analytical estimates and formulas in this case.Figure <ref> compares the linear absorption of ML MoTe_2 on EuO if Rashba SOC is neglected and if it is taken into account. In the absence of a magnetization breaking time-reversal symmetry there are only two optically active excitons, even for λ_R≠0 [Fig. <ref>(a)]. Both for an out-of-plane [Fig. <ref>(b)] and in-plane magnetization [Fig. <ref>(c)], Rashba SOC does also not qualitatively change the behavior of the absorption. The main quantitative change due to Rashba SOC is that the absorption spectra are just slightly blue-shifted as compared to the absorption spectra in Fig. 2 in the main text.§ COMPUTATIONAL DETAILS In our model, where intervalley coupling is neglected and excitons can be calculated for each valley separately, Eq. (<ref>) [Eq. (3) in the main text] contains two conduction and two valence bands for a given valley τ. In order to diagonalize Eq. (<ref>) [Eq. (3) in the main text], we use a coarse uniform N× N k-grid with a spacing of Δ k=2π/(Na_0) and a_0=3.52 Å for MoTe_2 and a_0=3.29 Å for WSe_2 in each direction as well as an upper energy cutoff E_cu. The Coulomb matrix elements W(k-k'), however, are not evaluated at the grid points of the N× N k-grid, but instead are averaged over a square centered around the coarse grid point k-k' with side widths Δ k on a fine N_int× N_int grid [with a corresponding spacing of Δ k_int=Δ k/N_int=2π/(NN_inta_0)]. To ensure convergence in our numerical calculations, we have used N=60, N_int=100 and energy cutoffs of 500 meV above the band gap, E_cu=E_g/2+500 meV, for air/MoTe_2/EuO and 750 meV above the band gap, E_cu=E_g/2+0.5 eV, for air/WSe_2/EuS. This procedure ensures that our numerical calculations converge reasonably fast and we find that the 1s binding energy changes by less than 1% when going from N=60 to N=100.§ ABSORPTION PROFILE OF WSE2 As noted in the main text, the absorption of WSe_2/EuS exhibits a qualitatively different behavior than the one predicted for MoTe_2/EuO due to the significantly smaller exchange splittings in WSe_2/EuS. In Fig. 4 of the main text, we have focused on the A peaks of the σ_+ and σ_- absorption for ML WSe_2 on EuS. To fully illustrate the differences between the two systems, Fig. <ref> shows the absorption of ML WSe_2 on EuS as M is rotated from out of plane (ϕ=0) to in plane (ϕ=π/2) and out of plane, but with reversed M (ϕ=π). In contrast to MoTe_2/EuO (compare to Fig. 3 of the main text), there is no conversion from dark to bright excitons and vice versa. Instead, the dark excitons become bright as M is rotated in plane and turn dark again as M is rotated out of plane, but with reversed M.§ MAGNETIZATION EFFECTS ON THE BETHE-SALPETER EQUATION The effect of the magnetization M is taken into account by the single-particle Hamiltonian (2) in the main text. Hence, the M-dependence in our approach enters the BSE (<ref>) on the right-hand side (RHS) via the single-particle energies ϵ^τ_c/v(k) and on the left-hand side (LHS) via the form factors f^τ_nn'(k,k') in the kernel [see Eq. (<ref>)].As a consequence of these two effects, the exciton peak is, in general, not just following the evolution of the single-particle excitation spectrum, but is subject to additional corrections. These corrections can be traced by following the binding energy E_b(ϕ)=[ϵ^τ_c(ϕ)-ϵ^τ_v(ϕ)]-Ω^τ_S(ϕ). We have noted in the main text that, for the used parameters of ML TMDs on magnetic substrates, these corrections are small and the full BSE results can be well described by a rigid shift of the excitonic peaks given by the single-particle spin splitting (see Fig. 3 of the main text). However, the single-particle description gives neither position nor the shape of the absorption peaks.In order to illustrate that this `single-particle' rigid shift agreement is not universal, Figs. <ref>(a) and (b) show the binding energies E_b(ϕ) and their relative change |E_b(ϕ)-E_b(0)|/E_b(0) as M is rotated for a different set of parameters. Here, we have chosen v_F=3.28×10^5 m/s, λ_c=-18 meV, λ_c=110 meV, ε=12.45, and r_0=6.3 nm, similar to MoTe_2/EuO, but with J_c=-J_v=200 meV and E_g=1.0 eV. The binding energy changes by as much as 30% and exhibits a more pronounced dependence on M than for the typical parameters shown in the main text of the paper.Table <ref> contains E_b(ϕ) for out-of-plane and in-plane M and different combinations of J_c, J_v and E_g. It illustrates that, in principle, there are parameter regimes where rigidly shifting the single-particle excitation energies by a constant, M-independent binding energy is not a good description of the exciton peaks. | http://arxiv.org/abs/1704.07984v2 | {
"authors": [
"Benedikt Scharf",
"Gaofeng Xu",
"Alex Matos-Abiague",
"Igor Žutić"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170426065017",
"title": "Magnetic Proximity Effects in Transition-Metal Dichalcogenides: Converting Excitons"
} |
Computer Science Dr. Vishal Patel, Dr. Ahmed Elgammal 3 2017 MayIn this thesis, we study two problems based on clustering algorithms. In the first problem, we study the role of visual attributes using an agglomerative clustering algorithm to whittle down the search area where the number of classes is high to improve the performance of clustering. We observe that as we add more attributes, the clustering performance increases overall. In the second problem, we study the role of clustering in aggregating templates in a 1:N open set protocol using multi-shot video as a probe. We observe that by increasing the number of clusters, the performance increases with respect to the baseline and reaches a peak, after which increasing the number of clusters causes the performance to degrade. Experiments are conducted using recently introduced unconstrained IARPA Janus IJB-A, CS2, and CS3 face recognition datasets.I would like to first thank my thesis advisor, Dr. Vishal Patel who provided gave me inspiration and encouragement throughout. I would also like to thank my thesis co-advisor, Dr. Ahmed Elgammal who helped me navigate through this journey. I would also like to thank all my lab-mates for the exchange of ideas, academic and otherwise. Finally, I express my profound gratitude for my family who has helped me arrive at this point in my academic career.I dedicate this thesis to my family.CHAPTER: FACE RECOGNITION§ INTRODUCTIONFace recognition has been actively studied over the past few decades which has led to satisfactory performances in recognition rates in controlled scenarios. But, in an unconstrained environment, face recognition is still a hard problem. A number of datasets have been thus developed to study face recognition in these scenarios that include LFW <cit.>, PubFig<cit.> and IJBA<cit.>. The intuitive pipeline<cit.> is shown infigure <ref> for face recognition, that includes face detection and tracking, face alignment, feature extraction and matching, described in sections below.§.§ Acquisition There are a few challenges that hinder the progress of face recognition in an unconstrained environment, which include challenges such as pose, illumination, and expression (PIE). Some of the other notable challenges include aging, cosmetics and resolution of the image. A lot of datasets have been developed that provide challenging media(images, videos, templates) so that algorithms can be developed to deal with these issues. Yale and YaleB<cit.> was introduced in 1997 that highlighted the challenges in illumination conditions, AR dataset<cit.> in 1998 highlighted occlusion apart from different emotions, and illuminations. Some of the more notable datasets in recent past are LFW<cit.> and PubFig<cit.> that contain huge amounts of images and deal with the face representation in the wild. One of the most challenging datasets as of now is the recently introduced unconstrained IARPA Janus IJB-A, CS2 and CS3 face recognition datasets<cit.>.§.§ Normalization and AlignmentSome of the pose and illumination artifacts are removed by normalization. There has been a lot of work that deals with this task. Depending on the applications, the issue of normalization can either be handled during the acquisition phase, where during the collection of the database, the acquisition parameters, such as capture device, ambient light are fixed. But, in the case where we want to develop algorithms invariant to these artifacts in unconstrained settings, learning from data in such preferential environment is averse to learning in the real-world settings. In such a case, post-processing of the collected data is done.§.§.§ Illumination NormalizationWe can handle illumination normalization during the acquisition phase, by making sure that the illumination remains the same throughout. As some of the datasets are collected in the real world settings, the natural illumination affects the final dataset. In such a case active devices such as thermal infra-red images, near infra-red images etc. can be used that provides its own light source to illuminate the object. In case this is not possible, such as images in the wild, normalization is done during post-processing to generate illumination invariant features. This can be done by using methods such as linear subspace, illumination cone, generalized photometric stereo, photometric normalization, reflectance model <cit.> etc. Some of the most studied models include, Self Quotient Images <cit.>,Logarithmic Total Variation <cit.>, Gradient Faces<cit.>, Robust Albedo Estimation<cit.>. §.§.§ Pose NormalizationThe images captured during the data acquisition phase can be constrained such that the pose of the captured images are consistent. But, is not the best solution, as even a slight error in capturing would result in a completely different image vector. Therefore, in such a case pose normalization is done during post-processing. The approach, in this case, is to find landmarks in the image that would remain consistent throughout, no matter how much shifted the image is. Some of these landmarks include the eyes, the nose, and the lips. Once these landmarks are detected, the image can be normalized based on these set of points. One such method that takes into account such an approach is called Geometric warping<cit.> where in-place pose normalization can be achieved. But, this approach cannot help in the case where there is an out-of plane rotation, for a case more robust methods are required. This follows from the fact that in an out-of plane rotation, pitch, roll, and yaw all have to be normalized. Some of the more used methods studied are, Incremental face alignment<cit.>, Deep Face Alignment<cit.>,Face Frontalization<cit.>.§.§ Features/RecognitionFeatures are distinct and unique properties of an entity, that can be used to distinguish it from others. These features are important as they form the framework for recognition of these entities. In a face recognition system, facial features could include, the shape of a person's face, eye color, the distance between eyes, etc. These features could either be hand-crafted, or they could be learned features.§.§.§ Hand-Crafted features Hand-crafted features as the name suggests is created manually by observing uniqueness in some aspect of the object. At the lowest levels, edges, lines, and corners form features, in a complex object, such as a face, a combination of these low-level features by hand is known as hand-crafted features. There are a few hand-crafted features that have been used extensively, such as SIFT<cit.>, HOG<cit.>, LBP<cit.>, etc. In such a framework, a classifier is trained using the hand-crafted features.The classification/recognition can be done usingSVM<cit.> , SRC<cit.> and Subspace methods such as PCA<cit.>, LDA <cit.> etc.§.§.§ Learned FeaturesInstead of coding the features by hand, features can also be learned from the data. This ensures an optimal representation given the data. At the end, a simple classifier can be used for classification. There are a few methods that are used in such a scenario, which include Dictionary Learning, Neural Networks etc. § PROTOCOLSRecognition is a term with wide scope when it comes to Face biometrics, as it encompasses a lot of authentication protocols, there are a few widely used authentication types that have been described below. §.§ IdentificationIn an identification problem, the question asked is, whether a given person exists in our system or not. The output from such a system is either Identified or Not-identified depending on whether that person exists in the given database.§.§ VerificationIn a verification protocol, given an instance of a user, we check if it matches the sample of the same user in our system. The output in such a scenario is a similarity score which defines how closely the new sample matches to the one already in the system§.§ SearchIn a search scenario, given a query image, we need to find all the instances of that person in the database. The output, in this case, top-k hits of the subject§ METRICS §.§ Error StatisticsA few of the more commonly used error statistics are False Match(Type I Error), False Non-Match (Type II Error), True-positive Identification Rate(TPIR),False-positive Identification-error Rate (FPIR). True-positive Identification Rate(TPIR)The True-positive Identification Rate (TPIR) is the proportion of identifications by enrolled subjects in which the subject’s correct class is returned. <cit.>False-positive Identification-error Rate (FPIR)The False-positive Identification-error Rate (FPIR) is the proportion of identifications by users not present in the system, which is returned. FPIR cannot be computed in closed-set identification, as all users are enrolled in the system <cit.>§.§ Decisions §.§ Metric curves There are a few metric curves that are used to plot the decisions, that include Reciever Operating Characteristics(ROC), Detection Error Tradeoff(DET), Cumulative Match Curve(CMC). CMCA CMC curve plots the Probability of identification versus the Rank as shown in figure <ref>§.§ Result interpretationThe result interpretation depends on the type of face recognition application. Some of the more used interpretations include Accuracy, Precision and Recall and F-Measure. F-measureThe F-measure is given in equation <ref> where P is Precision and R is RecallF_β = (β^2+1)P.R/β^2P + RThe F-1 measure is widely used where β=1, such that F-1 measure is the harmonic mean between precision and recall. The value of F-measure, therefore, is always between 0 and 1, and the higher the value, better is the performance of the recognition algorithm. Precision and RecallPrecision is defined as the ratio of True positives(TP) to the sum of True positives and False Negatives(FN) as shown in figure <ref> Precision=TP/TP+FP Recall is similarly defined as the ratio of true positives over the sum of true positives and false positives(FP) as in figure <ref>Recall=TP/TP+FNCHAPTER: FACE CLUSTERING§ INTRODUCTIONClustering is an unsupervised classification of patterns such as data items, feature vectors, or observations. In such a setting, given unlabelled data points, we have to group them based on a metric(ℓ_2,ℓ_p, Mahalanobis etc.). Clustering is a difficult problem, as we need to know a priori about the number of clusters or the stopping criterion. Clustering has a lot of applications such as exploration, segmentation in cases where the prior information about the data is not available. The pipeline<cit.> for clustering is given in figure <ref>A good representation the given data points/patterns is achieved by feature extraction. Once these features are computed, the clusters are merged/divided based on the inter-pattern similarity and the type of clustering. This process goes on until a stopping criterion such as a distance threshold or max number of clusters is met.§ CLUSTERING TECHNIQUES §.§ HierarchicalHierarchical clustering seeks to build a hierarchy of clusters such that it yields a dendrogram that represents the nested grouping of patterns and similarity levels<cit.>. These fall into two categories, agglomerative clustering, and divisive clustering. §.§.§ Agglomerative This is a bottom-up approach where each observation starts as an independent cluster, and pairs of clusters are merged based on the hierarchy and a stopping criterion. The merging of the clusters is based on certain linkage criterion, such as Single Link where the minimum distance between the points is used to merge the cluster. In the case of complete-link clustering, the clusters are merged based on the maximum distance between the data points of the two clusters. There are other order statistics that are used such as mean, centroid, group average, etc. to perform these linkages as well.§.§.§ DivisiveIn a divisive clustering framework, a top-down approach is followed such that all the data points start out in a single cluster,and they are split into different clusters moving down the hierarchy.§.§ PartitionalIn the case where construction of dendrograms is computationally inefficient/impossible, partitional methods are employed where a single partition of the data is obtained instead of a structure. The issue with using partitional clustering techniques is the fact that we need to know a priori the number of clusters/ partitions we need to perform. Partitional clustering is produced by optimizing a criterion function defined either locally or globally<cit.>. Some of the most common criterion used are squared error method as represented in equation <ref> <cit.>, whereX is the patterns set of the clustering L, whichcontains K clusters, such that x_i^(j) is the i^th pattern belonging to the j^th cluster and c_j is the centroid of the j^th cluster e^2(X,L)=∑_j=1^K ∑_i=1^n_j ||x_i^(j)-c_j||^2 A widely used method that uses squared error criterion is the k-means algorithm, where k points are randomly picked as the centroid and the cluster's center are recomputed until convergence by assigning each point to the cluster with the closest centroid. § EVALUATIONThe ultimate aim for clustering algorithm is to attain high intra-cluster similarity and low inter-cluster similarity. There are few evaluation metrics that are widely used to access the quality of the clustering. Some of these are Purity, Precision, and Recall, F-measure and compactness<cit.>. In our work, we use pair-wise precision and recall as defined in <cit.>Pairwise Precision is the same class fraction of pairs of data points within a cluster over the total number of same cluster pairs within the dataset. <cit.>. Pairwise Recall is the fraction of within class pairs of data points, that are placed in the same cluster, over the total number of same-class pairs in different clusters. <cit.>. § RECENT WORKS AND MOTIVATIONIn the problem of clustering faces, given unlabelled face images, we need to divide them into clusters, using a good feature space representation and a distance metric as shown in figure <ref><cit.>. There has been a lot of work in the area of face clustering that tries to improve the clustering accuracy. Zhu et al.<cit.> came up with Rank-Order Distance that is robust to both noise and outliers and can handle non-uniform cluster distribution like varying densities, shapes, and sizes of clusters. It calculates the dissimilarity between two faces based on their neighbouring information using ℓ_1 distance motivated by the fact that the same person shares top neighbours.The sub-clusters formed due to variation in pose illumination and expression, are subsequently merged agglomeratively using rank-order distance using a certain threshold or cluster level rank order distance to avoid the problem of too many high-precision, tight sub-clusters in the case of just using rank-order distance. Otto et al<cit.> used the same idea as Zhu et al. <cit.> on a larger scale, and therefore modified the algorithm to work on a large data setting. The effective and efficient Rank-order clustering algorithm used k-d tree algorithms to compute a small list of nearest neighbour, as the input size of data in order of millions, generating all the neighbours, as in the case of Zhu et al. <cit.> would be computationally hard. It used a single linkage agglomerative clustering algorithm based on a threshold to further compute the clusters and uses a pairwise F-measure to report the results on LFW dataset<cit.>. Zhu et al<cit.> came up with an algorithm to iteratively merge high precision clusters based on heterogeneous context information such as common-scene, people co-occurrence, human attributes and clothing information,such that the resulting clusters also have high recall. Clustering is hard as the performance decreases as the number of classes increases as it is evident in figure <ref>. Therefore our work is motivated by this challenge to whittle down the search domain in clustering using visual attributes to improve the clustering accuracy.§ EXPERIMENTIn our work, media averaging is done on CNN features that are computed from the IJBA CS2<cit.> samples in order to obtain templates. As the CS2<cit.> follows a template to template matching protocols, we perform clustering on these media averaged templates. Media averaging is shown in figure <ref>, where the video frames with the same media ID are averaged, and the resultant is then averaged with the images that belong to the same template ID. Once these templates are obtained, we use agglomerative clustering as defined in section <ref> where each template starts out as a different cluster are merged based on the stopping criterion of max number of clusters, as we have prior information of classes from the dataset. We use the average linkage with the cosine metric for merging these clusters based on the inter-pattern similarity. The templates are further divided into disjoint sets based on ground truth attributes from CS2 <cit.>. The template is divided into a Male subset, a female subset. The male subset is further divided into two different disjoint subsets based on the skin color attribute.The accuracy of the algorithm is reported based on pairwise F-1 score described in section <ref> § RESULTSThe algorithm is evaluated on IJBA CS2 dataset <cit.> that contains 500 subjects with 5,397 images and 2,042 videos split into 20,412 frames. The IJBA CS2 evaluation protocol consists of 10 random splits that contain 167 gallery templates and 1763 probe templates. The algorithm is evaluated on these 10 splits on JC's<cit.> and Swami's<cit.> deep features. The evaluated results on Swami's <cit.> features are shown in table <ref> and figure <ref>. The evaluated results on JC's<cit.> features are shown in table <ref> and figure <ref> § CONCLUSIONWe observein table <ref> and table <ref> that as we use more attributes, the clustering result improves. We can, therefore, assert that by using visual attributes we are narrowing down the search domain of the algorithm to boost the performance of clustering. CHAPTER: VIDEO BASED FACE TRACKING AND IDENTIFICATION§ INTRODUCTIONIn this work, we focus on a face identification task where the target is a multi-shot video and is annotated only once in one of the frames, and we need to search the annotated subject in a given gallery of images.The advantage over image to image retrieval in this case is that with a probe video, we have a lot more information and exemplars of the subject of interest and we can leverage this information to come up with a more robust representation that is invariant to the PIE challenges in face recognition. Traditionally, for a video to image retrieval task, the probe video is single shot where frame by frame bounding box of the subject of interest is provided as in the case of Youtube Faces <cit.>. For our work, we study an open set 1:N protocol using full motion video as probe where the probe video is multi-shot. In this setting, the subject of interest is annotated only for one of the frames, and the subject may or may not reappear in the subsequent shots. Therefore, matching a subject of interest from multi-shot video to gallery is a difficult task as we cannot use the traditional methods of a frame by frame bounding box tracking for the target face, because tracking algorithms are prone to drifting.A baseline approach to this problem is just to use the initial representation of the user annotated face of the subject to search for the subject in the gallery. But, the initial representation may not always be full frontal and devoid of any pose, illumination and expression variations. Hence, finding the subsequent appearance of the subject in the video is required to come up with a very robust representation of the subject. This is relatively easy in a single-shot video, where the entire video is a single shot, and there is no break in continuity. This can be achieved by making use of the temporal information and tracking the subject throughout in the video. But, in the case of multi-shot video, this task is relatively hard in a multi-shot video.§ MOTIVATION AND RECENT WORKSThe problem of face recognition as described in section <ref>, can be looked at in the terms of face verification and face identification. In face verification protocol one-to-one similarity is computed between the probe and the reference image. In face identification, on the other hand, one-to-many similarity between the probe and gallery is computed. With LFW<cit.> the face there were attempts to solve the face identification in the case where the dataset was unconstrained. Even so, there was a near-frontal selection bias while constructing the LFW<cit.>, hence the results are not representative of the set containing large in-the-wild pose variation. Also, because recent studies, <cit.> suggest the algorithmic performance of Face recognition algorithms is sub-par to humans, performance on unconstrained datasets with extreme pose, illumination, and expression are still lagging. One such challenging dataset is IJBA<cit.> that provides protocols for template-based verification and identification. The dataset consists of images and videos of subjects that are manually annotated and the performance evaluation is over a template, such that set of all media is combined into a single representation. Generating a robust representation in the formof a template is of utmost importance due to the large variation in pose, illumination, and expression. In our work, we improve an existing algorithm by template aggregation using clustering.There has been some work on templates and multi-shot video to gallery retrieval that has motivated our work in this direction. N. Crosswhite et al. <cit.> presented template adaptation, a type of transfer learning that works on the IJBA dataset <cit.> on one-to-many face identification protocol using CNN features,and a template specific one-vs-rest linear SVM. In their work, they learned a transfer learning mapping such that the source domain is the CNN features learned, and the target domain is the template of new subjects. This work uses encoding from the penultimate layer of VGG-Face<cit.> using an anisotropically scaled face crop of 224x223x3, followed by learning an L2-regularized L2-loss primal SVM with class weighted hinge loss objective<cit.> as expressed in equation <ref>. min_w 1/2 w^Tw +C_p ∑_i=1^N_pmax[0,1-y_iw^Tx_i]^2 + C_n ∑_j=1^N_nmax[0,1-y_jw^Tx_j]^2 such that C_p is the regularization constant for N_p positive observations obtained via average media encodings in the template, and C_nfor negative observation obtained via large external negative features. Ching Hui et al. <cit.> combine the work of Template Adaptation <cit.> and context-assisted clustering <cit.> to propose a Target Face Association(TFA) technique <cit.> that retrieves a set of representative face images from multi-shot video that is likely to have the same identity as the target face which is then used as a robust representation based on which the subject is looked up in a gallery of images.An OTS tracking technique<cit.> is used to track the target face. These images are treated as the initial positive training set(S_p). The faces are pre-associated <cit.> by selecting highest Intersection over Union(IOU)<cit.> of face detection bounding box with thewith tracking bounding boxes for the first k-frames. Ching Hui et al. learns a target specific linear SVM iteratively from pre-associated face images(positive samples) from the target video and negative samples(S_n) obtained from the cannot-link constraint<cit.>. In the cases where the target video cannot establish cannot-link constraints, due non-existence, an external dataset (S_b) is prepared for negative instances of the SVM.Their work uses two different models, wherein model one, the linear SVM is solved using the max-margin framework, where the training data is the union of all the three sets, i.e {(x_i,y_i)|i ∈ (S_p ∪ S_n ∪ S_b)} are used for training. In model 2, the set S_b is used only when there are no within-video negative instances.The robust face representation<cit.> is given in equation <ref> , x^fa=1/|A|∑_i ∈ Ax_i § METHODOnce the TFA <cit.> algorithm outputs the positive samples from the SVM, it simply averages these features as shown in equation <ref> to obtain the robust representation. In our work, however (TFA-C), we leverage a clustering algorithm to aggregate the features at the end of TFA. We use an Approximate Nearest Neighbour k-means++ using VLFeat library <cit.> algorithmsuch that the k data points that are picked greedily are maximally different. It is optimized using Approximate Nearest Neighbour algorithm that uses a randomized k-d tree. The max number of comparisons is limited to 100 and the number of trees is limited to 2 to trade off between speed and accuracy. The clustering is done by varying the number of clusters between 1 and 20. In the case where the number of samples is less than the number of clusters, the maximum cluster value is clipped to the maximum number of samples.§ RESULTS JANUS CS3 is an extended version of IJBA dataset <cit.> that contains 11,876 images and 7245 video clips of 1870 subjects. CS3 provides 11 different protocols, that include Identification, Verification and clustering tasks. In our work, we focus on Protocol 6, i.e CS3 1:N Multi Open Set (Video). In Protocol 6 there are 7195 probe templates, where each template is evaluated with respect to two disjoint galleries. There are 940 and 930 templates in Gallery 1 and Gallery 2, respectively. In this case given a video and the annotation of the subject of interest in the first frame, we need to search for a mated template in the gallery for a given probe template. As protocol 6 is an open-set identification problem, there exist some probe templates for which there are no mated templates in the gallery. Therefore, the ranking accuracy is evaluated only for those probe templates that have a mated template in the gallery, demonstrating the closed-set search. For these 20 clusters, the Rank-1, Rank-5, Rank-10, Rank-25, TPIR results are plotted for both JC<cit.> and Swami<cit.> features. These results are shown in figure 3.1 to figure 3.6 On an average k=7 clusters work best in respect to Rank-k accuracy and TPIR rate. The computed results for k=7 for JC<cit.> are given in table <ref> and the output on Swami's<cit.> features are given in table <ref>. As Ching Hui et al.<cit.> report their results on the average of these two features, we also report the average output in table <ref>. As we can clearly see, the results for TFA-C in table <ref> is better than the original TFA algorithm in table <ref> we can state that TFA-C performs better than TFA<cit.>§ CONCLUSIONWe observe that as the number of clusters(k) are increased for the template aggregation, the identification rate increases to a point and deprecates after that. Based on the averages, we observe cluster numbers, k=7 works the best for identification rate in closed set search as shown by the CMC Rank curves and also in the open set search as shown by the CMC TPIR curves. We conclude that our method TFA-C outperforms the existing TFA algorithm by a significant margin.§ ACKNOWLEDGEMENTThis research is based upon work supported by the Office of the Director ofNationalIntelligence(ODNI),IntelligenceAdvancedResearchProjects Activity(IARPA),viaIARPAR&DContractNo.2014-14071600012.The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, eitherexpressedorimplied,oftheODNI,IARPA,ortheU.S.Government. TheU.S.Governmentisauthorizedtoreproduceanddistributereprintsfor Governmental purposes notwithstanding any copyright annotation thereonCHAPTER: APPENDIX § PHOTO-SKETCHFacial sketches are an essential part of forensics in law enforcement, particularly in those cases where the only evidence is in the form of eye-witness testimony. Facial sketches are of two types, Forensic Sketches that are drawn by forensic artists, and Composite Sketches that are created using computer software <cit.>.Once the sketches are drawn from either of these methods, it allows the law enforcement to apprehend the person of interest based on it. Several works have tried to automate this process by automatically matching<cit.> the sketches to the criminal database. Figure <ref> shows composite and forensic sketches corresponding to the mugshot images as developed by Klum et al.<cit.>. They also show that the matching accuracy of composite sketches is higher than that of the forensic sketches. As evident from the figure <ref>, composite sketches are more close to the mugshot images in the domain, and hence they have a better matching accuracy.Motivated by the fact, that at the end the ultimate aim of sketches is matching, we wanted to develop automatic sketches in the mugshot domain. For our work, we used the PubFig dataset to develop single average template faces for the attributes using one attribute and two attributes as shown in figure <ref> and figure <ref>As evident from figure <ref> and figure <ref> the average template suffer high illumination artifacts and there is a bias across not only the subjects but across the attributes. So a trade-off needs to met so that the dataset is balanced not only in the subjects but also, attributes. Due to the lack of such a curated dataset and the ill-posed problem, we will like to work further on this problem by either developing a dataset in the future, or utilizing a dataset if any is created that balances classes across not only subjects, but attributes as well.The author of my thesisxxxxx-xxxxx 2017 M.S in Computer Science, Rutgers University, USA2014 B.E in Instrumentation and Control, University of Delhi, India xxxxx-xxxxx 2016-2017 Graduate assistant, Department of Computer Science, Rutgers University2015-2016 Teaching assistant, Department of Computer Science, Rutgers University2015-2016 Grader, Department of Computer Science, Rutgers University2011-2015 Visiting Researcher, IIT-Delhi, IndiatocchapterBibliography9HandbookFRS.Li and A.Jain, (ed). Handbook of Face Recognition, Springer-Verlag, 2005pubfigNeeraj Kumar, Alexander C. Berg, Peter N. Belhumeur, and Shree K. 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"authors": [
"Atul Dhingra"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170426195028",
"title": "Face Identification and Clustering"
} |
Joint Quantum Institute, NIST/University of Maryland, College Park, MD 20742, USA School of Physics and Astronomy, Monash University, Victoria 3800, AustraliaJoint Quantum Institute, NIST/University of Maryland, College Park, MD 20742, USA Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, MD 20742, USAJoint Quantum Institute, NIST/University of Maryland, College Park, MD 20742, USA We study out-of-time order correlators (OTOCs)of the form ⟨(t)(0)(t)(0)⟩ for a quantum system weakly coupled to a dissipative environment. Such an open system may serve as a model of, e.g., a small region in a disordered interacting medium coupled to the rest of this medium considered as an environment.We demonstrate that for a system with discrete energy levelsthe OTOC saturates exponentially ∝∑ a_i e^-t/τ_i+const to a constant value at t→∞, in contrast with quantum-chaotic systems which exhibit exponential growth of OTOCs. Focussing on the case of a two-level system, we calculate microscopically the decay times τ_i and the value of the saturation constant. Because some OTOCs are immune to dephasing processes and some are not, such correlators may decay on two sets of parametrically different time scales related to inelastic transitions between the system levels and to pure dephasing processes, respectively. In the case of a classical environment, the evolution of the OTOC can be mapped onto the evolution of the density matrix of two systems coupled to the same dissipative environment. Out-of-time-order correlators in finite open systems V. Galitski December 30, 2023 ==================================================== Quantum information spreading in a quantum systemis often described by out-of-time-order correlators (OTOCs) of the form K(t)=<(t)(0)(t)(0)>,where , ,andareHermitian operators, and ⟨…⟩ is the average with respect to the initial state of the system. Correlators of such form have been firstintroduced by A. Larkin and Y.N. Ovchinnikov<cit.> in the context of disordered conductors, where the correlator ⟨[p_z(t),p_z(0)]^2⟩ of particle momenta p_z has been demonstrated to grow exponentially ∝ e^2λ t for sufficiently long times t.The Lyapunov exponent λcharacterises the rate of divergence of two classical electron trajectories with slightly different initial conditions and serves as a measure of quantum chaotic behaviour in a system. The concept of OTOC has revived<cit.> recently in the context of quantum information scrambling and black holes, motivating further studies of such quantities (see, e.g., Refs. Maldacena:bound,SwingleChowdhury:LocScrambling,AleinerFaoroIoffe,Rosenbaum:rotor,PatelSachdev:FermiSurf). Despite not being measurableobservables[ Because an OTOC involves evolution backwards in time, measuring itrequires either using a second copy of the system<cit.> or effectively reverting the sign of the Hamiltonian<cit.>, possible, e.g., in spin systems using spin-echo-type techniques or ancilla qubits], OTOCs (<ref>) characterise the spreading of quantum information and the sensitivity of the system to the change of the initial conditions. It is also expected that OTOCs may be used<cit.> to distinguish between many-body-localised and many-body-delocalised states<cit.> of disordered interacting systems. So far the studies of quantum chaos and information scrambling have been focussing on closed quantum systems. In reality, however, each system is coupled to a noisy environment, which leads to decoherence and affects information spreading. Moreover, a sufficiently strongly disordered interacting system may be separated into a small subsystem, of the size of the single-particle localisation length or a region of quasi-localised states, coupled to the rest of the system considered as environment. In this paper we analyse out-of-time order correlators in a quantum system weakly coupled to a dissipative environment. Phenomenological picture in a strongly disordered material. A system with localised single-particle states and weak short-range interactions exhibits insulating behaviour at low temperatures<cit.>. Local physical observables in sucha system are strongly correlated only on short length scales, and their properties may be understoodby considering a single “localisaton cell”, particle states in a region of space of the size of order of the localisation length ξ, which may be considered weakly coupled to the rest of the system. The energy spectrum of the localisation cell may be probed via response functionsof local operators in the cell, e.g., the response function χ(ω)=∑_α,β(f_α-f_β)|Q_αβ|^2/E_α-E_β+ω+i0 of the charge Q in a region inside the cell to the voltage in this region, where E_α and E_β are the energies of many-body states and f_α is their distribution function. For temperatures smaller than a critical value,quasiparticles in the system have zero decayrate[When studying response functions, a system in the insulating state should be assumed coupled to an external bath, with the value of the coupling sent to zero at the end of the calculation<cit.>. The quasiparticle decay rate is then given by the bath strength.](“superinsulating” regime<cit.>), and the system thus responds only at a discrete set of frequencies ω=E_i-E_j, determined by the energy gaps between many-body states, as shown in Fig. <ref>. The OTOC (<ref>) in this regime oscillates K(t)∝∑_n a_n e^iω_n t with a discrete set of frequencies ω_n=E_i_n+E_i_n^'-E_j_n-E_j_n^'.When the temperature (or the interaction strength at a given temperature) exceeds a critical value, the levels and response functions get broadened (“metallic” phase<cit.>), as illustrated in Fig. <ref>, becoming smoother with increasing temperature and/or interactions. Near the superinsulator-metal transition the characteristic level width Γ is significantly smaller than the gaps between levels, and the localisation cell may be considered as an open system weakly coupled to a dissipative environment. The same model may be applied also to a strongly disordered material with an external bath, such as a system of phonons, which provide a finite level width Γ at all finite temperatures. The local operators , ,andin Eq. (<ref>) do not necessarily act on states in one localisation cell, but may involve states in several cells close to each other. These cells may still be considered as a single quantum dot in a noisy environment so long as the level spacing in the dot exceeds the level width. Such a model of an open quantum dot may be also realised directly, e.g., using superconducting qubits or trapped cold atoms. Summary of the results. We demonstrate that, for a system with discrete non-degenerate levels E_n, correlator (<ref>) at long times t exponentially saturates to a constant value, K(t)∝∑ a_n e^iω_n te^-t/τ_n+const, and calculate microscopically the value of the constant and the relaxation times τ_n as a function of the environment spectral function and the matrix elements of the system-environment coupling. Depending on the choice of the operators , ,and , the saturation value may be finite or zero. OTOCs relax due to both inelastic transitions between the system's levels and pure dephasing processes, which are caused by slow fluctuations of the energies E_n. While some OTOCs are immune to dephasing processes, a generic correlator has components both sensitive and insensitive to dephasing and thus decays on two sets of parametrically different scales related to dephasing and relaxation respectively, as shown in Fig. <ref>. Our results indicate, in particular, that a disordered system of interacting particlescannot exhibit quantum chaotic behaviourif the typical single-particle level splitting δ_ξin a volume of linear size ξ (localisation length) exceeds the dephasing rate and the rate of inelastic transitions due to interactions and/or phonons. Correlators (<ref>) in this system can onlysaturate to constant values at t→∞, in contrast with quantum-chaotic systems which display exponential growth of OTOCs with time. Our results thus suggest that chaotic behaviour in a disordered interacting system requires either the presence of delocalised single-particle states or sufficiently strong interactionsor, e.g., a phonon bath, which would lead to the quasiparticle decay rate exceeding the level spacing δ_ξ.For a classical environment, the evolution of an OTOC (<ref>) in an open systemmay be mapped onto the evolution of the density matrix of two systems coupled to the same environment, which allows one to measure OTOCs by observing the correlations between two systems in a noisy environment, such as spins in a random time-dependent magnetic field.Model. We consider a system with discrete non-degenerate energy levels E_n coupled to a dissipative environment and described by the Hamiltonian=_0 + V̂X̂ + _bath(X̂),where _0=∑_n E_n|n⟩⟨n| is the Hamiltonian of the system, _bath(X̂)– the Hamiltonian of the environment, and V̂X̂ is the coupling between the system and the environment, where the operator =∑_n,mV_nm|n⟩⟨m| acts on the system degrees of freedom, andis an environment variable which commutes with the system degrees of freedom. To compute the OTOC (<ref>), where the operators , ,andact on the system variables, it is convenient to decompose it as K=K_m_1m_2,n_1n_2A_n_1m_1C_n_2m_2 (summation over repeated indices implied), where A_n_1m_1 and C_n_2m_2, are the matrix elements of the operatorsand , andK_m_1m_2,n_1n_2=<|n_1⟩⟨m_1|(t)(0)|n_2⟩⟨m_2|(t)(0)>,where ⟨…⟩ is the averaging with respect to both the system and environment states.In the limit of a vanishing system-environment coupling , the correlators (<ref>) oscillate with time, K_m_1m_2,n_1n_2∝ e^i(E_n_1+E_n_2-E_m_1-E_m_2)t. A finite coupling between the system and the environment leads to dissipation and relaxation processes and thus to the decay of the elements K_m_1m_2,n_1n_2. For a weak coupling considered in this paper, the characteristic decay times of the OTOCs significantly exceed the correlation time of the environment degrees of freedom, i.e. of the function S(t-t^')=⟨(t)(t^')⟩_env, and the evolution of the elements is described by a system of Markovian Bloch-Redfield<cit.> equations (see Supplemental Material for the microscopic derivation) of the form∂_t K_m_1m_2,n_1n_2 = i(E_n_1+E_n_2-E_m_1-E_m_2)K_m_1m_2,n_1n_2 -∑_m_1^', m_2^',n_1^', n_2^'Γ_m_1m_2,n_1n_2^m_1^' m_2^',n_1^' n_2^' K_m_1^' m_2^',n_1^' n_2^'.From the definition of the elements (<ref>) it follows that∑_m,n K_nm,nm=<(0)(0)> =const. Eq. (<ref>) may be also derived from the microscopic equations of evolution, as shown in Supplemental Material. Due to the smallness of the decay rates Γ_m_1m_2,n_1n_2^m_1^' m_2^',n_1^' n_2^' in Eq. (<ref>), the evolution of each element K_m_1 m_2,n_1 n_2 is affected only by the elements K_m_1^' m_2^',n_1^' n_2^' with the same oscillation frequency E_n_1+E_n_2-E_m_1-E_m_2 (secular approximation). In this paper we consider systems with sufficiently non-degenerate energy spectra; if two elements oscillate with the same frequency, they may be different only by permutations of indices n_1 and n_2 and/or m_1 and m_2.For a generic N-level system there are 2N^2-N elements (<ref>) with zero energy gaps E_n_1+E_n_2-E_m_1-E_m_2 (with m_1=n_1, m_2=n_2 and/or m_1=n_2, m_2=n_1). These elements are immune to dephasing, i.e. to the accumulation of random phases caused by slow fluctuations of the energies E_n_i. Such vanishing of dephasing is similar to that in decoherence-free subspaces<cit.> of multiple-qubit systems. We emphasise, however, that even dephasing-immune correlators in general decay at long times due to the environment-induced inelastic transitions between the levels (relaxation processes).A generic OTOC (<ref>) includes components both sensitive and insensitive to dephasing, as well as a component independent of time, which exists due tothe conservation law (<ref>).For an environment with a smooth spectral function on the scale of the characteristic level splitting, the characteristic decay rate of the dephasing-immune components may be estimated as 1/τ_rel∼ V_^2S(Δ E), where V_ is the typical off-diagonal matrix element of the perturbationand Δ E is the characteristic level spacing. The other components decay with the characteristic rate 1/τ_deph+1/τ_rel, where 1/τ_deph∼ V_∥^2S(0) is the characteristic dephasing rate, where V_∥ is the typical diagonal matrix element of the perturbation . As a result, the decay of the OTOC consist of three stages, corresponding to these characteristic times, as illustrated in Fig. <ref>.Two-level system. In order to illustrate the meaning of these time scales and the related phenomena, we focus below on the case of a two-level system, equivalent to a spin-1/2 in a random magnetic field (for the microscopic analysis of OTOCs in the generic case of a multi-level system see Supplemental Material), described by the Hamiltonian =1/2B_z + 1/2 +_bath(),whereis a vector of Pauli matrices andis a constant unit vector, the direction of the fluctuations of the magnetic field.The dissipative environment induces transitions |↑⟩→|↓⟩ with the rate Γ_↓=1/4(n_x^2+n_y^2)S(B), as well as the opposite transitions|↓⟩→|↑⟩ with the rate Γ_↑=1/4(n_x^2+n_y^2)S(-B), where S(ω) is the environment spectrum, the Fourier-transform of S(t-t^')=⟨(t)(t^')⟩_env. Weak fluctuations of the magnetic field in the longitudinal direction lead to dephasing with the rate Γ^ϕ=1/2n_z^2S(0).We focus below on the long-time dynamics of the system and assume for simplicity that the rate Γ^ϕ of pure dephasing significantly exceeds the rates Γ_↑ and Γ_↓ of inelastic transitions between the levels of the spin; in the opposite case, all OTOC decayrates are of the same order of magnitude.The OTOCs K_↑↑,↓↓ and K_↓↓,↑↑ oscillate with frequencies ± 2(E_↓-E_↑)=∓ 2B and have dephasing rate 4Γ^ϕ, the same as ±1-projection states of a spin-1 in magnetic field B,K_↑↑,↓↓,K_↓↓,↑↑∝ e^∓ 2i Bte^-4Γ^ϕ t,where we have neglected the small relaxation rates Γ_↑,↓≪Γ^ϕ.There are 8 elements (<ref>) which correspond to 3 spin indices pointing in one direction and one spin index pointing in the opposite direction. These elements oscillate with frequencies ± B and have the same dephasing rate as a spin-1/2,K_↑↓,↓↓,K_↓↑,↑↑, K_↓↓,↑↓,…∝ e^-Γ^ϕ t. The behaviour of OTOCs at long times t≫ 1/Γ^ϕ is determined by the components with a vanishing frequency E_n_1+E_n_2-E_m_1-E_m_2 of coherent oscillations, because such components are insensitive to dephasing. For a spin-1/2, their evolution is described by the system of equations (as follows from the generic master equations for a multi-level system derived in Supplemental Material) ∂_t ( [ K_↓↑,↑↓; K_↑↓,↓↑; K_↑↓,↑↓; K_↓↑,↓↑; K_↑↑,↑↑; K_↓↓,↓↓ ]) = ( [ -Γ_↓-Γ_↑0 -Γ_↓ -Γ_↓Γ_↓Γ_↓;0 -Γ_↓-Γ_↑ -Γ_↑ -Γ_↑Γ_↑Γ_↑; -Γ_↑ -Γ_↓ -Γ_↓-Γ_↑0Γ_↓Γ_↑; -Γ_↑ -Γ_↓0 -Γ_↓-Γ_↑Γ_↓Γ_↑;Γ_↑Γ_↓Γ_↑Γ_↑-2Γ_↓0;Γ_↑Γ_↓Γ_↓Γ_↓0-2Γ_↑ ]) ( [ K_↓↑,↑↓; K_↑↓,↓↑; K_↑↓,↑↓; K_↓↑,↓↑; K_↑↑,↑↑; K_↓↓,↓↓ ]). The rates of the long-time decay of OTOCs are given by the eigenvalues of the matrix in Eq. (<ref>) (with minus sign) and are shown (except for the zero eigenvalue) in Fig. <ref>. Such a matrix always has a zero eigenvalue, due to the conservation law (<ref>). The system also has a triply degenerate decay rate Γ_↑+Γ_↓. The other two decay rates are given by 1/2[3Γ_↑+3Γ_↓±(Γ_↑^2+34Γ_↑Γ_↓ +Γ_↓^2)^1/2]. At long times t→∞ the correlator (<ref>) saturates to a constant value determined by the projection of the OTOC (<ref>) on the zero-decay-rate mode,K(t→∞) = 1/√(2Γ_↑^2+2Γ_↓^2) ( Γ_↓ A_↑↓C_↓↑ +Γ_↑ A_↓↑C_↑↓. . +Γ_↑ A_↑↑C_↑↑ +Γ_↓ A_↓↓C_↓↓).While we assumed a small inelastic relaxation rate in comparison with the dephasing rate, we emphasise that the result (<ref>) for the saturation value of the OTOC holds for an arbitrary ratio of dephasing and relaxation rates. Mapping to the evolution of two systems for a classical environment. The evolution of the OTOCs (<ref>) is similar to that of the density-matrix elementsρ_m_1m_2,n_1n_2 =<<|m_1⟩⟨n_1|(t)>_Sys_1<|m_2⟩⟨n_2|(t)>_Sys_2>_Xof a compound system consisting of two identical subsystems (“Sys_1” and “Sys_2”) coupled to the same dissipative environment, where m_1 and n_1 and m_2 and n_2 in Eq. (<ref>) are the states of the first and the second subsystems respectively,|m_i⟩⟨n_i|(t) is an operator in the interaction representation, and ⟨…⟩_X is the averaging with respect to the environment degrees of freedom. The Hamiltonian of such a compound system is given by=_0⊗ + ⊗_0 +(⊗ + ⊗)+_bath(X̂),where …⊗… is the product of the subsystem subspaces; _0 andare the Hamiltonian of each subsystem and its coupling to the environment, and the environment variablecommutes with all degrees of freedom of subsystems “Sys_1” and “Sys_2”.The evolution of the elements (<ref>) and (<ref>) is described by similar Markovian master equations (see Supplemental Material for microscopic derivation). In particular, in the limit of a classical environment ((t)(t^')=(t^')(t)), the evolution of OTOCs (<ref>) can be mapped exactly onto that of the density matrix (<ref>) of two systems coupled to this environment, as follows from thedefinitions of these quantities. The conservation law (<ref>) is mapped thenonto the conservation of the trace of the density matrix of a compound system consisting of two subsystems.In the limit of a classical environment, the spectral function is even, S(ω)=S(-ω), the relaxation rate i→ j for each pair of levels i and j in a system matches the reverse rate j→ i. In particular, in the case of a two-level system Γ_↑=Γ_↓=Γ, and the OTOC has three decay rates at long times t≫Γ^ϕ^-1: Γ_1=6Γ, Γ_2=2Γ (triply degenerate) and Γ_3=0 (doubly degenerate), as shown in Fig. <ref>. Due to the mapping, these rates match the decayrates of pair-wise correlators of observables in, e.g., an ensemble of spins in a uniform random magnetic field and thus may be conveniently measured in such ensembles. We emphasise that the mapping between an OTOC and the evolution of two subsystems coupled to the same classical environment holds for an arbitrary system-environment coupling but not only in the limit of a weak coupling considered in this paper. This mapping suggests a way for measuring OTOCs in generic systems in the presence of classical environments through observing correlators<<(t)>_Sys_1<(t)>_Sys_2>_X of observablesandbetween two systems.Discussion. We computed OTOCs in a system weakly coupled to a dissipative environment and demonstrated that they saturate to a constant value at long times. Because such an open system may serve as a model of a small region in a disordered interacting medium (in the presence or in the absence of a phonon bath), this suggests the absence of a chaotic behaviour in strongly disordered materials. While our result applies to weakly-conducting and insulating materials, for which the system-environment coupling may be considered small, we leave it for a future study whether non-chaotic behaviour persists in systems strongly coupled to the environment (corresponding to an effectively continuous energy spectrum of a localisation cell). For a classical environment, the evolution of an OTOC matches the evolution of correlators of observables between two identical systems coupled to the same environment, which may be used for measuring OTOCs in open systems in classical environments. The possibility to develop a similar measurementmethod for the case of a quantum environment is another question which deserves further investigsation. We have benefited from discussions with Yidan Wang. V.G. and S.V.S. were supported byUS-ARO (contract No. W911NF1310172), NSF-DMR 1613029 and Simons Foundation; A.V.G. and S.V.S. acknowledge support by NSF QIS, AFOSR, NSF PFC at JQI, ARO MURI, ARO and ARL CDQI.S.V.S. also acknowledges the hospitality of School of Physics and Astronomy at Monash University, where a part of this work was completed.26 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Larkin and Ovchinnikov(1969)]LarkinOvchnnikov author author A. Larkin and author Y. N. Ovchinnikov, title title Quasiclassical method in the theory of superconductivity, @noopjournal journal Sov. Phys. JETP volume 28, pages 960 (year 1969)NoStop [Kitaev(2015)]Kitaev:talk author author A. 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Tezuka, title title Creating and probing the Sachdev-Ye-Kitaev model with ultracold gases: Towards experimental studies of quantum gravity, @noopjournal journal ArXiv e-prints(year 2016), http://arxiv.org/abs/1606.02454 arXiv:1606.02454 NoStop [Tsuji et al.(2017)Tsuji, Werner, and Ueda]Tsuji:measurement author author Naoto Tsuji, author Philipp Werner,and author Masahito Ueda,title title Exact out-of-time-ordered correlation functions for an interacting lattice fermion model,@noopjournal journal Phys. Rev. Avolume 95, pages 011601 (year 2017)NoStop [Li et al.(2016)Li, Fan, Wang, Ye, Zeng, Zhai, Peng, andDu]Li:NMRmeas author author J. Li, author R. Fan, author H. Wang, author B. Ye, author B. Zeng, author H. Zhai, author X. Peng,and author J. Du, title title Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator, @noopjournal journal ArXiv e-prints(year 2016), http://arxiv.org/abs/1609.01246 arXiv:1609.01246 NoStop Supplemental Material for “Out-of-time-order correlators in finite open systems”§.§ Master equations for the density matrix in an open system Off-diagonal elements. For a system with non-degenerate energy levels weakly coupled to a dissipative environment, with the Hamiltonian given by Eq. (<ref>), the off-diagonal entries ρ_mn of the density matrix satisfy Bloch-Redfield master equations (see, e.g., Ref. Slichter:book)∂_t ρ_mn= i(E_mn+iΓ_mn)ρ_mn,where E_mn=E_m-E_n is the frequency of coherent oscillations for an isolated system, and the complex quantityΓ_mn=-i∫dω/2π∑_k (S(ω)|V_mk|^2/ω-E_mk-i0 +S(-ω)|V_nk|^2/ω-E_kn-i0) +iV_mmV_nn∫dω/2πS(ω)+S(-ω)/ω-i0accounts for the effects of the environment, where S(ω) is the Fourier-transform of the correlation function S(t-t^')=⟨(t)(t^')⟩_env =∫dω/2πe^-iω(t-t^')S(ω) of the environment degree of freedom (t). The quantity Γ_mn, given by Eq. (<ref>), may be decomposed asΓ_mn= 1/2∑_k≠ mΓ_m→ k^rel +1/2∑_k≠ nΓ_n→ k^rel -iδ E_m + iδ E_n +Γ_mn^deph,whereΓ_n→ k^rel=|V_nk|^2 S(E_n-E_k)is the rate of environment-induced transitions (relaxation) from level n to level k,Γ_mn^deph=1/2(V_nn-V_mm)^2S(0)is the pure dephasing rate, andδ E_m=∑_k≠ m|V_mk|^2∫dω/2πS(ω)/E_m-E_k-ωis the shift of the energy of the m-th level due to the interaction with environment (Lamb shift).The relaxation rate between two levels n and k, Eq. (<ref>), is determined by the environment spectrum S(ω) at frequency ω=E_nk equal to the energy gap between these levels, while the dephasing rate (<ref>) is determined by the low-frequency properties of the environment.Diagonal elements. The dynamics of the diagonal elements of the density matrix is described by the equations∂_t ρ_nn=-ρ_nn∑_kΓ_n→ k^rel +∑_kρ_kkΓ_k→ n^rel, where the transition rates Γ_n→ k are given by Eq. (<ref>). Lindblad form.Eqs. (<ref>) and (<ref>) for the evolution of the density matrix can be rewritten in the Lindblad form∂_t =-i[_eff,] -1/2∑_i,j( _ij^†_ij +_ij^†_ij -2_ij^†_ij),where the summation runs over all pairs of indices i=1,…,N and j=1,…,N in an N-level system; the effective Hamiltonian of coherent evolution is given by_eff=∑_i |i⟩⟨i|(E_i+δ E_i),and the Lindblad operators_ij=(1-δ_ij)√(Γ_j→ i^rel)|i⟩⟨j| +δ_ij√(S(0)/N)∑_l V_ll|l⟩⟨l|account for the effects of dephasing and dissipation.§.§ Master equations for OTOCs In what follows we derive microscopically the Bloch-Redfield-type master equations for the out-of-time-order correlator (<ref>), following a procedure similar to the derivation (see, e.g., Ref. <cit.>) of the master equations for the density matrix.Due to the weakness of the system-environment coupling, the OTOCs decay on long times significantly exceeding the characteristic correlation time of the environment.It follows directly from Eq. (<ref>) that∂_t K_m_1m_2,n_1n_2= i<[_0+_coupl(t),|n_1⟩⟨m_1|(t)](0)|n_2⟩⟨m_2|(t)(0)> +i<|n_1⟩⟨m_1|(t)(0)[_0+_coupl(t),|n_2⟩⟨m_2|(t)](0)>,where_0 is the Hamiltonian of the system (without the environment) and _coupl=∑_n,mV_nm|n⟩⟨m| is the coupling between the system and the environment. By expanding all Heisenberg operators in Eq. (<ref>) to the first order in the perturbation _coupl and neglecting the change of the density matrix of the system during the characteristiccorrelation time of the environment, we arrive at the equations for the evolution of the elements K_m_1m_2,n_1n_2 in the form∂_t K_m_1m_2,n_1n_2= i(E_n_1+E_n_2-E_m_1-E_m_2)K_m_1m_2,n_1n_2-<∫_-∞^t[_coupl(t^'),[_coupl(t),|n_1⟩⟨m_1|(t)]]dt^'(0)|n_2⟩⟨m_2|(t)(0)> -<|n_1⟩⟨m_1|(t) (0)∫_-∞^t[_coupl(t^'), [_coupl(t),|n_2⟩⟨m_2|(t)]]dt^'(0)> -<[_coupl(t),|n_1⟩⟨m_1|(t)](0) ∫_-∞^t[_coupl(t^'),|n_2⟩⟨m_2|(t)]dt^'(0)> -<∫_-∞^t[_coupl(t^'),|n_1⟩⟨m_1|(t)]dt^'(0) [_coupl(t),|n_2⟩⟨m_2|(t)](0)>,where only the terms up to the second order in the system-environment coupling have been kept and the lower time integration limit has been extended to -∞ in view of the short correlation time of the environment degrees of freedom, i.e. the correlation time between _coupl(t^')∝(t^') and _coupl(t)∝(t). Using Eq. (<ref>), we derive below the master equations for the evolution of the OTOCs in the form (<ref>).Due to the weakness of the system-environment coupling, the characteristic energy gaps between system levels significantly exceed the decay rates of the OTOCs, which are determined by the last four lines in Eq. (<ref>); the elements K_m_1m_2,n_1n_2 quickly oscillate with frequencies E_n_1+E_n_2-E_m_1-E_m_2 and decay with rates significantly exceeded by these frequencies. Thus, the evolution of each element K_m_1m_2,n_1n_2 depends only on other elements corresponding to the same energy splitting E_n_1+E_n_2-E_m_1-E_m_2. Below we consider separately the cases of finite and zero values of the splitting.§.§.§ Finite energy splitting For each combination of different m_1, m_2, n_1 and n_2 there are four elements K which correspond to the same energy splitting and differ from each other by permutations of indices. We assume for simplicity that there is no additional degeneracy of the quantities E_n_1+E_n_2-E_m_1-E_m_2 when all of the indices m_1, m_2, n_1 and n_2 are different. Eq. (<ref>) in that case gives∂_t K_m_1m_2,n_1n_2= i(E_n_1+δ E_n_1+E_n_2+δ E_n_2 -E_m_1-δ E_m_1-E_m_2-δ E_m_2)K_m_1m_2,n_1n_2 -1/2(∑_k≠ m_1Γ_m_1→ k^rel +∑_k≠ n_1Γ_n_1→ k^rel +∑_k≠ m_2Γ_m_2→ k^rel +∑_k≠ n_2Γ_n_2→ k^rel)K_m_1m_2,n_1n_2-Γ_n_2→ n_1^rel K_m_1m_2,n_2n_1 -Γ_m_1→ m_2^rel K_m_2m_1,n_1n_2 -Γ^ϕ_m_1n_1,m_2n_2 K_m_1m_2,n_1n_2,where the transition rates Γ_i→ j^rel are given by Eq. (<ref>); δ E_i is the renormalisation of the i-th level by environment, given by Eq. (<ref>); andΓ^ϕ_m_1n_1,m_2n_2=1/2(V_n_1n_1+V_n_2n_2-V_m_1m_1-V_m_2m_2)^2S(0)is the dephasing rate in a compound system consisting of two copies of the original system coupled to the same bath.§.§.§ Zero energy splitting Elements K_m_1m_2,n_1n_2 with zero splitting E_n_1+E_n_2-E_m_1-E_m_2 have a greater degeneracy and require separate analyses.“Diagonal” elements. Let us first consider the elements with n_1=m_1 and n_2=m_2. These elements satisfy the same equations of evolution as the diagonal elements of the density matrix of a compound system consisting of two copies of the original system. For n_1=n_2=n≠ m=m_1=m_2 we obtain from Eq. (<ref>)∂_t K_nm,nm= -K_nm,nm∑_k≠ nΓ_n→ k^rel -K_nm,nm∑_k≠ mΓ_m→ k^rel +∑_k≠ nΓ_k→ n^relK_km,km +∑_k≠ mΓ_k→ m^relK_nk,nk-Γ_m→ n^relK_nm,mn -Γ_n→ m^relK_mn,nm.In the case n=m Eq. (<ref>) gives∂_t K_nn,nn= -2K_nn,nn∑_k≠ nΓ_n→ k^rel +∑_k≠ n(Γ_k→ n^relK_kn,kn +Γ_k→ n^relK_nk,nk) +∑_k≠ n(Γ_k→ n^relK_nk,kn +Γ_n→ k^relK_kn,nk).From Eqs. (<ref>) and (<ref>) it follows immediately that∑_m,n K_mm,nn=const,which corresponds to the conservation of the sum of the diagonal elements of the density matrix of a compound system.“Non-diagonal” elements. The other set of elements with zero energy splitting, different from the “diagonal” elements, correspond to m_1=n_2 and m_2=n_1. Their evolution is described by the equations∂_t K_mn,nm= -(∑_k≠ mΓ_m→ k^rel+∑_k≠ nΓ_n→ k^rel)K_mn,nm -(K_mn,mn+K_nm,nm)Γ_m→ n^rel+∑_k≠ mK_kn,nkΓ_m→ k^rel +∑_k≠ nK_mk,kmΓ_k→ n^rel. §.§ Master equation for the density matrix for two copies of a system coupled to the same environment The equations for the evolution of the elements K_m_1m_2,n_1n_2 are similar to the equations of evolution of the density-matrix elements ρ_m_1m_2,n_1n_2=<|n_1n_2⟩⟨m_1m_2|(t)> of a compound system consisting of two copies of the original system coupled to the same environment, where n_i and m_i label the states of the i-th subsystem; i=1,2. The Hamiltonian of such a compound system is given by Eq. (<ref>). To the second order in the system-environment couplingthe evolution of the density matrix elements is described by the equation∂_t ρ_m_1m_2,n_1n_2= i(E_n_1+E_n_2-E_m_1-E_m_2)ρ_m_1m_2,n_1n_2-<∫_-∞^t[(t^')(t^')⊗ + ⊗(t^')(t^'), [(t)(t)⊗ + ⊗(t)(t), |n_1n_2⟩⟨m_1m_2|(t)]]dt^'>,The form of the coupling =∑_n,mV_nm|n⟩⟨m| and Eq. (<ref>) give, when all of the indices n_1, n_2, m_1 and m_2 are different,∂_t ρ_m_1m_2,n_1n_2= i(E_n_1+δ E_n_1+E_n_2+δ E_n_2 -E_m_1-δ E_m_1-E_m_2-δ E_m_2)ρ_m_1m_2,n_1n_2 -1/2(∑_k≠ m_1Γ_m_1→ k^rel +∑_k≠ n_1Γ_n_1→ k^rel +∑_k≠ m_2Γ_m_2→ k^rel +∑_k≠ n_2Γ_n_2→ k^rel)ρ_m_1m_2,n_1n_2-1/2(Γ_n_2→ n_1^rel+Γ_n_1→ n_2^rel +iE_n_1n_2^flip)ρ_m_1m_2,n_2n_1-1/2(Γ_m_2→ m_1^rel+Γ_m_1→ m_2^rel -iE_m_1m_2^flip)ρ_m_2m_1,n_1n_2-Γ^ϕ_m_1n_1,m_2n_2ρ_m_1m_2,n_1n_2,where the quantityE_n_1n_2^flip=|V_n_1n_2|^2∫dω/2πS(ω)-S(-ω)/ω+E_n_1n_2gives the rate of the flip-flop processes, i.e. the rate of the coherent interchange n_1 ↔ n_2, and the dephasing rate Γ^ϕ_m_1n_1,m_2n_2 is defined by Eq. (<ref>).Lindblad form. The master equations for the evolution of the density matrix of two systems in the same environment may may be also rewritten in the Lindblad form (<ref>) with the effective Hamiltonian_eff=∑_i |i⟩⟨i|[ (E_i+δ E_i)⊗ +⊗(E_i+δ E_i) ] +1/2∑_i,jE_ij^flip|i⟩⟨j|⊗|j⟩⟨i|and the Lindblad operators_ij=(1-δ_ij)√(Γ_j→ i^rel)(|i⟩⟨j|⊗+⊗|i⟩⟨j|) +δ_ij√(S(0)/N)∑_l V_ll(|l⟩⟨l|⊗+⊗|l⟩⟨l|).Mapping between OTOCs and two-system density matrix. Eq. (<ref>), which described the evolution of OTOCs for an open system in a dissipative environment, resembles Eq. (<ref>), which describes the evolution of the density matrix elements for two copies of the system coupled to this environment. Indeed, both equations have the same diagonal part, i.e. the part which relates the evolution of the element ρ_m_1m_2,n_1n_2 or K_m_1m_2,n_1n_2 to itself. Both equations alsohave terms with interchanged indices n_1↔ n_2 or m_1↔ m_2. While two systems coupled to an environment allow for a coherent (“flip-flop”) as well as inelastic interchange, the respective processes for OTOCs are purely inelastic.As discussed in the main text, in the limit of a classical environment the evolutions of the OTOC and two systems coupled to this environment may be mapped onto each other.Classical environment corresponds to the odd spectrum S(ω)=S(-ω), which leads to the vanishing of the flip-flop rates (<ref>) and identical relaxation ratesΓ_n_1→ n_2^rel=Γ_n_2→ n_1^rel of the transitionsn_1→ n_2 and n_2→ n_1 for each pair of states n_1 and n_2. The equations (<ref>) and (<ref>) for the evolution of the OTOC and the two systems become identical in this limit. | http://arxiv.org/abs/1704.08442v1 | {
"authors": [
"S. V. Syzranov",
"A. V. Gorshkov",
"V. Galitski"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci",
"cond-mat.quant-gas",
"cond-mat.str-el",
"quant-ph"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170427060922",
"title": "Out-of-time-order correlators in finite open systems"
} |
Asymptotical properties of social network dynamics on time scales Aleksey Ogulenko Accepted 2017 XXX. Received 2017 Apr; in original form 2016 Feb =================================================================== We present a construction of algebras of generalized functions of Colombeau-type which, instead of asymptotic estimates with respect to a regularization parameter, employs only topological estimates on certain spaces of kernels for its definition. MSC2010 Classification: 46F30, 46F05Keywords: Nonlinear generalized functions, Colombeau algebras, asymptotic estimates, elementary Colombeau algebra, diffeomorphism invariance§ INTRODUCTION Colombeau algebras, as introduced by J. F. Colombeau <cit.>, today represent the most widely studied approach to embedding the space of Schwartz distributions into an algebra of generalized functions such that the product of smooth functions as well as partial derivatives of distributions are preserved. These algebras have found numerous applications in situations involving singular objects, differentiation and nonlinear operations (see, e.g., <cit.>).All constructions of Colombeau algebras so far incorporate certain asymptotic estimates for the definition of the spaces of moderate and negligible functions, the quotient of which constitutes the algebra. There is a certain degree of freedom in the asymptotic scale employed for these estimates; while commonly a polynomial scale is used, generalizations in several directions are possible. For an overview we refer to works on asymptotic scales <cit.>, (, , )-algebras <cit.>, sequences spaces with exponent weights <cit.> and asymptotic gauges <cit.>.In this article we will present an algebra of generalized functions which instead of asymptotic estimates employs only topological estimates on certain spaces of kernels for its definition. This is a direct generalization of the usual seminorm estimates valid for distributions.We will first develop the most general setting in the local scalar case, namely that of diffeomorphism invariant full Colombeau algebras. We will then derive a simpler variant, similar to Colombeau's elementary algebra. Finally, we give canonical mappings into the most important Colombeau algebras, which points to a certain universality of the construction offered here. § PRELIMINARIESand _0 denote the sets of positive and non-negative integers, respectively, and ^+ the set of nonnegative real numbers. Concerning distribution theory we use the notation and terminology of L. Schwartz <cit.>.Given any subsets K,L ⊆^n (with n ∈) the relation KL means that K is compact and contained in the interior L^∘ of L.Let Ω⊆^n be open. C^∞(Ω) is the space of complex-valued smooth functions on Ω. For any K,L Ω, m,l ∈_0 and any bounded subset B ⊆ C^∞(Ω) we setf_K,m sup_x ∈K, α ≤m ^αf(x) (f ∈C^∞(Ω)), φ⃗_K,m; L, l sup_x ∈K, α ≤m y ∈L, β ≤l_x^α_y^βφ⃗(x)(y)(φ⃗∈C^∞(Ω, (Ω))), φ⃗_K,m; B sup_x ∈K, α ≤m f ∈B⟨f(y), _x^αφ⃗(x)(y) ⟩ (φ⃗∈C^∞(Ω, '(Ω))).Note that ·_K,m, ·_K,m; L, l and ·_K,m; B are continuous seminorms on the respective spaces.We define δ⃗∈ C^∞(Ω, '(Ω)) by δ⃗(x) δ_x for x ∈Ω, where δ_x is the delta distribution at x._L(Ω) is the space of test functions on Ω with support in L. For two locally convex spaces E and F, (E,F) denotes the space of linear continuous mappings from E to F, endowed with the topology of bounded convergence. By _x(Ω) we denote the filter base of open neighborhoods of a point x in Ω, and by _K(Ω) the filter base of open neighborhoods of K. By (E) we denote the set of continuous seminorms of a locally convex space E. B_r(x) { y ∈^n : y-x < r } is the open Euclidean ball of radius r>0 at x ∈^n.Our notion of smooth functions between arbitrary locally convex spaces is that of convenient calculus <cit.>. In particular, ^k f denotes the k-th differential of a smooth mapping f.§ CONSTRUCTION OF THE ALGEBRA Throughout this section let Ω⊆^n be a fixed open set. Letbe the category of locally convex spaces with smooth mappings in the sense of convenient calculus as morphisms.Consider C^∞(, (Ω)) and C^∞() as sheaves with values in . We define the basic space of nonlinear generalized functions on Ω to be the set of sheaf homomorphisms(Ω)( C^∞(, (Ω)), C^∞()).Hence, an element of (Ω) is given by a family (R_U)_U of mappingsR_U ∈ C^∞( C^∞(U, (Ω)), C^∞(U)) (U ⊆Ω open)satisfying R_U(φ⃗)|_V = R_V(φ⃗|_V) for all open subsets V ⊆ U and φ⃗∈ C^∞(U, (Ω)). We will casually write R in place of R_U.The basic space (Ω) can be identified with the set of all mappings R ∈ C^∞( C^∞(Ω, (Ω)), C^∞(Ω)) such that for any open subset U ⊆Ω and φ⃗, ψ⃗∈ C^∞(Ω, (Ω)) the equality φ⃗|_U = ψ⃗|_U implies R(φ⃗)|_U = R(ψ⃗)|_U (cf. <cit.>).(Ω) is a C^∞(Ω)-module with multiplication(f · R)_U(φ⃗) = f|_U · R_U(φ⃗)for R ∈(Ω), f ∈ C^∞(Ω), U ⊆Ω open and φ⃗∈ C^∞(U, (Ω)). Moreover, it is an associative commutative algebra with product (R · S)_U(φ⃗)R_U(φ⃗) · S_U(φ⃗).A distribution u ∈'(Ω) defines a sheaf morphism from C^∞(, (Ω)) to C^∞(). In fact, for U ⊆Ω open and φ⃗∈ C^∞(U, (Ω)) the function x ↦⟨ u, φ(x) ⟩ is an element of C^∞(U) (see <cit.> or <cit.>). More abstractly, this can be seen using the theory of topological tensor products <cit.> as follows:C^∞(U, (Ω)) ≅ C^∞(U) ⊗(Ω) ≅ ( '(Ω), C^∞(U)),where C^∞(U) ⊗(Ω) denotes the completed projective tensor product of C^∞(U) and (Ω). The assignment φ⃗↦⟨ u, φ⃗⟩ is smooth, being linear and continuous <cit.>. Hence, we have the following embeddings of distributions and smooth functions into (Ω): We define ι'(Ω) →(Ω) and σ C^∞(Ω) →(Ω) by (ιu)(φ⃗)(x) ⟨u, φ⃗(x) ⟩ (u ∈'(Ω))(σf)(φ⃗)(x) f(x)(f ∈C^∞(Ω))for φ⃗∈ C^∞(U, (Ω)) with U ⊆Ω open and x ∈ U.Clearly ι is linear and σ is an algebra homomorphism. Directional derivatives on (Ω) then are defined as follows:Let X ∈ C^∞(Ω, ^n) be a smooth vector field and R ∈(Ω). We define derivatives _X (Ω) →(Ω) and _X (Ω) →(Ω) by(_X R)(φ⃗)_X ( R_U ( φ⃗)) (_X R)(φ⃗) -R_U ( φ⃗) (_X φ⃗) + _X ( R_U ( φ⃗)) for φ⃗∈ C^∞(U, (Ω)) with U ⊆Ω open, where we set_X φ⃗ _X φ⃗+ _X^w ∘φ⃗. Here, (_X φ⃗)(x) is the directional derivative of φ⃗ at x in direction X(x) and (_X^ω∘φ⃗)(x) is the Lie derivative of φ⃗(x) considered as a differential form, given by _X^ω ( φ⃗(x)) = _X ( φ⃗(x)) + ( X)(x) ·φ⃗(x).Note that both _X and _X satisfy the Leibniz rule. We have_x ∘σ = σ∘_X, _X ∘σ = σ∘_X, _X ∘ι = ι∘_X.While _X is C^∞(Ω)-linear in X, _X is only -linear in X. We refer to <cit.> for a discussion of the role of these derivatives in differential geometry.For k ∈_0 we set_k^+ [y_0, …, y_k],_k{λ∈^+ [y_0, …, y_k, z_0, …, z_k] | λ(y_0, …, y_k, 0, …, 0) = 0 }.More explicitly, _k is the commutative semiring of polynomials in the k+1 commuting variables y_0, …, y_k with coefficients in ^+. Similarly, _k is the commutative semiring in the 2(k+1) commuting variables y_0, …, y_k, z_0, …, z_k with coefficients in ^+ and such that, if λ∈_k is given by the finite sumλ = ∑_α,β∈_0^k+1λ_αβ y^α z^β,then λ_α 0 = 0 for all α. Note that _k is a subsemiring of _k+1 and _k a subsemiring of _k+1. Furthermore, _k is an ideal in _k if _k is considered as a subsemiring of ^+ [y_0, …, y_k, z_0, …, z_k]. Given λ∈_k and y_i ≤ y_i' for i=0 … k we have λ(y) ≤λ(y'). For λ, μ∈_k we write λ≤μ if λ(y) ≤μ(y) for all y ∈ (^+)^k+1, and similarly for λ, μ∈_k.We can now formulate the following definitions of moderateness and negligibility, not involving any asymptotic estimates:An element R ∈(Ω) is called moderate if(∀ x ∈Ω) (∃ U ∈_x(Ω)) (∀ K,LU) (∀ m,k ∈_0)(∃ c,l ∈_0) (∃λ∈_k)(∀φ⃗_0,…,φ⃗_k ∈ C^∞(U, _L(U))): ^k R(φ⃗_0)(φ⃗_1,…,φ⃗_k)_K, m≤λ ( φ⃗_0_K,c; L, l, …, φ⃗_k_K,c; L, l).The subset of all moderate elements of (Ω) is denoted by (Ω). An element R ∈(Ω) is called negligible if(∀ x ∈Ω) (∃ U ∈_x(Ω)) (∀ K,LU) (∀ m,k ∈_0) (∃ c,l ∈_0)(∃λ∈_k) (∃ B ⊆ C^∞(Ω) bounded) (∀φ⃗_0, …, φ⃗_k ∈ C^∞(U, _L(U))): ^kR(φ⃗_0)(φ⃗_1,…,φ⃗_k)_K, m ≤λ ( φ⃗_0_K,c; L, l, …, φ⃗_k_K,c; L, l, φ⃗_0 - δ⃗_K, c; B, φ⃗_1_K, c; B, …, φ⃗_k_K, c; B).The subset of all negligible elements of (Ω) is denoted by (Ω). It is worthwile to discuss possible simplifications of these definitions, which at this stage should be considered more as a proof of concept than as the definite form they should have. First, we note that we cannot replace (∀ x ∈Ω) (∃ U ∈_x(Ω)) (∀ K,LU) by (∀ K,L Ω). In fact, in the second case K and L can be distant from each other, while in the first case it suffices to control the situation where K and L are close to each other. However, the following result shows that we can always assume KL and that the φ⃗_0,…,φ⃗_k are given merely on an arbitrary open neighborhood of K, i.e., as elements of the direct limit C^∞(K, _L(Ω)) _V ∈_K(Ω) C^∞(V, _L(Ω)): Let R ∈(Ω). Then R is moderate if and only if(∀ x ∈Ω) (∃ U ∈_x(Ω)) (∀ K,LU: KL) (∀ m,k ∈_0)(∃ c,l ∈_0) (∃λ∈_k)(∀φ⃗_0,…,φ⃗_k ∈ C^∞(K, _L(U))): ^k R(φ⃗_0)(φ⃗_1,…,φ⃗_k)_K, m≤λ ( φ⃗_0_K,c; L, l, …, φ⃗_k_K,c; L, l).Similarly, R is negligible if and only if(∀ x ∈Ω) (∃ U ∈_x(Ω)) (∀ K,LU: KL) (∀ m,k ∈_0) (∃ c,l ∈_0)(∃λ∈_k) (∃ B ⊆ C^∞(U) bounded) (∀φ⃗_0, …, φ⃗_k ∈ C^∞(K, _L(U))): ^kR(φ⃗_0)(φ⃗_1,…,φ⃗_k)_K, m ≤λ ( φ⃗_0_K,c; L, l, …, φ⃗_k_K,c; L, l, φ⃗_0 - δ⃗_K, c; B, φ⃗_1_K, c; B, …, φ⃗_k_K, c; B).Obviously each of these conditions is weaker than the corresponding one of <Ref> or <Ref>.Suppose we are given R ∈(Ω) such that the condition stated for moderateness holds. Given x ∈Ω there hence exists some U ∈_x(Ω). Now given arbitrary K, LU we choose a set L'U such that K ∪ LL'. Fixing m, k ∈_0 for the moderateness test, for (K,L') we hence obtain c,l ∈_0 and λ∈_k. Now fix some φ⃗_0, …, φ⃗_k ∈ C^∞(U, _L(U)); each of those represents an element of C^∞(K, _L'(U)), whence we have the estimate^k R(φ⃗_0)(φ⃗_1, …, φ⃗_k)_K,m ≤λ ( φ⃗_0_K,c; L', l, …, φ⃗_k_K,c; L', l)= λ ( φ⃗_0_K,c; L, l, …, φ⃗_k_K,c; L, l).where the last equality follows because the φ⃗_0,…,φ⃗_k take values in _L(U). This shows that R is moderate.For the case of negligibility we proceed similarly until we obtain c,l ∈_0, λ∈_k and B ⊆ C^∞(U). Let χ∈(U) be such that χ≡ 1 on a neighborhood of L' and set B' {χ f | f ∈ B }⊆ C^∞(Ω), which is bounded. For any φ⃗_0, …, φ⃗_k we then obtain^k R(φ⃗_0)(φ⃗_1, …, φ⃗_k)_K,m≤ ≤λ ( φ⃗_0_K,c; L', l, …, φ⃗_k_K,c; L', l, φ⃗_0 - δ⃗_K, c; B, φ⃗_1_K, c; B, …, φ⃗_k_K, c; B) = λ ( φ⃗_0_K,c; L, l, …, φ⃗_k_K,c; L, l, φ⃗_0 - δ⃗_K, c; B', φ⃗_1_K, c; B', …, φ⃗_k_K, c; B')which proves negligibility of R.If the test of <Ref>, <Ref> or <Ref> holds on some U then clearly it also holds on any open subset of U. The following characterization of moderateness and negligiblity is obtained by applying polarization identities to the differentials of R:Let R ∈(Ω).*R is moderate if and only if(∀ x ∈Ω) (∃ U ∈_x(Ω)) (∀ K,LU) (∀ m,k ∈_0)(∃ c,l ∈_0) (∃λ∈_min(1,k))(∀φ⃗, ψ⃗∈ C^∞(U, _L(U))): ^k R(φ⃗)(ψ⃗,…,ψ⃗)_K, m≤{ λ( φ⃗_K,c; L, l ) if k = 0, λ( φ⃗_K,c; L, l, ψ⃗_K,c; L, l ) if k ≥ 1. .*R is negligible if and only if(∀ x ∈Ω) (∃ U ∈_x(Ω)) (∀ K,LU) (∀ m,k ∈_0) (∃ c,l ∈_0)(∃λ∈_min(1,k)) (∃ B ⊆ C^∞(Ω) bounded) (∀φ⃗, ψ⃗∈ C^∞(U, _L(U))): ^kR(φ⃗)(ψ⃗,…,ψ⃗)_K, m ≤{ λ( φ⃗_K,c; L, l, φ⃗- δ⃗_K, c; B ) if k = 0, λ( φ⃗_K,c; L, l, ψ⃗_K,c; L, l, φ⃗- δ⃗_K, c; B, ψ⃗_K, c; B) if k ≥ 1. .We assume k ≥ 1, as for k=0 the statements are identical.<ref> “⇒”: One obtains λ∈_k such that^k R(φ⃗)(ψ⃗, …, ψ⃗)_K,m ≤λ ( φ⃗_K,c; L, l, ψ⃗_K,c; L, l, …, ψ⃗_K,c; L ,l)= λ' ( φ⃗_K,c;L,l, ψ⃗_K,c;L, l)with λ' ∈_1 given by λ'(y_0, y_1) = λ(y_0, y_1, …, y_1).“⇐”: One obtains λ∈_1. We then use the polarization identity <cit.>^k R (φ⃗_0)(φ⃗_1, …, φ⃗_k) = 1/n!∑_a=1^k (-1)^k-a∑_J ⊆{1 … k} J = aΔ^*(^k R(φ⃗_0))(S_J)where S_J ∑_i ∈ Jφ⃗_i and we have set Δ^*(^k R ( φ⃗_0))(ψ⃗) = ^k R (φ⃗_0)(ψ⃗, …, ψ⃗). Hence,^k R(φ⃗_0)(φ⃗_1, …, φ⃗_k)_K,m≤1/n!∑_a=1^k ∑_J = aΔ^*(^k R(φ⃗_0))(S_J)_K,m ≤1/n!∑_a=1^k ∑_J = aλ ( φ⃗_0_K,c; L, l, S_J_K,c;L,l)≤1/n!∑_a=1^k ∑_J = aλ ( φ⃗_0_K,c;L,l, ∑_i ∈ Jφ⃗_i_K,c;L,l) = λ' ( φ⃗_0_K,c;L,l, …, φ⃗_k_K,c;L,l)with λ' ∈_k given byλ'(y_0, …, y_k) = 1/n!∑_a=1^k ∑_J = aλ ( y_0, ∑_i ∈ J y_i ). <ref> “⇒”: We have λ∈_k such that^k R(φ⃗)(ψ⃗, …, ψ⃗)_K,m≤λ ( φ⃗_K,c; L, l, ψ⃗_K,c;L,l, …, ψ⃗_K,c;L,l, φ⃗- δ⃗_K,c;B,ψ⃗_K,c;B, …, ψ⃗_K,c;B) = λ' ( φ⃗_K,c;L;l, ψ⃗_K,c;L,l, φ⃗- δ⃗_K,c;B, ψ⃗_K,c;B)with λ' ∈_k given byλ' ( y_0, y_1, z_0, z_1) = λ(y_0, y_1, …, y_1, z_0, z_1, …, z_1). “⇐”: We obtain λ∈_1 such that, as above,^k R(φ⃗_0)(φ⃗_1, …, φ⃗_k)_K,m ≤1/n!∑_a=1^k ∑_J=aλ ( φ⃗_0_K,c;L,l, S_J_K,c;L,l, φ⃗_0 - δ⃗_K,c; B, S_J_K,c; B)≤1/n!∑_a=1^k ∑_J=aλ ( φ⃗_0_K,c;L,l, ∑_i ∈ Jφ⃗_i_K,c;L,l, φ⃗_0 - δ⃗_K,c;B, ∑_i ∈ Jφ⃗_i_K,c;B) = λ' ( φ⃗_0_K,c;L,l, …, φ⃗_k_K,c;L,l, φ⃗_0 - δ⃗_K,c;B, φ⃗_1_K,c;B, …, φ⃗_k_K,c;B)with λ' ∈_k given byλ'(y_0, …, y_k, z_0, …, z_k) = 1/n!∑_a=1^k ∑_J=aλ(y_0, ∑_i ∈ Jy_i, z_0, ∑_i ∈ Jz_i). Note that the polarization identities could be applied also in the formulation of <Ref>.(Ω) ⊆(Ω). Let R ∈(Ω) and fix x ∈Ω for the moderateness test. By negligibility of R there exists U ∈_x(Ω) as in <Ref>. Let K,LU and m,k ∈_0 be arbitrary. Then there exist c,l,λ and B such that the estimate of <Ref> holds. We know that λ∈_k is given by a finite sumλ(y_0, …, y_k, z_0, …, z_k) = ∑_α, βλ_αβ y^α z^β.It suffices to show that there are λ_1, λ_2 ∈_0 such that for any φ⃗∈ C^∞(U, _L(U)) we have the estimatesφ⃗- δ⃗_K,c; B ≤λ_1 ( φ⃗_K,c; L, l ),φ⃗_K,c; B ≤λ_2 ( φ⃗_K,c; L, l ).In fact, these inequalities imply^k R (φ⃗_0) (φ⃗_1, …, φ⃗_k)_K,m≤∑_α,βλ_αβφ⃗_0^α_0_K,c; L, l·…·φ⃗_k^α_k_K,c; L, l·φ⃗_0 - δ⃗^β_0_K,c; B·φ⃗_1^β_1_K,c; B·…·φ⃗_k^β_k_K,c; B≤∑_α,βλ_αβφ⃗_0^α_0_K,c; L, l·…·φ⃗_k^α_k_K,c; L, l·λ_1 ( φ⃗_0_K,c; L, l ) ^β_0·λ_2 ( φ⃗_1_K,c; L, l )^β_1…λ_2 ( φ⃗_k_K,c; L, l)^β_k = λ' ( φ⃗_0_K,c; L, l, …, φ⃗_k_K,c; L, l )with λ' ∈_k given byλ'(y_0, …, y_k) = ∑λ_αβ y^αλ_1(y_0)^β_0λ_2(y_1)^β_1…λ_2(y_k)^β_k.Inequality (<ref>) is seen as follows:φ⃗- δ⃗_K,c; B = sup_x ∈ K, α≤ c f ∈ B∫_L f(y)_x^αφ⃗(x)(y)y - ^α f(x) ≤L·sup_f ∈ Bf_L,0·φ⃗_K,c; L, l + sup_f ∈ Bf_K,c = λ_1 ( φ⃗_K,c; L, l)with λ_1(y_0) = L·sup_f ∈ Bf_L,0· y_0 + sup_f ∈ Bf_K,c, where L denotes the Lebesgue measure of L. Similarly, inequality (<ref>) results fromφ⃗_K,c; B= sup_x ∈ K, α≤ c f ∈ B∫_L f(y) _x^αφ⃗(x)(y)y ≤L·sup_f ∈ Bf_L,0·φ⃗_K,c; L, l= λ_2 ( φ⃗_K,c; L, l)with λ_2(y_0) = L·sup_f ∈ Bf_L,0· y_0.(Ω) is a subalgebra of (Ω) and (Ω) is an ideal in (Ω). This is evident from the definitions. Let u ∈'(Ω) and f ∈ C^∞(Ω). Then *ι u is moderate,*σ f is moderate,*ι f - σ f is negligible, and*if ι u is negligible then u=0.<ref>: Fix x for the moderateness test and let U ∈_x(Ω) be arbitrary. Fix any K,LU and m ∈_0. Then there are constants C = C(L) ∈^+ and l = l(L) ∈_0 such that ⟨ u, φ⟩≤ C φ_L,l for all φ∈_L(Ω). Hence, we see that(ι u)(φ⃗_0)_K,m = ⟨ u, φ⃗_0 ⟩_K,m = sup_x ∈ K, α≤ m⟨ u, _x^αφ⃗_0(x) ⟩ ≤ C ·sup_x ∈ K, α≤ m y ∈ L, β≤ l_x^α_y^βφ⃗_0(x)(y) = C φ⃗_0_K,m; L, l = λ ( φ⃗_0_K,m; L, l).with λ ( y_0 ) = C y_0. Moreover, we have( ι u )(φ⃗_0)(φ⃗_1)_K,m≤ C φ⃗_1_K,m; L, l = λ ( φ⃗_0_K,m; L, l, φ⃗_1_K,m; L, l)with λ(y_0, y_1) = C y_1. Higher differentials of ι u vanish and the moderateness test is satisfied with λ = 0 for k ≥ 2.<ref>: Fix x and let U ∈_x(Ω) be arbitrary. For any K,LU and m ∈_0 we have ( σ f)(φ⃗_0) _K,m = f_K, m = λ(φ⃗_0_K, 0; L, 0)with λ(y_0) = f_K,m. Differentials of σ f vanish, i.e., λ = 0 for k≥ 1.<ref>: Fix x and let U ∈_x(Ω) be arbitrary. For any K,LU and m,k ∈_0 we have(ι f - σ f)(φ⃗_0)= ⟨ f, φ⃗_0 - δ⃗⟩, ( ι f - σ f)(φ⃗_0)(φ⃗_1)= ⟨ f, φ⃗_1 ⟩,^k ( ι f - σ f)(φ⃗_0)(φ⃗_1, …, φ⃗_k)= 0for k≥ 2.Hence, with c = m, l = 0 and B = { f } the negligibility test is satisfied with λ(y_0, z_0) = z_0 for k=0, λ(y_0, y_1, z_0, z_1) = z_1 for k=1 and λ=0 for k ≥ 2.<ref>: We show that every point x ∈Ω has an open neighborhood V such that u|_V = 0, which implies u=0.Given x ∈Ω, let U ∈_x(Ω) be as in the characterization of negligibility in <Ref>. Choose an open neighborhood V of x such that K V U and r>0 such that L B_r(K) U. With k=m=0, <Ref> gives c,l ∈_0, λ∈_0 and B ⊆ C^∞(U), where λ has the formλ(y,z) = ∑_α∈_0^n, β∈λ_αβ y^α z^β.Choose φ∈(^n) with φ⊆ B_1(0), ∫φ(x) x = 1 and ∫ x^γφ(x) x = 0 for γ∈_0^n with 0 < γ≤ q, where q is chosen such that β(q+1) > α(n+c+l) for all α,β with λ_αβ 0 (e.g., take q = (n+c+l) _y λ, where _y λ is the degree of λ with respect to y). For >0 set φ_(y) = ^-nφ(y/). Then for <r, φ⃗_(x)(y) φ_(y-x) defines an element φ⃗_∈ C^∞(K, _L(Ω)) because φ_(.-x) = x + φ_⊆ B_(x) ⊆ B_r(K) ⊆ L for x ∈ B_r-(K). Consequently, we have(ι u)(φ⃗_)_K,0≤λ ( φ⃗__K,c; L, l, φ⃗_ - δ⃗_K,c; B).Because of the estimatesφ⃗__K,c; L, l= O(^-(n+l+c))φ⃗_ - δ⃗_K,c;B= O(^q+1),which may be verified by a direct calculation, we have(ι u)(φ⃗_)_K,0≤∑_α, βλ_α, β· O(^-α(n+c+l)) · O(^β(q+1)) → 0by the choice of q, which means that (ι u)(φ⃗_)|_V → 0 in C(V) and hence also in '(V). On the other hand, we have⟨ u, φ⃗_⟩|_V → u|_Vin '(V), as is easily verified. This completes the proof. For X ∈ C^∞(Ω, ^n) we have*_X ( (Ω)) ⊆(Ω) and _X ( (Ω)) ⊆(Ω),*_X ( (Ω)) ⊆(Ω) and _X ( (Ω)) ⊆(Ω). The claims for _X are clear because^k (_X R)(φ⃗)(ψ⃗, …, ψ⃗)_K,m = _X(^k R (φ⃗)(ψ⃗, …, ψ⃗))_K,m ≤ C ^k R(φ⃗)(ψ⃗, …, ψ⃗)_K,m+1for some constant C depending on X. As to _X, we have to deal with terms of the form^k+1R(φ⃗)(_Xφ⃗, ψ⃗, …, ψ⃗)and^kR(φ⃗)(_Xψ⃗, ψ⃗, …, ψ⃗)for which we use the estimate_Xφ⃗_K,c; L,l≤ C φ⃗_K,c,+1; L, l+1for some constant C depending on X. We now come to the quotient algebra.We define the Colombeau algebra of generalized functions on Ω by (Ω) (Ω) / (Ω). (Ω) is a C^∞(Ω)-module and an associative commutative algebra with unit σ(1). ι is a linear embedding of '(Ω) and σ an algebra embedding of C^∞(Ω) into (Ω) such that ι f = σ f in (Ω) for all smooth functions f ∈ C^∞(Ω). Furthermore, the derivatives _X and _X are well-defined on (Ω).Finally, we establish sheaf properties of . Note that for Ω' Ω open, the restriction R|_Ω'(φ⃗)R(φ⃗) is well-defined because for U ⊆Ω' open we have C^∞(U, (Ω')) ⊆ C^∞(U, (Ω)).Let R ∈(Ω) and Ω' ⊆Ω be open. If R is moderate then R|_Ω' is moderate; if R is negligible then R|_Ω' is negligible.Suppose that R ∈(Ω). Fix x ∈Ω', which gives U ∈_x(Ω). Set U'U ∩Ω' ∈_x(Ω') and let K,LU' and m,k ∈_0 be arbitrary. Then there are c,l,λ as in <Ref>. Let now φ⃗_0', …, φ⃗_k' ∈ C^∞(U', _L(U')) be given. Choose ρ∈(U') such that ρ≡ 1 on a neighborhood of K. Then ρ·φ⃗'_i ∈ C^∞(U, _L(U)) (i=0 … k) and^k R|_Ω' (φ⃗_0')(φ⃗_1', …,φ⃗_k')_K, m = ^k R|_Ω' ( ρφ⃗_0')(ρφ⃗'_1, …, ρφ⃗'_k)_K, m= ^k R(ρφ⃗'_0)(ρφ⃗'_1,…,ρφ⃗'_k)_K,m≤λ ( ρφ⃗_0'_K,c; L, l, …, ρφ⃗_k'_K,c; L, l) = λ ( φ⃗_0'_K,c; L, l, …, φ⃗_k'_K,c; L, l).Hence, the moderateness test is satisfied for R|_Ω'.Now suppose that R ∈(Ω). For the negligibility test fix x ∈Ω', which gives U ∈_x(Ω). Set U'U ∩Ω' and let K,LU' and m,k ∈_0 be arbitrary. Then ∃ c,l,B,λ as in <Ref>. Let now φ⃗_0', …, φ⃗_k' ∈ C^∞(U', _L(U')) be given. Choose ρ∈(U') such that ρ≡ 1 on a neighborhood of K. Then ρ·φ⃗'_i ∈ C^∞(U, _L(U)) (i=0 … k) and^k R|_Ω' (φ⃗_0')(φ⃗_1',…,φ⃗_k')_K, m = ^k R|_Ω' ( ρφ⃗_0')(ρφ⃗'_1, …, ρφ⃗'_k)_K, m = ^k R(ρφ⃗'_0)(ρφ⃗'_1,…,ρφ⃗'_k)_K,m ≤λ ( ρφ⃗_0'_K,c; L, l, …, ρφ⃗_k'_K,c; L, l, ρφ⃗_0' - δ⃗_K, c; B, …, ρφ⃗_k'_K,c; B) = λ ( φ⃗_0'_K,c; L, l, …, φ⃗_k'_K,c; L, l, φ⃗_0' - δ⃗_K,c; B, …, φ⃗_k'_K,c; B)which shows negligibility of R|_Ω'. () is a sheaf of algebras on Ω. Let X ⊆Ω be open and (X_i)_i be a family of open subsets of Ω such that ⋃_i X_i = X.We first remark that if R ∈(X) satisfies R|_X_i∈(X_i) for all i then R ∈(X), as is evident from the definition of negligibility.Suppose now that we are given R_i ∈(X_i) such that R_i|_X_i ∩ X_j - R_j|_X_i ∩ X_j∈(X_i ∩ X_j) for all i,j with X_i ∩ X_j ∅. Let (χ_i)_i be a partition of unity subordinate to (X_i)_i, i.e., a family of mappings χ_i ∈ C^∞(X) such that 0 ≤χ_i ≤ 1, (χ_i)_i is locally finite, ∑_i χ_i(x) = 1 for all x ∈ X and χ_i ⊆ X_i. Choose functions ρ_i ∈ C^∞(X_i, (X_i)) which are equal to 1 on an open neighborhood of the diagonal in X_i × X_i for each i. For V ⊆ X open and φ⃗∈ C^∞(V, (X)) we define R_V(φ⃗) ∈ C^∞(V) byR_V(φ⃗) ∑_i χ_i|_V · (R_i)_V ∩ X_i ( ρ_i|_V ∩ X_i·φ⃗|_V ∩ X_i).For showing smoothness of R_V consider a curve c ∈ C^∞(, C^∞(V, (X))). We have to show that t ↦ R_V(c(t)) is an element of C^∞(, C^∞(V)). By <cit.> it suffices to show that for each open subset W ⊆ V which is relatively compact in V the curve t ↦ R_V(c(t))|_W = R_W(c(t)|_W) is smooth, but this holds because the sum in (<ref>) then is finite. Hence, (R_V)_V ∈(Ω).Fix x ∈ X for the moderateness test. There is a finite index set F and an open neighborhood W ∈_x(X) such that W ∩χ_i ∅ implies i ∈ F. We can also assume that x ∈⋂_i ∈ FX_i. Let Y be a neighborhood of x such that ρ_i ≡ 1 on Y × Y for all i ∈ F. For each i ∈ F let U_i ∈_x(X_i) be obtained from moderateness of R_i as in <Ref>. Set U ⋂_i ∈ F U_i ∩ W ∩ Y ∈_x(X), and let K,LU as well as m,k ∈_0 be arbitrary. For each i ∈ F there are c_i, l_i,λ_i such that for any φ⃗_0, …, φ⃗_k ∈ C^∞(U, _L(U)) we have^k R_i ( φ⃗_0)(φ⃗_1, …, φ⃗_k)_K,m≤λ_i ( φ⃗_0_K,c_i; L, l_i, …, φ⃗_k_K,c_i; L, l_i).Now we have, for φ⃗∈ C^∞(U, _L(U)),R(φ⃗)|_W = ∑_i ∈ Fχ_i|_W · (R_i)_W ∩ X_i ( ρ_iφ⃗|_W ∩ X_i)and hence, for φ⃗_0, …, φ⃗_k ∈ C^∞(U, _L(U)),^k R ( φ⃗_0)(φ⃗_1, …, φ⃗_k)|_W= ∑_i ∈ Fχ_i|_W ·^k ((R_i)_W ∩ X_i) ( ρ_i φ⃗_0|_W ∩ X_i)(ρ_i φ⃗_1|_W ∩ X_i, …, ρ_i φ⃗_k|_W ∩ X_i).We see that^k R(φ⃗_0)(φ⃗_1, …, φ⃗_k)_K,m ≤∑_i ∈ F C(m) ·χ_i_K,m·λ_i ( φ⃗_0_K,c_i; L, l_i, …, φ⃗_k_K,c_i; L, l_i)= λ ( φ⃗_0_K,c; L, l, …, φ⃗_k_K,c; L, l)with c = max_j ∈ F c_j, l = max_j∈ F l_j, some constant C(m) coming from the Leibniz rule, and λ∈_k given byλ = ∑_i ∈ F C(m) χ_i_K,m·λ_i.This shows that R is moderate. Finally, we claim that R|_X_j - R_j ∈(X_j) for all j. For this we first note that(R|_X_j - R_j)(φ⃗) = ∑_i χ_i|_X_j· ( R_i ( ρ_iφ⃗|_X_i ∩ X_j) - R_j(φ⃗))for φ⃗∈ C^∞(X_j, (X_j)). Again, for x ∈ X_j there is a finite index set F and an open neighborhood W ∈_x(X) such that W ∩χ_i ∅ implies i ∈ F, and we can assume that x ∈⋂_i ∈ FX_i. Let Y be a neighborhood of x such that ρ_i ≡ 1 on Y × Y for all i ∈ F and let U_i ∈_x ( X_i ∩ X_j) be given by the negligibility test of R_i|_X_i ∩ X_j - R_j|_X_i ∩ X_j according to <Ref>. Set U ⋂_i ∈ F U_i ∩ W ∩ Y. Fix any K,LU and m,k ∈_0. For each i ∈ F there are c_i, l_i, λ_i, B_i such that for φ⃗_0, …, φ⃗_k ∈ C^∞(U, _L(U)) we have^k ( R_i|_X_i ∩ X_j - R_j|_X_i ∩ X_j)(φ⃗_0)(φ⃗_1,…,φ⃗_k)_K,m ≤λ_i ( φ⃗_0_K,c_i; L, l_i, …, φ⃗_0 - δ⃗_K, c_i; B_i, φ⃗_1_K,c_i; B_i, …, φ⃗_k_K,c_i; B_i).As above, we then have^k ( R|_X_j - R_j)(φ⃗_0)(φ⃗_1, …, φ⃗_k)_K,m ≤∑_i ∈ F C(m) ·χ_i_K,m·λ_i ( φ⃗_0_K,c_i; L, l_i, …, φ⃗_0 - δ⃗_K, c_i; B_i, φ⃗_1_K,c_i; B_i, …)≤λ ( φ⃗_0_K,c; L, l, …, φ⃗_0 - δ⃗_K, c; B, φ⃗_0_K,c; B, … )with c = max_i ∈ F c_i, l = max_i ∈ F l_i, B = ⋃_i∈ F B_i, and λ∈_k given byλ = ∑_i ∈ F C(m) χ_K,m·λ_i.This completes the proof. § AN ELEMENTARY VERSION We will now give a variant of the construction of <Ref> similar in spirit to Colombeau's elementary algebra <cit.>: if we only consider derivatives along the coordinate lines of ^n we can replace the smoothing kernels φ⃗∈ C^∞(U, _L(Ω)) by convolutions. This way, one can use a simpler basic space which does not involve calculus on infinite dimensional locally convex spaces anymore:Let Ω⊆^n be open. We setU(Ω) { ( φ, x) ∈(^n) ×Ω | φ + x ⊆Ω}.and define (Ω) to be the set of all mappings RU(Ω) → such that R(φ, ·) is smooth for fixed φ. Note that this is almost the basic space used originally by Colombeau (see <cit.> or <cit.>) but with (^n) in place of the space of test functions whose integral equals one. We now introduce a notation for the convolution kernel determined by a test function. For φ∈(^n) we define φ∈ C^∞(^n, (^n)) by φ(x)(y) φ(y-x). In fact, with this definition we have ⟨ u, φ⟩ = u * φ̌, where as usually we set φ̌(y) φ(-y). Furthermore, for c ∈_0 we writeφ_c sup_x ∈^n, α≤ c^αφ(x) (φ∈(^n)).The direct adaptation of <Ref> then looks as follows:Let R ∈(Ω). Then R is called moderate if(∀ x ∈Ω) (∃ U ∈_x(Ω)) (∀ K,LU: KL) (∀ m ∈_0)(∃ c ∈_0) (∃λ∈_0)(∀φ∈(^n): K + φ⊆ L):R(φ, .)_K, m≤λ ( φ_c).The subset of all moderate elements of (Ω) is denoted by (Ω).Similarly, R is called negligible if(∀ x ∈Ω) (∃ U ∈_x(Ω)) (∀ K,LU: KL) (∀ m ∈_0) (∃ c ∈_0)(∃λ∈_0) (∃ B ⊆ C^∞(U) bounded) (∀φ∈(^n): K + φ⊆ L): R(φ, .)_K, m≤λ ( φ_c, φ - δ⃗_K, c; B).The subset of all negligible elements of (Ω) is denoted by (Ω). It is convenient to work with the following simplification of these definitions.R ∈(Ω) is moderate if and only if (∀ K Ω) (∃ r>0: B_r(K)Ω) (∀ m ∈_0) (∃ c ∈_0)(∃λ∈_0)(∀φ∈(^n): φ⊆ B_r(0)):R(φ, .)_K, m≤λ ( φ_c).Similarly, R ∈(Ω) is negligible if and only if (∀ K Ω) (∃ r>0: B_r(K)Ω) (∀ m ∈_0) (∃ c ∈_0)(∃λ∈_0) (∃ B ⊆ C^∞(Ω) bounded) (∀φ∈(^n): φ⊆ B_r(0)): R(φ, .)_K, m≤λ ( φ_c, φ - δ⃗_K, c; B).Suppose R is moderate and fix K Ω. We can cover K by finitely many open sets U_i obtained from <Ref> and write K = ⋃_i K_i with K_iU_i. Choose r>0 such that L_i B_r(K_i) U_i for all i. Fixing m, by moderateness there exist c_i and λ_i for each i. Set c = max_i c_i and choose λ with λ≥λ_i for all i. Now given φ∈(^n) with φ⊆ B_r(0) we also have K_i + φ⊆ L_i and we can estimateR(φ, .)_K,m≤sup_i R(φ,.)_K_i, m≤sup_i λ_i ( φ_c_i ) ≤λ ( φ_c ). Conversely, suppose the condition holds and fix x ∈Ω for the moderateness test. Choose a>0 such that B_a(x)Ω. By assumption there is r>0 with B_r+a(x)Ω. Set UB_r/2(x). Then, fix KLU and m for the moderateness test. There are c and λ by assumption. Now given φ with K + φ⊆ L, we see that for y ∈φ and an arbitrary point z ∈ K we have y≤y+z-x + z-x < r, which means that φ⊆ B_r(0). But then R(φ,.)_K,m≤λ(φ_c) as desired.If R is negligible we proceed similarly until the choice of K_iL_iU_i and m gives c_i, λ_i and B_i. Choose χ_i ∈(U_i) with χ_i ≡ 1 on a neighborhood of L_i, and define B ⋃_i {χ_i f | f ∈ B_i }, which is bounded in C^∞(Ω). Then with c = max_i c_i and λ≥λ_i for all i we haveR(φ, .)_K,m≤sup_i λ_i ( φ_c_i, φ - δ⃗_K_i, c_i; B_i ) ≤λ ( φ_c, φ - δ⃗_K,c; B). The converse is seen as for moderateness by restricting the elements of B ⊆ C^∞(Ω) to U. The embeddings now take the following form.We define '(Ω) →(Ω) and C^∞(Ω) →(Ω) by (u)(φ, x) ⟨u, φ(.-x) ⟩ (u ∈'(Ω))(f)(φ, x) f(x)(f ∈C^∞(Ω)).Partial derivatives on (Ω) then can be defined via differentiation in the second variable: Let R ∈(Ω). We define derivatives _i (Ω) →(Ω) (i=1, …, n) by(_i R)(φ, x)/ x_i ( x ↦ R ( φ, x)).We have _i ( (Ω)) ⊆(Ω) and _i ( (Ω)) ⊆(Ω), This is evident from the definitions.We have _i ∘ι = ι∘_i and _i ∘σ = σ∘_i. _i ( ι u)(φ, x) = / x_i⟨ u(y), φ(y-x) ⟩ = ⟨ u(y), - (_i φ)(y-x) ⟩ = ⟨_i u(y), φ(y-x) ⟩ = ι( _i u)(φ, x). The second claim is clear. (Ω) ⊆(Ω). The result follows fromφ - δ⃗_K,c; B≤λ_1 ( φ_c_1 )for suitable λ_1 and c_1, which is seen as in the proof of <Ref>. Similarly to <Ref> we have:(Ω) is a subalgebra of (Ω) and (Ω) is an ideal in (Ω). Let u ∈'(Ω) and f ∈ C^∞(Ω). Then *u is moderate,*f is moderate,*f -f is negligible, and*if u is negligible then u=0.The proof is almost identical to that of <Ref> and hence omitted. We define the elementary Colombeau algebra of generalized functions on Ω by (Ω) (Ω) / (Ω). As before, one may show thatis a sheaf.§ CANONICAL MAPPINGS In this section we show that the algebraconstructed above is near to being universal in the sense that there exist canonical mappings from it into most of the classical Colombeau algebras which are compactible with the embeddings.We begin by constructing a mapping (Ω) →(Ω). Given R ∈(Ω) we define R∈(Ω) byR(φ, x)R ( φ⃗)(x) ((φ,x) ∈ U(Ω) )where φ⃗∈ C^∞(Ω, (Ω)) is chosen such that φ⃗= φ in a neighborhood of x. This definition is meaningful: given (φ,x) in U(Ω) we have φ(.-x') ⊆Ω for x' in a neighborhood V of x. Choosing ρ∈(Ω) with ρ⊆ V and ρ≡ 1 in a neighborhood of x, we can take φ⃗(x) ρφ. Obviously, R(φ,x) does not depend on the choice of φ⃗(x) and R(φ, .) is smooth, so indeed we have R∈(Ω).Let R ∈(Ω). Then the following holds:*.ι u. =u for u ∈'(Ω).*.σ f. =f for f ∈ C^∞(Ω).*R∈(Ω) for R ∈(Ω).*R∈(Ω) for R ∈(Ω). <ref>: For u ∈'(Ω) we have.ι u.(φ,x) = (ι u)(φ⃗)(x) = ⟨ u, φ⃗(x) ⟩ = ⟨ u, φ(x) ⟩ = ⟨ u(y), φ(y-x) ⟩ = ( u)(φ, x).<ref> is clear.<ref>: Suppose that R ∈(Ω). Fixing x ∈Ω, we obtain U as in <Ref>. Let KLU and m be given, set k=0, and choose L' such that LL'U. Then <Ref> gives c,l,λ such that for φ⃗∈ C^∞(K, _L'(U)),R(φ⃗)_K,m≤λ ( φ⃗_K,c; L', l ).Now for φ∈(^n) with K + φ⊆ L we have φ∈ C^∞(K, _L'(U)), which givesR(φ,.)_K,m = R(φ)_K,m≤λ ( φ_K,c; L', l) ≤λ ( φ_c+l )which proves that R∈(Ω).<ref>: Similarly, if R ∈(Ω) then for x ∈Ω we have U as in <Ref>. For KLU, m given, k=0, and L' such that LL'U, we obtain c,l,λ, B as in <Ref> such thatR(φ⃗) _K,m≤λ ( φ⃗_K,c; L', l, φ⃗- δ⃗_K,c; B )and henceR(φ, .)_K,m=R ( φ)_K,m≤λ ( φ_K,c; L', l, φ - δ⃗_K,c; B) ≤λ ( φ_c+l, φ - δ⃗_K,c; B ).which gives negligibility of R.§.§ The special algebraWe define the special Colombeau algebra ^s with the embedding as in <cit.>: fix a mollifier ρ∈(^n) with∫ρ(x) x = 1, ∫ x^αρ(x)x = 0∀α∈_0^n ∖{0}.Choosing χ∈(^n) with 0 ≤χ≤ 1, χ≡ 1 on B_1(0) and χ⊆ B_2(0) we setρ_(y) ^-nρ(y/),θ_(y) ρ_(y) χ ( y ln )(>0).Moreover, withK_ = { x ∈Ω | d(x, ^n ∖Ω) ≥}∩ B_1/(0) Ω (>0)we choose functions κ_∈(Ω) such that 0 ≤κ_≤ 1 and κ_≡ 1 on K_. Then the special algebra ^s(Ω) is given by^s(Ω) C^∞(Ω)^I with I(0,1],^s_M(Ω){ (u_)_∈^s(Ω) | ∀ K Ω ∀ m∈_0 ∃ N ∈: u__K,m = O(^-N) },^s(Ω){ (u_)_∈^s(Ω) | ∀ K Ω ∀ m∈_0 ∀ N ∈: u__K,m = O(^N) },^s(Ω)^s_M(Ω) / ^s(Ω), (ι^s u)_ ⟨ u, ψ⃗_⟩ (u ∈'(Ω)), (σ^s f)_f(f ∈ C^∞(Ω)),ψ⃗_(x)(y)θ_(x-y) κ_(y). For R ∈(Ω) we define R^s = (R^s_)_∈^s(Ω) by R^s_ (x)R ( ψ⃗_ ) (x).*(ι u)^s = ι^s u for u ∈'(Ω).*(σ f)^s = σ^s f for f ∈ C^∞(Ω).*R^s ∈^s_M(Ω) for R ∈(Ω).*R^s ∈^s(Ω) for R ∈(Ω). <ref> and <ref> are clear.For <ref> it suffices to show the needed estimate locally. Fix x ∈Ω, which gives U ∈_x(Ω) as in <Ref>. Choose any K,L such that x ∈ KLU, fix m, and set k=0. Then there are c,l,λ as in <Ref>. Because ψ⃗_(x) ⊆ B_2 ln^-1(x) we have ψ⃗_∈ C^∞(K, _L(U)) forsmall enough, which givesR^s__K,m≤λ ( ψ⃗__K,c; L, l ).Consequently, (R^s_)_∈^s_M(Ω) follows fromψ⃗__K,c; L, l = sup_x, α, y, β_x^α_y^β( ρ_(x-y) χ ( ( x-y) ln) κ_(y))= O(^-n-c-l).For negligibility we proceed similarly; the claim then follows by using that for a bounded subset B ⊆ C^∞(U) we have ψ⃗_ - δ⃗_K,c; B = O(^N) for all N ∈, which is seen as in <cit.> and actually merely a restatement of the fact that ι^s f - σ^s f = O(^N) for all N uniformly for f ∈ B.§.§ The diffeomorphism invariant algebraThere are several variants of the diffeomorphism invariant algebra ^d; we will employ the following formulation <cit.>:^d(Ω) C^∞( (Ω), C^∞(Ω))_M^d(Ω){ R ∈ C^∞((Ω)) | ∀ K Ω ∀ k,m ∈_0 ∀ (φ⃗_)_∈ S(Ω) ∀ (ψ⃗_1,)_, …, (ψ⃗_k,)_∈ S^0(Ω) ∃ N ∈: ^k R(φ⃗_)(ψ⃗_1,, …, ψ⃗_k, )_K,m = O(^-N ) },^d(Ω){ R ∈ C^∞((Ω)) | ∀ K Ω ∀ k,m ∈_0 ∀ (φ⃗_)_∈ S(Ω) ∀ (ψ⃗_1,)_, …, (ψ⃗_k,)_∈ S^0(Ω) ∀ N ∈: ^k R(φ⃗_)(ψ⃗_1,, …, ψ⃗_k, )_K,m = O(^N ) },^d(Ω)_M^d(Ω) / ^d(Ω),(ι^d u) (φ)(x)⟨ u, φ⟩,(σ^d f)(φ)(x) f(x). The spaces S(Ω) and S^0(Ω) employed in this definition are given as follows: Let a net of smoothing kernels (φ⃗_)_∈ C^∞(Ω, (Ω))^I be given and denote the corresponding net of smoothing operators by (Φ_)_∈ ( '(Ω), C^∞(Ω))^I. Then (φ_)_ is called a test object on Ω if*Φ_→𝕀 in ('(Ω), '(Ω)),*∀ p ∈ ( ('(Ω), C^∞(Ω)) ) ∃ N ∈: p ( Φ_ ) = O (^-N),*∀ p ∈ ( (C^∞(Ω), C^∞(Ω))) ∀ m ∈: p( Φ_|_C^∞(Ω) - 𝕀 ) = O(^m),*∀ x ∈Ω ∃ V ∈_x(Ω) ∀ r>0 ∃_0>0 ∀ y ∈ V ∀ < _0: φ_(y) ⊆ B_r(y). We denote the set of test objects on Ω by S(Ω). Similarly, (φ⃗_)_ is called a 0-test object if it satisfies these conditions with <ref> and <ref> replaced by the following conditions:* Φ_→ 0 in ('(Ω), '(Ω)),enumi1 *∀ p ∈ ( (C^∞(Ω), C^∞(Ω))) ∀ m ∈: p( Φ_|_C^∞(Ω) ) = O(^m).The set of all 0-test objects on Ω is denoted by S^0(Ω).For R ∈(Ω) we define R^d ∈^d(Ω) by R^d(φ)(x)R ( [x' ↦φ ] ) (x). *(ι u)^d = ι^d u for u ∈'(Ω).*(σ f)^d = σ^d u for f ∈ C^∞(Ω).*R^d ∈^d_M(Ω) for R ∈(Ω).*R^d ∈^d(Ω) for R ∈(Ω).<ref> and <ref> are clear from the definition. <ref> and <ref> follow directly from the estimatesφ⃗__K,c; L, l = O(^-N) for some N, φ⃗_ - δ⃗_K,c; B = O(^N) for all N,which hold by definition of the spaces S(Ω) and S^0(Ω).§.§ The elementary algebra For Colombeau's elementary algebra we employ the formulation of <cit.>, Section 1.4. For k ∈_0 we let _k(^n) be the set of all φ∈(^n) with integral one such that, if k ≥ 1, all moments of φ order up to k vanish.U^e(Ω){ (φ, x) ∈_0(^n) ×Ω | x + φ⊆Ω} ^e(Ω){ RU^e(Ω) → | ∀φ∈_0(^n): R(φ,.) is smooth} ^e_M(Ω){ R ∈^e(Ω)| ∀ K Ω ∀ m ∈_0 ∃ N ∈ ∀φ∈_N(^n):R(S_φ, .) _K,m = O(^-N) } ^e(Ω){ R ∈^e(Ω)| ∀ K Ω ∀ m ∈_0 ∀ N ∈ ∃ q ∈ ∀φ∈_q(^n):R(S_φ, .) _K,m = O(^N) } ^e(Ω)^e_M(Ω) / ^e(Ω) (ι^e u)(φ,x)⟨ u, φ(.-x) ⟩(σ^e f)(φ,x) f(x) For R ∈(Ω) we define R^e ∈^e(Ω) by R^e(φ, x)R(φ, x).*( u)^e = ι^e u for u ∈'(Ω).*( f)^e = σ^e u for f ∈ C^∞(Ω).*R^e ∈^e_M(Ω) for R ∈(Ω).*R^e ∈^e(Ω) for R ∈(Ω).Again, <ref> and <ref> are clear from the definition. For <ref>, fix K Ω and m ∈_0. From <Ref> we obtain r, c and λ such that for φ⊆ B_r(0), R(φ, .)_K,m≤λ ( φ_c). For φ∈_0(^n) andsmall enough, S_φ⊆ B_r(0), so we only have to take into account that S_φ_c = O(^-N) for some N ∈. Similarly, <ref> is obtained from the fact that given any N, for q large enough we have (S_φ)^* - δ⃗_K,c; B = O(^N) for all φ∈_q(^n).Acknowledgments. This research was supported by project P26859-N25 of the Austrian Science Fund (FWF). | http://arxiv.org/abs/1704.08167v1 | {
"authors": [
"Eduard A. Nigsch"
],
"categories": [
"math.FA",
"46F30"
],
"primary_category": "math.FA",
"published": "20170426154423",
"title": "Colombeau algebras without asymptotics"
} |
Vibrational-ground-state zero-width resonances for laser filtration: An extended semiclassical analysis. Osman Atabek December 30, 2023 ======================================================================================================== The Generalized Uncertainty Principle (GUP) is a modification of Heisenberg's Principle predicted by several theories of Quantum Gravity. It consists of a modified commutator between position and momentum. In this work we compute potentially observable effects that GUP implies for the harmonic oscillator, coherent and squeezed states in Quantum Mechanics. In particular, we rigorously analyze the GUP-perturbed harmonic oscillator Hamiltonian, defining new operators that act as ladder operators on the perturbed states. We use these operators to define the new coherent and squeezed states. We comment on potential applications. § INTRODUCTIONOne of the most active research areas in theoretical physics is the formulation of a quantum theory of gravity that would reproduce the well-tested theories of Quantum Mechanics (QM) and General Relativity (GR) at low energies. The large energy scales necessary to test proposed theories of Quantum Gravity make this investigation extremely challenging. Nonetheless, it is important to propose phenomenological models and tests of such theories at low energies, and the Generalized Uncertainty Principle (GUP) offers precisely this opportunity.Many theories of QG suggest that there exists a momentum-dependent modification ofthe Heisenberg Uncertainty Principle (HUP), and the consequent existence of a minimal measurable length <cit.>. This modification is universal and affects every Hamiltonian, since it will affect the kinetic term. In the last couple of decades several investigations have been conducted on many aspects and systems of QM,such as Landau levels <cit.>, Lamb shift, the case of a potential step and of a potential barrier <cit.>, the case of a particle in a box <cit.>, and the theory of angular momentum <cit.>. Furthermore, potential experimental tests have been proposed considering microscopic <cit.> or macroscopic Harmonic Oscillators (HO) <cit.>, or using Quantum Optomechanics <cit.>. Therefore our analysis is motivated by the fact that while very accurate systems withvery little noise can be constructed, the energy perturbations and other results derived in our paper will always be there, and potentially observable for highlysensitive systems.Similarly, any such deviations from the standard uncertainty profiles if observed, would also be ascribed to new physics such as the GUP.The most general modification of the HUP, was proposed in <cit.> and includes linear and quadratic terms in momentum of the following form [q,p] = i ħ (1 - 2 δγ p + 4 ϵγ^2 p^2)in 1 dimension, where δ and ϵ are two dimensionless parameters defining the particular model, andγ = γ_0/M_P c ,where M_P and c are Planck mass and the speed of light, respectively, and γ_0 is a dimensionless parameter. The presence of the quadratic term is dictated by string theory <cit.> and gedanken experiments in black hole physics <cit.>, whereas the linear term is motivated by doubly special relativity <cit.>, by the Jacobi identity of the corresponding q_i,p_j commutators, and as a generalization of the quadratic model. To incorporate both possibilities, we will leave the two parameters δ and ϵ undetermined, unless otherwise specified. While it is possible to absorb γ intoredefinitions of δ and ϵ, it is in practice useful to keep γ distinct so that it can function as an expansion parameter under various circumstances.In the present work,we revisit the problem of quantizing the HO <cit.> by incorporating the GUP(<ref>), and rigorously investigate the implications of this for coherent and squeezed states. Unlike previous investigations <cit.>, our focus is on a rigorous algebraic approach, which not only allows for a more efficient definition of the HO energy spectrum but also defines coherent and squeezed states of the HO with GUP (<ref>) in a consistent way. Furthermore, we consider for the first timesimultaneous perturbations in p^3 and p^4;previous studies focused on a p^4 perturbation only. We will also consider perturbation theory up to second order in γ, finding finite results up to that order. We are thus able to derive potentially observable deviations from the standard HO due to GUP or Planck scale effects. In particular, we find a model-dependent modification of the HO energy spectrum, as well as modified position and momentum uncertainties for coherent and squeezed states. Similar analyses concerning coherent and squeezed states of the HO for noncommutative spaces were carried out in <cit.>. A similar problem was also addressed in <cit.>, although a number of terms in the Hamiltonian arising from perturbation were neglected.This paper is organized as follows. In Sec. <ref>, we consider a GUP-induced perturbation of the HO, computing perturbed energy eigenstates and eigenvalues. Moving beyond the previous treatment of the HO with the GUP <cit.>, the algebraic method adopted here turns out to be more concise, allowing for definitions that will be useful in the rest of the paper. A new set of ladder operators for the perturbed states is defined in Sec. <ref>, and in Sec. <ref>, we consider coherent states, focusing on their position and momentum uncertainties, and their time evolution. A similar analysis for squeezed states for a HO is performed in Sec. <ref>,with special reference to specific GUP models <cit.> and <cit.>. In Sec. <ref> we summarize our work, and comment on potential applications.§ HARMONIC OSCILLATOR IN GUPConsider the Hamiltonian for the one-dimensional HOH = p^2/2m + 1/2mω^2 q^2 =-ħω/4 (a^† - a)^2 + ħω/4(a + a^†)^2 = ħω(N + 1/2[a,a^†]) ,where N = a^† a is the usual number operator. Though in the standard theory [a,a^†]=1, it is easy to show that this relation is no longer valid once GUP is incorporated.Using the model in (<ref>) and expanding the physical position q and momentum p in terms of the low-energy quantities, q_0 and p_0 respectively, we obtain <cit.>,q = q_0 ,p = p_0 ( 1 - δγ p_0 + 2 γ^2 δ^2 + 2ϵ/3 p_0^2 )where [q_0,p_0] = i ħ. In terms of these quantities, we can write the Hamiltonian for the harmonic oscillator asH = p_0^2/2m + 1/2mω^2 q_0^2 - δγp_0^3/m + 7 δ^2 + 8 ϵ/3γ^2 p_0^4/2 mto leading order in the parameters. Notice that we consider at the same time perturbations proportional to p^3 and p^4, in contrast to previous studies along these lines<cit.>.Using perturbation theory with perturbation parameter γ,we can define the state|n^(K)⟩ = ∑_k=0^K σ_k γ^k |n_k ⟩to order γ^K, where |n_k ⟩ is the k-th order perturbation of the eigenvectors |n_0 ⟩ =|n⟩ of the unperturbed HO, whose eigenvalues areE^(0) = ħω (n + 1/2), and the σ_k are constant coefficients that are calculable from perturbation theory. It is tedious but straightforward to obtain the following expression for thenormalized perturbed Hamiltonian eigenstates up to second order in γ|n^(2)⟩ = - δ^2/72γ^2 ħ m ω/2√(n^6) |n-6⟩ + γ^2/16ħ m ω/2[ δ^2 ( 4n - 2/3) + 8 ϵ/3] √(n^4) |n-4⟩ - i δγ/6√(ħ m ω/2)√(n^3)|n-3⟩ + - γ^2/8ħ m ω/2[ δ^2 ( 7n^2 - 29/3 n - 11/3) + 16 ϵ/3 (2n - 1) ] √(n^2) |n-2⟩ + i 3/2δγ√(ħ m ω/2) n √(n^1) |n-1⟩ + + [1 - δ^2 γ^2/72ħ m ω/2 (164 n^3 + 246 n^2 + 256 n + 87)]|n⟩ + i 3/2δγ√(ħ m ω/2) (n+1) √((n+1)^1) |n+1⟩ + - γ^2/8ħ m ω/2[ δ^2 ( 7n^2 + 71/3 n + 13 ) - 16 ϵ/3 (2n + 3) ] √((n+1)^2) |n+2⟩ - i δγ/6√(ħ m ω/2)√((n+1)^3) |n+3⟩ + + γ^2/16ħ m ω/2[ δ^2 ( 4n + 14/3) - 8 ϵ/3] √((n+4)^4) |n+4⟩ - δ^2/72γ^2ħ m ω/2√((n+1)^6) |n+6⟩ ,where we used the notation(n+1)^k= (n+k)!/n! , n^k= n!/(n-k)!k ≤ n , n^k= 0k > n .The perturbed energy eigenvalue isE^(2) = ħω{( n + 1/2) - ħ m ω/2γ^2 [ (4 n^2 + 4 n + 1) δ^2 - (2 n^2 + 2 n + 1) 2 ϵ ] } = E^(0) + Δ E ,whereΔ E = - ħωħ m ω/2γ^2 [ (4 n^2 + 4 n + 1) δ^2 - (2 n^2 + 2 n + 1) 2 ϵ] .Furthermore, the spacing of the energy levels isE^(2)(n+1) - E^(2)(n) = ħω - 8 ħω (n + 1) ħ m ω/2γ^2 [ δ^2 - ϵ ] .Note that the linear and the quadratic contributions to (<ref>), identifiable through the parameters δ and ϵ, make opposing contributions to the energy.It is therefore worth noting that, for the class of models with δ^2 = ϵ, the correction Δ E is independent of n, resulting in an equally spaced energy spectrum. For δ^2 < ϵ, the correction is a positive function of the number n. In particular, for δ = 0 and ϵ = 1/4, we obtain the results presented in <cit.> and <cit.>. Finally, we see that for δ^2 > ϵ the spacing between energy levels decreases and we find the value of n corresponding to the maximal energyn_max = ⌊1/8 ħ m ω/2γ^2 (δ^2 - ϵ) - 1 ⌋ .Notice that this depends on the Planck scale parameters, and the energy corresponding to the above n signals the breakdown of the GUP model, and necessitates higher order terms.To estimate the magnitude of these corrections, we compute|Δ E|/E^(0) = ħ m ω/2γ^2 |(4 n^2 + 4 n + 1) δ^2 - (2 n^2 + 2 n + 1) 2 ϵ|/n + 1/2n≫1≃ 4 n ħ m ω/2γ^2 |δ^2 - ϵ| ≃ 2 m E^(0)γ^2 |δ^2 - ϵ| ,Conversely, if we performed an experiment with a sensitivity Δ = |Δ E|/ E^(0), we could detect a deviation from the unperturbed energies atE^(0) = Δ/2 m γ^2 |δ^2 - ϵ| = 21.279 Δ/m γ_0^2 |δ^2 - ϵ|Jor n = Δ/4 ħ m ω/2γ^2 |δ^2 - ϵ| = 20.169 Δ/ m ωγ_0^2 |δ^2 - ϵ|× 10^34for m and ω measured in Kg and in Hz, respectively. Although this valueappears huge, as we show below these Planck scale effects are potentially accessible to a number of current and future experiments. We in particular see that massive and/or rapidly oscillating systems most significantly enhance GUP effects.As particular examples, we considered the cases of the mechanical oscillator considered in <cit.>, the oscillators considered in <cit.>, the resonant-mass bar AURIGA in its first longitudinal mode <cit.>, and the mirrors in LIGO oscillating at a frequency in the middle of its detection band <cit.>. § NEW GUP MODIFIED LADDER OPERATORSGiven the form of the perturbed eigenstates in terms of the standard number states (<ref>), it is easy to show that the standard annihilation and creation operators do not act anymore as ladder operators. We therefore define a new set of operators, useful for constructing coherent and squeezed states, such thatã | n^(2)⟩ =√(n) | (n-1)^(2)⟩ ,a^† | n^(2)⟩ =√(n+1) | (n+1)^(2)⟩ ,Ñ | n^(2)⟩ = n | n^(2)⟩ .Note that these operators obey the following relationsa^† =ã^† ,Ñ =ã^†ã , [ã,ã^†] = 1 .Using these definitions, we find the following expressions for theoperators in (<ref>) in terms of the standard annihilation, creation and number operators:ã = a - δ i γ/2√(ħ m ω/2)[ 3 a^2 + 3 (2N+1) - a^†^2 ] +- 2 γ^2 ħ m ω/2[ ( 5/3δ^2 - 2/3ϵ) a^3 + δ^2 a N + ( δ^2 + 2 ϵ) N a^† - ( 2/3δ^2 + 1/3ϵ) a^†^3 ] , a^† = a^† + δ i γ/2√(ħ m ω/2)[ 3a^†^2 + 3(2N+1) - a^2 ] +- 2 γ^2 ħ m ω/2[ ( 5/3δ^2 - 2/3ϵ) a^†^3 + δ^2 N a^† + ( δ^2 + 2 ϵ ) a N - (2/3δ^2 + 1/3ϵ) a^3 ] , Ñ = N - δ i γ/2√(ħ m ω/2)[ a^3 - 3 ( a N - N a^† ) - a^†^3 ] + + γ^2/2ħ m ω/2{7 δ^2 + 8 ϵ/6 [ a^4 - 2 (2 a N a + a^2) - 2 (2 a^† N a^† + a^†^2) + a^†^4 ] + δ^2/2 (30 N^2 + 30 N + 11) } To obtain these relations, we used the following procedure. Consider, for example, the annihilation operator ã defined by (<ref>). Applying this operator on the perturbed number state (<ref>), we notice that to the 0-th order in γ we get ã |n⟩ = a |n⟩, as expected. Requiring (<ref>) to hold to order γ,we compute (ã - a) |n^(1)⟩ = √(n)|(n-1)^(1)> - a |n^(1)> = - δ i γ/2√(ħ m ω/2)[ 3 a^2 + 3 (2N+1) - a^†^2 ] |n^(1)>where |n^(1)⟩ is obtainedfrom the order-γ terms in (<ref>). Therefore ã = a - δ i γ/2√(ħ m ω/2)[ 3 a^2 + 3 (2N+1) - a^†^2 ]to first order in γ.Repeating this procedure to order γ^2 yields {ã - a + δ i γ/2√(ħ m ω/2)[ 3 a^2 + 3 (2N+1) - a^†^2 ] }| n^(2)> = - 2 γ^2 ħ m ω/2[ ( 5/3δ^2 - 2/3ϵ) a^3 + δ^2 a N + ( δ^2 + 2 ϵ) N a^† - ( 2/3δ^2 + 1/3ϵ) a^†^3 ] | n^(2)>which gives (<ref>). A similar procedure can be used for ã^̃†̃ and Ñ. Continuing this procedure, one can extend the relations in (<ref>) to any arbitrary order.Using these expressions, the Hamiltonian of an harmonic oscillator with GUP can be written in a simple form:H = ħω{( Ñ + 1/2) - ħ m ω/2γ^2 [ 4 (Ñ^2 + Ñ) (δ^2 - ϵ) + δ^2 - 2 ϵ] }as well asq = (ã^† + ã) √(ħ/2 m ω) - 2 i (ã^†^2 - ã^2 ) δħ/2γ - 2 [ ( ã^3 + ã^†^3 ) ( 2 δ^2 + ϵ) + (ãÑ + Ñã^†) (3 δ^2 - 2 ϵ) ] ħ/2√(ħ m ω/2)γ^2 p = i (ã^† - ã) √(ħ m ω/2) + 2 (ã^†^2 + 2 Ñ + 1 + ã^2) δħ m ω/2γ + 2 i [( ã^3 - ã^†^3 ) ( 2 δ^2 + ϵ) + 3 (ãÑ - Ñã^† ) δ^2 ] (ħ m ω/2)^3/2γ^2 for the physical position and momentum operators. It is then easy to find the expectation values and the uncertainties in position and momentum for the perturbed number states. Defining ⟨ A ⟩ = ⟨n^(2)| A | n^(2)⟩ for any operator A, we find ⟨ q ⟩ = 0 , ⟨ q^2 ⟩ =ħ/2 m ω (2 n + 1) - 4 γ^2 ħ^2/4[ δ^2 (2n + 1)^2 - 2 ϵ (2 n^2 + 2 n + 1) ] , (Δ q)^2 =⟨ q^2 ⟩ , ⟨ p ⟩ = 2 δγħ m ω/2 (2n + 1) , ⟨ p^2 ⟩ =ħ m ω/2 (2n + 1) , (Δ p)^2 =ħ m ω/2 (2n + 1) [ 1 - 4 δ^2 γ^2 ħ m ω/2 (2n + 1) ] , (Δ q)^2 (Δ p)^2 =ħ^2/4 (2n + 1)^2 - 8 γ^2 ħ^2/4ħ m ω/2 (2n + 1) [ δ^2 (2n + 1)^2 - ϵ (2n^2 + 2n + 1)]The presence of δ in (<ref>) and (<ref>) leads to interesting features. First, the expectation value of the momentum does not vanish whereas the expectation value of the position does. This is a consequence of the linear part of the GUP (<ref>), which singles out a preferred direction and so breaks translation invariance. Furthermore, this same term implies thatthere exists a value for n such that (Δ p)^2 < 0, showing that a linear model leads to critical results for high energy systems, i.e. when the energy is close to the Planck scale. We expect that at this energy scale a full theory of Quantum Gravity has to be considered, with non-negligible higher order corrections.To conclude this section,we observe that the definitions in (<ref>) and the methods developed in this section can be easily extended to any perturbationthat is of the form of a polynomial of q and p and to any order in perturbation theory. Using this method, the full Hamiltonian can be written in the following formH = ħω{(Ñ + 1/2) + ∑_i=0^S c_i Ñ^i } ,S = {[ 0de = 1; ⌊ (d + e - 1)/2 ⌋ ].where d is the degree of the polynomial representing the perturbation and e the order of the perturbation theory. The coefficients c_i are constants that depend on the parameters of the perturbation. In this paper we consider the case with d = 4 , e = 2 , S = 2 ,c_0 = -ħ m ω/2γ^2 (δ^2 - 2ϵ) , c_1 = c_2 = - 4 ħ m ω/2γ^2 (δ^2 - ϵ) . Furthermore, we emphasize that |n^(2)⟩ are the eigenstates of the full Hamiltonian with GUP (<ref>) up to 𝒪(γ^2), while the standard states |n⟩ are not. Therefore, we will consider the former as the physical number states. Furthermore, the ∼-operators that we have defined have the same properties, in terms of commutators and actions on number states, as the analogous operators of the standard theory. For these reasons, we apply the above new operators to define coherent and squeezed states.§ COHERENT STATESAs in the standard theory, we define a coherent state as the eigenstate of the annihilation operator ãã | α^(2)⟩ = α |α^(2)⟩to order γ^2. This is because |n^(2)⟩ is the physical number state and ã is its operator. Following the results of the previous section, we then have|α^(2)⟩ = e^- |α|^2/2∑_n=0^∞α^n/√(n!) |n^(2)⟩ ,i.e. a Poisson distribution of number states in ∼-coherent states[We can also define a displacement operator of the form 𝒟̃(α) = e^αã^† - α^* ã.]. Furthermore, given the expansions in (<ref>), we can easily obtain the following results for the uncertainties⟨ q ⟩ =( α^⋆ + α) √(ħ/2 m ω) - 2 i δ( α^⋆^2 - α^2) ħ/2γ + + [ (2 ϵ - 3 δ^2) (α^⋆ + α) ( 1 + |α|^2) - ( ϵ + 2 δ^2) ( α^⋆^3 + α^3 ) ] ħ/2√(ħ m ω/2)γ^2 , (Δ q)^2 =ħ/2 m ω - 4 i δ( α^⋆ - α) ħ/2√(ħ/2 m ω)γ + - 2 [ ϵ(- 4 + α^2 - 8 |α|^2 + α^⋆^2) + δ^2 ( 2 + 9 α^2 + 4 |α|^2 + 9 α^⋆^2 ) ] ħ^2/4γ^2 , ⟨ p ⟩ = i (α^⋆-α) √(ħ mω/2) + 2 δ( α^⋆^2 + 2 |α|^2 + α^2 + 1 ) ħ m ω/2γ + - i [ 3 δ^2 (1 + |α|^2) (α^⋆ - α) + (ϵ + 2 δ^2) ( α^⋆^3 - α^3) ] ( ħ m ω/2)^3/2γ^2 , (Δ p)^2 =ħ m ω/2 + 2 (ħ m ω/2)^2 γ^2 [ ( α^⋆^2 + α^2) (5 δ^2 - 3 ϵ) + (4 |α|^2 - 2)δ^2 ] , (Δ q)^2 (Δ p)^2 =ħ^2/4{ 1 - 4 i δ( α^⋆ - α) √(ħ m ω/2)γ +8 [ ϵ( 1 + α - α^⋆) ( 1 - α + α^⋆) - δ^2 ( 1 + α^2 + α^⋆^2 ) ] ħ m ω/2γ^2 } where we see that the last term exhibits interesting properties. If ϵ > 0 and δ=0 then the uncertainty (Δ q)^2 (Δ p)^2 is greater than ħ^2/4 whereas if ϵ = 0 and δ>0 then this quantity is smaller than ħ^2/4.The same situation holds for the minimal uncertainty product, which is the smallest uncertainty product predicted by QM as found using the Schrödinger-Robertson relation. For the model in (<ref>), we obtain [(Δ q)^2 (Δ p)^2]_min =| ⟨ qp + pq ⟩/2 - ⟨ q ⟩⟨ p ⟩|^2 +|⟨[q,p]⟩|^2/4 = = ħ^2/4{ 1 - 4 i δ( α^⋆ - α) √(ħ m ω/2)γ +8 [ ϵ( 1 + α - α^⋆) ( 1 - α + α^⋆) - δ^2 ( 1 + α^2 + α^⋆^2 ) ] ħ m ω/2γ^2 } and again nonzero ϵmakes this quantity larger than the standard QM value whereas non-zero δ makes it smaller.Finally we note that (Δ q)^2 (Δ p)^2 - [(Δ q)^2 (Δ p)^2]_min =0 ∀α, γ, δ, ϵ .and so coherent states are also minimal uncertainty states when the GUP is included. The quantity (Δ q)^2 (Δ p)^2 is the actual product of the uncertainties in position and momentum.Therefore, we see that coherent states as defined in (<ref>) satisfy all the known properties of the standard theory, i.e. they follow a Poisson distribution, they are eigenstates of the annihilation operator, and are minimal uncertainty states. The uncertainty product (Δ q)^2 (Δ p)^2 in (<ref>) is negative for some values of α and of √(ħ m ω/2)γ. This is due to the linear term in GUP and will go negative forδ^2 > 1 + 4 ^2(α)/1 + 2 ^2 (α)ϵ .In this case, the model will present anomalies when the characteristic momentum of the system fulfills the following relation√(ħ m ω/2)γ =-(α) δ±√( 2 [1 + 2 ^2 (α) ] δ^2 - 2 [1 + 4 ^2 (α) ] ϵ)/ 2 [1 + 2(α^2) ] δ^2 - 2 [ 1 + 4 ^2 (α) ] ϵ .Notice that this quantity depends on the phase of α and that one cannot a priori choose the sign in the expression.§.§ Time EvolutionTo consider the time-evolved coherent states, we analyze them in Heisenberg picture. Since we seek to understand the evolution of the variances (<ref>), we first write the time-evolved annihilation operator ã(t) = ã(0) exp{ - i ω t [ 1 - 8 Ñ( δ^2 - ϵ) ħ m ω/2γ^2 ] }≡ã(0) exp[ - i ω t ( 1 - 8 Ñχ) ] ,where we have defined the dimensionless quantityχ = ( δ^2 - ϵ) ħ m ω/2γ^2 .Notice that for the number operator we have Ñ(t) = Ñ(0), and therefore the statistical properties of the coherent states do not change.Using the time-evolved operators (<ref>), we find( Δ q )^2 =ħ/2 m ω{ 1 + 2 |α|^2 [ 1 - e^- 4 |α|^2 sin^2 [4 χω t]] } + α^2 ħ/2 m ω{ e^-|α|^2 [1 - e^16 i χω t] - 2 i ω t [1 - 12 χ] -e^- 2 |α|^2 [1 - e^8 i χω t] - 2 i ω t [1 - 8 χ] } + α^⋆^2 ħ/2 m ω{ e^-|α|^2 [1 - e^-16 i χω t] + 2 i ω t [1 - 12 χ] -e^-2 |α|^2 [1 - e^- 8 i χω t]+ 2 i ω t [1 - 8 χ] } + 4 i √(ħ/2 m ω)ħ/2( α e^- i ω t- α^⋆ e^i ω t) δγ - 2 ħ^2/4[ 2 (δ^2 - 2 ϵ) ( 1 - 2 |α|^2) + (9 δ^2 + ϵ) ( α^2 e^- 2 i ω t + α^⋆^2 e^2 i ω t) ] γ^2 , ( Δ p )^2 =ħ m ω/2{1 + 2|α|^2[1 - e^ - 4 |α|^2 sin^2 [4 χω t] ]} + α^2 ħ m ω/2{ e^-2 |α|^2 [1 - e^8 i χω t] - 2 i ω t [1 - 8 χ] - e^ - |α|^2 [1 - e^16 i χω t ] - 2 i ω t [1 - 12 χ]} + α^⋆^2 ħ m ω/2{ e^- 2 |α|^2 [1 - e^- 8 i χω t ] + 2 i ω t [1 - 8 χ t ] - e^- |α|^2 [1 - e^- 16 i χω t ] + 2 i ω t [1 - 12 χ t ] } + 2 (ħ m ω/2)^2 [ (5 δ^2-3 ϵ) (α^2 e^ - 2 i ω t+ α^⋆^2 e^2 i ω t) - 2 δ^2 (1 - 2 |α|^2) ] γ^2 for the uncertainties in position and momentum.These two expressions indicate that the GUP imputes important effects on the uncertainties of position and momentum. Consider the first line of each expression. We notice an oscillation in the uncertainties, governed by the term sin^2 [4 χω t], with amplitude proportional to the standard uncertainties multiplied by |α|^2. This term, initially 0 for t=0, varies with time with a periodT = | π/4 χω| . It is worth noticing that the period depends on m,ω, and the GUP parameters through the quantity χ. The subsequent two terms in each expression present an oscillation with a similar period. This oscillation is present only for models with δ^2 ≠ϵ. For the systems considered at the end of Sec. <ref>, we have the Table <ref>.We see that in all cases we have a period several orders of magnitude larger than the age of the universe.As for the uncertainty product, we have(Δ q)^2 (Δ p)^2 = ħ^2/4{1 + 4 |α|^2 [1 - e^- 2 |α|^2 sin^2 (4 χω t)] + 2 |α|^4 [2 + e^- |α|^2 [3 - 2 e^8 i χω t - e^- 16 i χω t] - 8 i χω t+ . . . . + e^- |α|^2 [3 - 2 e^-8 i χω t - e^16 i χω t] + 8 i χω t - e^- 2 |α|^2 sin^2 [8 χω t] + e^- 4 |α|^2 sin^2 [4 χω t] - 4 e^- 2 |α|^2 sin^2 [4 χω t]] } - α^4 ħ^2/4{e^- |α|^2 [1 - e^16 i χω t] - 2 i ω t [1 - 12 χ]- e^- 2 |α|^2 [1 - e^ 8 i χω t] - 2 i ω t [1 - 8 χ]}^2 + - α^⋆^4 ħ^2/4{e^- |α|^2 [1 - e^- 16 i χω t] + 2 i ω t [1 - 12 χ]- e^- 2 |α|^2 [1 - e^ - 8 i χω t] + 2 i ω t [1 - χ]}^2 + - 8 ħ m ω/2ħ^2/4[ (δ^2-ϵ) - 2 |α|^2 ϵ + (δ^2+ϵ) (α^2 e^- 2 i ω t+ α^⋆^2 e^2 i ω t) ] γ^2 + 4 i ħ^2/4√(ħ m ω/2)δ[α e^ - i ω t- α^⋆ e^i ω t] γand shows variations similar to those previously analyzed. We conclude this section by showing that similar results can be obtained to any order in perturbation theory for a large class of perturbations. In this case it is convenient to use the Schrödinger picture, in whicheach number state component of a coherent state acquires a different phase. In fact, let us rewrite the Hamiltonian in (<ref>) asH = ħω( ∑_i=0^S κ_i Ñ^i ) ,where κ_0 = c_0 + 1/2, κ_1 = c_1 + 1, and κ_i = c_i with i≥ 2. A time-evolved coherent state can then be written as|α,t⟩ = exp[- i ω t ( ∑_i=0^S κ_i Ñ^i )] |α,0⟩∝∑_n=0^∞(α e^-i κ_1 ω t)^n/√(n!)exp[- i ω t ( ∑_i=2^S κ_i n^i )] |n^(e)⟩ ,where |n^(e)⟩ is the perturbed energy eigenstate up to order π such that Ñ |n^(e)⟩ = n |n^(e)⟩. One can then prove that ⟨α,t|q|α,t⟩∝ ∑_n=0^∞α^2n/n!cos{ t [ ωκ_1 + (ω_n+1 - ω_n) ] } , ⟨α,t|p|α,t⟩∝ ∑_n=0^∞α^2n/n!sin{ t [ ωκ_1 + (ω_n+1 - ω_n) ] } , where ω_n = ω( ∑_i=2^S κ_i n^i ). We then see that the coherent state spreads in phase-space, since each component moves with a different frequencyΩ_n = ω{κ_1 + ∑_i=2^S κ_i ∑_j=1^i ij n^i-j} .In a frame rotating at the frequency of the ground state, the other states move with a frequencyΔΩ_n = Ω_n - Ω_0 = ω∑_i=2^S κ_i ∑_j=1^i-1ij n^i-jWe have two cases: * Only one coefficient κ_i is different from 0: then all the frequencies are multiple of ω_1 = ωκ_i and the time interval over which the coherent state spreads and restores is given byT = 2π/|κ_i| ω .* κ_i = κμ_i, with κ∈ℝ and μ_i ∈ℚ: in this case we haveT = 2 π/ω |κ|ζ/ξ ,where ζ is the least common denominator of μ_i's and ξ is the greatest common divisor of the numerator of μ_i's.In many cases, this is an over estimation, since when i is a prime number, the actual period is T' = T/i, or, if i is a power of a prime number k, then one can take T' = T/k as the period. Finally, notice that the result in (<ref>) corresponds to the first case with κ_2=-4χ and that in general the period depends on the HO frequency ω and on the HO mass m and the GUP parameters included in κ_i.§ SQUEEZED STATESNext we construct squeezed states with GUPup to second order in γ. Following <cit.>, we define a new set of operatorsã_r =ãcosh r - e^iθã^†sinh r ,ã^†_r = ã^†cosh r - e^- iθãsinh r ,where r is the squeeze parameter and θ a phase angle. We can proceed as in the standard theory, defining a squeezed vacuum state throughã_r |0⟩_r = 0and we find that|0⟩_r can be expanded in terms of even numbered states only|0⟩_r = ∑_n=0^∞ b_2n |2n^(2)⟩ ,with[Alternatively, to define squeezed states is through the squeeze operator , where z=r e^iθ. In this case we haveã_z =𝒮̃^† (z) ã 𝒮̃(z) |0^(2)⟩_z = 𝒮̃(z) |0^(2)⟩ ]b_2n = (tanh r e^iθ)^n √((2n-1)!!/2n!!) b_0 , b_0 =[ 1 + ∑_n=1^∞ (tanh r)^2n(2n-1)!!/2n!!]^-1/2 .Considering the case θ = 0, we can invert the relations (<ref>) and study the properties of squeezed states using (<ref>), obtaining the following expressions for the uncertainties (Δ q)^2 =ħ/2 m ω e^- 2 r - 4 i (α^⋆ - α) δħ/2√(ħ/2 m ω) e^- 2 rγ + ϵħ^2/4 (5 + 3 e^- 4 r - 2 α^⋆^2 e^-2 r + 16 |α|^2 e^-2 r - 2 α^2 e^-2 r ) γ^2 + + δ^2 ħ^2/4[ 11 - 15e^ - 4 r + 2 α^2 (2 e^2 r - 11 e^-2 r) + 8 |α|^2 (e^2 r - 2 e^-2 r) + 2 α^⋆^2 (2 e^2 r - 11 e^-2 r) ] γ^2 (Δ p)^2 =ħ m ω/2 e^2 r + (ħ m ω/2)^2 { 2 (α^⋆^2 + α^2) [ 8 e^-2 rδ^2 - 3 e^2 r (δ^2 + ϵ)] + 8 |α|^2 δ^2 (4 e^-2 r - 3 e^2 r) - 3 e^4 r (δ^2 - ϵ) . + . - 3 (3 δ^2 + ϵ)+ 8 δ^2 e^-4r}γ^2 , (Δ q)^2 (Δ p)^2 =ħ^2/4{ 1 - 4 i (α^⋆ - α) δ√(ħ m ω/2)γ + 4 ħ m ω/2{ 2 ϵ (e^r - α^⋆ + α) (e^r + α^⋆ - α) + . . + δ^2 [ α^⋆^2 (e^4 r - 7 + 4 e^-4r) + 2 |α|^2 (e^4 r - 5 + 4e^-4 r) . . . . + α^2 (e^4 r - 7 + 4 e^-4r) + 2 (e^2 r - 3 e^-2r + e^- 6 r) ] }γ^2 } to order γ^2.As expected, these reduce to quantities in the standard theory for γ→ 0. In this case too, we can compute the theoretical minimal uncertainty through the Schrödinger-Robertson relation[(Δ q)^2 (Δ p)^2]_min = ħ^2/4{ 1 - 4 i (α^⋆ - α) δ√(ħ m ω/2)γ+ 4 ħ m ω/2{ 2 ϵ(e^r - α^⋆ + α) (e^r + α^⋆ - α) γ^2 + . . . ħ m ω/2. - δ^2 [ 3 α^⋆^2 + 2 |α|^2 + 3 α^2 + 2 e^-2 r - (α^⋆ + α)^2 (e^2r - 2 e^-2r)^2 ] }γ^2 }and thus find (Δ q)^2 (Δ p)^2 - [(Δ q)^2 (Δ p)^2]_min = 32 δ^2 ħ^2/4ħ m ω/2 e^- 2 rsinh^2 (2r) γ^2for the difference with the minimal uncertainty product.Several features are noteworthy.First, the above difference is positive for all values of r and does not depend on α. Second, the difference with the minimal product is of the second order in the GUP parameter. Finally, it is present only for models with a non-zero linear term; it is not present in the model introduced in <cit.>. Finding evidence for the relation (<ref>) could therefore help distinguish the models. §.§ Minimal Uncertainties for Squeezed States In the standard theory, squeezed states saturate the uncertainty relation allowing for reduced uncertainties, with respect to coherent states, in either position or momentum. In principle, the uncertainty in position (momentum) can be made arbitrarily small for HUP-saturating states. However theGUPintroducesa minimal uncertainty in position. In this section we study the implications of this fact.To better observe the features of optimal squeezing in GUP, we consider the particular models examined in <cit.>. This model corresponds to δ = 0 and ϵ = 1/4. The position uncertainty for this model is(Δ q)^2 = ħ/2 m ω e^- 2 r + ħ^2/4 (5 + 3 e^- 4 r - 2 α^⋆^2 e^-2 r + 16 |α|^2 e^-2 r - 2 α^2 e^-2 r ) γ^2/4 .We obtain a minimal position uncertainty for e^-r = 0 ⇒ r = + ∞. It is(Δ q)^2 = ħ^2/45/4γ^2> (Δ q)^2_min = ħ^2 γ^2/4whereas the latter quantity has been previously computed <cit.>. Therefore, the maximally squeezed uncertainty in position is always larger than the minimal uncertainty predicted by the model.The momentum uncertainty for the same model is(Δ p)^2 = ħ m ω/2 e^2 r -3/4(ħ m ω/2)^2 { 2 (α^⋆^2 + α^2) e^2 r - e^4 r + 1 }γ^2 ,and is minimized fore^2r = 0 ⇒ r = - ∞, the minimal uncertainty being(Δ p)^2 = - 3/4(ħ m ω/2)^2 γ^2which is negative. This means that this model cannot describeinfinite squeezing. Rather, by inverting the reasoning, there must exist a lower value of r for the interval of squeezing in which GUP can be used. In fact, when we havee^2r =- 4 + 3 (α^⋆^2 + α^2) ħ m ωγ^2 + √( 9ħ^2 m^2 ω^2 γ^4 + [ 4 - 3 (α^2 + α^⋆^2) ħ m ωγ^2 ]^2 )/3 ħ m ωγ^2≃3 ħ m ωγ^2/8 ,the squeezed uncertainty in momentum vanishes.We next turn to the consequences for the model in <cit.>, for which δ=1 and ϵ=1, a case motivated from doubly special relativity as noted in the introduction. For simplicity, we consider a squeezed vacuum state, i.e. α=0. The position uncertainty in this case is(Δ q)^2 = ħ/2 m ωe^-2r + 4 ħ^2/4 (4 - 3 e^-4r) γ^2 .It has a minimum at e^-r = 0 ⇒ r = + ∞, corresponding to(Δ q)^2 = 16 ħ^2/4γ^2 >(Δ q)^2_min = 8 ħ^2/4γ^2where the latter quantity isthe minimal length allowed by the model in <cit.>. As for the uncertainty in momentum,we find(Δ p)^2 = ħ m ω/2 e^2 r - 4 (ħ m ω/2)^2 (2 e^-4r - 3) γ^2for this model. Given the presence of both decreasing and increasing exponentials in r, there is no minimal value for the momentum uncertainty. Instead we see that Δ p vanishes fore^2r_min = 4 ħ m ω/2γ^2 + 32/3(ħ m ω/2)^2/3γ^4/3 - 2 ( ħ m ω/2)^1/3γ^2/3 + 𝒪(γ^8/3) .As with the previous model, this value r_min of the squeeze parameter can be interpreted as the smallest physical value of rpermitted in this model, since smaller values yield (Δ p)^2 < 0.Finally, in both model we observe similar features: the uncertainties in position have a non-vanishing minimal value larger than the minimal length allowed by the models. This uncertainties are achieved for infinite squeezing. On the other hand, minimal uncertainties in momentum, as computed by the GUP models, can be negative, signaling the break down of GUP for high energies. Therefore, GUP con describe squeezing in momentum uncertainties only up to particular limits for the squeeze parameter.§ APPLICATIONS AND OUTLOOKVarious theories of Quantum Gravity predict a modification of the Heisenberg principle and the Heisenberg algebra to include momentum dependent terms. We consider its implications on the HO from the GUP (<ref>) by looking at perturbations from the γ=0 caseup to second order in γ. We were thus able to find normalized eigenstates (new number states) and eigenvalues, and a new set of ladder operators. We then focused our attention on coherent and squeezed states.Our most notable results are the new spacing of the energy eigenvalues and the new energy eigenstatesas functions of the linear and quadratic terms δ and ϵ in the GUP. There are three distinct cases. When δ^2 < ϵ, the spacing of the energy ladder increases with the number n. For the case δ^2 = ϵ, the spacing does not depend on n, obtaining a regular ladder, as in the standard theory. Finally, for δ^2 > ϵ the correction is negative. Therefore the spacing of the energy ladder decreases with the number n till a maximum number n_max, corresponding to the maximum value of the energy. This simply indicates the GUP breaks down for large enough energies, where Planck scale effects become relevant and a full theory of Quantum Gravity is necessary since higher order terms cannot be neglected. We were then able to define a set of modifiedannihilation, creation, and number operators for the perturbed states. The algebra of these new operators is identical to that of the standard ones. Furthermore, we show that with their aid, the Hamiltonian can be written in a compact form, from which expectation values, uncertainties, etc. can be computed quite easily. For coherent states, defined as eigenstates of the new annihilation operator, we find that they still are minimal uncertainty states. Squeezed states introduce additional interesting features. The position uncertainty has a lower bound– not present in the standard theory – that depends on the particular GUP model under consideration. As for the momentum uncertainty, its maximal squeezing depends on the model and on the choice of the coefficients δ and ϵ. That is, for some choices of the coefficients, a lower bound for the squeeze parameter exists, since beyond this limit higher order Planck scale effects cannot be neglected.Our results are potentially testable. For example, direct application to mechanical oscillators can be tested, e.g. as in <cit.>. In fact, mainly motivated by the results in <cit.>, many groups already proposed experiments to test them. So far this class of experiments focused only on the energy spectrum of the HO. On the other hand, experiments on resonant mass detectors could in principle also test the results concerning coherent and squeezed states <cit.>.Actual laboratory based systems, such as those referred above may already be near the threshold of observing such minute corrections.A different class of tests could be performed for larger systems that can be treated quantum mechanically. In fact, as we showed in Sec. <ref> and in particular in (<ref>), massive systems constitute ideal observational tools to probe GUP effects. However it is somewhat debatable whether the GUP applies only to fundamental constituents of matter (e.g. <cit.>), in which case Planck scale effects on mesoscopic and macroscopic objects are small, or if it can be applied to the center of mass of an optomechanical oscillator (and other systems) (e.g. <cit.>), in which case Planck scale effects may be observable withcurrent accuracies. For the latter, it will be a challenge to isolate GUP effects from the rest.In other words the GUP, being itself a modification of the Heisenberg Uncertainty Principle, should only be applied to systems that exhibit quantum properties.Indeed, we apply our results in Tables <ref> and <ref> only for systems that require a quantum description.It is interesting to comment on the choice of mass scale in (<ref>). It is generally expected that this is the Planck mass, but on empirical grounds we could replace M_Pl in (<ref>) with any mass scale that has yet to be probed by experiment.A conservative value would be something just above the LHC scale, e.g. E_LHC∼ 10 TeV. This might come from some hitherto undiscovered intermediate scale, or one deriving from a large 5-th dimension with order TeV Planck energy in five dimensions. Our estimated correctionswould then be amplified by a factor of M_Pl c^2 / E_LHC∼ 10^15, or equivalently by a different value of γ_0 in (<ref>). Since this ratio appears squared in (<ref>), the most interesting scenario would be the optomechanical oscillator experiments, for which the periods would vary from seconds to hours, potentially rendering the calculated effects observable.Alternatively, a scale E∼ 10^11 GeV would yield oscillation times of seconds for the LIGO detector, again observable at least in principle.Depending on the temporal resolution of the detectors, these arguments can set lower bounds to the relevant energy scale for the GUP (see Table <ref>). Finally, applications to Quantum Optics of the results shown in this paper can lead to important and potentially observable features <cit.>. Indeed, starting from a description of the electromagnetic field in terms of amplitude and phase quadratures, it is possible to include the GUP commutation relation to obtain a perturbed Hamiltonian. Following similar steps and definitions of the present paper, one can then construct a GUP-modified theory of Quantum Optics. This modification can have important implications on many optical experiments involving squeezed states, in particular future evolution of gravitational wave interferometers. This is specially relevant as the use of squeezed states have been planned for LIGO interferometers <cit.>. Therefore, while on one hand further study is still necessary to understand some of the features highlighted in this paper, on the other hand we may be one step closer to experimental evidences of Planck scale Physics.§ ACKNOWLEDGEMENTSThis work was supported in part by the Natural Sciences and Engineering Research Council of Canada, Perimeter Institute of Theoretical Physics through their affiliate program, and Quantum Alberta. 10Gross1988_1 D. Gross and P. Mende, “String theory beyond the Planck scale,” Nuclear Physics B, vol. 303, pp. 407–454, jul 1988.Amati1989_1 D. Amati, M. Ciafaloni, and G. 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"authors": [
"Pasquale Bosso",
"Saurya Das",
"Robert B. Mann"
],
"categories": [
"gr-qc",
"hep-th",
"quant-ph"
],
"primary_category": "gr-qc",
"published": "20170426163718",
"title": "Planck scale Corrections to the Harmonic Oscillator, Coherent and Squeezed States"
} |
Secure Precise Wireless Transmission with Random-Subcarrier-Selection-based Directional Modulation Transmit Antenna Array Feng Shu, Xiaomin Wu, Jinsong Hu, Riqing Chen, and Jiangzhou Wang This work was supported in part by the National Natural Science Foundation of China (Nos. 61472190, and 61501238), the Open Research Fund of National Key Laboratory of Electromagnetic Environment, China Research Institute of Radiowave Propagation (No. 201500013). Feng Shu, Xiaomin Wu, Jinsong Hu are with School of Electronic and Optical Engineering, Nanjing University of Science and Technology, 210094, CHINA. E-mail:{shufeng, xiaoming.wu}@njust.edu.cn Feng Shu and Riqing Chen are with the College of Computer and Information Sciences, Fujian Agriculture and Forestry University, Fuzhou 350002, China. E-mail: [email protected]. Feng Shu is also with the National Key Laboratory of Electromagnetic Environment, China Research Institute of Radiowave Propagation, Qingdao 266107, China Jiangzhou Wang iswiththe School of Engineering and Digital Arts, University of Kent, Canterbury Kent CT2 7NT, United KingdomDecember 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== We consider the family of problems, where the input consists of an instance I of size N over a universe U_I of size n and the task is to check whether the universe contains a subset with property Φ (e.g., Φ could be the property of being a feedback vertex set for the input graph of size at most k). Our main tool is a simple randomized algorithm which solves in time (1+b-1/c)^n N^O(1), provided that there is an algorithm for the problem with running time b^n-|X| c^k N^O(1). Here, the input for is an instance I of size N over a universe U_I of size n, a subset X⊆ U_I, and an integer k, and the task is to check whether there is a set Y with X⊆ Y ⊆ U_I and |Y∖ X|≤ k with property Φ. We also derandomize this algorithm at the cost of increasing the running time by a subexponential factor in n, and we adapt it to the enumeration setting where we need to enumerate all subsets of the universe with property Φ. This generalizes the results of Fomin et al. [STOC 2016] who proved them for the case b=1. As case studies, we use these results to design faster deterministic algorithms for * checking whether a graph has a feedback vertex set of size at most k,* enumerating all minimal feedback vertex sets,* enumerating all minimal vertex covers of size at most k, and* enumerating all minimal 3-hitting sets.We obtain these results by deriving new b^n-|X| c^k N^O(1)-time algorithms for the corresponding problems (or the enumeration variant). In some cases, this is done by simply adapting the analysis of an existing algorithm, in other cases it is done by designing a new algorithm. Our analyses are based on Measure and Conquer, but the value to minimize, 1+b-1/c, is unconventional and leads to non-convex optimization problems in the analysis. § INTRODUCTION In exponential-time algorithmics <cit.>, the aim is to design algorithms for NP-hard problems with the natural objective to minimize their running times. In this paper, we consider a broad class of subset problems, where for an input instance I on a universe U_I, the question is whether there is a subset S of the universe satisfying certain properties. For example, in the Feedback Vertex Set problem, the input instance consists of a graph G=(V,E) and an integer k, the universe is the vertex set and the property to be satisfied by a subset S is the conjunction of “|S|≤ k” and “G-S is acyclic”.More formally, and using definitions from <cit.>, an implicit set system is a function Φ that takes as input a string I ∈{0,1}^* and outputs a set system (U_I, _I), where U_I is a universeand _I is a collection of subsets of U_I.The string I is referred to as an instanceand we denote by |U_I| = n the size of the universe and by |I|= the size of the instance.We assume that ≥ n. The implicit set system Φ is polynomial time computable if (a) there exists a polynomial time algorithm that given I produces U_I, and (b) there exists a polynomial time algorithm that given I, U_I and a subset S of U_I determines whether S ∈_I. All implicit set systems discussed in this paper are polynomial time computable.An instance IA set S ∈ F_I if one exists.An instance I, a set X ⊆ U_I, and an integer k. Does there exists a subset S ⊆ (U_I ∖ X) such that S ∪ X ∈ F_I and |S| ≤ k?In recent work, Fomin et al. <cit.> showed that c^k N^O(1) time algorithms (c∈ O(1)) for lead to competitive exponential-time algorithms for many problems. The main tool was a simple randomized algorithm which solves in time (2-1/c)^n N^O(1) if there is an algorithm that solves in time c^k N^O(1). A derandomization was also given, turning the randomized algorithm into a deterministic one at the cost of a 2^o(n) factor in the running time. The method was also adapted to enumeration algorithms and combinatorial upper bounds. This framework, together with a large body of work in parameterized algorithmics <cit.>, where c^k N^O(1) time algorithms are readily available for many subset problems, led to faster algorithms for around 30 decision and enumeration problems.In this paper, we extend the results of Fomin et al. <cit.> and show that a b^n-|X| c^k N^O(1) time algorithms (b,c∈ O(1)) for lead to randomized (1+b-1/c)^n N^O(1) time algorithms for . Our result can be similarly derandomized and adapted to the enumeration setting. Observe that for b=1, the results of <cit.> coincide with ours, but that ours have the potential to be more broadly applicable and to lead to faster running times. The main point is that if we use a c^k N^O(1) time algorithm as a subroutine to design an algorithm exponential in n, we might as well allow a small exponential factor in n in the running time of the subroutine. Similar as in <cit.>, the problem can often be solved by preprocessing the elements in X in a simple way and then using an algorithm for a subset problem. In the case of Feedback Vertex Set, the vertices in X can simply be deleted from the input graph. Whereas the literature is rich with c^k N^O(1) time algorithms for subset problems, algorithms with running times of the form b^n c^k N^O(1) with b>1 are much less common. [One notable exception is by Eppstein <cit.>, who showed that all maximal independent sets of size at most k in a graph on n vertices can be enumerated in time (4/3)^n (81/64)^k n^O(1).] One issue is that there is, in general, no obviously best trade-off between the values of b and c for such algorithms. However, the present framework gives us a precise objective: we should aim for values of b and c that minimize the base of the exponent, (1+b-1/c).Our applications consist of three case studies centered around some of the most fundamental problems considered in <cit.>, feedback vertex sets and hitting sets. For the first case study, we considered the Feedback Vertex Set problem: given a graph G and an integer k, does G have a feedback vertex set of size at most k? For this problem, we re-analyze the running time of the algorithm from <cit.>. In <cit.>, the algorithm was analyzed using Measure and Conquer: using a measure that is upper bounded by α n and aiming for a running time of 2^α n n^O(1) the analysis of the branching cases led to constraints lower bounding the measure and the objective was to minimize α subject to these constraints. In our new analysis, we add an additive term w_k· k to the measure and adapt the constraints accordingly. If all constraints are satisfied, we obtain a running time of 2^α n+w_k k n^O(1). Our framework naturally leads us to minimize 2^α-2^-w_k. This approach leads to a O(1.5422^n · 1.2041^k) time algorithm, which, combined with our framework gives a deterministic time algorithm for Feedback Vertex Set. This improves on previous results giving O(1.8899^n) <cit.>, O(1.7548^n) <cit.>, O(1.7356^n) <cit.>, O(1.7347^n) <cit.>, and O(1.7216^n) <cit.> time algorithms for the problem. We note that adapting the analysis of other existing exact and parameterized algorithms did not give faster running times. Also, if we allow randomization, the O(1.6667^n) time algorithm by <cit.> (which can also be achieved using our framework) remains fastest.Our second case study is more involved. Simply using an existing algorithm and adapting the measure was not sufficient to improve upon the best known enumeration algorithms (and combinatorial upper bounds) for minimal feedback vertex sets. Here, the task is, given a graph G, to output each feedback vertex set that is not contained in any other feedback vertex set. We design a new algorithm for enumerating all minimal feedback vertex sets. We also need a new combinatorial upper bound for the number of minimal vertex covers of size at most k to handle one special case in the enumeration of minimal feedback vertex sets. [Previous work <cit.> along these lines focused on small maximal independent sets in the context of graph coloring, whose bounds were insufficient for our purposes. Here, we need better bounds on large maximal independent sets or small minimal vertex covers.] We obtain a O(1.7183^n · 1.1552^k) time algorithm for enumerating all minimal feedback vertex sets. Our framework thus leads to a running time of , improving on the previous best bound of O(1.8638^n) <cit.>.The current best lower bound for the number of minimal feedback vertex sets is O(1.5926^n) <cit.>. We would like to highlight that the enumeration of minimal feedback vertex sets is completely out of scope for the more restricted framework of <cit.>: the number of minimal feedback vertex sets of size at most k cannot be upper bounded by c^k n^O(1), as evidenced by a disjoint union of k cycles of length n/k.Our last case study gives a new algorithm for enumerating all minimal 3-hitting sets, also known as minimal transversals of rank-3 hypergraphs. These are minimal sets S of vertices of a hypergraph where each hyperedge has size at most 3 such that every hyperedge contains at least one vertex of S. We re-analyze an existing algorithm <cit.> for this enumeration problem, adapting the measure in a similar way as in the first case study, and we obtain a multivariate running time of O(1.5135^n · 1.1754^k), leading to an time enumeration algorithm. This breaks the natural time bound of O(1.6667^n) of the previously fastest algorithm <cit.>. The current best lower bound gives an infinite family of rank-3 hypergraphs with Ω(1.5848^n) minimal transversals <cit.>.§ PRELIMINARIESLet G = (V,E) be a graph with a set of vertices V and a set of edges E ⊆{uv : u, v ∈ V}. The degree d(u) of a vertex u is the number of neighbors of u in G. The degree of a graph Δ(G) is the maximum d(u) across all u ∈ V. A graph G' = (V', E') is a subgraph of G if V' ⊆ V and E' ⊆ E and G' is an induced subgraph of G if, in addition, G has no edge uv with u,v∈ V' but uv∉ E'. In this case, we also denote G' by G[V']. A forest is an acyclic graph. A subset F⊆ V is acyclic if G[F] is a forest. An acyclic subset F⊆ V is maximal in G if it is not a subset of any other acyclic subset. For an acyclic subset F⊆ V, we denote the set of maximal acyclic supersets of F as _G(F) and the set of maximum (i.e., largest) acyclic supersets of F as ^*_G(F).Let T be a subgraph of G. Define Id(T,t) as an operation on G which contracts all edges of T into one vertex t, removing induced loops. This may create multiedges in G. Define Id^*(T,t) as the operation Id(T,t) followed by removing all vertices connected to t by multiedges. A non-trivial component of a graph G is a connected component on at least two vertices. The following propositions from <cit.> will be useful. <cit.>Let G = (V, E) be a graph, F ⊆ V be an acyclic subset of vertices and T be a non-trivial component of G[F]. Denote by G' the graph obtained from G by the operation Id^*(T,t) and let F' = F ∪{t} T. Then for X' = X ∪{t} T where X, X' ⊆ V* X ∈_G(F) if and only if X' ∈_G'(F'), and * X ∈^*_G(F) if and only if X' ∈^*_G'(F').Using operation Id^* on each non-trivial component of G[F], results in an independent set F'. <cit.>Let G = (V , E) be a graph and F be an independent set in G such that VF = N(t) for some t ∈ F. Consider the graph G' = G[N(t)] and for every pair of vertices u,v ∈N(t) having a common neighbor in F {t} add an edge uv to G'. Denote the obtained graph by H and let I ⊆ N(t). Then F ∪ I ∈_G(F) if and only if I is a maximal independent set in H . In particular, F ∪ I ∈^*_G(F) if and only if I is a maximum independent set in H. For an acyclic subset F, a so-called active vertex t∈ F and a neighbor v∈ N(t)∖ F, we will now define the concept of generalized neighbors of v, as well as their generalized neighbors. Denote by K the set of vertices of F adjacent to v other than t. Let G' be the graph obtained after the operation Id(K ∪{v}, u). A vertex w ∈ V(G') \{t} is a generalized neighbor of v in G if w is a neighbor of u in G'. Denote by gd(v) the generalized degree of v which is its number of generalized neighbors. For a given generalized neighbor x of v, denote by K' the set of vertices in F adjacent to x. Denote G” as the graph obtained after the operation Id(K' ∪{x}, u'). A generalized neighbor of x is any vertex y ∈ V(G”) {v} which is adjacent to u' in G”. Also use the notation gd(x) to represent the generalized degree of x, which is a very similar notion to that of gd(v). Lastly, all randomized algorithms in this paper are Monte Carlo algorithms with one-sided error. On -instances they always return , and on -instances they return (or output a certificate) with probability >1/2.§ RESULTSOur first main result gives exponential-time randomized algorithms for based on single-exponential multivariate algorithms for with parameter k.theoremthmMainOneIf there is an algorithm for with running time b^n - |X| c^k^O(1) then there is a randomized algorithm for with running time (1+b-1/c)^n^O(1). The next main result derandomizes the algorithm of Theorem <ref> at a cost of a subexponential factor in n in the running time.theoremthmMainTwoIf there is an algorithm for with running time b^n - |X| c^k^O(1) then there is an algorithm for with running time (1+b-1/c)^n + o(n)^O(1).We require the following notion of (b,c)-uniform to describe our enumeration algorithms. Let c,b ≥ 1 be real valued constants and Φ be an implicit set system. Then Φ is (b,c)-uniform if for every instance I, set X ⊆ U_I, and integer k ≤ n - |X|, the cardinality of the collection ℱ_I,X^k = { S ⊆ U_IX : |S| = kandS ∪ X ∈ℱ_I } is at most b^n - |X| c^k n^O(1). Then the following theorem provides new combinatorial bounds for collections generated by (b,c)-uniform implicit set systems.theoremthmMainThreeLet c,b ≥ 1 and Φ be an implicit set system. If Φ is (b,c)-uniform then |ℱ_I| ≤(1 + b - 1/c)^n n^O(1) for every instance I. We say that an implicit set system is efficiently (b,c)-uniform if there exists an algorithm that given I, X and k enumerates all elements of _X,I^k in time b^n - |X|c^k ^O(1). In this case, we enumerate _I in the same time, up to a subexponential factor in n.theoremthmMainFourLet c,b ≥ 1 and Φ be an implicit set system. If Φ is efficiently (b,c)-uniform then there is an algorithm that given as input I enumerates _I in time ( 1 + b - 1/c)^n+o(n)^O(1). § RANDOM SAMPLING AND MULTIVARIATE SUBROUTINESIn this section, we prove Theorem <ref>. To do this, we first need the following lemmas.If b, c ≥ 1 then b ·c^1/bc≤ 1 + b - 1/c As both sides of the inequality are positive, it suffices to show that log (b c^1/bc) ≤log(1 + b - 1/c). So we let y = log(1 + b - 1/c) - log b - 1/bclog c and prove that y ≥ 0 for all b,c ≥ 1. When c = 1 we have that y = 0 for all b. We will show that for any fixed b ≥ 1 we have that y ≥ 0 by showing that y increases with c≥ 1. For fixed b, the partial derivative with respect to c is∂ y/∂ c = (bc + c - 1) log c - c + 1/bc^2(bc + c - 1). When c = 1 then for all b, ∂ y/∂ c = 0. As the denominator is positive for b,c ≥ 1 it is sufficient to show that the numerator z = (bc + c - 1) log c - c + 1 is non-negative. To show that z ≥ 0, we consider the partial derivative again with respect to c:∂ z/∂ c = (b + 1) log c + b - 1/c For b,c≥ 1, we have that b - 1/c≥ 0 and (b+1)log(c) ≥ 0. Since ∂ z/∂ c≥ 0, we conclude that z is increasing and non-negative which implies y is also increasing and non-negative, for all b, c ≥ 1. This proves the lemma. The proof of the next lemma follows the proof of Lemma 2.2 from<cit.>, who proved it for b=1. Let b, c ≥ 1, n and k ≤ n be non-negative integers. Then, there exists t ≥ 0 such thatnt/kt b^n-t c^k-t = ( 1 + b - 1/c)^n n^O(1) We consider two cases. First suppose k ≤n/bc. Then for t = 0 the LHS (left-hand side) is at most b^n c^k ≤ b^n c^n/bc≤( 1 + b - 1/c )^n by Lemma <ref>.Now if k > n/bc then we rewrite the LHS asnt/kt b^n-t c^k-t= nkb^n-k/n-tk-t( 1/bc)^k-tLet us lower bound the denominator. For any x ≥ 0 and an integer m ≥ 0,∑_i ≥ 0m + ii x^i = ∑_i ≥ 0m + im x^i = 1/(1-x)^m+1,by a known generating function. For m = n - k and x = 1/bc, the summand at i = k - t equals the denominator n-tk-t( 1/bc)^k-t. Since n/k < bc we have that m+k/k < 1/x and the terms of this sum decay exponentially for i > k. Thus, the maximum term (m+i)(m+i-1)… (m+1)/i(i-1) … 1 x^i for this sum occurs for i ≤ k, and its value is Ω( ( 1/1-x)^m ) up to a lower order factor of O(k). So by the binomial theorem the expression is at mostnkb^n-k(1-x)^n-k n^O(1) = ( 1 + b - 1/c)^n n^O(1)Specifically, the maximum term for Equation (<ref>) occurs when m + i/i = 1/x, that is when n - t/k - t = cb, and therefore, t=cbk-n/cb-1. If there exist constants b, c ≥ 1 and an algorithm for with running time b^n-|X|c^k ^O(1) then there exists a randomized algorithm for with running time ( 1 + b - 1/c)^n - |X|^O(1) Our proof is similar to Lemma 2.1 in <cit.>. Let ℬ be an algorithm for Φ-Extension with running time b^n-|X| c^k ^O(1). We now present a randomized algorithm 𝒜, for the same problem for an input instance (I,X,k') with k' ≤ k. * Choose an integer t ≤ k' depending on b, c, n, k' and |X|, the choice of which will be discussed later. Then select a random subset Y of U_I \ X of size t.* Run Algorithm ℬ on the instance (I, X ∪ Y, k' - t) and return the answer.Algorithm 𝒜 has a running time upper bounded by b^n-|X|-t c^k'-t^O(1). Algorithm 𝒜 returns yes for (I,X,k') when ℬ returns yes for (I, X ∪ Y, k' - t). In this case there exists a set S ⊆ U_I \ (X ∪ Y) of size at most k'- t ≤ k - t such that S ∪ X ∪ Y ∈ℱ_I. This, Y ∪ S witnesses that (I, X, k) is indeed a yes-instance.Next we lower bound the probability that 𝒜 returns yes if there exists a set S ⊆ U_I \ X of size exactly k' such that X ∪ S ∈ℱ_I. The algorithm 𝒜 picks a set Y of size t at random from U_I \ X. There are n - |X|t possible choices for Y. If 𝒜 picks one of the k't subsets of S as Y then 𝒜 returns yes. Thus, given that there exists a set S ⊆ U_I \ X of size k' such that X ∪ S ∈ℱ_I, we have that[ 𝒜 returns yes] ≥[Y ⊆ S] = k't / n - |X|tLet p(k') =k't / n - |X|t.For each k' ∈{0 ,... , k}, our main algorithm runs 𝒜 independently 1/p(k') times with parameter k'. The algorithm returns yes if any of the runs of 𝒜 return yes. If (I, X, k') is a yes-instance, then the main algorithm returns yes with probability at least min_k' ≤ k{ 1 - (1 - p(k'))^1/p(k')}≥ 1 - 1/e > 1/2.Next we upper bound the running time of the main algorithm, which is ∑_k' ≤ k1/p(k') b^n - |X| - t c^k' - t^O(1) ≤max_k' ≤ kn - |X|t/k't b^n - |X| - t c^k' - t^O(1)≤max_k' ≤ n - |X|n - |X|t/kt b^n - |X| - t c^k - t^O(1). The choice of t in algorithm 𝒜 is chosen to minimize the value of n - |X|t/kt b^n - |X| - t c^k - t. For fixed n and |X| the running time of the algorithm is upper bounded bymax_0 ≤ k ≤ n - |X|{min_0 ≤ t ≤ k{n - |X|t/kt b^n - |X| - t c^k - t^O(1)}}By application of Lemma <ref> we choose t = cbk - (n - |X|)/cb - 1 to obtain the upper bound( 1 + b - 1/c)^n - |X| (n - |X|)^O(1), which, combined with n < N, completes the proof. Running algorithm 𝒜 with X = ∅ and for each value of k ∈{0,....,n} results in an algorithm forwith running time ( 1 + b - 1/c)^n ^O(1), proving Theorem <ref>.§ DERANDOMIZATIONIn this section we prove Theorem <ref>, by derandomizing the algorithm in Theorem <ref>.*Given a set U and an integer q ≤ |U| let Uq represent the set of sets which contain q elements of U. From <cit.> we define a pseudo-random object, the set-inclusion-family, as well as an almost optimal sub-exponential construction of these objects. Let U be a universe of size n and let 0 ≤ q ≤ p ≤ n. A family 𝒞⊆Uq is an (n, p, q)-set-inclusion family, if for every set S ∈Up, there is a set Y ∈𝒞 such that Y ⊆ S. Let κ(n,p,q) = nq / pq. We also make use of the following theorem.There is an algorithm that given n, p and q outputs an (n,p,q)-set-inclusion-family 𝒞 of size at most κ(n,p,q) · 2^o(n) in time κ (n,p,q) · 2^o(n). We are now ready to prove Lemma <ref>, by a very similar proof to Lemma <ref>.If there exists constants b, c ≥ 1 and an algorithm for with running time b^n-|X| c^k^O(1) then there exists a deterministic algorithm for with running time ( 1 + b - 1/c)^n - |X|· 2^o(n)·^O(1). Let ℬ be an algorithm for with running time b^n-|X| c^k ^O(1). We can then adapt Algorithm 𝒜 from the proof of Lemma <ref>. Let 𝒜' be a new algorithm which has an input instance (I, X, k') with k' ≤ k. Choose t = cbk' - (n - |X|)/cb - 1.* Compute an (n - |X|, k', t)-set-inclusion-family 𝒞 using the algorithm from Theorem <ref> of size at most κ(n-|X|, k', t) · 2^o(n), in κ(n-|X|, k', t) · 2^o(n) time.* For each set Y in the set-inclusion-family 𝒞 run algorithm ℬ on the instance (I, X ∪ Y, k' - t) and return Yes of at least one returns Yes and No otherwise.The running time of 𝒜' is upper bounded by κ(n - |X|, k', t) · 2^o(n)· b^n-|X|-t c^k'-t^O(1), a term encountered in Equation <ref> with a new subexponential factor in n,max_k' ≤ kn - |X|t/k't· b^n-|X|-t c^k'-t^O(1)· 2^o(n).From here the proof follows that of Lemma <ref>.The proof of Theorem <ref> follows by inclusion of the factor 2^o(n).§ ENUMERATIONWe now proceed to prove Theorem <ref>, and <ref> on combinatorial upper bounds and enumeration algorithms.Consider the following random process.* Choose an integer t based on b, c, n and k, then randomly sample a subset X of size t from U_I.* Uniformly at random pick a set S from _I, X^k-t, and output W = X ∪ S. In the special case where _I,X^k-t is empty output the empty set. * Let I be an instance, k ≤ n. We will prove that the number of sets in _I of size exactly k is upper bounded by |_I| ≤( 1 + b - 1/c)^n n^O(1), where k is chosen arbitrarily. We follow the random process described above, which picks a set W of size k from _I.For each set Z ∈_I of size exactly k, let E_Z denote the event that the set W output in step 2 is equal to Z. We then have the following lower bound on the probability of the event E_Z: [E_Z]= [X ⊆ ZS = Z \ X] = [X ⊆ Z] ×[S = Z \ Z | X ⊆ Z] = kt/nt·1/|_I,X^k-t| Since Φ is (b,c)-uniform then |_I,X^k-t| ≤ b^n - |X|c^k n^O(1) and X is selected such that |X| = t, this results in the lower bound[E_Z] ≥kt/nt b^-(n - t)c^-(k-t) n^-O(1).A choice of t is made to minimize the lower bound, and this choice is given by Lemma <ref> which states that for every k ≤ n there exists a t ≤ k such that we obtain a new lower bound[E_Z] ≥( 1 + b - 1/c)^-n· n^O(1)for every Z ∈_I of size k. For every individual set Z ∈_I, the event E_Z occurs disjointly, and we have that ∑_Z ∈_I, |Z| = k[E_Z] ≤ 1.This fact with the lower bound of [E_Z] implies an upper bound on the number of sets in _I of (1 + b - 1/c)^n n^O(1), completing the proof.* We alter the random process used to prove Theorem <ref> to a deterministic one: * Construct a (n, k, t)-set inclusion family 𝒞 using Theorem 6 from <cit.>. Loop over X ∈𝒞.* For each X ∈𝒞, loop over all sets S ∈_I,X^k-t.Then we output W = X ∪ S from these two loops. Looping over 𝒞 instead of random sampling for X incurs a 2^o(n) overhead in the running time. As Φ is efficiently (b,c)-uniform, the inner loop requires (1 + b - 1/c)^n N^O(1) time. In order to avoid enumerating duplicates, we save the sets which have been output in a trie and check first in linear time if a set has already been output. The product of the running times for these two nested loops results in the running time claimed by the theorem statement. § FEEDBACK VERTEX SETFirst we describe the extension variant of Feedback Vertex SetA graph G = (V,E), vertex subset X ⊆ V and an integer k Does there exist subset S ⊆ V \ X such that S ∪ X is a FVS and |S| ≤ k? Instead of directly finding the feedback vertex set in a graph, we present algorithm (G,F,k) <cit.> which computes for a given graph G and an acylic set F the maximum size of an induced forest F' containing F with |F'| ≥ n - k. This means that G - F is a minimal feedback vertex set of size at most k. This algorithm can easily be turned into an algorithm computing at least one such set. During the execution ofone vertex t ∈ F is called an active vertex. Algorithmthen branches on a chosen neighbor of t. Let v ∈ N(t). Let k be the set of all vertices of F \{t} that are adjacent to v and parameter k which represents a bound on the size of the feedback vertex set.As well as describing the algorithm we simultaneously perform the running time analysis which uses the Measure and Conquer framework and Lemma <ref> at its core.<cit.>Let A be an algorithm for a problem P, B be an algorithm for a class C of instances of P, c ≥ 0 and r > 1 be constants, and μ(·), μ_B(·), η(·) be measures for P, such that for any input instance I from C, μ_B(·) ≤μ (I), and for any input instance I, A either solves P on I ∈ C by invoking B with running time O(η(I)^c+1r^μ B(I)), or reduces I to k instances I_1,...,I_k, solves these recursively, and combines their solutions to solve I, using time O(η(I)^c) for the reduction and combination steps (but not the recursive solves),(∀ i) η(I_i) ≤η (I) - 1,and ∑_i=1^k r^μ(I_i)≤ r^μ(I).Then A solves any instance I in time O(η(I)^c+1 r^μ(I)). Branching constraints of the form∑_i = 1^j 2^-δ_i≤ 1 are given as branching vectors (δ_1, ..., δ_j).§.§.§ Measure To upper bound the exponential time complexity of the algorithmwe use the measureμ = |N(t) \ F|w_1 + |V \ (F ∪ N(t))|w_2 + k · w_k.In other words, each vertex in F has weight 0, each vertex in N(t) has weight w_1, each other vertex has weight w_2 and each unit of budget for the feedback vertex set has weight w_k, in the measure with an active vertex t.§.§.§ Algorithm The description ofconsists of a sequence of cases and subcases. The first case which applies is used, and inside a given case the hypotheses of all previous cases are assumed to be false. Preprocessing procedures come before main procedures.*Preprocessing* If G consists of j ≥ 2 connected components G_1, G_2, ... ,G_j, then the algorithm is called on each component. For F_i = G_i ∩ F for all i ∈{1, 2, ..., j} and ∑_i = 1^j k_i ≤ k then(G, F, k) = ∑_i=1^j(G_i, F_i, k_i) * If F is not independent, then apply operation Id^*(T, v_T) on an arbitrary non-trivial component T of F. If T contains the active vertex then v_T becomes active. Let G' be the resulting graph and let F' be the set of vertices of G' obtained from F. Then(G, F, k) = (G', F', k) + |T| - 1*Main Procedures* If k < 0 then(G, F, k) = 0. * If F = ∅ and Δ(G) ≤ 1 then _G(F) = {V} and(G, F, k) = |V|. * If F = ∅ and Δ(G) ≥ 2 then the algorithm chooses a vertex t ∈ G of degree at least 2. Then t is either contained in a maximum induced forest or not. The algorithm branches on two subproblems and returns the maximum:(G, F, k) = max{(G, F ∪{t}, k), (G \{t}, F, k-1) }.The first branch reduces the weight of t to zero, as it is in F, and at least 2 neighbors have a reduced degree from w_2 to w_1. In the second branch we remove t from the graph, meaning it will be in the feedback vertex set. We thus also gain a reduction of w_k in the measure. Hence this rule induces the branching constraint(w_2 + 2(w_2 - w_1), w_2 + w_k).* If F contains no active vertex then choose an arbitrary vertex t ∈ F as an active vertex. Denote the active vertex by t from now on. * If V \ F = N(t) then the algorithm constructs the graph H from Proposition <ref> and computes a maximum independent set I in G of maximum size n - k. Then(G, F, k) = |F| + |I|. * If there is v ∈ N(t) with gd(v) ≤ 1 then add v to F.(G,F,k) = (G, F ∪{v}, k).* If there is v ∈ N(t) with gd (v) ≥ 4 then either add v to F or remove v from G.(G, F, k) = max{(G, F∪{v}, k), (G \{v}, F, k-1) }.The first case adds v to F reducing the measure by w_1, and a minimum of 4(w_2 - w_1) for each of the generalized neighbors. The other case removes v this decreasing the measure by w_k. Hence this rule induces the branching constraint(w_1 + 4(w_2 - w_1), w_1 + w_k). *If there is v ∈ N(t) with gd(v) = 2 then denote its generalized neighbors by u_1 and u_2. Either add v to F or remove v from G but add u_1 and u_2 to F. If adding u_1 and u_2 to F induces a cycle, we just ignore the last branch.(G, F, k) = max{(G, F∪{v}, k), (G \{v}, F ∪{u_1, u_2}, k-1) }.Let i ∈{0, 1, 2} be the number of vertices adjacent to v with weight w_2. The first case adds v to F, and hence all i w_2-weight neighbors of v reduce to w_1, and the other 2-i vertices of weight w_1 induce a cycle, hence we remove them from G and reduce the measure by (2-i)w_k. The second case removes v and adds both u_1 and u_2 to F. This causes a reduction of i w_2 for the relevant vertices and (2-i)w_1 for the other vertices, and a single w_k reduction due to the removal of v. This rule induces the branching constraint(w_1 + i (w_2 - w_1) + (2 - i)w_1 + (2-i)w_k, w_1 + iw_2 + (2-i)w_1 + w_k).* If all vertices in N(t) have exactly three generalized neighbors then at least one of these vertices must have a generalized neighbor outside N(t), since the graph is connected and the condition of the case Main 6 does not hold. Denote such a vertex by v and its generalized neighbors by u_1, u_2 and u_3 in such a way that u_1 ∉N(t). Then we either add v to F; or remove v from G but add u_1 to F; or remove v and u_1 from G and add u_2 and u_3 to F. Similar to the previous case, if adding u_2 and u_3 to F induces a cycle, we just ignore the last branch.(G,F) = max{ (G, F ∪{v}, k), (G \{v}, F ∪{u_1}, k - 1), (G \{v, u_1}, F ∪{u_2, u_3}, k - 2)}.Let i ∈{1, 2, 3} be the number of vertices adjacent to v with weight w_2. The first and last cases are analogous to the analysis done in Main <ref>. The second case removes v from the forest hence adding it to the minimum feedback vertex set and reducing the measure by w_1 + w_k. A reduction of w_2 is gained by adding u_1 to F. Then this rule induces the branching constraint(w_1 + i(w_2-w_1) + (3-i) w_1 + (3-i) w_k, w_1 + w_2 + w_k, w_1 + i w_2 + (3-i) w_1 + 2 w_k ). §.§.§ Results Let G be a graph on n vertices. Then a minimal feedback vertex set in G can be found in time . Using the algorithm above along with the measure μ, the following values of weights can be shown to satisfy all the branching vector constraints generated above.w_1 = 0.2775w_2 = 0.6250w_k = 0.2680These weights result in an upper bound for the running time of as O(1.5422^n · 1.2041^k) for computing a maximally induced forest of size a least n - k, and hence we have the running time for of O(1.5422^n-|X|· 1.2041^k). By Theorem <ref> this results in a algorithm for computing a minimal feedback vertex set. § MINIMAL VERTEX COVERS theoremthmMVCresultLetbe a constant with 0.169925 ≈ 2 log_2 3-3 ≤≤ 1. For every n≥ k≥ 0, and every graph G on n vertices, the number of minimal vertex covers of size at most k of G is at most 2^ n +k, where = (1-)/2.The proof is by induction on n. For the base case, a graph on at most one vertex has one minimal vertex cover – the empty set – and 2^ n +k≥ 1 since n +k≥ 0.Suppose the statement holds for graphs with fewer than n vertices. We will repeatedly use the observation that for every vertex v, no minimal vertex cover of G contains N[v]. Let v be a vertex of minimum degree in G.If v has degree 0, then no minimal vertex cover contains v. Thus, G has as many minimal vertex covers as G-v. The number of minimal vertex covers of G is therefore upper bounded by2^ (n-1) +k≤ 2^ n +k. If v has degree 1, then every minimal vertex cover either excludes v but includes its neighbor u, or it includes v but excludes u. The number of minimal vertex covers of G is therefore upper bounded by2· 2^ (n-2) +(k-1)≤ 2^ n +k -(2 +) +1 = 2^ n +ksince 2 + = 1.If v has degree 2, then every minimal vertex cover excludes a vertex among N[v], but includes its neighbors who all have degree at least 2. The number of minimal vertex covers of G is therefore upper bounded by3· 2^ (n-3) +(k-2)≤ 2^ n +k -(3 +2) +log_2 3≤ 2^ n +ksince 3 +2 = 3+/2≥log_2 3.If v has degree at least 3, every minimal vertex cover includes v or excludes v but includes all its neighbors. The number of minimal vertex covers of G is therefore upper bounded by2^ (n-1) +(k-1) + 2^ (n-4) +(k-3) ≤ 2^ n +k· (2^--+2^-4 -3)= 2^ n +k· (2^-1+/2+2^-2-) ≤ 2^ n +ksince 2^-1+/2+2^-2-≤ 0.89. The upper bound of Theorem <ref> is tight for everywithin the constraints of the theorem, as shown by 1-regular graphs with k=n/2. For = 2 log_2 3-3, the disjoint union of triangles also matches the upper bound for k=2n/3.We note that the proof of Theorem <ref> can easily be turned into an algorithm enumerating all minimal vertex covers of G in time 2^ n +k n^O(1). Alternatively, a polynomial-delay algorithm, such as the one by <cit.>, could be used for the enumeration. § MINIMAL FEEDBACK VERTEX SETSIn this section, we apply our framework to enumerating all minimal feedback vertex sets of an undirected graph on n vertices. We will modify the algorithm from <cit.>, and conduct a multivariate branching analysis.When combined with Theorem <ref> we obtain an algorithm for enumerating all minimal feedback vertex sets in time . §.§ MeasureFollowing <cit.>, we show that for any acyclic subset F of G = (V,E), |_G(∅)| ≤ .We assume F is independent by Proposition <ref>. For a graph G, an independent set F, and an active vertex t ∈ F, we use the measure:μ(G,F,t) = |A|+ |N(t)(F ∪ A)|+ |V(F ∪ N(t))|+ k ·where the set A ⊆ N(t)F consists of vertices which have generalized degree at least 3. We apply positive weights ,andto the three sets defined, with 0 ≤≤≤. A weight ofis applied to the each vertex in the feedback vertex set. §.§ Algorithm Similar to Feedback Vertex Set in Subsection <ref>, we perform an algorithm description and a running time analysis using a Measure and Conquer framework simultaneously. Let f(G,F, k)= |M_G (F )| be the number of maximal induced forests containing F of size at least n - k. Let f (μ, k) be a maximum f(G,F, k) among all four-tuples (G,F,t, k) of measure at most μ.For the algorithm denote t ∈ F to be the active vertex. If F ≠∅ contains no active vertex then we choose an arbitrary vertex as active, reducing the measure.§.§.§ Cases * If k < 0 then f(G, F, k) = 0. * If k = 0 then f(G, F, k) = 1 if G = F otherwise f(G,F,k) = 0. * F = ∅. If Δ(G) ≤ 1 then _G(F) = {V} so f(G,F,k) = 1. Otherwise choose an active vertex t ∈ V of degree at least 2. Every maximal forest either contains t or does not, meaning that the number of maximal forests isf(G, {t}, k) + f(G {t}, ∅, k - 1)which results in the branching vector( + 2( - ),+ ).From now on, denote t ∈ F as the active vertex. Let G_t = (V_t, E_t) be the connected component of G which contains t.* V_tF = N(t). By Proposition <ref>, f(μ, k) is equal to the number of maximal independent sets in the graph H of size at least n - k. By Theorem <ref>, we have an upper bound on the number of minimal vertex covers of size at most k giving us an upper bound also on the maximal independent sets of size at least n - k. We ensure that this computation is not worse than that of enumerating feedback vertex sets. f(μ, k) ≤ 2^ n +k≤ 2^μfor 2 log_2 3 - 3 ≤≤ 1 and = (1 - )/2. * gd(v) = 0. In this case every forest X ∈_G(F) contains v thusf(G, F, k) = f(G, F ∪{c}, k)which does not induce a branching vector.From this point on, pick a vertex v ∈ N(t). If there is no such vertex then t is no longer an active vertex and if F ≠∅ then we choose an arbitrary vertex in F as active. * gd(v) = 1. In this case every forest X ∈_G(F) either contains v or does not contain v and contains its generalized neighbor u. This means that the number of maximal induced forests is at mostf(G, F ∪{v}, k) + f(G {v}, F ∪{u}, k-1).If we have that u ∈ N(t), in the worst case we have the branching vector( + ( + ),++ ) otherwise if u ∉N(t) we have the branching vector ( + ( - ),++ ).* gd(v) = 2. Denote the generalized neighbors of v by u_1 and u_2, and assume that u_1 ∉N(t). If u_2 ∈ N(t) and v belongs to a maximal induced forest X then u_2 does not belong to X. Then every forest X from _G(F) satisfies one of the following conditions:* either X contains v, but not u_2,* or X does not contain v, and contains u_1,* or X does not contain v and u_1 but contains u_2. So the number of maximal forests is at mostf(G {u_2}, F ∪{v}, k) + f(G {v}, F ∪{u_1}, k-1) + f(G {v, u_1}, F ∪{u_2}, k-2).In the worst case, where u_2 has weight , then this results in the branching vector ( + ( - ) ++ ,++ ,+++ 2). However, if u_1, u_2 ∉N(t), assume gd(u_1) ≤ gd(u_2). If not, swap u_1 and u_2. We consider new subcases and rules based on d(u_1), the generalized degree of the vertex, and the structure of the local graph near the vertex u_1. If gd(u_1) = 2, let x_1 and x_2 be the two generalized neighbors of u_1. The weights of v, u_1, u_2 are , , respectively. We also note that when v is selected, u_1 and u_2, if not removed from the graph or already considered in the branching analysis, will result in a reduction in measure of at least ( - ) for each of u_1 and u_2. The branching analysis below has different weights for x_1 and x_2 depending on the subcase of the algorithm which is applied.* gd(u_1) = 0. Since every maximal forest X ∈_G(F) will contain u_1, then f(G, F, k) = f(G, F ∪{u_1}, k). This does not induce a branching vector. * gd(u_1) = 1. Let the generalized neighbor of u_1 be x. Then every forest X from _G(F) satisfies one of the following conditions: * either X contains v;* or X does not contain v but contains u_1;* or X does not contain v and u_1 but contains u_2 and xwhich means the number of maximal induced forests is at mostf(G, F ∪{v}, k) + f(G {v}, F ∪{u_1}, k-1) + f(G {v, u_1}, F ∪{u_2, x}, k - 2)which with worst case weights will induce the branching vector( + 2( - ),++ ,+ 2+ 2+ ).* gd(u_1) = 2, x_1 ∈ N(t) and x_1 is generalized neighbor of u_1 and u_2. If swapping x_1 and x_2 results in this case occurring, then do so. Then every forest X from _G(F) satisfies one of the following conditions: * either X contains v and x_1, but does not contain u_1 and u_2;* or X contains v and not x_1;* orX does not contain v and contains u_1;* orX does not contain v and u_1 but contains u_2.This means that the number of maximal induced forests is at mostf(G {u_1, u_2}, F ∪{v, x_1}, k - 2) + f(G {x_1}, F ∪{v}, k - 1) + f(G {v}, F ∪{u_1}, k- 1) + f(G {v, u_1}, F ∪{u_2}, k - 2). Since it is possible that x_1 has weightthen in the worst case then this results in the branching vector( ++ 2 + 2,+ 2( - ) ++ ,++ ,+ 2 + 2).* gd(u_1) = 2, x_1 ∈ N(t) and x_2 ∈ N(t).hen every forest X from _G(F) satisfies one of the following conditions: * either X contains v and u_1, but does not contain x_1 and x_2;* or X contains v and not u_1;* or X does not contain v and contains u_1;* or X does not contain v and u_1 but contains u_2. This means that the number of maximal induced forests is at mostf(G {x_1, x_2}, F ∪{v, u_1}, k - 2) + f(G {u_1}, F ∪{v}, k - 1) + f(G {v}, F ∪{u_1}, k- 1) + f(G {v, u_1}, F ∪{u_2}, k - 2). In the worst case we have x_1, x_2 obtaining a weight of , the branching vector is( ++ 2 + 2 + ( - ),+++ ( - ),++ ,+ 2+ 2).* gd(u_1) = 2, and previous subcases do not apply. At least one of x_1 and x_2 has weightotherwise we would be in case (d). Let x_1 have weight , and if not we can swap x_1 and x_2. Then every forest X from _G(F) satisfies one of the following conditions: * either X contains v;* or X does not contain v and contains u_1;* or X does not contain v and u_1 but contains u_2 and x_1;* or X does not contain v, u_1 and x_1 but contains u_2 and x_2. The number of maximal induced forests is thus at mostf(G, F ∪{v}, k)+ f(G {v}, F ∪{u_1}, k - 1) + f(G {v, u_1}, F ∪{u_2, x_1}, k - 2)+ f(G {v, u_1, x_1}, F ∪{u_2, x_2}, k - 3). We now consider the weight of x_2, which is only ever selected into the forest. If x_2 is of weightthe measure reduces by . If x_2 is of weightsince x_2 doesn't have both u_1 and u_2 as generalized neighbors due to case (c), in the worst case when x_2 is selected we also obtain a ( - ) reduction. If x_2 is of weight , we now have at least 2 unaccounted generalized neighbors which obtain a 2( - ) reduction.We induce the following constraints to simplify the analysis so that in the worst case, we obtain a reduction ofwhenever x_2 is selected≤ + 2( - )and ≤ + ( - ) which results in the branching vector( + 2( - ),++ ,+ 3 + 2,+ 4 + 3). * gd(u_1) ≥ 3. This means that gd(u_2) ≥ 3 as well.hen every forest X from _G(F) satisfies one of the following conditions: * either X contains v;* or X does not contain v and contains u_1;* or X does not contain v and u_1 but contains u_2.This means the number of maximal induced forests is at mostf(G, F ∪{v}, k) + f(G {v}, F ∪{u_1}, k - 1) + f(G {v, u_1}, F ∪{u_2}, k - 2). Since both u_1 and u_2 are generalized neighbors of v with gd ≥ 3 meaning they both obtain a weight ofwhen v is selected into the forest X. This establishes the branching vector( + 2( - ),++ ,+ 2 + 2). * gd(v) = 3. Denote the generalized neighbors of v by u_1, u_2 and u_3 according to the rule that u_j ∉N(t) and u_k ∈ N(t) if and only if j < k. Let i be the number of generalized neighbors of v that are not adjacent to t. For i = 1,2 we have that every forest X from _G(F) satisfies one of the following conditions: * either X contains v;* or X does not contain vmeaning the number of maximal induced forests is at mostf(G, F ∪{v}, k) + f(G {v}, F, k-1). Each of the 3 - i generalized neighbors which are a neighbor of t induces a cycle when v is selected so we instead remove the vertex. This results in the branching vector( + i( - ) + (3 - i)( + ),+ ). When i = 3 we take care of each case separately depending on the generalized degree of u_1 and the structure of its neighbors. Let gd(u_1) ≤ gd(u_2) ≤ gd(u_3). If gd(u_2) = 2 then let x_1 and x_2 be the two generalized neighbors.* gd(u_1) = 0. Every maximal forest X ∈_G(F) will contain u_1, so f(G,F,k) = f(G, F ∪{u_1}, k). This doesn't induce a branching vector. * gd(u_1) = 1. Let x be the generalized neighbor of u_1. Then every forest X from _G(F) satisfies one fo the following conditions: * either X contains v;* or X does not contain v but contains u_1;* or X does not contain v and u_1 but contains x.This means the number of maximal induced forests is at mostf(G, F ∪{v}, k) + f(G {v}, F ∪{u_1}, k-1) + f(G {v, u_1}, F ∪{x}, k - 2).Since x has weight at least , then this will induce the following branching vector( + 3( - ),++ , 2 ++ 2 ).* gd(u_1) = 2, x_1 ∈ N(t) and x_1 is also a generalized neighbor of either u_2 or u_3 (or both). We then have that every forest X from _G(F) satisfies one of the following conditions: * either X contains v and x_1 but does not contain u_1 and either u_2 or u_3 (or both);* or X contains v and does not contain x_1;* or X does not contain v. This means the number of maximal induced forests is at mostf(G {u_1, u_2}, F ∪{v, x_1}, k-2) + f(G {x_1}, F ∪{v}, k-1) + f(G {v}, F, k - 1). Since x_1 is a neighbor of t, and was chosen instead of v, then this means that t has weight , and has 3 generalized neighbors, two of which are u_1 and either u_2 or u_3. If at least three generalized neighbors of x_1 is u_1, u_2 and u_3 we have a more desirable branching, hence we assume that at most 2 generalized neighbors of v are also generalized neighbors of x_1. But since at least 1 generalized neighbor of x_1 is not a generalized neighbor of v, then we gain at least a - reduction when x_1 is chosen into the forest X.This case results in the branching vector( + ( + ( - )) + 2( + ) , ++ 3( - ) + ,+)* gd(u_1) = 2, and x_1 ∈ N(t) and x_2 ∈ N(t). We note that due to previous cases, x_1 and x_2 are only generalized neighbors of u_1 and not u_2 or u_3. Then every forest X from _G(F) satisfies one of the following conditions: * either X contains v and u_1 but not x_1 and x_2;* or X contains v but not u_1;* or X does not contain v. This means the number of maximal induced forests is at mostf(G {x_1, x_2}, F ∪{v, u_1}, k-2) + f(G {u_1}, F ∪{v}, k-1) + f(G {v}, F, k - 1).In the first case, selecting v of weightalso reduces the measure by 2( - ), one for each of u_2 and u_3. Selection of u_1 reduces the measure byand removing x_1 and x_2 results in a reduction of at least 2( + ). The second case also selects v, but removes u_1 for a measure decrease of +. The third case just removes v, for a total decrease of +. which results in the branching vector( + 2( - ) ++ 2 ( + ),+ 2( - ) ++,+ ).* gd(u_1) = 2, and previous subcases don't apply. At least one of x_1 and x_2 has weightotherwise we would be in case (b). Let x_1 has weight . Then every forest X from _G(F) satisfies one of the following conditions: * either X contains v;* or X does not contain v but contains u_1;* or X does not contain v and u_1 but contains u_2 and x_1;* or X does not contain v, u_1 and x_1 but contains u_2 and x_2;* or X does not contain v, u_1 and u_2 but contains u_3 and x_1;* or X does not contain v, u_1, u_2 and x_1 but contains u_3 and x_2.Now we consider the weight of x_2, which is only ever selected into the forest. In these cases, if x_2 has weightwe simply reduce the measure by . Now x_2 cannot has weightsince case 6 did not occur. So if x_2 is of weight , it has at least 3 generalized neighbors of which only 1 is u_1 and u_2 and u_3, are not generalized neighbors of x_2. This means that when x_1 is selected into the forest X we have another 2 ( - ) reductions in the worst case.To simplify our analysis into a single branching vector, we enforce that theweight reduction is the worst case≤ + 2( - ).This effectively means that if vertex x_2 is ever selected, then in the worst case there is a reduction of measure of at least . We thus obtain the branching vector( + 3( - ),++ ,+ 3 + 2, + 4 + 3,+ 4 + 3,+ 5 + 4).* gd(u_1) ≥ 3. Every forest X from _G(F) satisfies one of the following conditions * either X contains v* or X does not contain v.This means the number of maximal induced forests is at mostf(G, F ∪{v}, k) + f(G {v}, F, k - 1). Since all of u_i for i = 1,2,3 have at least 3 generalized neighbors, then we obtain the branching vector( + 3( - ),+ ).* gd(v) ≥ 4. Every forest X from _G(F) either contains v or doesn't contain v. Hence the number of forests is upper bounded byf(G, F ∪{v}, k) + f(G {v}, F, k - 1)which results in the branching vector( + 4( - ),+ ). §.§ ResultstheoremthmMFVSresultFor a graph G with n vertices, all minimal feedback vertex sets can be enumerated in time .We evaluate the running time of the proposed algorithm above using the stated measure μ. It can be shown that the weights= 0.4506859777,= 0.4233244855,= 0.7809613776,= 0.2081356098,satisfy all stated branching factors and constraints necessary.The number of maximal induced forests containing F of size at least n - k is upper bounded by f(μ, k) ≤ 2^· 2^≤ 1.7183^n 1.1552^kThis results in a O(1.7183^n 1.1552^k) algorithm for enumerating the maximal induced forests of size at least n-k, and also enumerating minimal feedback vertex sets of size k.Consider now the enumeration of the collection_I,X^k = {S ⊆ V \ X : |S| = kand S ∪ X is a minimal FVS}.By running the new algorithm just described on the subgraph G[VX] that remains after removing the vertices of X, we enumerate all minimal feedback vertex sets of size k in time 1.7183^n · 1.1552^k · N^O(1). For every minimal feedback vertex set S that was just enumerated, we can check in polynomial time if S ∪ X is also a minimal feedback vertex set. This means that the collection _I,X^k can thus also be enumerated in time 1.7183^n - |X|· 1.1552^k · N^O(1). Combined with Theorem <ref> this results in a deterministic algorithm for the number of minimal feedback vertex sets in a graph G.§ MINIMAL HITTING SETS Based on <cit.> we once again apply a multivariate analysis to enumerating all minimal hitting sets on a hypergraph of rank 3. §.§ Measure We conduct a new analysis using the algorithm described in <cit.>, by first deriving a similar measure. Let H be a hypergraph of rank 3 and k be an upper bound on the size of the hitting set S ⊆ V. Denote by n_i the number of vertices of degree i ∈ℕ and m_i the number of hyperedges of size i ∈{0,...,3}. Also denote m_≤ i := ∑_j = 0^i m_j. Then a measure for H and a given k isμ(H, k) = (m_≤ 2) + ∑_i = 0^∞ w_i n_i + w_k · kwhere : ℕ→ℝ_≥ 0 is a non-increasing non-negative function independent of n, and ω_i are non-negative reals. Clearly μ(H, k) ≥ 0. We make the same simplifying assumptions as <cit.> which provides the constraintsω_i:= ω_5,(i):= 0 for eachi ≥ 6Δω_i:= ω_i - ω_i-1, Δ(i):= (i) - (i-1) for eachi ≥ 1 0≤δω_i+1≤Δω_i, and0≥Δ(i+1) ≥Δ(i) for eachi ≥ 1 Further branching rules will add constraints on the measure. Denote T(μ) := 2^μ as an upper bound on the number of leaves of the search tree modelling the recursive algorithm for all H with μ(H) ≤μ. §.§ AnalysistheoremthmMHSresult For a hypergraph H with n vertices and rank 3, all minimal hitting sets can be enumerated in time .We follow the algorithm and rules from <cit.>, but adding an additional weight w_k to the decrease in measure every time a vertex is to be selected into S, the partial hitting set for the hypergraph H. This is applied across all cases and constraints as outlined in the algorithm. All constraints are satisfied with the given weights for the measure μ. The number of leaves in the search tree is upper bounded T(μ) ≤ 2^w_6 · n + w_k · k. These weights result in a multivariate running time of O(1.5135^n · 1.1754^k) for enumerating minimal hitting sets of size at most k in rank 3 hypergraphs. Then the collection_I,X^k = {S ⊆ V \ X : |S| = kand S ∪ X is a minimal 3-HS}can be enumerated in time 1.5135^n - |X|· 1.1754^k · N^O(1). Combined with Theorem <ref> this results in an algorithm for enumerating minimal hitting sets in rank 3 hypergraphs in . § CONCLUSIONThe main contribution of this paper is a framework allowing us to turn many b^n c^k ^O(1) time algorithms for subset and subset enumeration problems into (1+b-1/c)^n N^O(1) time algorithms, generalizing a recent framework of Fomin et al. <cit.>.The main complications in using the framework are, firstly, that new algorithms or running-time analyses are often needed, and, secondly, that such analyses need solutions to non-convex programs in the Measure and Conquer framework. In the usual Measure and Conquer analyses <cit.>,the objective is to upper bound a single variable (α) which upper bounds the exponential part of the running time (2^α n) subject to convex constraints. Thus, it is sufficient to solve a convex optimization problem to minimize the running time <cit.> resulting from the constraints derived from the analysis. Here, the objective function (2^α-2^-w_k) is non-convex. While experimenting with a range of solvers, either guaranteeing to find a global optimum (slow and used for optimality checks) or only a local optimum (faster and used mainly in the course of the algorithm design), we experienced on one hand that the local optima found by solvers are often the global optimum, but on the other hand that weakening non-tight constraints can sometimes lead to a better globally optimum solution. *Acknowledgments We thank Daniel Lokshtanov, Fedor V. Fomin, and Saket Saurabh for discussions inspiring some of this work. Serge Gaspers is the recipient of an Australian Research Council (ARC) Future Fellowship (FT140100048) and acknowledges support under the ARC's Discovery Projects funding scheme (DP150101134).plain | http://arxiv.org/abs/1704.07982v1 | {
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[email protected] ^1Laboratory of Experimental Condensed Matter Physics, The Rockefeller University. New York City, NY, USAOrganic material in anoxic sediment represents a globally significant carbon reservoir that acts to stabilize Earth's atmospheric composition. The dynamics by which microbes organize to consume this material remain poorly understood. Here we observe the collective dynamics of a microbial community, collected from a salt marsh, as it comes to steady state in a two-dimensional ecosystem, covered by flowing water and under constant illumination. Microbes form a very thin front at the oxic-anoxic interface that moves towards the surface with constant velocity and comes to rest at a fixed depth. Fronts are stable to all perturbations while in the sediment, but develop bioconvective plumes in water. We observe the transient formation of parallel fronts. We model these dynamics to understand how they arise from the coupling between metabolism, aerotaxis, and diffusion. These results identify the typical timescale for the oxygen flux and penetration depth to reach steady state. Localization and dynamics of sulfur-oxidizing microbes in natural sediment Alexander Petroff^1*, Frank Tejera^1 & Albert Libchaber^1 December 30, 2023 ==========================================================================When organic material is buried in sediment <cit.>, its decay by microbes is slowed by the limited diffusive flux of oxygen from the surface <cit.>, thus sequestering a large reservoir of fixed carbon <cit.>.The burial of organic material contributed to the rise of oxygen in the ancient atmosphere <cit.> and its decay may further destabilize the modern climate <cit.>.Predicting the quantity carbon sequestered and respired requires an understanding of how complex microbial communities organize and move in nutrient gradients.Although the collective dynamics of simple microbial communities are well studied <cit.>, the collective dynamics of complex microbial communities—typical of natural sediment— remain poorly understood.Here we observe the dynamics by which a microbial community, collected from a salt-marsh microbial mat and brought into the lab, organizes to fix the oxygen penetration depth.At this depth, sulfur-oxidizing bacteria consume oxygen and H_2S (sulfide) <cit.>, the waste product of sulfur-reducing bacteria <cit.>. As these reducing and oxidizing bacteria exchange sulfur compounds, they maintain a sulfur cycle <cit.>.Periodic observations by Garcia-Pichel et al have shown that sulfur-oxidizing bacteria organize into a front that moves through the sediment <cit.>.We first describe a new experiment that allows one to continuously observe these dynamics in a two-dimensional chamber.We find that microbes self-organize into a thin stable front that moves with constant velocity to a steady-state depth.We then present a model describing how these dynamics arise from the coupling between aerotaxis, metabolism, and diffusion.These results allow us to identify the typical timescale required for the oxygen penetration depth to come to steady-state in natural sediment.We begin by collecting a cyanobacterial mat from Little Sippewissett Marsh (41.575268^∘N 70.638406^∘W) near Woods Hole Massachusetts <cit.>.Bringing this material into the lab, we observe the typical metabolic stratification. A green stratum of photosynthetic microbes lays above purple, white, and dark bands of purple sulfur, sulfur-oxidizing, and sulfur-reducing bacteria, respectively <cit.>.Details regarding the collection and maintenance of this mat in the lab are described in reference <cit.>.To continuously observe the motion of the sulfur-oxidizing microbes in a constant environment, we developed a two-dimensional microbial ecosystem in a Hele-Shaw chamber <cit.>.Figure 1(a) shows a schematic of this experiment.We place sediment between two 40.5 × 75 clear acrylic walls separated by a 2 mm (∼10sand grains) rubber spacer, held together by bolts at the edges of the chamber.We continuously flow fresh saltwater media over the surface of the sediment at a flow rate of Q=0.44 cm^3/min (oxygen Peclet number ∼ 5000), which maintains a constant chemical environment at the surface.The oxygen concentration in the fresh media is in equilibrium with the atmosphere and contains 20mM SO_4^2-, typical of natural salt marshes <cit.>. These oxidants maintain populations of aerobic, sulfur-oxidizing, and sulfur-reducing bacteria in the sediment.To mimic the natural environment, we focus a light sheet (produced from a high powered Thorlabs MCWHL5 led) onto the top surface of the sediment while keeping the edges of the chamber in darkness. The way the chamber is loaded is critical to the experiment.We take sediment from a microbial mat by making two parallel slices into the sediment 5mm apart with clean metal knives. We take the sediment from between the slices by pressing the knives together and lifting.We lay this sediment on one of the acrylic walls surrounded on three sides by the rubber spacer.Extra sediment is removed to the height of the rubber spacer. We then place the second acrylic wall onto the sediment.Finally, we bolt the walls together to avoid leaks.This compresses the sediment.Although we are careful to preserve the vertical structure of the sediment as we lay it on the chamber wall, the topmost several millimeters of the mat mix. The thickness of the mixed layer varies between experiments.As a result, there is spurious oxygen in the top most layer of sediment.In these experiments, we observe the dynamics by which this oxygen reservoir is consumed as the oxic-anoxic interface comes to steady state.5 to 8 hours after loading the chamber, we observe the formation of an extraordinarily thin front of microbes (fig 1b), with a typical thickness of only 120±60 μm. It extends the entire 40.5mm width of the chamber.Due to the presence of sulfur globules <cit.> in the constituent bacteria, this band is clearly visible to the eye, appearing as a thin white band.This allows us to follow the dynamics of this front by photographing its position at 15minute intervals.We observe a wide range of initial positions of the front (fig. 2), ranging from 1.9± .1mm to 6.1± .1mm.We attribute this range of values to the variability in initial conditions imposed by loading the chamber.To confirm that this front forms at the oxic-anoxic interface, we use the oxygen-sensitive fluorescent dye Tris(4,7-diphenyl-1,10-phenanthroline) ruthenium(II) Dichloride (99% pure, American Elements RU-OM-02).The fluorescence of this dye is reversibly quenched by oxygen, allowing one to visualize the distribution of oxygen <cit.>.The dye is fixed within a porous plastic matrix of polyethylene terephthalate on a 4cm ×2cm Mylar sheet, as described in ref. <cit.>. To improve the signal strength, the detector is coated with 0.08mm layer of PDMS containing 2% TiO_2, similar to ref <cit.>.This sheet is placed flush against the chamber wall. The top of the detector is even with the sediment surface. During a measurement, the side of the chamber is illuminated with a second high powered LED (455nm Thorlabs M455L3, not shown in fig 1) for 1 sec. Photographing the illuminated sheet through a 530nm high pass filter allows one to visualize the gradient in dye fluorescence and thus the oxygen gradient.Despite the TiO_2 insulation, scattering of light off of sediment behind the detector introduces small variations in the measured oxygen concentrationMeasuring the oxygen gradient with a Clark type microelectrode <cit.> (Unisense OX-100) provides independent confirmation of the gradients. Indeed, we observe the front forms (fig 1c) near the oxic-anoxic interface at an oxygen concentration of c=6.7%±2% atmospheric concentration.Given a diffusion coefficient of oxygen D=4× 10^-6 cm^2, the observed slope corresponds to an oxygen flux j_o=25±1 μm μM/sec. After 40 hours, the increase in microbial activity reduces the penetration depth to only 1 mm.The gradient then remains constant.The scales of these gradients are typical of natural sediments <cit.>. The front, which extends the width of the chamber, moves towards the surface with a constant velocity of U=0.056± 0.01 μm sec^-1.Remarkably, the entire front moves with the same speed and does not develop fingering instabilities <cit.>..During the motion towards the surface, we observe transient fluctuations in the shape of the front.The typical amplitude of these fluctuation is ∼ 300 μm, equivalent to ∼ 3 front widths. The front comes to rest at a depth of d_0=1.27± 0.33 mm regardless of the depth at which it formed.The trajectories of fronts in six different experiments are shown in figure 2. Notice that front in one experimental run overshot its steady state depth despite moving without inertia (Reynolds number Re=7.2× 10^-6±3.5× 10^-6).In this overshot experiment, we observed the transient formation of a second, much fainter front parallel to the first (figs. 2 and 3b).Notably, the second front appeared 0.180± 0.09mm above the stable front.It persisted for 7± .25hours before vanishing.The return of the original front to its steady-state depth was coincident with the disappearance of the transient front. The presence of these double fronts in field observations has previously been attributed to two populations of sulfur-oxidizing microbes, a fast population that moves with the changing gradients and a slow population that is left behind <cit.>.As the transient front appeared discontinuously above the moving front, rather than splitting from it, this explanation does not explain our result.The flow of fresh media over the sediment surface is critical to the stability of the front.When the flow stops, the front moves from the sediment into the water and becomes unstable, immediately developing bioconvective plumes (figure <ref>c) <cit.>.Such plumes form when the density of the upward swimming microbes in the front exceeds a critical value to develop gravity-driven flows<cit.>.Viscous forces stabilize the front in the sediment<cit.>.It is surprising that a front can form and move coherently through this complex medium.Past work has shown that populations of identical bacteria form fronts in response to nutrient gradients <cit.>.To account for this behavior, we consider how gradients of oxygen c and H_2S (sulfide) s change in response to microbial metabolism and the resulting motion of the sulfur-oxidizing community. We begin by considering the time t evolution of these gradients around an arbitrary distribution of microbes.First-order kinetics for metabolism with oxygen and sulfide require c=D ∇^2 c - σ k(𝐱,t) s cands=D_s ∇^2 s - k(𝐱,t) s c.We take the diffusion coefficients of oxygen and sulfide in the sediment (compressed in the chamber) to be D=4× 10^-6 cm^2/sec and D_s=2× 10^-6 cm^2/sec, respectively, slightly smaller than their values in loose sand.k is the metabolic rate per microbe per oxygen molecule. It varies in space with the local density of bacteria.The stoichiometry coefficient σ is the number of oxygen molecules required to oxidize a sulfide <cit.>. To close these equations, one must include a model of aerotaxis to describe how the local metabolic rate constant k(𝐱,t) change with the moving chemical gradients.Two approximations make this possible for an arbitrary community of sulfur-oxidizing microbes.First—because the front velocity U≈ 0.06 μm/sec is much slower than the range of speeds v∼ 1-600 μm/sec of swimming and gliding microbes <cit.>— the density of microbes and thus the metabolic rate k evolves quasistatically.This separation of scales allows us to characterize the density and metabolic rate of microbes in the front with their steady-state values.Second, we assume that the steady-state distribution of microbes is only determined by the concentration of oxygen.This approximation is justified as sulfur-oxidizing microbes (and the associated protists) typically concentrate themselves near oxic-anoxic interface, at oxygen concentrations ranging from 0 to 10% atmospheric <cit.>.We describe this diversity of steady-state positions with a distribution p(c), which is the fraction of microbes that concentrate at an oxygen concentration of c.To mimic the natural range of tolerable oxygen concentrations in this model, we take p(c) to have mean c_0=5% atmospheric and variance ∼ c_0^2.Combining these approximations, k(𝐱,t)=k_0 p(c(𝐱,t)), where k_0 is the effective metabolic rate per oxygen molecule.This quasistatic approximation removes any explicit dependence of the front dynamics on the motion of bacteria in the gradients.We observed that the front width, velocity, and steady-state depth are similar across all experiments despite differences in their initial conditions.We seek to understand how this surprising consistency arises from the length scales and timescales imposed by diffusion and metabolism.We begin by presenting scaling relations to estimate the width, velocity, and final front depth. We find these estimates to be roughly consistent with observations. We then solve equations (1) and (2) numerically to investigate the sensitivity of these dynamics to p(c).We begin with the front width. How is it that such a thin front can form despite being composed of diverse microbes?The quasistatic approximation allows one to understand how diverse microbes are forced together into a thin front by competition for oxygen (figure 4a inset).Metabolism sharpens oxygen gradients, leading to shorter distance between the highest and lowest tolerable oxygen concentrations; consequently the front thins to a width similar to the diffusive length scale ℓ=√(D/K), where K=σ k_0 s is the rate at which oxygen is consumed in the front.To estimate K, we balance the flux of oxygen j_o=25±0.1 μmμM/sec, measured from the oxygen profile in figure 1c, with metabolic flux K w c_0. We find K=0.011±0.006sec^-1.The corresponding diffusive length ℓ=190±55 μm is indeed similar to the measured front width w=120± 60 μm. Next, we estimate the front velocity from the aerotactic motion of microbes in the front.The front moves as a result of bacterial metabolism (fig. 4a inset), which depletes oxygen near the front, and the resulting quasistatic reorganization of bacteria in the gradient. As oxygen levels around the front fall, the bacterial front moves to remain concentrated at c=c_0.Given a constant rate of oxygen consumption, the front moves with a constant velocity U towards the surface, where the concentration is c^*=170 μM (measured from the profile in figure 1c).This propagation is analogous to the front dynamics described by the Fisher-Kolmogorov equation, in which microbes consume a diffusing resource at a constant rate <cit.>.However, unlike the microbial fronts described by this equation, the metabolic rate of the sulfur-oxidizing bacteria requires the presence of two nutrients, oxygen and sulfide, in opposing gradients.Balancing the oxygen consumed by the moving front as U c^* with the metabolic flux K c_0 w, we expect a velocity of ∼ (c_0/c^*) K w=0.240±0.15 μm/sec.This value is a factor of 4 larger than the observed velocity U=0.056±0.01 μm/s. Finally, we consider the equilibrium position of the sulfur-oxidizing bacteria.As the front approaches the surface, the increasing flux of oxygen causes it to slow.The front reaches a steady-state depth d_0 where the fluxes of oxygen j_c≈ D (c^*-c_0)/d_0 and sulfide j_s to the bacteria are both balanced by the metabolism.Requiring the ratio of these fluxes to be the stochiometric ratio σ and solving for d_0, the steady-state depth is d_0∼ D (c^*-c_0)/σ j_s.We measure the sulfide flux from the concentration of sulfide s=300± 30 μM measured below the front (Lamotte colorimetric sulfide assay 4456-01). This concentration corresponds to a flux of j_s= 25± 2 μM μm/sec. Taking σ=2 (oxidation of H_2S to SO_4^2-), we expect the front to come to rest at a depth 2.0±.2mm. This estimate is roughly consistent with the observed depth 1.27± 0.33mm.This closes our scaling analysis.To evaluate the sensitivity of these dynamics to the composition of the microbial community, characterized by the distribution p(c) of preferred oxygen levels, we integrate equations (1) and (2) numerically.We initially take p(c) to be a Maxwell-Boltzmann distribution with mean c_0= 5% atmospheric.We take the unknown parameters from the results of the scaling analysis with no fitted parameters.Comparing this solution (fig 4a) to the observed motion of fronts (fig 2), shows that this solution captures the motion of the front to the surface, its timescales, and length scales.Figure 4b shows the distribution of oxygen, sulfide, and bacteria at steady state.The front dynamics depend very weakly on the distribution p(c) of steady-state oxygen concentrations of the bacteria. The trajectories are similar if one takes p(c) to be normal, exponential, or uniformly distributed. They are similarly unchanged if one imposes conservation of cell number. The front width w∼ℓ depends very weakly on the range of tolerable oxygen concentrations. Varying c_0 by a factor of 100—from 0.003 atmospheric to .3 atmospheric—results in fronts that vary in width by a factor of 4.5.Thus we conclude that these dynamics are relatively insensitive to p(c). Rather the quasistatic motion of the front is determined by the sulfide flux and surface oxygen concentration.In conclusion, we have observed the formation of a thin front of microbes at the oxic-anoxic interface and its constant-velocity motion to a steady-state depth.The front moves as the initial reservoir of oxygen (introduced to the sediment by loading the chamber) is consumed by the microbial population.The front width, velocity, and steady-state position vary little across experiments. We have shown their magnitudes are fixed by the metabolic timescale K, diffusive length scale √(D/K), sulfide flux j_s, and concentrations of oxygen at the surface c^*.We conclude that these dynamics arise from the coupling between diffusion, metabolism, and aerotaxis.The simplicity of this phenomenon is counterintuitive given the complexity of this system.Future work should proceed in three directions. First, our observations have identified the transient formation of parallel fronts (fig 3b) that cannot currently be explained.Our model allows us to propose a hypothesis for this phenomenon: they arise from metabolic shifts in the constituent microbes.Notice that the steady-state depth of a front d_0∼ D (c^*-c_0)/σ j_s depends explicitly on stochiometric coefficient σ, the ratio of oxygen molecules consumed per sulfide molecule.The value of this coefficient is quantized by the oxidation state of the waste product.It takes values of 1/2, 3/2, or 2 for oxidation to S^0, SO_3^2-, and SO_4^2-, respectively.A front of bacteria that completely oxidize sulfur (σ=2) comes to rest at a shallower depth than those that partially oxidize sulfur (e.g., to S^0).Thus, if a sub-population in the front switch from partial to complete sulfur oxidation (or between intermediates), they would move discontinuously to a shallower depth.These two fronts would then compete for sulfur and oxygen.This hypothesis can be tested by extracting bacteria from each of the fronts and measuring the concentration of mRNA coding for the enzymes responsible for the different steps of sulfur oxidation.We leave this analysis for future work. Second, to connect these results to natural environments, we must account for environmental fluctuations.Our results identifies a fundamental timescale τ=d_0/U=6.3± 1.6hr for microbial community to come to steady state after a perturbation.Comparing this timescale to the frequency f of environmental perturbations (e.g., by tides and storms), we find an important dimensionless number fτ.If fτ∼1, as in salt marshes regularly disturbed by tides f=1/12 hr^-1, the sediment never reaches steady state. We therefore expect that the decay of organics is strongly coupled to environmental fluctuations.To measure the relationship between organic decay and environmental fluctuations, we will modify this experiment (fig 1a) to include a variable flow of fresh media over the surface and measure resulting variability in gross metabolic rate. Finally, we have only examined the dynamics of the oxic-anoxic interface in a fixed quantity of sediment.Including a sedimentation rate of U_s at the surface, the sediment-water interface moves away from the microbial front.To examine the influence of this parameter, we will modify our experiment to include a flux of sediment and measure the gross metabolic rate.These dynamics may explain the observed variability of cell density with sedimentation rate in the sea <cit.>.This work was supported by HFSP RGP0037. The authors declare that they have no competing financial interests. Correspondence and requests for materials should be addressed to A.P.P (email: [email protected]). 40 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Hayes and Waldbauer(2006)]hayes2006carbon author author J. M. 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Source File Set Search for Clone-and-Own Reuse Analysis Takashi Ishio12, Yusuke Sakaguchi1, Kaoru Ito1, Katsuro Inoue1 1 Graduate School of Information Science and Technology, Osaka University, Osaka, Japan 2 Graduate School of Information Science, Nara Institute of Science and Technology, Nara, Japan Email: {ishio, s-yusuke, ito-k, inoue}@ist.osaka-u.ac.jp December 30, 2023 =======================================================================================================================================================================================================================================================================================================================Sequence-to-sequencemodels have shown strong performance across a broad range of applications. However, their application to parsing and generating text using Abstract Meaning Representation (AMR) has been limited,due to the relatively limited amount of labeled data and the non-sequential nature of the AMR graphs. We present a novel training procedure that can lift this limitation using millions of unlabeled sentences and careful preprocessing of the AMR graphs.For AMR parsing, our model achieves competitive results of 62.1 SMATCH, the current best score reported without significant use of externalsemantic resources. For AMR generation, our model establishes a new state-of-the-art performance of BLEU 33.8. We present extensive ablative and qualitative analysis includingstrong evidence that sequence-based AMR models are robust against ordering variations of graph-to-sequence conversions. § INTRODUCTIONAbstract Meaning Representation (AMR) is a semantic formalism to encode the meaning of natural language text. As shown in Figure <ref>, AMR represents the meaning using a directed graph while abstracting away the surface forms in text. AMR has been used as an intermediate meaning representation for several applications including machine translation (MT) <cit.>, summarization <cit.>, sentence compression <cit.>, and event extraction <cit.>. While AMR allows for rich semantic representation, annotating training data in AMR is expensive, which in turn limits the use of neural network models<cit.>. In this work, we present the first successful sequence-to-sequence (seq2seq) models that achieve strong results for both text-to-AMR parsing and AMR-to-text generation. Seq2seq models have been broadly successful in many other applications <cit.>. However, their application to AMR has been limited,in part because effective linearization (encoding graphs as linear sequences) and data sparsity were thought to pose significant challenges.We show that these challenges can be easily overcome, by demonstrating that seq2seq models can be trained using any graph-isomorphic linearization and that unlabeled text can be used to significantly reduce sparsity.Our approach is two-fold.First, we introduce a novel paired training procedure that enhances both the text-to-AMR parser and AMR-to-text generator.More concretely, first we use self-training to bootstrap a high quality AMR parser from millions of unlabeled Gigaword sentences <cit.> and then use the automatically parsed AMR graphs to pre-train an AMR generator.This paired training allows both the parser and generator to learn high quality representations of fluent English textfrom millions of weakly labeled examples, that are then fine-tuned using human annotated AMR data.Second, we propose a preprocessing procedure for the AMR graphs, which includes anonymizing entities and dates, grouping entity categories, and encoding nesting information in concise ways, as illustrated in Figure <ref>(d).This preprocessing procedure helps overcoming the data sparsity while also substantially reducing the complexity of the AMR graphs. Under such a representation, we show that any depth first traversal of the AMR is an effective linearization,and it is even possible to use a different random order for each example.Experiments on the LDC2015E86 AMR corpus (SemEval-2016 Task 8) demonstrate the effectiveness of the overall approach.For parsing, we are able to obtain competitive performance of 62.1 SMATCH without using any external annotated examples other than the output of a NER system, an improvement of over 10 points relative to neural models with a comparable setup.For generation, wesubstantially outperform previous best results, establishing a new state of the art of 33.8 BLEU.We also provide extensive ablative and qualitative analysis, quantifying the contributions that come from preprocessing and the paired training procedure. § RELATED WORKAlignment-based Parsing flanigan-EtAl:2014:P14-1 (JAMR) pipeline concept and relation identification with a graph-based algorithm.zhou-EtAl:2016:EMNLP20163 extend JAMR by performing the concept and relation identification tasks jointly with an incremental model.Both systems rely on features based on a set of alignments produced using bi-lexical cues and hand-written rules. In contrast, our models train directly on parallel corpora, and make only minimal use of alignments to anonymize named entities. Grammar-based Parsing wang-EtAl:2016:SemEval (CAMR) perform a series of shift-reduce transformations on the output of an externally-trained dependency parser, similar to damonte-cohen-satta:2017:EACLlong, brandt-EtAl:2016:SemEval, puzikov-kawahara-kurohashi:2016:SemEval, and goodman-vlachos-naradowsky:2016:SemEval.artzi-lee-zettlemoyer:2015:EMNLP use a grammar induction approach with Combinatory Categorical Grammar (CCG),which relies on pre-trained CCGBank categories, like bjerva-bos-haagsma:2016:SemEval. pust-EtAl:2015:EMNLP recast parsing as a string-to-tree Machine Translation problem, using unsupervised alignments <cit.>,and employing several external semantic resources.Our neural approach is engineering lean, relying only on a large unannotated corpus of English and algorithms to find and canonicalize named entities. Neural ParsingRecently there have been a few seq2seq systems for AMR parsing <cit.>.Similar to our approach, peng:2017:EACL deal with sparsity by anonymizing named entities and typing low frequency words, resulting in a very compact vocabulary (2k tokens). However, we avoid reducing our vocabulary by introducing a large set of unlabeled sentences from an external corpus, therefore drastically lowering the out-of-vocabulary rate (see Section <ref>). AMR Generation flanigan-EtAl:2016:N16-1 specify a number of tree-to-string transduction rules based on alignments and POS-based features that are used to drive a tree-based SMT system.pourdamghani-knight-hermjakob:2016:INLG also use an MT decoder; they learn a classifier that linearizes the input AMR graph in an order that follows the output sentence, effectively reducing the number of alignment crossings of the phrase-based decoder. song-EtAl:2016:EMNLP2016 recast generation as a traveling salesman problem, after partitioning the graph into fragments and finding the best linearization order.Our models do not need to rely on a particular linearization of the input, attaining comparable performance even with a per example random traversal of the graph.Finally, all three systems intersect with a large language model trained on Gigaword.We show that our seq2seq model has the capacity to learn the same information as a language model, especially after pretraining on the external corpus. Data Augmentation Our paired training procedure is largely inspired by sennrich-haddow-birch:2016:P16-11. They improve neural MT performance for low resource language pairs by using a back-translation MT system for a large monolingual corpus of the target language in order to create synthetic output, and mixing it with the human translations. We instead pre-train on the external corpus first, and then fine-tune on the original dataset. § METHODSIn this section, we first provide the formal definition of AMR parsing and generation (section <ref>). Then we describe the models we use (section <ref>), graph-to-sequence conversion (section <ref>), and our paired training procedure(section <ref>).§.§ Tasks We assume access to a training dataset D where each example pairs a natural language sentence s with an AMRa. The AMR is a rooted directed acylical graph. It contains nodes whose names correspond to sense-identified verbs, nouns, or AMR specific concepts, for example , , and in Figure <ref>. One of these nodes is a distinguished root, for example, the nodein Figure <ref>. Furthermore, the graph contains labeled edges, which correspond to PropBank-style <cit.> semantic roles for verbs or other relations introduced for AMR, for example,orin Figure <ref>. The set of node and edge names in an AMR graph is drawn from a set of tokens C, and every word in a sentence is drawn from a vocabulary W.We study the task of training an AMR parser, i.e., finding a set of parameters θ_P for model f, that predicts an AMR graph â, given a sentence s: â = _a f( a | s; θ_P ) We also consider the reverse task, training an AMR generator by finding a set of parameters θ_G, for a model f that predicts a sentence ŝ, given an AMR graph a: ŝ = _s f( s | a; θ_G ) In both cases, we use the same family of predictors f, sequence-to-sequence models that use global attention, but the models have independent parameters, θ_P and θ_G.§.§ Model For both tasks, we use a stacked-LSTM neural architecture employed in neural machine translation <cit.>.[We extended the Harvard NLP seq2seq framework from <http://nlp.seas.harvard.edu/code>.] Our model uses a global attention decoder and unknown word replacement with small modifications<cit.>.The model uses a stacked bidirectional-LSTM encoder to encode an input sequence and a stacked LSTM to decode from the hidden states produced by the encoder. We make two modifications to the encoder: (1) we concatenate the forward and backward hidden states at every level of the stack instead of at the top of the stack, and (2) introduce dropout in the first layer of the encoder.The decoder predicts an attention vector over the encoder hidden states using previous decoder states. The attention is used to weigh the hidden states of the encoder and then predict a token in the output sequence. The weighted hidden states, the decoded token, and an attention signal from the previous time step (input feeding) are then fed together as input to the next decoder state. The decoder can optionally choose to output an unknown word symbol, in which case the predicted attention is used to copy a token directly from the input sequence into the output sequence.§.§ Linearization Our seq2seq models require that both the input and target be presented as a linear sequence of tokens.We define a linearization order for an AMR graph as any sequence of its nodes and edges. A linearization is defined as (1) a linearization order and (2) a rendering function that generates any number of tokens when applied to an element in the linearization order (see Section <ref> for implementation details).Furthermore, for parsing, a valid AMR graph must be recoverable from the linearization. [1]▹ #1 §.§ Paired Training Obtaining a corpus of jointly annotated pairs of sentences and AMR graphs is expensive and current datasets only extend to thousands of examples.Neural sequence-to-sequence models suffer from sparsity with so few training pairs. To reduce the effect of sparsity, we use an external unannotated corpus of sentences S_e, and a procedure which pairs the training of the parser and generator.Our procedure is described in Algorithm <ref>, and first trains a parser on the dataset D of pairs of sentences and AMR graphs. Then it uses self-training to improve the initial parser. Every iteration of self-training has three phases: (1) parsing samples from a large, unlabeled corpus S_e, (2) creating a new set of parameters by training on S_e, and (3) fine-tuning those parameters on the original paired data.After each iteration, we increase the size of the sample from S_e by an order of magnitude.After we have the best parser from self-training, we use it to label AMRs for S_e and pre-train the generator. The final step of the procedure fine-tunes the generator on the original dataset D.§ AMR PREPROCESSING We use a series of preprocessing steps, including AMR linerization, anonymization, and other modifications we make to sentence-graph pairs.Our methods have two goals: (1) reduce the complexity of the linearized sequencesto make learning easier while maintaining enough original information, and (2) address sparsity from certain open class vocabulary entries, such as named entities (NEs) and quantities. Figure <ref>(d) contains example inputs and outputs with all of our preprocessing techniques.Graph Simplification In order to reduce the overall length of the linearized graph, we first remove variable names and therelation () before every concept.In case of re-entrant nodes we replace the variable mention with its co-referring concept.Even though this replacement incurs loss of information, often the surrounding context helps recover the correct realization, e.g., the possessive rolein the example of Figure <ref> is strongly correlated with the surface form his. Following pourdamghani-knight-hermjakob:2016:INLG we also remove senses from all concepts for AMR generation only. Figure <ref>(a) contains an example output after this stage. §.§ Anonymization of Named EntitiesOpen-class types including NEs, dates, and numbers account for 9.6% of tokens in the sentences of the training corpus, and 31.2% of vocabulary W. 83.4% of them occur fewer than 5 times in the dataset. In order to reduce sparsity and be able to account for new unseen entities, we perform extensive anonymization.First, we anonymize sub-graphs headed by one of AMR's over 140 fine-grained entity types that contain arole.This captures structures referring to entities such as , , miscellaneous entities marked with , and typed numerical values, .We excludeentities (see the next section). We then replace these sub-graphs with a token indicating fine-grained type and an index, i, indicating it is the ith occurrence of that type.[In practice we only used three groups of ids: a different one for NEs, dates and constants/numbers.] For example, in Figure <ref> the sub-graph headed bygets replaced with .On the training set, we use alignments obtained using the JAMR aligner <cit.> and the unsupervised aligner of pourdamghani-EtAl:2014:EMNLP2014 in order to find mappings of anonymized subgraphs to spans of text and replace mapped text with the anonymized token that we inserted into the AMR graph. We record this mapping for use during testing of generation models. If a generation model predicts an anonymization token, we find the corresponding token in the AMR graph and replace the model's output with the most frequent mapping observed during training for the entity name. If the entity was never observed, we copy its name directly from the AMR graph. Anonymizing Dates For dates in AMR graphs, we use separate anonymization tokens for year, month-number, month-name, day-number and day-name, indicating whether the date is mentioned by word or by number.[We also use three date format markers that appear in the text as: YYYYMMDD, YYMMDD, and YYYY-MM-DD.]In AMR generation, we render the corresponding format when predicted. Figure <ref>(b) contains an example of all preprocessing up to this stage. Named Entity Clusters When performing AMR generation, each of the AMR fine-grained entity types is manually mapped to one of the four coarse entity types used in the Stanford NER system <cit.>: person, location, organization and misc. This reduces the sparsity associated with many rarely occurring entity types. Figure <ref> (c) contains an example with named entity clusters.NER for Parsing When parsing, we must normalize test sentences to match our anonymized training data. To produce fine-grained named entities, we run the Stanford NER system and first try to replace any identified span with a fine-grained category based on alignments observed during training.If this fails, we anonymize the sentence using the coarse categories predicted by the NER system, which are also categories in AMR.After parsing, we deterministically generate AMR for anonymizations using the corresponding text span. §.§ LinearizationLinearization Order Our linearization order is defined by the order of nodes visited by depth first search, including backward traversing steps. For example, in Figure <ref>, starting atthe order contains , , , , , , , , ,.[Sense,and variable information has been removed at the point of linearization.] The order traverses children in the sequence they are presented in the AMR. We consider alternative orderings of children in Section <ref> but always follow the pattern demonstrated above. Rendering Function Our rendering function marks scope, and generates tokens following the pre-order traversal of the graph:(1) if the element is a node, it emits the type of the node.(2) if the element is an edge,it emits the type of the edge and then recursively emits a bracketed string for the (concept) node immediately after it. In case the node has only one child we omit the scope markers (denoted with left “”, and right “” parentheses), thus significantly reducing the number of generated tokens. Figure <ref>(d) contains an example showing all of the preprocessing techniques and scope markers that we use in our full model. § EXPERIMENTAL SETUP We conduct all experiments on the AMR corpus used in SemEval-2016 Task 8 (LDC2015E86), which contains 16,833/1,368/1,371 train/dev/test examples. For the paired training procedure of Algorithm <ref>, we use Gigaword as our external corpus and sample sentences that only contain words from the AMR corpus vocabulary W. We sub-sampled the original sentence to ensure there is no overlap with the AMR training or test sets. Table <ref> summarizes statistics about the original dataset and the extracted portions of Gigaword.We evaluate AMR parsing with SMATCH <cit.>, and AMR generation using BLEU <cit.>[We use the multi-BLEU script from the MOSES decoder suite <cit.>.].We validated word embedding sizes and RNN hidden representation sizes by maximizing AMR development set performance (Algorithm <ref> – line 1).We searched over the set {128, 256, 500, 1024} for the best combinations of sizes and set both to 500. Models were trained by optimizing cross-entropy loss with stochastic gradient descent, using a batch size of 100 and dropout rate of 0.5. Across all models when performance does not improve on the AMR dev set, we decay the learning rate by 0.8. For the initial parser trained on the AMR corpus, (Algorithm <ref> – line 1), we use a single stack version of our model, set initial learning rate to 0.5 and train for 60 epochs, taking the best performing model on the development set.All subsequent models benefited from increased depth and we used 2-layer stacked versions, maintaining the same embedding sizes. We set the initial Gigaword sample size to k=200,000 and executed a maximum of 3 iterations of self-training.For pre-training the parser and generator, (Algorithm <ref> – lines 4 and 9), we used an initial learning rate of 1.0, and ran for 20 epochs. We attempt to fine-tune the parser and generator, respectively, after every epoch of pre-training, setting the initial learning rate to 0.1. We select the best performing model on the development set among all of these fine-tuning attempts. During prediction we perform decoding using beam search and set the beam size to 5 both for parsing and generation.§ RESULTSParsing Results Table <ref> summarizes our development results for different rounds of self-training and test results for our final system, self-trained on 200k, 2M and 20M unlabeled Gigaword sentences. Through every round of self-training, our parser improves.Our final parser outperforms comparable seq2seq and character LSTM models by over 10 points.While much of this improvement comes from self-training, our model without Gigaword data outperforms these approaches by 3.5 points on F1. We attribute this increase in performance to different handling of preprocessing and more careful hyper-parameter tuning. All other models that we compare against use semantic resources, such as WordNet, dependency parsers or CCG parsers (models marked with * were trained with less data, but only evaluate on newswire text; the rest evaluate on the full test set, containing text from blogs). Our full models outperform the original version of JAMR <cit.>, a graph-based model but still lags behind other parser-dependent systems (CAMR[Since we arecurrently not using any Wikipedia resources for the prediction of named entities, we compare against the no-wikification version of the CAMR system.]), and resource heavy approaches (SBMT). Generation Results Table <ref> summarizes our AMR generation results on the development and test set.We outperform all previous state-of-the-art systems by the first round of self-training and further improve with the next rounds. Our final model trained on outperforms TSP and TreeToStr trained on LDC2015E86, by over 9 BLEU points.[We also trained our generator on and fine-tuned on LDC2014T12 in order to have a direct comparison with PBMT, and achieved a BLEU score of 29.7, i.e., 2.8 points of improvement.]Overall, our model incorporates less data than previous approaches as all reported methods train language models on the whole Gigaword corpus.We leave scaling our models to all of Gigaword for future work. Sparsity ReductionEven after anonymization of open class vocabulary entries, we still encounter a great deal of sparsity in vocabulary given the small size of the AMR corpus, as shown in Table <ref>. By incorporating sentences from Gigaword we are able to reduce vocabulary sparsity dramatically, as we increase the size of sampled sentences: the out-of-vocabulary rate with a threshold of 5 reduces almost 5 times for .Preprocessing Ablation Study We consider the contribution of each main component of our preprocessing stages while keeping our linearization order identical. Figure <ref> contains examples for each setting of the ablations we evaluate on. First we evaluate using linearized graphs without parentheses for indicating scope, Figure <ref>(c),then without named entity clusters, Figure <ref>(b), and additionally without any anonymization, Figure <ref>(a).Tables <ref> summarizes our evaluation on the AMR generation. Each components is required, and scope markers and anonymization contribute the most to overall performance.We suspect without scope markers our seq2seq models are not as effective at capturing long range semantic relationships between elements of the AMR graph. We also evaluated the contribution of anonymization to AMR parsing (Table <ref>).Following previous work, we find that seq2seq-based AMR parsing is largely ineffective without anonymization <cit.>. § LINEARIZATION EVALUATIONIn this section we evaluate three strategies for converting AMR graphs into sequences in the context of AMR generation and show that our models are largely agnostic to linearization orders.Our results argue, unlike SMT-based AMR generation methods <cit.>, that seq2seq models can learn to ignore artifacts of the conversion of graphs to linear sequences. §.§ Linearization OrdersAll linearizations we consider use the pattern described in Section <ref>, but differ on the order in which children are visited.Each linearization generates anonymized, scope-marked output (see Section <ref>), of the form shown in Figure <ref>(d). Human The proposal traverses children in the order presented by human authored AMR annotations exactly as shown in Figure <ref>(d).Global-Random We construct a random global ordering of all edge types appearing in AMR graphs and re-use it for every example in the dataset.We traverse children based on the position in the global ordering of the edge leading to a child.Random For each example in the dataset we traverse children following a different random order of edge types.§.§ ResultsWe present AMR generation results for the three proposed linearization orders in Table <ref>. Random linearization order performs somewhat worse than traversing the graph according to Human linearization order.Surprisingly, a per example random linearization order performs nearly identically to a global random order, arguing seq2seq models can learn to ignore artifacts of the conversion of graphs to linear sequences. Human-authored AMR leaks information The small difference between random and global-random linearizations argues that our models are largely agnostic to variation in linearization order. On the other hand, the model that follows the human order performs better, which leads us to suspect it carries extra information not apparent in the graphical structure of the AMR. To further investigate, we compared the relative ordering of edge pairs under the same parent to the relative position of children nodes derived from those edges in a sentence, as reported by JAMR alignments. We found that the majority of pairs of AMR edges (57.6%) always occurred in the same relative order, therefore revealing no extra generation order information.[This is consistent with constraints encoded in the annotation tool used to collect AMR. For example,edges are always ordered beforeedges.] Of the examples corresponding to edge pairs that showed variation, 70.3% appeared in an order consistent with the order they were realized in the sentence.The relative ordering of some pairs of AMR edges was particularly indicative of generation order. For example, the relative ordering of edges with typesand , was 17% more indicative of the generation order than the majority of generated locations before .[Consider the sentences “She went to school in New York two years ago”, and “Two years ago, she went to school in New York”, where “two year ago” is the time modifying constituent for the verb went and “New York” is the location modifying constituent of went.]To compare to previous work we still report results using human orderings. However, we note that any practical application requiring a system to generate an AMR representation with the intention to realize it later on, e.g., a dialog agent,will need to be trained either using consistent, or random-derived linearization orders. Arguably, our models are agnostic to this choice.§ QUALITATIVE RESULTS Figure <ref> shows example outputs of our full system.The generated text for the first graph is nearly perfect with only a small grammatical error due to anonymization.The second example is more challenging, with a deep right-branching structure, and a coordination of the verbsandin the subordinate clause headed by . The model omits some information from the graph, namely the conceptsand .In the third example there are greater parts of the graph that are missing, such as the whole sub-graph headed by .Also the model makes wrong attachment decisions in the last two sub-graphs (it is thethat is unimpeachable and irrefutable, and not the ), mostly due to insufficient annotation () thus making their generation harder.Finally, Table <ref> summarizes the proportions of error types we identified on 50 randomly selected examples from the development set. We found that the generator mostly suffers from coverage issues, an inability to mention all tokens in the input, followed by fluency mistakes, as illustrated above. Attachment errors are less frequent, which supports our claim that the model is robust to graph linearization, and can successfully encode long range dependency information between concepts. § CONCLUSIONS We applied sequence-to-sequence models to the tasks of AMR parsing and AMR generation, by carefully preprocessing the graph representation and scaling our models via pretraining on millions of unlabeled sentences sourced from Gigaword corpus. Crucially, we avoid relying on resources such as knowledge bases and externally trained parsers. We achieve competitive results for the parsing task (SMATCH 62.1) and state-of-the-art performance for generation (BLEU 33.8). For future work, we would like to extend our work to different meaning representations such as the Minimal Recursion Semantics (MRS; Copestake2005). This formalism tackles certain linguistic phenomena differently from AMR (e.g., negation, and co-reference), contains explicit annotation on concepts for number, tense and case, and finally handles multiple languages[A list of actively maintained languages can be found here: <http://moin.delph-in.net/GrammarCatalogue>] <cit.>. Taking a step further, we would like to apply our models on Semantics-Based Machine Translation using MRS as an intermediate representation between pairs of languages, and investigate the added benefit compared to directly translating the surface strings, especially in the case of distant language pairs such as English and Japanese <cit.>.§ ACKNOWLEDGMENTSThe research was supported in part by DARPA under the DEFT program through AFRL (FA8750-13-2-0019) and the CwC program throughARO (W911NF-15-1-0543), the ARO (W911NF-16-1-0121), the NSF (IIS-1252835, IIS-1562364, IIS-1524371), an Allen Distinguished Investigator Award, Samsung GRO, and gifts by Google and Facebook. The authors thank Rik Koncel-Kedziorski, the UW NLP group, and the anonymous reviewers for their thorough and helpful comments.acl_natbib | http://arxiv.org/abs/1704.08381v3 | {
"authors": [
"Ioannis Konstas",
"Srinivasan Iyer",
"Mark Yatskar",
"Yejin Choi",
"Luke Zettlemoyer"
],
"categories": [
"cs.CL"
],
"primary_category": "cs.CL",
"published": "20170426235334",
"title": "Neural AMR: Sequence-to-Sequence Models for Parsing and Generation"
} |
Bohmian MechanicsRoderich Tumulka[Fachbereich Mathematik, Eberhard-Karls-Universität, Auf der Morgenstelle 10, 72076 Tübingen, Germany. E-mail: [email protected]]August 15, 2019 ========================================================================================================================================================================We provide coordinate-free versions of the classical projection Theorem of Marstrand–Kaufman–Mattila. This allows us to generalize this Theorem to the complex setting; in restriction to complex spheres, we obtain further projection Theorems along so-called complex chains.§ INTRODUCTION§.§ Motivation§.§.§ BackgroundSince Marstrand's seminal paper <cit.>, several authors have sought to improve and generalize the so-called Marstrand projection Theorem.Let us recall the basic Euclidean setup in any dimension, as in <cit.> (Marstrand's paper dealt only with the plane). Let n ≥ 2 and 1 ≤ k ≤ n-1. Fix a Borel subset A ⊂^n of Hausdorff dimension s.Pick a vector subspace of dimension k at random (with respect to the Lebesgue measure on the space ofk–dimensional vector subspaces of ^n). Project A along the vector subspace, e.g.pushing it down the quotient mapping. The projected set (sitting in a space isometric to ^n-k) has,almost surely, Hausdorff dimension inf{s, n-k}.A Marstrand-type result is a Theorem of this kind: an almost sure equality for the dimension of a given Borel set projectedthrough a random projection.It could also be said that a Marstrand-type result deals with the almost sure dimension of a Borel set, transverse to a random foliation (according to a fixed measure on some space of foliations). This is our point of view in this paper. Fora definition, see <ref>.There are (at least) two natural ways to generalize this result: * To look at a restricted family of foliations, e.g. to consider, in ^3, vector lines spanned not by any vector but by a one-parameter family of vectors, as in <cit.>. * To look at foliations defined in the same fashion in non-Euclidean spaces, e.g. in Heisenberg group. See, for example, <cit.>. §.§.§ Description the of resultsIn this paper, we look at a quite obvious generalization: namely,we generalize Marstrand projection Theorem to the complex setting. The word complex sphere in the title of this paper refers to the Euclidean spheres of odd dimension; these spheres sit naturally in complex projective spaces, and the complex Marstrand projection Theorem we will obtain can be restricted to complex spheresto yield interesting projection Theorems with respect to some special families of so-called “small spheres”.The most notable feature of our approach is that everything happens in the projective space; this allows us to docoordinate-free computations, using extensively the Grassmann algebra. This adds some conceptual and notational difficulty.The first half of our results (dealing with linear foliations of projective spaces) could be obtained using coordinatesand standard computations as in <cit.> (where the real setting is handled). The other half of our results would be very awkward to formulate without the language of Grassmann algebra, especially in the complexsetting, which is of interest to us.Informally, a reason for this is the fact that there is no good analogue, for Heisenberg groups,of projective spaces associated to vector spaces. The projective space _^n can be defined at the space of“infinite circles of ^n+1 passing through the origin”. This definition would also make sense in Heisenberg group, replacing “infinite circles” with “infinite chains” (see <cit.> for the definitionof finite and infinite chains in Heisenberg group, or <ref> for the definitionof chains we will be using); the point is that there is only one infinite chain passingthrough the origin. This is why, in this paper, we have to consider “finite chains”; this also explains why it ismuch more efficient to work without coordinates. Throughout this paper, we will deal with the real and complex settings at the same time. In the real setting,none of the results we obtain is new: they are all essentially equivalent to the basic Theoremof Marstrand–Kaufman–Mattila. We state them nonetheless because they serve to provide some geometricintuition to the reader, and to convince them that we are indeed generalizing theclassical, real Euclidean, Marstrand Theorem – this may not be obvious at first. §.§.§ Plan of the paperIn <ref> we give a precise meaning to the notion of transverse dimension with respect to a foliation.In <ref> to <ref> we introduce the needed algebraic device: the Hermitian Grassmann (bi)algebra associated to a Hermitian space. In <ref> we state and prove useful properties of theinner product in the Hermitian Grassmann algebra. The distance formula in <ref>is a first hint of the geometric significance of the Hermitian structure on the Grassmann algebra.Section <ref> is the part of the paper that deals with linear foliations of projective spaces and this is where we apply the algebraic tools described previously. In <ref> we define the generalized radial projections we will use to parametrize our linear foliations. We then endow, in <ref>, the codomainof these radial projections with a canonical metric. The analysis of transversality of generalized radial projections is made easy by theproduct formula contained in <ref>. The result of <ref> is not needed in this paper; it is statedbecause it answers a question that arises naturally in this context. In <ref> we prove transversality of the basiclinear foliations of projective spaces, and this is applied to obtain a coordinate-free version of Marstrand's projection Theorem.We improve on this in <ref> by looking at a lower-dimensional family of foliations; in the real setting theresult we then obtain is equivalent to the classical Theorem of Marstrand–Kaufman–Mattila. In Section <ref> we look at the results of the previous Sections in restriction to spheres. The special case of 1–chains has to be dealt with separately in <ref>. In <ref> we describe our foliations in coordinates in order to help the reader get an idea of the geometry behind the algebra; a description of the geometry ofchains is out of the scope of this paper and we refer to <cit.> for details, and suggestive computer-generated pictures.§.§ Transverse dimensionWe give a precise meaning to the notion of dimension of a Borel set transverse toa given foliation. Let X be a locally compact metric space. A foliation on X is a partition ξ, all atoms of which are closed subsets of X. We denote by ξ(x) the atom of ξ a given point x ∈ X belongs to. The quotient space X/ξ has elements the atoms of ξ and is endowed with the final topology for the projection mapping X → X/ξ. In general the metric of X does not pass to the quotient, because two distinct atoms of ξ may be at zero distance from one another. On the other hand, for any compact subset K of X, the trace ξ|K of ξ on K has compact atoms; the metric of X, restricted to K, passes to the quotient, and the projection mapping K → K/ξ is, by definition, Lipschitz. In this situation, the transverse dimension of K with respect to ξ is the Hausdorff dimension of the quotient metric space K/ξ. If A is a Borel subset of X, the transverse dimension of A with respect to ξ is the supremum of the transverse dimensions of all compact subsets K ⊂ A, with respect to ξ.The transverse dimension of A with respect to ξ is at most equal to the Hausdorff dimension of A, A; indeed, A = sup_KK where K goes through the family of compact subsets of A; and for any compact K, the quotient mapping K → K/ξ is Lipschitz and cannot increase Hausdorff dimension. This definition of transverse dimension highlights the fact that in general, we do not have to be too concerned with the choice of the Lipschitz mapping we use to parametrize the foliation. In the most classical situa- tion, X is the Euclidean plane and ξ is the foliation of X by affine lines of some given angle θ. The orthogonal projection onto the vector line of angle θ + π/2 is a suitable Lipschitz mapping, and so is the quotient mapping with respect to the vector line of angle θ.In more complex situations, the adequate projection may have a less elementary description, and it may also not be Lipschitz on the whole space X, but only on every compact subspace of X. Our emphasis in this paper will be on the geometry of foliations. We will introduce suitable Lipschitz projections to work with but in our perspective the foliations come first.Let us provide a simple example to explain why it is useful to think in terms of foliations rather than projections. Let X be the Euclidean plane minus the origin, and let ξ be the foliation of X by vector lines with the origin removed. Now let X' be the Euclidean plane minus two points x and y, and let ξ' be the foliation of X' by circles passing through x and y (with x and y removed from the circle). From the point of view of transverse dimension, there is no difference between (X, ξ) and (X', ξ'). We may identify X' with X (we have to remove one more point from X) via a Möbius transformation f; now f maps ξ' onto ξ and it is locally biLipschitz, so any local dimension property is preserved by f.The fact that ξ can be parametrized by the radial mapping at 0 is irrelevant and there is no need to find a corresponding projection mapping for ξ.Of course the transverse dimension of a given subset A with respect to ξ depends of how A is sitting with respect to ξ; in general this is a very difficult problem. A Marstrand-type Theorem deals not with a fixed foliation but rather with almost every foliation in a given foliations space endowed with some version of the Lebesgue measure.We will usually make an abuse of language and speak of a foliation ℱ of a space X when in fact something has to be removed from both X and ℱ in order to get a genuine partition.For example we may say “look at the foliation of the plane by lines passing through the origin”. What we mean in this case is “look at the foliation of the plane minus the origin by lines passing through the origin with the origin removed”. Likewise, if some line L is fixed in ^3, the family of all affine planes containing L,with L removed, is a foliation of ^3 ∖ L, but we will actually write “Let ℱ be the foliation of ^3 by affine planes containing L”. It would be very unpleasant and quite pedantic to write down inevery case which subset should be removed from the space and the leaves, and we leave it to the reader to make the obvious corrections.§.§.§ Transversality and Kaufman's argumentTransversality of a family of foliations (or a family of projections) is the crucial property to look forwhen one sets out to prove a Marstrand-type result. In the presence of transversality, a quite general argument,due to Kaufman <cit.>, allows to prove a version of Marstrand's projection Theorem, as well as someimprovements which we do not discuss in this paper in order to keep things short. In this paper, we are not going to improve on Kaufman's argument. Our purpose is to introduce “good foliations” and transversality will follow quite naturally from the definition (and the product formula, see <ref>). A general exposition of Kaufman's argument in an abstract setting (dealing with parametrized families of projections) can be found in <cit.>. We write down a detailed proof of Corollary <ref> becausethere are some issues, requiring us to work “locally” (cutting the measureinto small pieces), that do not appear in Kaufman's usual argument. Our statements deal with dimension of sets; it would be equally sensible to concern ourselves with dimension of measures and in the proofs this is what we actually do, implicitly using Frostman's Lemma, as in <cit.>.§.§ Hermitian forms on the Grassmann algebra §.§.§ The Grassmann algebra We refer to <cit.> for a good elementary exposition of the Grassmann exterior (bi)algebra associated to a vector space. Another good reference is <cit.>, but beware of the conflicting notations. In this paper we will use the same notations as in <cit.>.We now recall some basic definitions and fix appropriate notations.Letbeorand fix a finite-dimensional -vector space E. The elements of E will usually be denoted by the letters u, v, w.We denote by _(E) the Grassmann algebra (over ) associated with E. The Grassmann algebra is also called the exterior algebra of E. The progressive product (also called exterior product) will be denoted by the symbol ∨ The regressive product (to be introduced later) will be denoted by ∧. Ifis , we may consider the Grassmann algebra over , _(E), as well as the Grassmann algebra over , _(E). In this paper we will always work with the Grassmann algebra over(when =). In a later paper, we will also consider the Grassmann algebra, over , of a complex space, and this will allow us to define and study a family of foliations (called real spheres or Ptolemy circles) different from the chains which are the main focus of this paper. See e.g. <cit.> for definitions. Henceforth, we drop the subscriptin the notation for the Grassmann algebra. The subspace of k–vectors will be denoted by ^k (E). If n is the dimension of E (over ),(E) = ⊕_k=0^n ^k(E)where ^0(E) is canonically isomorphic to , ^1(E) is canonically isomorphic to E, ^n-1(E) is non-canonically isomorphic to the algebraic dual of E, and ^n(E) is non-canonically isomorphic to(the choice of a basis of ^n(E) is equivalent to the choice of a non-degenerate alternating n–linear form).The above direct sum is graded: for U ∈^k(E) and V ∈^ℓ(E), the progressive product U ∨ V belongs to ^k+ℓ(E) (where by definition ^i(E) = 0 if i > n). The set of pure or decomposable k–vectors, i.e. k–vectors of the form u_1 ∨⋯∨ u_k (u_1,…, u_k ∈ E) will be denoted by ^k(E). We will often use capital letters U, V, W to denote k–vectors, and most of the time we will consider pure k–vectors only.Bear in mind that unless k = 1 or k = (E) - 1, ^k(E) is not a vector space.If U = u_1 ∨⋯∨ u_k is a non-zero element of ^k(E), we denote by (U) the vector subspace u_1⊕⋯⊕ u_k of E. This is the smallest vector subspace E' of E such that U belongs to ^k(E').Thus, if U and V are, respectively, a pure k–vector and a pure ℓ–vector, such that U ∨ V ≠ 0, then (U ∨ V ) = (U) ⊕(V).The basic fact that k–dimensional vector subspaces of E are in one-to-one correspondance with projective classes of elements of ^k+1(E) will be used at every moment throughout this paper.§.§.§ The regressive product We now recall briefly the definition of the regressive product. Let n be the dimension of E (over ). The choice of a non-degenerate alternating n-linear form ω on E yields a Hodge isomorphism* : (E) →(E^*)that identifies ^k(E) with ^n-k(E^*) (were E^* is the algebraic dual space of E in the usual sense).The pull-back, through this isomorphism, of the progressive product in (E^*) is, by definition, the regressive product in (E), denoted by ∧.The regressive product depends on the choice of ω; to put it differently, it depends on the choice of a basis of the 1–dimensional space ^n(E).By definition,U ∧ V = (U^* ∨ V^*)^* Let us also recall the definition of the Hodge isomorphism. The bilinear mapping^k(E) ×^n-k(E) →^n(E) defined by(u_1 ∨⋯∨ u_k, u_k+1∨⋯∨ u_n) ↦ u_1 ∨⋯∨ u_n(and extended by linearity) is composed with the isomorphism ^n(E) → associated to ω,u_1 ∨⋯∨ u_n ↦ω(u_1, …, u_n) and identifies ^k(E) with the dual of ^n-k(E); this dual is also canonically isomorphic to ^n-k(E^*). In this way, we obtain for every k an isomorphism ^k(E) →^n-k(E^*) which is, by definition, the Hodge isomorphism restrictedto ^k(E).For details, see <cit.> or <cit.>.The geometric significance of the regressive product should be clear: if U ∧ V ≠ 0, where U, V are, respectively, a pure k–vector and a pure ℓ–vector,and k+ℓ≥ n, then U ∧ V is a pure (k+ℓ-n)–vector such that(U ∧ V) = (U) ∩(V); if k+ℓ <n, U ∧ V=0. Endowed with ∨ and ∧, (E) is the Grassmann bialgebra of E. §.§.§ Grassmann extensions of Hermitian forms Let E be, as before, a finite-dimensional -vector space (= or ), now endowed with a sesquilinear form Φ; by definition Φ(α u, β v) = αβΦ(u, v) for α, β∈ and u, v ∈ E. Ifisthis is bilinearity in the usual sense.Denote by E the –vector space with the same underlying additive group as E and the –operation law defined by α· u= α u (where the right-hand side denotes the operation of α on u in E).Ifis , E is equal to E, whereas if = the identity mapping is an anti-isomorphism E→ E.We now recall, as in <cit.>, the canonical extension of the sesquilinear form Φ to the Grassmann algebra (E).Fix k ≥ 1. The mapping E^k × E^k → defined by(u_1, …, u_k; v_1, …, v_k) ↦ (Φ(u_i, v_j )) (whereis the usual determinant of a k × k matrix) is –multilinear.Since, also, the right-hand side is zero as soon as u_i = u_j or v_i = v_j for some i ≠ j, this mapping yields, by the universal property of the Grassmann algebra (see <cit.>, 7, Proposition 7), a –bilinear form ^k(E) ×^k(E) →.Using the canonical antilinear identication of ^k(E) with ^k(E), we obtain a sesquilinear form on ^k(E) which we denote by ^k(Φ).If Φ is Hermitian, meaning that Φ(v, u) = Φ(u, v), so is ^k(Φ). Also if Φ is non-degenerate, ^k(Φ) is non-degenerate as well; if Φ is definite, ^k(Φ) is definite, and of the same sign.The following result is basic. If U_1,U_2 ∈^k(E), and V ∈^ℓ(E), are such that (U_1) and (U_2) are both Φ–orthogonal to (V), ^k+ℓ (Φ) (U_1 ∨ V, U_2 ∨ V) = ^k (U_1,U_2) ×^ℓ (V,V)§.§.§ Basic properties of the Hermitian normFix n ≥ 1 and let again beor . We denote the canonical basisof ^n+1 by (e_0,…,e_n). For any u, v ∈^n+1, we denote the usual Hermitian inner product byuv = ∑_i=0^nx_i y_i(where u=(x_0,…,x_n) and v=(y_0,…,y_n)) and we use the same symbol for the canonical Grassmann extension, i.e.u_1 ∨⋯∨ u_kv_1 ∨⋯∨ v_k =(u_iv_j)(where the right-hand side is the determinant of the k × k matrix whose (i, j)–component is u_iv_j. This Grassmann extension is still a Hermitian inner product on ^k (^n+1) and the associated Hermitian norm is denoted, as usual, by ·.The n–dimensional projective space over is denoted by _^n ; this is the space of –vector lines in ^n+1. If =, respectively =, ^n_ is a Riemannian manifold of dimension n, respectively a Hermitian manifold of complex dimension n (and real dimension 2n). In general, if E is some –vector space, the symbol E denotes the projective space associated to E over .We will also use the notation ^k (^n+1) to denote the space of projective classes of elements of ^k (^n+1). (Note that ^k (^n+1) is not a vector space in general.)In this paper, the letter d will always denote the angular metric on ^n_ defined byd(u,v) = uvwhere u, v are non-zero elements of ^n+1. In the left-hand side we are abusing notations and denoting elements of ^n_ by corresponding elements of ^n+1. It seems preferable to slightly abuse notations rather than use the cumbersome notation [u],[v] when we are dealing with projective classes.From this definition, the following formula follows at once:d(u,v)^2 = 1 - |uv|^2/u^2 ·v^2 = sin^2 (θ)where θ is the (non-oriented) angle from u to v.The orthogonal complement with respect to the Hermitian inner product will be denoted by ⊥. For example, v^⊥ is the space of all vectors u ∈^n+1 such that uv = 0. For any U ∈^k(^n+1), V ∈^ℓ(^n+1), U ∨ V ≤ U · V and this is an equality if and only if (U) and (V) are orthogonal.This follows from the following Lemma which we state separately for future reference. Let V ∈^ℓ(^n+1) and denote * π_V^⊥ the orthogonal projection ^n+1→(V)^⊥ * Π_V^⊥ the orthogonal projection ^k (^n+1) →^k((V)^⊥). Then ^k (π_V^⊥)=Π_V^⊥ and for any U ∈^k (^n+1), U ∨ V= Π_V^⊥ (U) · V The notation ^k(f), where f is a linear mapping with domain E, stands for the extension of f to ^k(E), which is characterized by the relation ^k(f)(u_1 ∨⋯∨ u_k) = f(u_1) ∨⋯∨ f(u_k) for any u_1, …, u_k ∈ E. Recall the basic property that a linear projection π is orthogonal if and only if for any vector x in the image of π and any other vector y, xπ(y) is equal to xy. To show that ^k (π_V^⊥) is the orthogonal projection onto ^k ((V)^⊥), it is enough to check that for any U ∈^k (^n+1) and any U' ∈^k ((V)^⊥), UU' = ^k (π_V^⊥) (U)U' Now by definition the right-hand side is equal to (π_V^⊥(u_i)u_j') and π_V^⊥(u_i)u_j'=u_iu_j' by the basic property of orthogonal projections; this determinant is thus equal to (u_iu_j') = UU'. The formula for norms then follows from the fact that U ∨ V = ^k (π_V^⊥) (U) ∨ V by definition of the progressive product.§.§.§ First distance formula Let U= u_0 ∨⋯∨ u_k be a non-zero element of ^k+1(^n+1) and let w ∈^n+1 be non-zero. The quantity τ(U,w)=Uw is equal to the distance between w and the k–dimensional projective subspace _ ((U)) in _^n+1.We commit the usual abuse of language of denoting by w both a non-zero vector and its image in _^n. Let π be the orthogonal projection from ^n+1 onto (U). Then w - π(w) is orthogonal to (U), so τ(U,w)^2 is equalto w - π(w)^2/w^2 = 1 - π(w) ^2/w^2 = 1 - sup_v ∈(U)| wv |^2/w^2 ·v^2 (where we used the convexity of orthogonal projections and ). By using formula (<ref>), we see that the right-hand side of the last equation is equal to inf_v ∈(U) d(w,v)^2 = d(w, ((U)))^2§ LINEAR FOLIATIONS IN REAL AND COMPLEX PROJECTIVE SPACES In this section, we fix an integer n ≥ 2 and we work in the n–dimensional –projective space _^n.If U = u_0 ∨⋯∨ u_k is a non-zero element of ^k+1(^n+1) (k ≤ n), we will denote by L_U the projective subspace ((U)) of _^n. The mapping [U] ↦ L_U is a bijection from ^k+1(^n+1) to the space of k–dimensional –projective subspaces of ^n_. We will identify these spaces and say “let [u_0 ∨⋯∨ u_k] be a k–dimensional projective subspace of ^n_.”§.§ Generalized radial foliationsFix an integer k, 0 ≤ k ≤ n-2. For any k–dimensional projective subspace L of ^n_, and any x ∈^n_∖ L, there is one and only one (k+1)–dimensional projective subspace of ^n_ containing L ∪{x}.To L we may thus associate a foliation of ^n_ by (k+1)–dimensional projective subspaces. We exclude the case k = n - 1 because the foliation is then trivial (it has only one leaf).(Recall that when we say “a foliation of ^n_by projective subspaces” here we actually mean “a foliation of ^n_∖ L_U by projective subspaces containing L_U,with L_U removed”.)Let us describe this foliation algebraically, thanks to the Grassmann algebra, in order to perform computations.Fix a k–dimensional projective subspace [U] = [u_0∨⋯∨ u_k] ∈^k+1(^n+1). of ^n_ and denote by _U[ _U: _^n ∖ L_U →^k+1(^n+1); [w] ↦ [u_0 ∨⋯∨ u_k ∨ w]; ]By definition, the fibers of this mapping are exactly the (k + 1)–dimensional projective subspaces of _^n containing L_U.We are now going to endow ^k+1(^n+1) with a natural metric, in order to be able to prove the needed transversality properties for our linear foliations. §.§ The angular metric on the codomain We endowed earlier ^ℓ (^n+1) with a Hermitian structure for 0 ≤ℓ≤ n+1. To this Hermitian space we may associate its degree 2 Grassmann algebra, ^2(^ℓ(^n+1)) and this is in turn a Hermitian space in a natural way. We are now looking at the Grassman algebra arising from the vector space underlying a Grassmann algebra; its elements are of the form U ∨ V , where U and V belong to ^ℓ(^n+1), and the reader should be careful not to believe that, somehow, if U = u_1 ∨⋯∨ u_ℓ and V = v_1 ∨⋯∨ v_ℓ, the element U ∨ V of ^2(^ℓ(^n+1)) could be equal to the element U ∨ V of ^2 ℓ(^n+1). These elements do not sit in the same space. They are the same thing if and only if ℓ = 1.This construction allows to endow ^ℓ(^n+1)(the projective space associated to^ℓ (^n+1)) with the metric defined as in (<ref>).In turn, the restriction of this metric to ^ℓ(^n+1) endows the space of (ℓ - 1)-projective subspaces of _^n with a natural metric.Our aim in the paragraph to follow is to study the distance between _U(w_1) and _U(w_2) for w_1,w_2 ∈_^n. §.§ The product formula Let p ≥ 1. For any element V ∈^p-1 (^n+1) and any w_1,w_2 ∈_^n ∖ L_V, (V ∨ w_1) ∨ (V ∨ w_2)=V · V ∨ w_1 ∨ w_2 where (V ∨ w_1) ∨ (V ∨ w_2) belongs to ^2 (^p (^n+1)) and V ∨ w_1 ∨ w_2 belongs to ^p+1(^n+1). Denote by π the orthogonal projection ^n+1→(V)^⊥. By the basic properties of progressive product we have V ∨ w_1 = V ∨π(w_1) and similarly for w_2; we also have, for the same reasons, V ∨ w_1 ∨ w_2 = V ∨π(w_1) ∨π(w_2) Without loss of generality, we can thus assume that w_1 and w_2 are orthogonal to (V). Now, by definition, the square of the left-hand side in equation (<ref>) is equal to the 2 × 2 determinant | [ V ∨ w_1 ^2 V ∨ w_1V ∨ w_2; V ∨ w_2V ∨ w_1V ∨ w_2^2 ]| We can apply Lemma <ref> and the previous determinant is equal to V^4 ·| [w_1^2 w_1w_2; w_2w_1 w_2 ^2 ]| = V^4 · w_1 ∨ w_2 ^2 On the other hand, using again orthogonality of w_1 and w_2 with respect to (V), as well as Lemma <ref> we see that V ∨ w_1 ∨ w_2=V ·w_1 ∨ w_2 and the proof is over.For any elements U ∈^k (^n+1), V ∈^ℓ (^n+1), we denote by τ(U,V) the numberτ(U,V) = UVwhere U ∨ V ∈^k+ℓ(^n+1). If k (resp. ℓ) is equal to 1, this is the distance from U (resp. V) to ((V)) (resp. ((U))) (Theorem <ref>). Also, note that τ(U,V) ≤ 1.With this notation, the previous Theorem has the following consequence: d (_U (w_1),_U(w_2)) = τ(U,w_1 ∨ w_2)/τ(U,w_1) τ(U,w_2) d(w_1,w_2)which will play a crucial role in our analysis. §.§ Generalized distance formulaIn this paragraph, we elucidate the geometric significance of the number τ(U,V) introduced above; this is not needed in the rest of the paper.We start with some notations and a lemma. Fix q ≥ 2 and k,ℓ≥ 1. such that k + ℓ≤ q. If V is some non-zero decomposable ℓ-vector of ^q, i.e. V ∈^ℓ (^q), we let_0^k (V;^q) = { U ∈^k (^q) ; U ∨ V = 0 } This is the annihilator of V in ^k (^q), a vector subspace of ^k (^q). The orthogonal complement of ^k ((V)^⊥) in ^k (^q) is equal to _0^k (V;^q). Let π be the orthogonal projection ^q →(V)^⊥. Then ^k (π) is the orthogonal projection ^k (^q) →^k((V)^⊥). Here, we denote by ^k(π) the extension of π to ^k (^q). Also, it follows from the definition of progressive product that U ∨ V = ^k (π) (U) ∨ V. This (k+ℓ)-vector is zero if and only if ^k(π) (U)=0, which is equivalent to saying that U is orthogonal to ^k ((V)^⊥). For U ∈^k (^q) and V ∈^ℓ (^q), UV = d(U,_0^k(V;^q)) where U ∨ V belongs to ^k+ℓ(^q).In other words, the number τ(U,V) is equal to the distance, in ^k (^q), between U and the projective subspace _0^k (V;^q). Let π be the orthogonal projection from ^q onto (V)^⊥. We compute, taking into account the previous Lemma, 1 - ( UV)^2 = 1 - ^k (π) (U) ^2/U^2 = U-^k (π)(U)^2/U^2 and U - ^k (π)(U) is the image of U through the orthogonal projection onto ^k ((V)^⊥)^⊥. The convexity property of inner product then yields U-^k (π)(U)^2/U^2 = sup_W|UW|^2/U^2 ·W^2 where the supremum is relative to W ∈^k((V)^⊥)^⊥ = _0^k (V;^q). The right-hand side is equal to 1 - inf_W ( UW)^2 = 1 - inf_W d(U,W)^2 (where still W ∈^k_0 (V;^q) and U ∨ W ∈^2 (^k (^q))) and the Lemma is proved.§.§ Lipschitz property; transversality; Marstrand-type Theorem Recall the setting from the beginning of the section: n ≥ 2 is fixed, 0 ≤ k ≤ n-2 and U=u_0 ∨⋯∨ u_k ∈^k+1(^n+1) is a (momentarily fixed) decomposable (k+1)–vector of ^n+1.Introduce the Lipschitz modulus functionϕ_U (w_1,w_2) = d(_U(w_1),_U(w_2))/d(w_1,w_2)for any pair of distinct w_1,w_2 ∈_^n ∖ L_U.Recall the formula (<ref>)ϕ_U(w_1,w_2) = τ(U,w_1 ∨ w_2)/τ(U,w_1) τ(U,w_2) A basic fact is that _U is “locally Lipschitz”. The restriction of _U to any compact subspace of _^n ∖ L_U enjoys the Lipschitz property. By the above formula (<ref>), the Proposition follows from the fact that the function w ↦τ(w,U) is continuous and non-zero in _^n ∖ L_U.The space ^k+1 (^n+) is a Hermitian manifold and carries a natural Lebesgue measure . This measure can be defined in an elementary way: endow (_^n)^k+1 = _^n ×⋯×_^n with the product (k+1 times) of the Lebesgue measure of _^n, and push this product measure forward through the almost-everywhere defined mapping[ (_^n)^k+1 = _^n ×⋯×_^n→^k+1 (^n+1);([u_0],…,[u_k])↦[u_0 ∨⋯∨ u_k];] We now state our first transversality result. For any distinct w_1,w_2 ∈_^n and any r>0, { U ∈^k+1 (^n+1) ; ϕ_U(w_1,w_2) < r }≲ r^δ_ (n-k-1) uniformly in w_1,w_2, where δ_ is 1 if = and 2 if =. Since ϕ_U(w_1,w_2) ≥τ(U,W) by (<ref>), where we let W=w_1 ∨ w_2, it is enough to show that { U ∈^k+1 (^n+1); τ(U,W) < r }≲ r^n-k-1 If k=0 this is a special case of Lemma <ref> below. If k ≥ 1 we argue by induction using the inequality τ(U,W) ≥τ(u_k,U' ∨ W) τ(U',W) where U=u_0 ∨⋯∨ u_k and U' = u_0 ∨⋯∨ u_k-1. Lemma <ref> along with Fubini's Theorem yield { U ∈^k+1 (^n+1) ; τ(U,W) < r } ≲ r^δ_ (n-k-1)∫d(U')τ(U',W)^-δ_ (n-k-1) The induction hypothesis implies that ∫d(U')τ(U',W)^-δ_ (n-k-1) is finite, and the Proposition follows. For any ℓ-dimensional projective subspace L of _^n { u ∈_^n ; d(u,L) ≤ r}≲ r^δ_ (n-ℓ) uniformly in r.This Lemma and its proof are standard. Fix k, 0 ≤ k ≤ n-2. Let A be a Borel subset of _^n of Hausdorff dimension s. For almost every k–dimensional projective subspace L of _^n, the transverse dimension of A, with respect to the foliation of _^n ∖ L by (k+1)-dimensional projective subspace containing L, is equal to inf{δ_(n-k-1),s} We apply Kaufman's classical argument using the transversality property stated in Proposition <ref>. Let us provide some details. We assume s>0. First, note that if v_1,v_2 are two different points of _^n, the set of k–dimensional projective subspaces passing through v_1 and v_2 has Lebesgue measure 0 in ^k+1(^n+1). Taking this fact into account, pick 2disjoint closed balls B_1, B_2 such that the Hausdorff dimension of A ∩ B_i is s for each i, and small enough that the set of k–dimensional projective subspace of ^n_ meeting both B_1 and B_2 has very small Lebesgue measure. This is possible because the Hausdorff dimension of a finite union ∪ X_i is the supremum of the Hausdorff dimensions of the X_i. Let O_1,O_2 be open subsets of ^k+1(^n+1) such that for any U ∈O_i, the projective subspace L_U does not meet B_i, and that the complement of O_1 ∪ O_2 has very small Lebesgue measure in ^k+1(^n+1). Now fix i=1 or 2. Let σ < inf{ s, δ_ (n-k-1)} and μ be a Borel probability measure supported on A ∩ B_i such that the σ–energy of μ is finite: I_σ (μ) = ∫dμ (w_1) dμ(w_2)/d(w_1,w_2)^σ < ∞ (see <cit.> 8.8 and 8.9). We apply Fubini's Theorem: ∫_O_id (U)I_σ (_U (μ)) = ∫dμ (w_1) dμ(w_2)/d(w_1,w_2)^σ∫_O_id (U) ϕ_U (w_1,w_2)^-σ and we will show that the right-hand side is finite by checking that ∫_O d(U) ϕ_U(w_1,w_2)^-σ is bounded by a uniform constant for any distinct w_1,w_2 ∈B_i. A standard application of Fubini's Theorem followed by a change of variable yields ∫_O_id(U) ϕ_U(w_1,w_2)^-σ≲ 1+∫_0^1 { U ∈ O_i ; ϕ_U(w_1,w_2) < t} t^-(1+σ) dt (where the constant implied by the notation ≲ does not depend on w_1,w_2). Taking into account Proposition <ref>, we see that the right-hand side is bounded by a uniform constant as soon as σ < δ_ (n-k-1), which holds by assumption. All in all, we get ∫_O_id (U)I_σ (_U (μ)) ≲ I_σ (μ) < ∞ showing that for Lebesgue–almost every U ∈ O_i, the transverse dimension of A ∩ B_i, along the foliation by (k+1)–dimensional projective subspaces containing U, is at least equal to σ-ε. Thus for almost every U ∈ O_1 ∪ O_2, the transverse dimension of A, along the foliation by (k+1)–dimensional projective subspaces containing U, is at least equal to σ-ε. Since ε was arbitrary, we get the desired conclusion for almost every U ∈ O_1 ∪ O_2. The Theorem follows from this, because the complement of O_1 ∪ O_2 has arbitrarily small Lebesgue measure. The previous results will now be improved by looking at a special subfamily of foliations. §.§ Transversality of pointed foliationsAs before, n ≥ 2 is fixed and we work in _^n. Fix V ∈^n (^n+1). For any U ∈^k (^n+1) and W ∈^2 (^n+1) such that * (U) ⊂(V); * (W) ⊄(V) we have UW≥ U ∨ (W ∧ V)/U·W ∧ V Let V=v_1 ∨⋯∨ v_n and assume, as we may, that (v_1,…,v_n) is an orthonormal basis of (V). Also, choose w_2 ∈(V) and w_1 orthogonal to (V) such that W=w_1 ∨ w_2. It follows that W ∧ V is colinear to w_2; hence U ∨ (W ∧ V)/U·W ∧ V= Uw_2 On the other hand, UW = U ∨ w_2w_1×Uw_2× d(w_1,w_2)^-1 where the first term of the right-hand side is equal to 1 because w_1 was chosen to be orthogonal to (V) and both (U) and w_2 are inside (V); also, we know that d(w_1,w_2) ≤ 1 ((<ref>) and Lemma <ref>). Hence the Lemma. Fix V as in the Lemma, and let K be some compact subset of _^n ∖ L_V. For any distinct w_1,w_2 ∈ K { U ∈^k+1 ((V)) ; ϕ_U (w_1,w_2) < r }≲ r^δ_ (n-k-1) where as previously δ_ is 1 or 2 according asisor . Follow the line of the proof of Proposition <ref>, replacing the 2-vector W with the (genuine) vector V ∧ W. Fix k and V as before. Let A be a Borel subset of _^n ∖ L_V of Hausdorff dimension s. For almost every k–dimensional projective subspace L of _ ((V)), the transverse dimension of A with respect to the foliation of _^n ∖ L_V by (k+1)–dimensional projective subspaces containing L, is equal to inf{δ_ (n-k-1),s}This is very similar to the previous Corollary, but we are now looking at k–dimensional projective subspaces of a fixed projective hyperplane L_V, effectively lowering the dimension of the space of foliations.More precisely, the space of k–dimensional projective subspaces of _^n has dimension (k+1)(n-k) whereas the space of k–dimensional projective subspaces of L_V has dimension (k+1)(n-k-1).In restricting our space of foliations, we did not lose anything dimension-wise. We will see later that this is not really surprising, by showing how, when =, this Corollary is actually equivalent to the classical Marstrand–Kaufman–Mattila projection Theorem.§ LINEAR FOLIATIONS OF SPHERES §.§ General setup Let n ≥ 2. We will deal at the same time with the (n-1)-sphere in _^n andthe (2n-1)-sphere in _^n, so let us introduce suitable notations: denote by , ⊂_^nthe sets. [^n-1 = { [1:x_1:…:x_n] ∈_^n ; x_1^2+⋯+x_n^2 = 1};^n = { [1:x_1:…:x_n] ∈_^n ; x_1^2+⋯+x_n^2 < 1} ]}if= . [^2n-1= { [1:z_1:…:z_n]∈_^n ; |z_1|^2+⋯+|z_n|^2 = 1};^2n= { [1:z_1:…:z_n]∈_^n ; |z_1|^2+⋯+|z_n|^2 < 1} ]}if= If L is some k–dimensional projective subspace of _^n, let ℱ_L be the foliation ofthe leaves of which are the intersections ofwith (k+1)–dimensional projective subspaces of _^n containing L.(Remember that we are abusing the language and that what we really mean here is that ℱ_L is the foliation of ∖ (L ∩) the leaves of which are the intersections of ∖ (L ∩) with (k+1)–dimensional projective subspaces of _^n containing L.)If =, the case k=0 is essentially empty and should be removed from consideration.The leaves of ℱ_L are small k–spheres if =, respectively small (2k+1)-spheres if =.Our previous projection results in _^n (Corollaries <ref> and <ref>)translate without any further work to interesting projection results in . We need only remark that the restriction of d tois equal to the usual angular metric on . Fix k, 0≤ k ≤ n-2, and let A be a Borel subset ofof Hausdorff dimension s. For almost every k–dimensional projective subspace L of _^n, the transverse dimension of A with respect to ℱ_L is equal to inf{ s, δ_ (n-k-1)}.We are going to restrict this family of foliations in order to obtain foliations which can be described purely in terms of spherical geometry.For k ≥ 1, let ℒ_^k be the space of k–dimensional projective subspaces L of _^n that meet . If =, this is the same thing as thespace of small (k-1)–spheres of ^n-1. Assume =. For k ≥ 0, a k–chain is the intersection of ^2n-1 with a k–dimensional projective subspace of _^n that meets ^2n.A k–chain is the complex analogue of a small k–sphere; it is also a special case of small (2k-1)–sphere.For example, the space of all small 1-spheres (i.e. all small circles) of ^3 has dimension 6, whereas the space of all 1-chains of ^3 has dimension 4. If k ≥ 1, ℒ_^k is an open subset of ^k+1(^n+1), the space of all k–dimensional projective subspaces of _^n.In particular, the Lebesgue measure of this space is non-zero. Corollary <ref> thus implies the following Fix k, 1 ≤ k ≤ n-2 and let A be a Borel subset ofof Hausdorff dimension s. = : For almost every (k+1)–tuple (u_0,…,u_k) of points of =^n-1, the transverse dimension of A, with respect to the foliation of ^n-1 by small k–spheres passing through each of these points, is equal toinf{n-k-1,s} = : For almost every (k+1)–tuple (u_0,…,u_k) of points of =^2n-1, the transverse dimension of A, with respect to the foliation of ^2n-1 by (k+1)–chains passing through each of these points, is equal toinf{2(n-k-1),s}This follows from Corollary <ref> because the restriction ofthe Lebesgue measure to ℒ_^k is equivalent to the probability measure obtainedby picking k+1 points at random onand looking at the only k–dimensional projective subspacepassing through each of these points.In the same fashion, Corollary <ref> implies the followingFix k, 1 ≤ k ≤ n-2 and let L ⊂ be a small (n-2)–sphere if =,respectively a (n-2)–chain if =. Let A be a Borel subset ofof Hausdorff dimension s. = : For almost every (k+1)–tuple (u_0,…,u_k) of points of L, the transverse dimension of A, with respect to the foliation of ^n-1 by small k–spheres passing through each of these points, is equal toinf{n-k-1,s} = :For almost every (k+1)–tuple (u_0,…,u_k) of points of L, the transverse dimension of A, with respect to the foliation of ^2n-1 by (k+1)–chains passing through each of these points, is equal toinf{2(n-k-1),s} When =, the case k=0 is missing (because ^2n-1 is negligible for theLebesgue measure on _^n) and we have to handle it separately. §.§ Foliations by 1-chains We now fix =. For every u ∈^2n-1⊂^n_, the foliation of _^n by projective lines passing through u induces a foliation of ^2n-1 by 1-chains passing through u. Let A be a Borel subset of ^2n-1 of Hausdorff dimension s. For almost every u ∈^2n-1, the transversedimension of A with respect to the foliation of ^2n-1 by 1–chains passing through u is equal toinf{s, 2n - 2}The Hausdorff dimension of a Borel set A is the supremum of the Hausdorff dimensions of its compact subsets. Using this fact, we can assume, without loss of generality, that A is a compact subset of ^2n-1. Let O be an open subset of ^2n-1 that is non-empty and such that the closure O does not meet A. We first show that the conclusion holds for almost every u ∈ O.Introduce the set C_ε(A, O) of all projective lines [W] ∈^2(^n+1) that meet A and O_ε, where O_ε is the ε–neighbourhood of O in ^n_, i.e. Fix ε small enough that A does not meet O_ε. Let G = 𝐏𝐔(1, n) andfix as in Lemma <ref> a compact subset 𝒢 of G such that any 1-chain meeting A and O_ε is of the form gL_0, where g ∈𝒢 and L_0 is the 1-chain passing through, say, [e_0 + e_1] and [e_0 - e_1]. (Recall that (e_0,…,e_n) is the canonical basis of ^n+1.)I claim that for any r > 0, and any distinct w_1, w_2 ∈ A_^2n-1{u ∈ O ; τ (u, W ) ≤ r}≲ r^2n-2where W = w_1 ∨ w_2 and the constant implied by the notation ≲ is uniform in r, W and u. We can assume that r is small enough (with respect to the ε fixed above) that the projective line [W ] has to meet both A and O_ε in order for the left-hand side to be non-zero.Let g be an element of 𝒢 such that [W ] = [gW_0] where W_0 = e_0 ∨ e_1. This is possible because (e_0 + e_1) ∨ (e_0 - e_1) is a scalar multiple of e_0 ∨ e_1.Now_^2n-1{u ∈ O ; τ (u, gW_0) ≤ r}≲_^2n-1{u ∈ O ; τ (u, W_0) ≲ r} ≲_^2n-1{u ∈^2n-1 ; τ (u, W_0) ≤ r}≲ r^2n-2where the first inequality follows from the compacity of 𝒢 (and thesubsequent fact that the singular values of g belong to a compact subset of ]0, +∞[) and the last inequality is an easy computation.At this point we can apply Kaufman's argument; this yields that for Lebesgue-almost every u ∈ O, the transverse dimension of A with respect to the foliation of ^2n-1 by 1-chains passing through u is equal to inf{s, 2(n - 1)}.Now let x be any point of ^2n-1. For any ε > 0 we can find δ > 0 such that(∖ B(x, δ)) ≥ s - ε(whereis the Hausdorff dimension). Taking into account the previous statement,it follows that for Lebesgue-almost every u ∈ B(x, δ/2), the transverse dimension of K, with respect to the foliation by 1-chains passing through u, is at least equal to inf{s - ε, 2(n - 1)}.The compacity of ^2n-1 then implies that for Lebesgue-almost every u ∈^2n-1, the transverse dimension of K, with respect to the foliation by 1-chains passing through u, is at least inf{s - ε, 2(n - 1)}.The Theorem follows by letting ε go to 0 along a countable sequence.Let K^-, K^+ be disjoint non-empty compact subsets of ^2n-1 and let L_0 be a fixed 1–chain. There is a compact subset 𝒢 of G such that any 1–chain meeting both K^- and K^+ is of the form gL_0 for some g ∈𝒢. Fix ξ^-, ξ^+ two distinct elements of L_0 and let KAN be an Iwasawa decomposition of G in which the Cartan subgroup A fixes both ξ^- and ξ^+ (and, consequently, leaves L_0 globally invariant).The mapping g ↦ (gξ^-, gξ^+) defines, by passing to the quotient, a proper and onto mappingω : G/A →{(η^-, η^+) ∈^2n-1×^2n-1 ; η^- ≠η^+ }(Recall that a continuous mapping is proper if the inverse image of any compact subset is a compact subset.)Now, K^- × K^+ being compact, so must be its inverse image ω^-1(K^-× K^+).To conclude, use the fact that any compact subset of G/A is the image (through the quotient mapping G → G/A) of a compact subset of G. Let A be a Borel subset of ^2n-1 of Hausdorff dimension s. Fix a (n - 1)–chain L. For Lebesgue-almost every u ∈ L, the transverse dimension of A with respect to the foliation of ^2n-1 by 1-chains passing through u is at leastinf{s, 2n - 3}We argue as in the proof of the previous Theorem. Using transitivity of G=𝐏𝐔(1, n) in the same fashion as before, we can safely assume that L is the intersection of ^2n-1 with ( e_0 ⊕⋯⊕ e_n-1). Let V = e_0 ∨⋯∨ e_n-1.Now fix a compact subset K ⊂^2n-1 that does not meet L = L_V.For any distinct w_1, w_2 ∈ K, letting W = w_1 ∨ w_2, we know by Lemma <ref> that τ (u, W ) ≥τ (u, W ∨ V )for any u ∈^2n-1. It follows that_L { u ∈ L ; τ (u, W ) ≤ r}≤_L { u ∈ L ; d(u, W ∨ V ) ≤ r}and W ∨ V is just a point of the (n - 1)–chain L.The (n - 1)–chain L is equal to the (2n - 3)–sphere ^2n-3⊂_^n-1 where we identify _((V)) with _^n-1. The angular metric defined on ^n-1_ as in formula (<ref>) is equal to the restriction of d to ^n-1_. Its restriction toL is just the spherical metric of ^2n-3. The right-hand side in the previous equation is thus ≤ r^2n-3. The Theorem follows from this estimate, as above.§.§ A concrete look at these foliationsWe now look at some concrete examples in order to get a better idea of what is actually going on in the previous Theorems and how our results are related to Marstrand's classical projection Theorem.§.§.§ The real caseWe fix =.In affine coordinates. Let k = 0 and n ≥2. For any u ∈_^n, we consider the foliation of _^n by projective lines passing through u (with u removed). If we pick affine coordinates ^n ⊂_^n and send some projective hyperspace ^n-1 to infinity, we are looking, when u belongs to ^n,at the usual radial projection in ^n.On the other hand, if u belongs to ^n-1, the resulting foliation of ^n has leaves affine lines parallel to the vector line associated to u.We can now translate the conclusion of Corollary <ref> in affine terms: for Lebesgue-almost every u ∈^n-1, the dimension of A transverse to the foliation of ^n by affine lines parallel to the vector line associated to u, is equal to inf{s, n - 1}.This is exactly the statement of the usual Marstrand projection Theorem along lines.We see that this statement is equivalent to the following one: for any affine hyperspace L in ^n, for Lebesgue-almost every u ∈ L, the dimension of A transverse to the foliation of ^nby affine lines passing through u is equal to inf{s, n - 1}.We leave it to the reader to inspect the case when k ≥ 1 and come to the conclusion that Corollary <ref> is again essentially equivalent to Marstrand's classical projection Theorem (which, in this case, is due to Kaufman and Mattila). Foliations of the sphere. If k = 0, there is no interestingfoliation induced because a (real) projective line meets the sphere in at most two points.Assume 1 ≤ k ≤ n - 2. Let us fix already a (n - 2)–sphere L ⊂^n-1 and let A be a Borel subset of ^n-1 of Hausdorff dimension s. According to Theorem <ref> for almost every u_0, …, u_k in L, the transverse dimensionof A with respect to the foliation of ^n-1 by k–spheres passing through u_0, …, u_k is equal to inf{s, n - 1 - k}.Let us look at this result in the Euclidean space: send u_0 at infinity viathe stereographic projection, in such a way that L is the subspace ^n-2 of ^n-1.For any u_1, …, u_k ∈ L, the foliation of ^n-1 we obtain has leaves the affine k–spaces containing u_1, …, u_k. We thus see that Theorem <ref> is actually weaker than Corollary <ref> when =.§.§.§ The complex case In affine coordinates. Let us look already at the affine version of Corollary <ref>. First, fix k = 0. We are looking at foliations of ^n by complex affine lines parallel to a given complex vector line (associated to some point u ∈_^n-1). We can recast this in real terms: we are looking at foliations of ^2n with real affine 2-planes parallel to a given real vector 2-plane.Now this real vector 2-plane cannot be just any 2-plane: it has to be the real plane underlying some complex line.It becomes apparent that we are effectively improving on Marstrand's projection Theorem. This Theorem deals with the family of every real vector plane of ^2n, and we see that the conclusion still holds when we restrict to this subspace of foliations, which is is Lebesgue-negligible.It is perhaps enlightening to compare the dimension of the space of all foliations of ^2n by parallel affine 2-planes, which is equal to 2(n-1)× 2 = 4(n-1), to the dimension of the subspace of those special foliations coming from complex lines,which is equal to 2(n - 1) (this is of course the real dimension of _^n-1).For an arbitrary k ≥ 0 (and k ≤ n - 2), we are looking, in the complex case, at a family of foliations of ^2n by 2(k + 1)–dimensional real spaces coming from (k + 1)–dimensional complex spaces; the dimension of this space of foliations is equal to 2(n - 1 - k)(k + 1), whereas the dimension of the spaceof all 2(k + 1)–dimensional real vector subspaces of ^2n is equal to 4(n - 1 - k)(k + 1). Foliations of the sphere. A k–chain of ^2n-1 is a special case of a small (2k - 1)–sphere of this sphere. It is only natural to wonder what kind of object that is. In fact, chains appear naturally in complex hyperbolic geometry. The complex hyperbolic space of (complex) dimension n, 𝐇^n_, has ^2n-1 as its boundary at infinity. A totally geodesic submanifold S of 𝐇^n_ is one of two types:* S is isometric to a complex hyperbolic space 𝐇_^k, 1 ≤ k ≤ n;* Or S is isometric to a real hyperbolic space 𝐇_^k, 1 ≤ k ≤ 2(n - 1).In the first case, the boundary of S is a k–chain. In the second case, it is, in Cartan's terminology, a real k–sphere. This is quite an unfortunate term;a complex chain is a sphere just as much as a real sphere is.We will come back to so-called real spheres in a later paper.plain | http://arxiv.org/abs/1704.08010v2 | {
"authors": [
"Laurent Dufloux"
],
"categories": [
"math.MG",
"28A80, 53D10"
],
"primary_category": "math.MG",
"published": "20170426082908",
"title": "Linear foliations of complex spheres I. Chains"
} |
IEEE Transactions on Circuits and Systems for Video Technology Shell et al.: Bare Demo of IEEEtran.cls for JournalsTFDASH: A Fairness, Stability, and Efficiency Aware Rate Control Approach for Multiple Clients over DASH Chao Zhou, Chia-Wen Lin, Senior Member, IEEE, Xinggong Zhang, Zongming Guo^*Manuscript received Nov. 21, 2016; revised July 2, 2017, and Feb. 12, 2017; accepted October 29, 2017. This work was supported in part by KTP project, a Private Transport Protocol developed by Kuaishou for video Live, VOD, and Real-Time communication. This paper was recommended by Associate Editor Dr. Zhu Li Chao Zhou is with the Beijing Kuaishou Technology Co., Ltd., Beijing, China. (e-mail: [email protected]) Chia-Wen Lin is with the Department of Electrical Engineering and the Institute of Communications Engineering, National Tsing Hua University, Hsinchu, Taiwan (e-mail: [email protected]). Xinggong Zhang and Zongming Guo (corresponding author) are with the Institute of Computer Science & Technology, Peking University, Beijing, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. December 30, 2023 =========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== Dynamic adaptive streaming over HTTP (DASH) has recently been widely deployed in the Internet and adopted in the industry. It, however, does not impose any adaptation logic for selecting the quality of video segments requested by clients and suffers from lackluster performance with respect to a number of desirable properties: efficiency, stability, and fairness when multiple players compete for a bottleneck link. In this paper, we propose a throughput-friendly DASH (TFDASH) rate control scheme for video streaming with multiple clients over DASH to well balance the trade-offs among efficiency, stability, and fairness. The core idea behind guaranteeing fairness and high efficiency (bandwidth utilization) is to avoid OFF periods during the downloading process for all clients, i.e., the bandwidth is in perfect-subscription or over-subscription with bandwidth utilization approach to 100%. We also propose a dual-threshold buffer model to solve the instability problem caused by the above idea. As a result, by integrating these novel components, we also propose a probability-driven rate adaption logic taking into account several key factors that most influence visual quality, including buffer occupancy, video playback quality, video bit-rate switching frequency and amplitude, to guarantee high-quality video streaming. Our experiments evidently demonstrate the superior performance of the proposed method. Dynamic HTTP Streaming, Multiple Clients, Rate Adaptation, Quality of Experience§ INTRODUCTION Dynamic adaptive streaming over HTTP (DASH) has been recently widely adopted for providing uninterrupted video streaming services to users with dynamic network conditions and heterogeneous devices <cit.>. In contrast to the past RTP/UDP, the use of HTTP over TCP is easy to configure and, in particular, can greatly simplify the traversal of firewalls and network address translators. Besides, the deployment cost of DASH is relatively low since it employs standard HTTP servers and, therefore, can easily be deployed within content delivery networks. In DASH, a video clip is encoded into multiple versions at different bit-rates, each being further divided into small video segments containing seconds or tens of seconds worth of video. At the client side, a DASH client continuously requests and receives video segments from DASH servers that own the segments. To adapt the video bit-rate to a varying network bandwidth, DASH allows clients to request for video segments from different versions of a video, each of which being coded with a specific bit-rate. This is known as dynamic rate adaptation, which is one of the most important features for DASH since it can automatically throttle the visual quality to match the available bandwidth so that a user receives the video at the maximum quality possible.it is very challenging to provide satisfactory user experience during a whole video session under highly dynamic network conditions. Without an effective rate adaptation algorithm, a DASH client may suffer from frequent interruptions and degraded visual quality. For example, on one hand, a video bit-rate higher than the available bandwidth would cause network congestion. On the other hand, when the video bit-rate is lower than the available bandwidth, the visual quality cannot reach the maximum allowed by the available bandwidth. Besides, during the playback, smooth video bit-rate is preferred, and frequent bit-rate switching is annoying to users <cit.>. However, this is contradictory with the characteristics of the available bandwidth which, generally, is time-varying.Furthermore, when multiple clients are competing over a bottle-neck link, although the DASH clients employ HTTP over TCP which has proven to be fair, their estimated bandwidth can be significantly different which usually leads to several performance problems, such as inefficiency, instability, and unfairness. This phenomenon is mainly caused by the ON-OFF (ON means downloading a segment, and OFF means staying idle) in DASH <cit.>. In Section <ref>, we shall explain these problems in more details.Although there are quite a few works addressing rate adaptation problems for video transport over DASH, several research problems about video streaming for multiple clients over DASH are still open and challenging. For example, there is a trade-off between the stability and efficiency for video bit-rate adaptation. Generally, undertime-varying bandwidth, requesting a low video bit-rate gives more room for rate selection, and therefore well ensure high stability (smoothness) and continuous video playback. This, however, leads to low video quality with low bandwidth utilization (under-utilization or inefficiency). In addition, there is a trade-off between the sensitivity and the stability. Since channel bandwidth is inherently time-varying, high sensitivity of a rate control approach usually makes the video bit-rate match the bandwidth well, but may lead to high instability. These challenges become more troublesome when multiple clients are competing over a bottleneck link since each client will try to maximize its received video quality without considering the others. Besides, the fairness problem arises for multiple clients over DASH since the DASH client is deployed over HTTP/TCP. In this case, the trade-off among different performance factors becomes more challenging as will be detailed in Section <ref>.In this work, we propose a Throughput-Friendly DASH (TFDASH) rate control scheme for multiple clients over DASH, to well balance the trade-offs among efficiency, stability, and fairness. First, considering that the estimated bandwidth is generally not equal to the fair shared bandwidth as explained in Section <ref>, the probed bandwidth, which is obtained by our proposed Logarithmic Increase Multiplicative Decrease (LIMD) based bandwidth probing scheme, is adopted to guide the rate adaptation. For the fairness consideration, we focus on preventing the occurrence of OFF phenomenon during the downloading process, this is because only when the bandwidth is under over-subscription, the bandwidths estimated by the clients individually are approximately equal and fairness can therefore be reached. Besides, without OFF periods, i.e., there is no idle period during the downloading process, the bandwidth utilization is naturally high. However, in this case, how to maintain the system stability becomes challenging since the bandwidth is time-varying, as a too high video bit-rate leads to congestion and playback freeze, while a too low video bit-rate causes buffer overflow without OFF periods. Thus, considering that the available video bit-rates are limited and discrete, the actual video bit-rate must be switched up and down around the fairly shared bandwidth, leading to high instability. Therefore, we adopt a dual-threshold buffer model proposed in our previous work <cit.> that employ two thresholds (a low threshold for preventing buffer underflow and a high threshold for preventing buffer overflow) to smooth out the video bit-rate. With this model, a DASH client will continuously download video segments without the need of OFF periods. Specifically, if the available bandwidth is higher than the requested video bit-rate, the buffer occupancy will increase and then approach to the high threshold, and when the buffer occupancy exceeds the high threshold,the requested video bit-rate will be switched up to prevent buffer overflow, and vice versa. Therefore, without OFF periods, the fairness and efficiency problems are resolved. Furthermore, when the buffer occupancy is in between the two thresholds, a probability driven rate adaptation scheme to ensure high visual quality by jointly considering several key factors including buffer occupancy, video playback quality, video bit-rate switching frequency and amplitude.§ BACKGROUND AND MOTIVATION Although DASH is relatively new, due to its popularity, it has attracted much research effort recently. For example, Watson <cit.> systematically introduced the DASH framework of Netflix, which has been the largest DASH stream provider in the world.As mentioned above, dynamic rate adaptation is a key feature of DASH since it can automatically throttle the visual quality to match the available bandwidth so that each user receives the video with the maximum quality possible. Akhshabi et al. <cit.> compared the rate adaption schemes used for three popular DASH clients: Netflix client <cit.>, Microsoft Smooth Streaming <cit.>, and Adobe OSMF <cit.>. It was reported in <cit.> that none of the DASH client-based rate adaptation is good enough, as they are either too aggressive or too conservative. Some clients even just switch between the highest and lowest bitrates. Also, all of them lead to relatively long response time under the shift of network congestion level. Existing rate adaptation schemes for DASH, such as bandwidth-based schemes <cit.> and buffer-based schemes <cit.>, aim at either achieving a high bandwidth utilization efficiency by dynamically adapting the video bitrate to match an available bandwidth, or maintaining continuous video playback by smoothing the video bitrate to avoid buffer overflow/underflow. Nevertheless, due to unavoidable bandwidth variations, existing schemes usually cannot achieve a good trade-off between video bitrate smoothness and bandwidth utilization. In our previous work <cit.>, a control theoretic approach was proposed to achieve a good trade-off between video rate smoothness and bandwidth utilization. In <cit.>, a dual-threshold based buffer occupancy model was proposed to smooth out the short-term bandwidth variations so as to maintain the smoothness of video rate. The main objective of the rate adaption method in <cit.> is, however, to avoid buffer overflow and underflow without considering other factors, such as switching frequency and amplitude, which also can affect the perceived visual quality <cit.>. Furthermore, these rate control algorithms are designed with the underlying assumption that the TCP downloading throughput observed by a client is its fair share of the network bandwidth, and the fairness problem, which seriously affect the performance for multiple clients over DASH, has not been taken into account. However, due to the OFF intervals during the downloading process <cit.>, such schemes often lead to video bit-rate oscillations, under-utilization,and unfair bandwidth sharing when multiple clients compete over a common bottleneck link, as will be demonstrated in Section <ref>.Recently, the problems with video streaming for multiple DASH clients competing over a common bottleneck have been studied in <cit.>. In <cit.>, the authors studied bit-rate adaptation problems and identified the causes of several undesirable interactions that arise as a consequence of overlaying the video bit-rate adaptation over HTTP. Further, theydeveloped a suite of techniques that tried to systematically guide the trade-offs between stability, fairness, and efficiency. However, the approach did not well consider the factors that most influence visual quality, such as video bit-rate switching frequency and amplitude. Besides, it also did not take care of reducing the OFF periods which is the root causes of unfairness and underutilization in bandwidth allocation <cit.>, instead, it adopted a randomized scheduler to determine when to download a new segment, by which the client needs to wait for a short period. In <cit.>, a "probe and adapt" principle was proposed for video bit-rate adaptation where "probe" refers to trial increment of the data rate, instead of sending auxiliary piggybacking traffic, and "adapt" means to select a suitable video bit-rate based on the probed bandwidth. Its additive-increase multiplicative-decrease (AIMD) probing mechanism shares similarities with TCP congestion control <cit.>, which leads to a long convergence time and failure in properly tracking the time-varying bandwidth. While for the "adapt" component, the video bit-rate switching decision is mainly dependent on the probed bandwidth without considering the other factors which most influence visual quality, leading to low bandwidth utilization and high instability.Besides rate adaptation mechanisms for DASH, there are some other works addressing the fairness problem from several different angles <cit.>. The approach in <cit.>aims to achieve proportional fairness at the packet level by implementing weighted fair queuing. However, these approaches are conducted in a concentrated manner, which has limitation in supporting the large-scale multimedia delivery. A new protocol is proposed in the context of multi-server HTTP adaptive streaming, aiming to improve the user experience by providing better fairness, efficiency and stability in <cit.>. In <cit.>, the authors jointly optimized the network resource allocation and video quality adaptation by the cross-layer optimization method, in which fairness was also considered. In <cit.>, the fairness problem was addressed by applying a traffic shaping server.In <cit.>, we studied the instability, unfairness, and underutilization problems for multiple DASH clients competing over a common bottleneck, where we proposed to probe the bandwidth by a logarithmic law based increase probing scheme and a conservative back-off based decrease probing scheme. Compared with <cit.>, the probed bandwidth converges more quickly and tracks the time-varying bandwidth better. Besides, a dual-threshold buffer model was adopted to avoid buffer overflow/underflow. However, the method did not fully consider key factor that most impact visual quality, including buffer occupancy, video playback quality, video bit-rate switching frequency and amplitude. As an extension of <cit.>, the work achieves a better trade-off among the instability, unfairness, and underutilization. Besides the LIMD based bandwidth probing scheme and dual threshold buffer model, a probability driven rate control logic is also proposed to achieve better visual quality. Moreover, it can break out the balance that the bandwidth is in perfect-subscription with unfair share of the bandwidth.§ PROBLEMS WITH RATE ADAPTION FOR MULTIPLE CLIENTS OVER DASH §.§ Problem Statement To provide satisfactory performance for media streaming under the scenario of multiple users over DASH, the issues about stability, fairness, andbandwidth utilization need to be jointly considered. To explain it clearly, Fig. <ref> illustrates the temporal overlap of the ON-OFF periods among two competing users as an example. Suppose the available bandwidth is C and that a single active connection gets the whole available bandwidth, while two active connections share it fairly, i.e., each gets C/2. We further denote by C_1 and C_2 the throughput received by the two users, and C_1 = C_2 = C/2 ideally. Fig. <ref>(a) shows the case where there is no overlap of the ON periods for the two users, i.e., each user monopolizes the available bandwidth during its ON period such that we have C_1 = C_2 = C. When both users overestimate their fairly shared bandwidth, they may request new segments with higher bit-rates, thereby causing congestion. In the case of congestion, the users will find that their estimated bandwidth is less than their previous estimation, and thus will switch to a lower video bit-rate. This oscillation can repeat, leading to video playback instability. Fig. <ref>(b) shows the situation where the ON period of one user falls in the ON period of the other user. This may happen when the users request video segments with different video bit-rates. In this case, the throughput observed by the users is C_1 > C/2 and C_2 = C/2, i.e., user one overestimates the fair share of bandwidth. When only one user overestimates the fair share of bandwidth, the two users will converge to a stable but unfair equilibrium that the user, who overestimates the fair share of bandwidth, will request a higher video bit-rate, causing unfairness problem. At last, in Fig. <ref>(c) the ON periods of the two users are perfectly aligned, and both users observe that C_1 = C_2 = C/2. Though both users estimate the fair share of bandwidth correctly, it still may cause underutilization problem. This is mainly because the available bandwidth at the server side is discrete (quantized) and limited. For example, when the quantized available bandwidth cannot be C/2, then both users may request the maximum available video bit-rate which is smaller than C/2 to avoid playback interruptions (buffer underflow). In this case, OFF periods (sleeping mechanism) are adopted to prevent buffer overflow, and underutilization problem occurs.For bandwidth consumption, three different scenarios are considered: perfect-subscription, over-subscription,and under-subscription, denoting the situations that the total amount of bandwidth requested by the clients is equal to, larger than, and smaller than the available bandwidth, respectively.Suppose TCP is in ideal behavior, i.e., perfectly equal sharing of the available bandwidth when the transfers overlap. The unfairness among clients mainly comes from the ON-OFF phenomenon in DASH <cit.>. As demonstrated in <cit.>, only when the bandwidth is in over-subscription, i.e., the congestion occurs, the bandwidths estimated by the clients are approximately equal, and fairness can be obtained. This is because when all the clients are continuously downloading without any OFF period, the bandwidth is equally shared by the DASH clients since all DASH clients are built on top of TCP which is fair in nature. However, when congestion occurs, the video buffer at the client side will be drained, causing playback freeze, which can severely degrade quality of experience. When the bandwidth is in under-subscription, the requested video bit-rate is smaller than the available bandwidth, and OFF periods are needed to suspend the transmission so as to avoid buffer overflow. In this case, instability, unfairness, and underutilization problems may happen as explained above. Intuitively the objective of a rate adaption scheme is to control the bandwidth to stay in perfect-subscription, and all the clients request the same bit-rate as the fair-share bandwidth. However, it is very difficult, if not impossible, to achieve this objective since only when the bandwidth is in over-subscription, the fairness can be guaranteed for the DASH clients. On the other hand, even if the DASH clients can obtain the fair-share bandwidth, generally there is no available video bit-rate which exactly matches the fair-share bandwidth since the video sequences are only transcoded into several discrete bit-rates at the server side. §.§ Rate Adaption Approach Overview In this work, fairness is enforced on a long-term (not instantaneous) basis such that, for an certain moment, the clients may request different video bit-rates. While for relatively long time, the average video bit-rates for the clients are approximately equivalent. This can be achieved by alternately switching up and down the video bit-rate. However, from the viewpoint of quality of user experience, frequent bit-rate switching is annoying. Therefore, a dual-threshold buffered model is adopted, which was proposed and proven to be effective in our previous work <cit.>. To improve the visual quality, based on this model, we further propose a probability-driven rate switching logic, which takes into account several key factors that most impact visual quality, including buffer occupancy, video playback quality, video rate switching frequency and amplitude.To ensure fairness, we focus on preventing the occurrence of OFF phenomenon, i.e., the bandwidth is either in perfect-subscription or in over-subscription. In the case of over-subscription, the fairness is guaranteed, whereas in the case of perfect-subscription, there is an unlimited number of bandwidth sharing modes. It is worth to notice that though it is easy to avoid OFF phenomenon by keeping on downloading without interruption, it is even impossible to distinguish if the bandwidth is in perfect-subscription or in over-subscription. Moreover, since the available bit-rate is limited and discrete, instead of using the estimated bandwidth, we propose an effective bandwidth probing scheme to guide the rate adaptation. As a result, the dynamic equilibrium state is probed for each client such that the client's video bit-rate fluctuates around the fair-share bandwidth, which reaches long-term fairness.Fig. <ref> shows the flowchart of our proposed rate adaption approach. In DASH, the video bit-rate can be changed only when a segment is completely downloaded. Then, to determine the video bit-rate for the next segment to be downloaded, we first estimate the bandwidth and smooth it by a Kalman filter (Section <ref>). Then, together with the estimated bandwidth, we update the probed bandwidth which will be used to guide the rate adaption (Section <ref>). According to the buffer occupancy, the video bit-rate is adapted under three scenarios. When the buffer occupancy is in between the two predefined thresholds, a probability driven rate adaption is applied (Section <ref>); otherwise, the video bit-rate is selected to avoid buffer overflow and underflow (<ref>).The advantages of our proposed rate adaption scheme can be summarized as follows:Fairness: In our proposed rate adaptation approach, no OFF phenomenon happens unless the maximal available video bit-rate has been reached. Without OFF phenomenon, the bandwidth is either over-subscription or perfect-subscription. For the former, the fairness is guaranteed. While for the later, though there are unlimited number of bandwidth sharing modes in theory, depending on the starting time of downloads, instead of using the estimated bandwidth, we probe the video bit-rate by combining the probed bandwidth and probability driven rate adaption to break the balance, and bandwidth is transfered from perfect-subscription to over-subscription.Efficiency: Without OFF phenomenon, the available bandwidth can be shared by clients with high-efficiency, and the bandwidth utilization approaches to 100%. The experiments also show that our proposed rate adaptation approach achieves high average video bit-rate and high bandwidth utilization efficiency.Stability: With the proposed dual-threshold, the effect of buffer occupancy oscillations and bandwidth variations on video bit-rate adaptation can be mitigated. Furthermore, during the rate adaptation process, the smoothness (i.e., the number of previous segments which have the same video bit-rate) is also taken into account.High visual quality: Besides the smoothness, our method also consider several additional key factors that most influence visual quality, including buffer occupancy, video playback quality, video rate switching frequency and amplitude. Besides, buffer overflow and underflow are effectively mitigated.In the following sections, we give the details of the components in Fig. <ref> for the rate adaptation approach. Important symbols used in this paper are summarized in Table <ref>.§ RECEIVING BANDWIDTH ESTIMATION AND PROBING §.§ Smooth TCP Receiving Bandwidth Estimation Assume a video clip is encoded intoL different versions, with different playback video bit-rates V_1<V_2< ... <V_L. All versions of the video are partitioned into equal-length segments, each of which consuming the same playback time of τ. For each client, the streaming process is divided into sequential segment downloading steps k = 1, 2, 3, .... Without loss of generality, suppose a client starts downloading segment k at time instant t^s_k and the segment is downloaded completely at t^e_k. and the video bit-rate for segment k is v_k. Then, the smoothed segment-bias bandwidth, b̂_k, can be estimated as <cit.>:b̂_k = v_kτ/t_k^e - t_k^s Furthermore, to eliminate the noise and interference during the receiving bandwidth estimation, Kalman filter <cit.> is adopted. Then, we have3pt⌢ b_k = w b̂_k-1 + ( 1 - w )3pt⌢ b_k - 1where 3pt⌢ b_k is the amended bandwidth by Kalman filter, and is w a coefficient with 0 < w < 1. Eq. (<ref>) shows that the amended bandwidth for downloading segment k, 3pt⌢ b_k, is adjusted by w based on the previous amended bandwidth3pt⌢ b_k-1 and the currently measured bandwidth b̂_k-1. If w is small, the previous amended bandwidth may play a more important role in predicting the bandwidth, As w gets larger, the amended bandwidthis closer to the current measured value. Thus, if the variation of bandwidth is large, we should decrease w, and if the variation of bandwidth is small, we can increase w to more accurately reflect the change of network condition. Therefore, the value of coefficient w is given as follows:w = 1/1 + e^u - u_0where u_0 is a constant, and u is the normalized difference between the measured value and amended value thatu = | b̂_k - 1 - 3pt⌢ b_k - 1|/b̂_k - 1§.§ LIMD-based Bandwidth Probing Due to the ON-OFF phenomenon in DASH, the bandwidth estimated by a client is discrepant and cannot be used directly for rate adaptation <cit.>. Moreover, as demonstrated in <ref>, only when the bandwidth is oversubscribed, i.e., the congestion occurs, the bandwidth estimated by a client is equal to the fair-share bandwidth (all the clients see nearly the same available bandwidth). On the other hand, when congestion occurs, the requested video bit-rate cannot be supported by the bandwidth, and playback freeze may happen. Thus, how to obtain the fair-share bandwidth without congestion is critical for improving the rate adaptation performance for DASH.In this section, we propose a Logarithmic Increase Multiplicative Decrease (LIMD) based bandwidth probing scheme, which includes a logarithmic increase phase and a multiplicative decrease phase, tomake the probed bandwidth quickly converge to the fair-share bandwidth. The intuition behind this method is that the estimated bandwidth in <ref> by a client is always the upper bound of the fair-share bandwidth due to the off intervals <cit.>. Thus, during the increase probing phase, the probed bandwidth will continuously increase until it exceeds the estimated bandwidth in (<ref>). Then, when the probed bandwidth is higher than the estimated bandwidth, the client will switch to the decrease probing phase to avoid congestion. This is similar to the Additive Increase Multiplicative Decrease (AIMD) based congestion control of TCP, where the transport rate continuously increases until a packet loss happens.However, there are many problems with AIMD-based scheme in video streaming, such as its slow start and frequent fluctuations. Therefore,the granularity of probing has to be selected carefully <cit.>. Coarse granularity makes the probed bandwidth converge quickly, but it may over-probe and lead to congestion. In contrast, probing of too fine granularity takes a long time to converge and usually cannot track the time-varying bandwidth well.To address the above issues, an LIMD based probing scheme is proposed to increase the probed bandwidth since when the gap between the probed bandwidth and the fair-share bandwidth (we cannot obtain the fair-share bandwidth in practice, and instead its upper bound, i.e., the estimated bandwidth is used for approximation) is large, a coarse granularity probing scheme should be employed to increase the probed bandwidth quickly. On the other hand, when the gap is small, a fine granularity probing scheme should be adopted to avoid over-probing. Moreover, when the probed bandwidth is higher than the fair-share bandwidth, congestions may occur, thereby leading to playback freezing. In this case, a conservative back-off scheme is designed to decrease the probed bandwidth guaranteeing that no congestion happens.Specifically, we denote b̃_k the probed bandwidth which will be used for guiding the video bit-rate switching forsegment k, and it is initialized to zero that b̃_0 = 0. Whenever a segment is completely downloaded, the estimated bandwidth is update according to (<ref>). Then, the probed bandwidth is updated as follows:b̃_k= {[ b̃_k-1 + max (3pt⌢ b_k-1- b̃_k-1/2,△ ),;if b̃_k-1 < 3pt⌢ b_k-1; ;b̃_k-1 + α (3pt⌢ b_k-1- b̃_k-1),;if b̃_k-1≥3pt⌢ b_k-1 ].where △ is a constant to avoid slow convergence, andα is a positive constant satisfying that α > 1. By (<ref>), the probed bandwidth will quickly approach to the estimated bandwidth in the logarithmic law at first. However, when the gap between the b̃_k-1 and 3pt⌢ b_k-1 is small, i.e., 3pt⌢ b_k-1- b̃_k-1/2 < △, the probed bandwidth would be additively increased by △ rather than by logarithmic law. On the other hand, when the probed bandwidth exceeds the estimated, a conservative back-off scheme is adopted to control b̃_k-1 to be not higher than 3pt⌢ b_k-1.§ RATE ADAPTATION LOGIC §.§ Buffered Video Time Model To maintain continuous playback, a video streaming client normally contains a video buffer to absorb temporary mismatch between the video downloading rate and video playback rate. In conventional single-version video streaming, the buffered video playback time can be easily measured by dividing the buffered video size by the average video playback rate. In DASH, however, different video versions have different video playback rates. Since a video buffer contains segments from different versions, there is no longer a direct mapping between the buffered video size and the buffered video time. To tackle the problem, we use the buffered video time to measure the length of video playback buffer.The buffered video time process, represented as q(t), can be modeled as a queue with a constant service rate of unity, i.e., in each second, a piece of video with length of one second of playback time is dequeued from the buffer and then played. The enqueue process is driven by the video download rate and the downloaded video version. We adapt the video bit-rate when a segment has been downloaded completely. Without loss of generality, suppose a client starts downloading segment k at time instant t^s_k and the segment is downloaded completely at t^e_k.Then, we havet_k+1^e- t_k+1^s= v_ k+1/3pt⌢ b_k τand 3pt⌢ b_k is the estimated bandwidth given in (<ref>)). Then the buffered video time evolution becomesq( t_k+1^e ) = q( t_k+1^s ) + τ - v_k+1/3pt⌢ b_k τwhere the second term of (<ref>) is the added video time upon the completion of the downloading of segment k+1, and the third term reflects the fact that the buffered video time is consumed linearly at a rate of unity during the downloading process. However, if the bandwidth is too high, a sleep mechanism is needed to postpone the segment request so as to avoid buffer overflow <cit.>. Assume the sleeping time is τ _s, then (<ref>) is rewritten asq( t_k + 1^e ) = q( t_k+1^s )+ τ - v_k+1/3pt⌢ b_k τ- τ _s From the control system point of view, there is a fundamental conflict between maintaining stable video bit-rate and maintaining stable buffer occupancy, due to the unavoidable network bandwidth variations. Nevertheless, from the end user point of view, video bit-rate fluctuations are much more perceivable than buffer occupancy oscillations. The recent work in <cit.> reported that switching back-and-forth between different bit-rates will significantly degrade users viewing experience, whereas buffer size variations do not have direct impact on video streaming quality as long as the video buffer does not deplete. Thus, a dual-threshold buffer occupancy model is adopted to mitigate the effect of buffer occupancy oscillations on video bit-rate adaptation. The dual-threshold buffer occupancy model has two predefined thresholds, q_high and q_low.When the buffer occupancy is in between q_high and q_low, the video bit-rate is adapted in a probabilistic manner by taking into account several key factors that have critical impact on visual quality, including buffer occupancy, video playback quality, video rate switching frequency and amplitude. Otherwise, video bit-rate is switched to avoid buffer overflow/underflow. The details are given in the next subsections. §.§ Probability-Driven Rate Control When the buffer occupancy is in between the two thresholds, i.e., q_min≤ q_k-1≤ q_max, the continuous video playback can be guaranteed with high confidence. Then, we prefer to switch the video bit-rate to a more suitable level. Considering the fairness of all the competing clients, the video bit-rate is adapted in a probabilistic manner.The probability is given by (<ref>), which is determined by several factors, including the bit-rate of last segment v_k-1, the current buffer occupancy q_k-1, the number of consecutive segments which have the same video bit-rate with segment k-1, and the target video bit-rate for the segment to be requested (i.e., bit-rate v_k for segment k). The probability is designed by comprehensively considering the effects of key factors on quality of user experience, including the risk of buffer overflow and underflow, the video bit-rate switching frequency and amplitude, and the video quality. The probability in (<ref>) consists of four items. Now, we give the details of the motivation and explanation for each item as follows.Effect of buffer occupancy on video bit-rate switching decision, C_1. It is the sum of two sub-items, respectively denoting the switch-up and switch-down of video bit-rate, compared with the current video bit-rate. Each sub-item can be further roughly classified into three cases: i) the buffer occupancy is lower than the reference threshold q_ref, ii) the buffer occupancy is equal to q_ref, and iii) the buffer occupancy is higher than q_ref.Now, we take the first sub-item, i.e., switch-up of video bit-rate as an example to analyze the above three cases. When the buffer occupancy is lower than q_ref, denoting the risk of buffer underflow is higher than that of buffer overflow in the bias of buffer occupancy. Thus, switch-up of video bit-rate is inadvisable. Besides, the larger of the gap between the buffer occupancy and q_ref, the higher of the risk of buffer underflow, and also the smaller of the probability of video bit-rate switch-up, and vice versa. At last, when the buffer occupancy is equal to q_ref, the risk of buffer underflow is the same as that of buffer overflow, and the probabilities of switch-up and switch-down of video bit-rate should be equal. Moreover, the probability of switch-up of video bit-rate should not be linearly proportional to the buffer occupancy. Instead, the probability should decrease quickly when the buffer occupancy goes below q_ref to avoid buffer underflow, and increase quickly when the buffer occupancy goes above q_ref to avoid buffer overflow.Therefore, the following Sigmoid function is applied since it meets the above requirements:f( x ) = {[ 0 ifx < x_min; 1 ifx > x_max; 1/1 + e^x_0 - xelse ].where in C_1, f(q) = f(x) by setting x_min = q_min, x_max = q_max, x_0 = q_ref, and sgn(x) is given in (<ref>):sgn( x ) = {[ 1 ifx > 0; 0 ifx = 0; - 1 ifx < 0 ].denoting if the video bit-rate is switched up or down, or remains unchanged.The second sub-item in C_1 (i.e., switch-down of video bit-rate) has opposite characteristics with the case of video bit-rate switch-up, and the probability is given by 1-f(x).Effect of video quality on video bit-rate switching decision, C_2 . Generally, the higher the video bit-rate, the better the quality of experience. Thus, only in the bias of video bit-rate, a high video bit-rate is preferred during the rate switching process. On the other hand, as shown in <cit.>, user experience follows the logarithmic law, and QoE function can be modeled in a logarithmic form for applications of file downloading and web browsing. As such, we use the logarithmic function of the video bit-rate to present the effect of video bit-rate on the probability of video bit-rate switching. In C_2, εis a small positive number with ε≥ 1to ensure that the probability is nonnegative, typically we set ε = 1, and the term of ln( v_max - v_min + ε) is used to normalize the probability to range [0, 1].Effect of switching amplitude, i.e., the gap of the video bit-rate between two consecutive segments, on bit-rate switching decision, C_3. It was shown in <cit.> that gradually switching the video bit-rate is more preferred. Therefore, the probability of video bit-rate switching should be decreased as the gap of the video bit-rate increases. Also, the probability is modeled as the logarithmic function of the difference of the video bit-rate between two consecutive segments as shown in C_3, and parameter ε≥ 1.Effect of switching frequency (i.e., the smoothness of video bit-rate) on the bit-rate switching decision, C_4. Because frequent bit-rate switching is annoying <cit.>, the effect of bit-rate switching on visual quality decreases as the number of previous segments with the same video bit-rate increases. This is mainly because human is sensitive to frequent short-term video bit-rate fluctuations rather than long-term video bit-rate switchings. We use the number, which denotes the number of previous segments with the same video bit-rate, to present the smoothness of video bit-rate. The lager the number is, the higher the smoothness will be, and the less impact the video bit-rate switching will make on visual quality. Therefore, the probability is an increasing function of the number, and the Sigmoid function in (<ref>) is adopted so that f(n_k) = f(x) with x_min = n_min, x_max = n_max and x_0 = n_0. Typically, we set n_max = N = 15 which is a length of 30s of video content with segment length τ = 2s, n_min = 1 which is the minimum value of n_k, i.e., the latest two consecutive segments have different video bit-rates, and n_0 = 2*n_max/3 as smooth video bit-rate is more preferred when the buffer occupancy is in between q_min and q_max.After deriving the probabilities in (<ref>) for all candidate video bit-rates, the video bit-rate is switched to the target one for segment k according to the probabilities. §.§ Buffer Overflow and Underflow Control When the buffer occupancy is higher than q_high or lower than q_low, buffer overflow or underflow should be avoided. To this end, when q(t_k^s) > q_high or q(t_k^s) < q_low, an appropriate video bit-rate should be selected. Specifically, when buffer occupancy is low (lower than the threshold q_low), we will select the maximal available video bitrate which is lower than the estimated bandwidth. On the other hand, when when buffer occupancy is high (higher than the threshold q_high), the minimal available video bitrate which is higher than the estimated bandwidth is selected. Therefore, the video bit-rate for segment k is given as:v_k = {[ max_1 ≤ i ≤ L ( V_i| V_i ≤b̂_k. )ifq(t_k^s) < q_low; min_1 ≤ i ≤ L ( V_i| V_i ≥b̂_k. ) ifq(t_k^s) > q_high ].and when q_low≤ q(t_k^s) ≤ q_high, the video bit-rate is selected according to Section <ref>.§ EXPERIMENTS In this section, we evaluate our rate adaption algorithms in terms of efficiency, fairness, and stability under the scenario that multiple clients compete over a bottleneck link. §.§ Experiments SetupWe implement our TFDASH system in Linux/UNIX-based platforms. Our testbed consists of one web server handling media delivery, one router, and n (n ≥ 2) DASH clients. The server and clients run the standard Ubuntu of version 12.04.1. The server is installed with the Apache HTTP server of version 2.4.1, and Dummynet is used in the server to control the upload bandwidth <cit.>. In the testbed, the bottleneck is the bandwidth between the server and the router. In our experiments, for the content, we adopted the Big Buck Bunny sequence which is also part of the DASH dataset <cit.>.The video sequence is pre-transcoded into 11 different video bitrates with three resolutions that the resolutions is 480p for bitrate of 235, 375, 560 kbps, 720p for 750, 1050, 1750, 2350 kbps, and 1080p for 3000, 3850, 4300, 5800 kbps.Due to the complex network characteristics, it is hard to find the optimal values of the two thresholds q_high and q_low <cit.>. In our experiments, we set q_low = 5s since this start-up delay can be tolerated by most existing streaming systems, and q_high = 25s which is reasonable value for current existing devices, including mobile phones, to buffer such a length of media data. The maximum buffer size is set to 30s for all schemes in this work. However, generally a better performance can be obtained with a larger buffer size. This is because with a larger buffer size, the bandwidth variations can be better absorbed by the buffered video and a smoother video rate is guaranteed. Besides, we can also set relatively large q_low to maintain a higher buffer occupancy so as to ensure continuous video playback. The effects of these two parameters on system performance can be found in our previous work <cit.>. For the other parameters used in the experiments, there default values are α = 1.25, Δ = 32kbps, and u_0 = 0.5.For performance comparison, besides our TFDASH method, we also implement some popular DASH clients, including FESTIVE <cit.> and PANDA <cit.>. FESTIVE was proposed to balance the performance of efficiency, fairness, and stability for DASH clients by subtracting a random buffered video time from the target buffer occupancy for each segment request. In contrast, PANDA particularly considers the scenario when multiple clients are competing over a bottleneck link. It adopts the additive-increase multiplicative-decrease (AIMD) scheme to probe the available bandwidth, which is similar to the TCP congestion control scheme, and then adjust the video bit-rate according to the probed bandwidth. §.§ Video Bit-Rate Switching Performance In this experiments, the bandwidth between the server and the router is controlled by Dummynet such that, during the first 230s, the available bandwidth (avail_bw) is set to be 3000 kbps, and then it is limited to be 1500 kbps from 230s to 440s. At last, the avail_bw is switched to 4000 kbps until the experiment ends. While during the first phase, i.e., from 0s to 230s, four short-term bandwidth variations are added, including both positive and negative spikes.And for the second phrase (from 230s to 440s) and third phrase (from 440s to 650s), several positive and negative bandwidth spikes are added. Besides, two clients are competing for the avail_bw (for the case of more clients, the performance is shown in the next subsection), the first client begins to download the video segments at the beginning of the experiment until all the video segments have been downloaded completely, and then it leaves the system. The second client joints the system at 50s, and leaves when it has completely downloaded all video segments.The results of all DASH clients are shown in Fig. <ref>. First, the performances on rate and buffer occupancy with FESTIVE are shown in Fig. <ref> and Fig. <ref>, respectively. Since FESTIVE does not apply the probing scheme, and instead it directly utilize the estimated bandwidth for rate adaptation, we plot the estimated bandwidth (i.e., segment-level bandwidth, frag_bw for short, which is obtained as the segment size divided by the time to download the segment <cit.>) for each client. Due to the ON-OFF operations, the estimated bandwidth is generally higher than the fairly-shared bandwidth. Fig. <ref> shows the sum of the two clients' estimated bandwidths is generally higher than the available bandwidth, while both clients have close estimated bandwidth. Thus, the clients usually over-estimate the fairly-shared bandwidth and then request for video segments with too high video bit-rate, thereby causing congestions. When this happens, the clients will find that their estimated bandwidth is less than their previous estimation, and thus switch to a lower video bit-rate. This oscillation can repeat, thereby causing video playback instability as explained in Fig. <ref>. Fig. <ref> also demonstrates this the video bit-rates of clients switch frequently, which can significantly hurt user experience. Besides, the buffer occupancy fluctuates around a certain value (15s) for both clients. This is because in FESTIVE, the buffer occupancy threshold is preset to be 15s in the experiments. When the buffer occupancy of a client is higher than the threshold, the client will wait for some time before requesting the next segments. Otherwise, it will immediately send segment requests. Moreover, the threshold will be randomly adjusted when a segment is downloaded completely (before requesting the next segment), this also leads to small fluctuations of buffer occupancy. At 230s, the buffer occupancy decreases quickly and then increases to be around the threshold, this is mainly because the available bandwidth decreases dramatically from 3000 kbps to 1500 kbps at 230s.We than analyze the performance of PANDA as shown in Fig. <ref> and Fig. <ref>. Considering that the estimated bandwidth generally differs from (higher than most of the time) the fairly-shared bandwidth, the AIMD-based probing scheme is adopted to probe the fairly-shared bandwidth. As a result, a PANDA client switches its video bit-rate according to the probed bandwidth and its buffer occupancy. However, due to the characteristics of AIMD, such as slow start and frequent fluctuations, when the bandwidth changes, it takes long time to track the bandwidth well as Fig. <ref> shows. For example, when client-2 joins the system at 50s, it takes more than 50s to probe the bandwidth well. Also, when client-2 joins the system, client-1 does not detect that the the fair-shared bandwidth is decreasing quickly. Until longer than 50s later, both clients have nearly the same probed bandwidth. Compared with FESTIVE, we can find that, with the probing scheme, PANDA achieves much smoother video bit-rate for both clients. However, since the probed bandwidth is used as the main signal for rate adaptation in PANDA, the video bit-rate also fluctuates to match the probed bandwidth. Besides, as the video bit-rate is discrete, the video bit-rate selected is usually lower than the probed bandwidth to ensure continuous video playback, which leads to higher buffer occupancy as shown in Fig. <ref>. However, this also leads to low bandwidth utilization efficiency since the video bit-rate is generally lower than the bandwidth.Fig. <ref> and Fig. <ref> show the performance of TFDASH. Similar to PANDA, the probing scheme is also adopted and the probed bandwidth is used to guide the rate adaptation. However, there are several major differences from PANDA. First, the proposed LIMD scheme tracks the bandwidth much better and quicker than AIMD as illustrated in Fig. <ref>. From Fig. <ref> we can find that when client-2 joins the system, it takes only about 10s to track the fair-share bandwidth well, and client-1 detects the bandwidth change quickly such that the probed bandwidths for both clients converge soon. Another major difference is that, instead of solely relying on the probed bandwidth for rate adaptation, the proposed dual-threshold buffer model effectively avoids buffer overflow/underflow. Besides, the proposed probability driven rate control logic can adequately break out the balance in case that the bandwidth is in perfect-subscription with unfair bandwidth sharing. Thanks to the dual-threshold buffer model, we smooth out bitrate by relaxing the smoothness of buffer occupancy provided that no buffer underflow (playback freeze) happens. This is reasonable since from the control system point of view, there is a fundamental conflict between maintaining stable video bitrate and maintaining stable buffer occupancy, due to the unavoidable network bandwidth variations. Nevertheless, from the end user point of view, video bitrate fluctuations are much more annoying than buffer occupancy oscillations. Besides, our rate adaptation scheme takes into account several factors that most influence the quality of experience, including buffer occupancy, video playback quality, video bit-rate switching frequency and amplitude. Therefore, the video bit-rate in Fig. <ref> is much smoother (more stable) than that with FESTIVE and PANDA. Besides, TFDASH allows the bit-rate to be higher than the probed bandwidth, thereby achieving significantly a higher bandwidth utilization efficiency and higher average video bit-rate compared with FESTIVE and PANDA under the some network conditions. At last, the buffer occupancy results in Fig. <ref> shows that no buffer overflow/underflow happens, i.e., continuous video playback is guaranteed. From all the results, we can conclude that compared with FESTIVE and PANDA, TFDASH achieves much smoother and higher video bit-rate, and higher efficiency and stability.Moreover, for the short-term bandwidth spikes, FESTIVE and PANDA generally switch up/down their bitrates, which is often unnecessary since there is enough buffered media/buffer space to accommodate the short-term bandwidth variations. In contrast, TFDASH can tolerate these short-term bandwidth variations more adequately, thanks to the proposed dual-threshold buffer model. This demonstrates the high robustness of TFDASH to short-term network variations.§.§ Efficiency, Fairness ,and Stability PerformancesIn this subsection, we compare the efficiency, stability, and fairness performances of all three schemes based on the following metrics <cit.>: * Inefficiency: The inefficiency at time is measured by | ∑_i v_k,i/b|, where v_k,i is the video bit-rate of the k^th segment for client i, and b is the available bandwidth. A value close to zero implies that the clients in aggregate are using as high an average bit rate as possible to improve user experience.* Instability: Recent studies suggest that users are likely to be sensitive to frequent and significant bit-rate switches <cit.>. We define the instability metric as ∑_d = 0^d_0 - 1| v_k,k - d - v_k,k - d - 1| w( d )/∑_d = 1^d_0v_k,k - d w( d ), which is the weighted sum of all switch steps observed within the last d_0 = 10 segments divided by the weighted sum of bit-rates in the last d_0 segments. We use the weight function w(d) = k-d to add linear penalty to more recent bit-rate switch.* Unfairness: The unfairness at t is defined as √(1 - JainFair_t), where JainFair_t the Jain fairness index <cit.> calculated on the rates v_k,i at time t over all clients. We first compare the performance of all schemes with two competing clients, and the avail_bw varies from 1 Mbps to 10 Mbps. Fig. <ref> show that TFDASH always achieves the lowest inefficiency, instability, and unfairness, i.e., compared with FESTIVE and PANDA, TFDASH can provide higher and smoother video bit-rate, and besides, it can also better guarantee the fairness between the clients. As the avail_bw increases, the performance does not monotonically increase or decrease. This is mainly because the video bit-rate is discrete so that for a given avail_bw, if the fair-share bandwidth is equal (close) to an available video bit-rate, generally better performance can be achieved, and vice versa.Moreover, we also vary the number of competing clients from two to fifteen to evaluate the performance of the three schemes with avail_bw constraint of 10 Mbps as shown in Fig. <ref>. Similar to the results in Fig. <ref>, TFDASH always achieve the best performance in terms of efficiency, stability, and fairness. The performance improvement mainly come from the LIMD based bandwidth probing scheme, the proposed dual-threshold buffer model, and the probability-driven rate control logic. Then, we vary both the number of competing clients and avail_bw to evaluate the performances of all schemes. Specifically, the fairly shared bandwidth for each client is kept to be 1 Mbps, i.e., the avail_bw is set to be 2 Mbps for two competing clients, the avail_bw is set to be 3 Mbps for three competing clients, and so on. The results in Fig. <ref> show that TFDASH still achieves the best performance in all cases, demonstrating the high efficiency of TFDASH. As for the performance of efficiency and stability, Fig. <ref> and Fig. <ref> show that in different cases, all the schemes have insignificant performance fluctuations, this is because the fair-share bandwidth plays an important role in the decision of video bit-rate, which is always the same (equal to 1 Mbps) in this experiment. While in terms of fairness, TFDASH performs the best, but all schemes fluctuate as the number of clients increase. This is obvious since fairness is easy to be affected by the number of competing clients.At last, we conduct performance evaluation according to the suggested test cases for multiple clients in IETF RMCAT <cit.>, we conduct new experiments based on the path capacity variation pattern listed in Table <ref> with a corresponding end time of 125 seconds. We use background non-adaptive UDP traffics to simulate a time-varying bottleneck for congestion controlled media flows. In the experiments, the physical path capacity is 10 Mbps and the UDP traffic source rate changes over time as 10-x Mbps, where x is the bottleneck capacity specified in Table <ref>.As shown in Fig. <ref>, TFDASH always achieves the best performance in terms of efficiency, stability, and fairness. The performance improvement mainly comes from the LIMD based bandwidth probing scheme, the proposed dual-threshold buffer model based and probability driven rate control logic. Specifically, the LIMD-based bandwidth probing scheme can track the bandwidth significantly better and quicker, the proposed dual-threshold buffer model can achieve a good trade-off between the smoothness and high efficiency, and the probability-driven rate control logic can improve the fairness among clients competing for channel bandwidth. §.§ Performance with three types of competing clientsIn DASH, it is important to design an appropriate rate adaptation algorithm so that multiple clients can fairly share bandwidth with high stability and efficiency. However, in fact all the clients adapt their bitrate independently in the following aspects:i) for each client, the bitrate is adapted based on its own rate adaptation logic, ii) no information is exchanged among the clients, iii) the algorithm is distributedly executed in each client and no centralized controller/server is needed. However, since no DASH algorithms has been widely deployed, different in-house algorithms are used in commercial products from different companies, such as Apple?s HLS, Netflix, and YouTube. Thus, it is hard for a newly designed algorithm to achieve high performance, especially in terms of fairness performance, when competing with different DASH clients implementing different rate adaptation algorithms. In this section, we consider the scenario with different types of clients competing bandwidth over a bottle-neck link. In the experiment, we use a bandwidth trace collected from real networkswhich contains more variations as shown in Fig. <ref> shows. The trace is collected from the Plantlab with a server physically deployed in Hong Kong and clients deployed in Beijing, China. Considering that the bandwidth is not high enough to support too many clients, we implemented two TFDASH clients, two FESTIVE clients, and two PANDA clients at the same time. The six DASH clients compete the bandwidth based on their own rate adaptation algorithms and the results are show in Table <ref>. The results show that TFDASH achieves better performance than FESTIVE and PANDA in terms of fairness, stability, and efficiency. However, compared with the results in Sec. VI.C, all types of clients perform a bit worse than the scenario with only the same type of clients. Moreover, when comparing the overall performance of all six clients, we can observe that all metrics (fairness, stability, and efficiency) become worse, this is mainly because different types of clients are designed with different objectives without considering the other types of clients, making one type of clients difficult to cooperate with others fairly and efficiently. § CONCLUSIONIn this paper, we addressed the rate adaptation problem with multiple DASH clients competing bandwidth over a bottle-neck link by considering the efficiency, stability, and fairness among clients. We proposed a throughput-friendly DASH client to intelligently and dynamically switch the video bit-rate so as to reach a good trade-off among these objectives. Specifically, we proposed a Logarithmic Increase Multiplicative Decrease (LIMD) based bandwidth probing scheme to guide the rate adaptation, by which the fair shared bandwidth can be effectively and quickly detected. 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Before joining Kuaishou, he was a Senior Research Engineer with the Media Technology Lab, CRI, Huawei Technologies CO., LTD, Beijing, China.Dr. Zhou's research interests include HTTP video streaming, joint source-channel coding, and multimedia communications and processing. He has been the reviewer for IEEE Transactions on Circuits and Systems for Video Technology, IEEE Transactions on Multimedia, IEEE Transactions on Wireless Communication and so on. He received Best Paper Award presented by IEEE VCIP 2015, and Best Student Paper Awards presented by IEEE VCIP 2012. [ < g r a p h i c s > ]Chia-Wen Lin (S'94-M'00-SM'04) received his Ph.D. degree in electrical engineering from National Tsing Hua University (NTHU), Hsinchu, Taiwan, in 2000. He is currently Professor with the Department of Electrical Engineering and the Institute of Communications Engineering, NTHU. He was with the Department of Computer Science and Information Engineering, National Chung Cheng University, Taiwan, during 2000–2007. Prior to joining academia, he worked for the Information and Communications Research Laboratories, Industrial Technology Research Institute, Hsinchu, Taiwan, during 1992–2000. His research interests include image and video processing and video networking. Dr. Lin has served as an Associate Editor of IEEE Transactions on Image Processing, IEEE Transactions on Circuits and Systems for Video Technology, IEEE Transactions on Multimedia, IEEE Multimedia, and Journal of Visual Communication and Image Representation. He was a Steering Committee member of IEEE Transactions on Multimedia from 2014 to 2015. He was Chair of the Multimedia Systems and Applications Technical Committee of the IEEE Circuits and Systems Society from 2013 to 2015. He served as Technical Program Co-Chair of IEEE ICME 2010, and will be the General Co-Chair of IEEE VCIP 2018 and Technical Program Co-Chair of IEEE ICIP 2019. His papers won Best Paper Award of IEEE VCIP 2015, Top 10% Paper Awards of IEEE MMSP 2013, and Young Investigator Award of VCIP 2005. He received the Young Investigator Award presented by Ministry of Science and Technology, Taiwan, in 2006. [ < g r a p h i c s > ]Xinggong Zhang (M'11) received the B.S. degree from Harbin Institute of Technology and the M.S. degree from Zhejiang University in 1995 and 1998 respectively, and the Ph.D degree from Peking University, Beijing, China, in 2011. He is currently an Associate Professor with the Institute of Computer Science and Technology, Peking University. He has been a senior Research Staff with Founder Research China. His general research interests lie in multimedia networking and video communications. His current research directions include video conferencing, dynamic HTTP streaming, and content-centric networking. Dr. Zhang was a recipient of the First Prize of the Ministry of Education Science & Technology Progress Award in 2006, and the Second Prize of the National Science & Technology Award in 2007.[ < g r a p h i c s > ]Zongming Guo received the B.Sc. degree in mathematics, the M.Sc. and the Ph. D degrees in computer science from Peking University, China, in 1987, 1990 and 1994, respectively. He is currently Dean of Institute of Computer Science & Technology, Peking University. His current research interests include streaming media technology, IPTV and mobile multimedia, and image and video processing. He has published over 80 technical articles in refereed journals and conference proceedings in the areas of multimedia, image and video compression, image and video retrieval, and watermarking. Dr. Guo led the research & developing team, which won the first prize of The State Administration of Radio Film and Television, the first prize of Ministry of Education Science and Technology Progress Award and the second prize of National Science and Technology Award in 2004, 2006 and 2007, respectively. He received Government Allowance granted by the State Council in 2009. | http://arxiv.org/abs/1704.08535v2 | {
"authors": [
"Chao Zhou",
"Chia-Wen Lin",
"Xinggong Zhang",
"Zongming Guo"
],
"categories": [
"cs.MM",
"cs.NI"
],
"primary_category": "cs.MM",
"published": "20170427123530",
"title": "TFDASH: A Fairness, Stability, and Efficiency Aware Rate Control Approach for Multiple Clients over DASH"
} |
http://arxiv.org/abs/1704.08259v1 | {
"authors": [
"John Campbell",
"Marcela Carena",
"Roni Harnik",
"Zhen Liu"
],
"categories": [
"hep-ph",
"hep-ex"
],
"primary_category": "hep-ph",
"published": "20170426180004",
"title": "Interference in the $gg\\rightarrow h \\rightarrow γγ$ On-Shell Rate and the Higgs Boson Total Width"
} |
|
EEG-Based User Reaction Time EstimationUsing Riemannian Geometry Features Dongrui Wu1, Senior Member, IEEE, Brent J. Lance2, Senior Member, IEEE, Vernon J. Lawhern23, Member, IEEE, Stephen Gordon4,Tzyy-Ping Jung5, Fellow, IEEE, Chin-Teng Lin6, Fellow, IEEE 1DataNova, NY USA2Human Research and Engineering Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, MD USA3Department of Computer Science, University of Texas at San Antonio, San Antonio, TX USA4DCS Corp, Alexandria, VA USA5Swartz Center for Computational Neuroscience & Center for Advanced Neurological Engineering,University of California San Diego, La Jolla, CA6Centre for Artificial Intelligence, Faculty of Engineering and Information Technology, University of Technology Sydney, AustraliaE-mail: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]===================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== Riemannian geometry has been successfully used in many brain-computer interface (BCI) classification problems and demonstrated superior performance. In this paper, for the first time, it is applied to BCI regression problems, an important category of BCI applications. More specifically, we propose a new feature extraction approach for Electroencephalogram (EEG) based BCI regression problems: a spatial filter is first used to increase the signal quality of the EEG trials and also to reduce the dimensionality of the covariance matrices, and then Riemannian tangent space features are extracted. We validate the performance of the proposed approach in reaction time estimation from EEG signals measured in a large-scale sustained-attention psychomotor vigilance task, and show that compared with the traditional powerband features, the tangent space features can reduce the root mean square estimation error by 4.30-8.30%, and increase the estimation correlation coefficient by 6.59-11.13%.Brain-computer interface, EEG, reaction time estimation, Riemannian geometry, spatial filtering § INTRODUCTION Brain-computer interfaces (BCIs) can use brain signals such as the scalp electroencephalogram (EEG) to enable people to communicate or control external devices <cit.>. Thus, they can help people with devastating neuromuscular disorders such as amyotrophic lateral sclerosis, brainstem stroke, cerebral palsy, and spinal cord injury <cit.>. However, there are still many challenges in their transition from laboratory settings to real-life applications, including the reliability and convenience of the sensing hardware <cit.>, and the availability of high-performance and robust algorithms for signal analysis and interpretation <cit.>. This paper focuses on the latter, particularly, feature extraction for EEG-based BCIs.Riemannian geometry (RG) <cit.> is a very useful mathematical tool in machine learning and signal/image processing, due to its utility in generating smooth manifolds from intrinsically nonlinear data spaces. Recently it has also been introduced into the BCI community and demonstrated superior performance in a number of applications <cit.>.For example, Li, Wong, and de Bruin <cit.> used RG of the EEG power spectral density matrices for sleep pattern classification. They also proposed a closed-form weighting matrix for the power spectral density matrices to minimize the distance between similar features and to maximize the distance between dissimilar features, and demonstrated better performance than the Euclidian distance and the Kullback-Leibler distance. Barachant et al. <cit.> proposed two RG approaches for motor imagery classification. The first uses the spatial covariance matrices of the EEG signal as features and RG to directly classify them in the manifold of symmetric and positive definite (SPD) matrices. The second maps the covariance matrices onto the Riemannian tangent space, which is a Euclidean space, and then performs variable selection and classification. They achieved comparable or better performance than a multiclass Common Spatial Pattern (CSP) plus Linear Discriminant Analysis (LDA) approach. In <cit.>, Congedo, Barachant, and Andreev further used RG to build calibrationless BCI systems for applications based on event-related potentials, sensorimotor (mu) rhythms, and steady-state evoked potential. It outperformed several state-of-the-art approaches, including xDAWN, stepwise LDA, CSP+LDA, and blind source separation plus logistic regression. Barachant <cit.> also proposed a spatial filter to increase the signal to signal-plus-noise ratio of magnetoencephalography (MEG) signals before constructing a special form of a covariance matrix for RG feature extraction, and a k-means clustering like unsupervised learning algorithm in the Riemannian manifold to improve the offline classification performance. This approach outperformed 266 other approaches and won the Kaggle “DecMeg2014 – Decoding the Human Brain" competition[https://www.kaggle.com/c/decoding-the-human-brain.], which aimed to predict visual stimuli from MEG recordings of human brain activity. Kalunga et al. <cit.> proposed an online classification approach in the Riemannian space and showed that it outperformed Canonical Correlation Analysis in Steady-State Visually Evoked Potential classification. Yger, Lotte, and Sugiyama <cit.> empirically compared several covariance matrix averaging methods for EEG signal classification. They showed that RG for averaging covariance matrices improved performances for small dimensional problems, but as the dimensionality of the covariance matrix increased, RG became less efficient. Lotte <cit.> also proposed a framework to combine transfer learning, ensemble learning, and RG for calibration time reduction, which outperformed CSP+LDA. The Riemannian distance was used in regularization to emphasize auxiliary users whose covariance matrices are close to the target user. Navarro-Sune et al. <cit.> proposed a BCI to automatically detect patient-ventilator disharmony from EEG signals. RG of EEG covariance matrices was used in semi-supervised learning for effective classification of respiratory state, and it outperformed the Euclidean distance. Waytowich et al. <cit.> proposed an approach to integrate RG with transfer learning and spectral meta-learner <cit.>, an offline ensemble fusion approach, for user-independent BCI, and demonstrated in single-trial event-related potential classification that it can significantly outperform existing calibration-free techniques and traditional within-subject calibration techniques when limited data is available.All above approaches focused on EEG classification problems in BCI, whereas BCI regression problems have been largely overlooked. In theory a regression problem is equivalent to a classification problem with infinitely many classes, and hence the output has much finer granularity than a traditional two-class or multi-class classification problem, which provides richer information in decision making. There are at least two types of BCI regression problems in the literature and practice. The first type is behavioral or cognitive status prediction, e.g., estimating the continuous value of a driver's drowsiness from the EEG <cit.>, and estimating a subject's response speed in a psychomotor vigilance task (PVT) from the EEG <cit.>. The second type is direct control applications, e.g., controlling the movement of a mouse cursor using BCI <cit.>, and controlling the continuous movement of a hand in the 3D space using EEG <cit.>.Once the EEG signal is acquired, the regression problem involves three steps: 1) signal processing to increase the signal-to-noise ratio. Frequency domain filters, such as band pass filters and notch filters <cit.>, and spatial filters, such as independent component analysis <cit.> and CSP <cit.>, are frequently used here. 2) feature extraction to construct meaningful predictors, e.g., standardized difference of the EEG voltage <cit.>, and EEG power band features <cit.>. 3) regression algorithms to estimate the continuous output, e.g., ordinary linear regression<cit.>, ridge regression <cit.>, LASSO <cit.>, k-nearest neighbors (kNN) <cit.>, fuzzy neural networks <cit.>, transfer learning <cit.>, active learning <cit.>, etc.In this paper, we apply RG and tangent space features to supervised BCI regression problems. To overcome the limitation pointed out by Yger, Lotte, and Sugiyama <cit.>, i.e., RG is less efficient when the dimensionality of the covariance matrix is large, we adopt an approach similar to what Barachant used in <cit.>: we first use a spatial filter proposed in <cit.> to reduce the dimensionality of the covariance matrices and also to increase the EEG signal quality, and then extract the RG features in the Riemannian tangent space. We validate the performance of the proposed approach in reaction time (RT) estimation from EEG signals measured in a large-scale sustained-attention PVT <cit.>, which collected 143 sessions of data from 17 subjects in a 5-month period. To our knowledge, this is the first time that RG has been used in BCI regression problems.The remainder of this paper is organized as follows: Section <ref> describes the spatial filter we proposed earlier for supervised BCI regression problems. Section <ref> introduces RG and the tangent space features for BCI regression problems. Section <ref> describes the experimental setup, RT and EEG data preprocessing techniques, and the procedure to evaluate the performances of different feature extraction methods. Section <ref> presents the results of the comparative studies. Section <ref> provides parameter sensitivity analysis and additional discussions. Finally, Section <ref> draws conclusions and outlines a future research direction. § SPATIAL FILTERING FOR SUPERVISED BCI REGRESSION PROBLEMSRecently we <cit.> proposed two spatial filters for supervised BCI regression problems, which were extended from the common spatial pattern (CSP) algorithm for supervised classification problems. They have similar performance and computational cost. One of them, CSP for regression - one versus the rest (CSPR-OVR), is briefly introduced in this section, as the RG features are better extracted from the spatially filtered EEG data than the raw EEG data.Let 𝐗_n∈ℝ^C× S (n=1,...,N) denote the nth EEG trial in the training data, where C is the number of channels and S the number of time samples. We assume that the mean of each channel measurement has been removed, which is usually performed by band-pass filtering. Let y_n∈ℝ be the corresponding RT of the nth trial. CSPR-OVR first constructs K fuzzy sets <cit.>, which partition the training samples into K fuzzy classes. To do that, it partitions the interval [0, 100] into K+1 equal intervals, and denotes the partition points as {p_k}_k=1,...,K. It is easy to obtain thatp_k=100· k/K+1, k=1,...,KFor each p_k, CSPR-OVR then finds the corresponding p_k percentile value of all training y_n and denotes it as P_k. Next we define K fuzzy classes from them, as shown in Fig. <ref>.Then, for each fuzzy class, CSPR-OVR computes its mean spatial covariance matrix as:Σ̅_k=∑_n=1^N μ_k(y_n)𝐗_n𝐗_n^T/∑_n=1^N μ_k(y_n),k=1,...,Kwhere μ_k(y_n) is the membership degree of y_n in Fuzzy Class k.Next CSPR-OVR designs a spatial filtering matrix 𝐖_k^*∈ℝ^C× F, where F is the number of individual vector filters, to maximize the variance difference between Fuzzy Class k and the rest, i.e.,𝐖_k^* =max_𝐖∈ℝ^C× FTr(𝐖^TΣ̅_k𝐖)/Tr[𝐖^T(∑_i≠ kΣ̅_i)𝐖]where Tr(·) is the trace of a matrix. (<ref>) is a generalized Rayleigh quotient <cit.>, and the solution 𝐖_k^* is the concatenation of the F eigenvectors associated with the F largest eigenvalues of the matrix (∑_i≠ kΣ̅_i)^-1Σ̅_k.The final spatial filtering matrix 𝐖^*∈ℝ^C× KF is the concatenation of all 𝐖_k^*, i.e.,𝐖^*=[𝐖_1^*, …, 𝐖_K^*]and the spatially filtered trial for 𝐗_n is:𝐗_n'=𝐖^*^T𝐗_n, n=1,...,N. In summary, the complete CSPR-OVR algorithm for supervised BCI regression problems is shown in Algorithm <ref>. § RG AND THE TANGENT SPACE FEATURESThis section introduces the basics of RG, and an approach to extract the Riemannian tangent space features. §.§ Riemannian Geometry The RG approach for BCI works on the covariance matrices of EEG trials, which are symmetric positive-definite and form a differentiable Riemannian manifold ℳ <cit.> with dimensionality R(R+1)/2, where R is the number of rows (columns) of the covariance matrices. As a result, we need to use Riemannian metrics, instead of the traditional Euclidean metrics, which are more appropriate for flat spaces of vectors. Particularly, we are interested in the distance measure between two covariance matrices, as many machine learning methods rely on such distances.The Riemannian distance δ(Σ̅,Σ_n) between two covariance matrices Σ̅∈ℝ^R× R and Σ_n∈ℝ^R× R, called the geodesic, is the minimum length of a curve connecting them on the manifold ℳ. It can be computed as <cit.>:δ(Σ̅,Σ_n)=log(Σ̅^-1Σ_n)_F =[∑_r=1^Rlog^2λ_r]^1/2where the subscript _F denotes the Frobenius norm, and λ_r, r=1,...,R, are the real eigenvalues of Σ̅^-1Σ_n.At Σ̅∈ℳ, a scalar product can be defined in the associated tangent space 𝒯_Σ̅ℳ. This tangent space is Euclidean and locally homomorphic to the manifold. So, Riemannian distance computations in the manifold can be approximated by Euclidean distance computations in the tangent space <cit.>.The logarithmic map projects locally a Σ_n∈ℳ onto the tangent space 𝒯_Σ̅ℳ of Σ̅ by:Σ̂_n=Log_Σ̅(Σ_n)= Σ̅^1/2logm(Σ̅^-1/2Σ_nΣ̅^-1/2)Σ̅^1/2where logm(·) denotes the logarithm of a matrix <cit.>. The logarithm of a diagonalizable matrix 𝐀=𝐕𝐃𝐕^-1 is defined as logm(𝐀)=𝐕𝐃'𝐕^-1, where 𝐃' is a diagonal matrix with elements 𝐃'_i,i=log(𝐃_i,i).The exponential map projects an element Σ̂_n on the tangent space 𝒯_Σ̅ℳ back to the manifold ℳ by:Σ_n=Exp_Σ̅(Σ̂_n)= Σ̅^1/2expm(Σ̅^-1/2Σ_nΣ̅^-1/2) Σ̅^1/2where expm(·) denotes the exponential of a matrix <cit.>. The exponential of a diagonalizable matrix 𝐀=𝐕𝐃𝐕^-1 is defined as expm(𝐀)=𝐕𝐃'𝐕^-1, where 𝐃' is a diagonal matrix with elements 𝐃'_i,i=exp(𝐃_i,i).Fig. <ref> illustrates a Riemannian manifold ℳ, the tangent space 𝒯_Σ̅ℳ at Σ̅, the geodesic between Σ̅ and Σ_n, and the corresponding logarithmic and exponential maps.The Riemannian distance δ(Σ̅,Σ_n) between two covariance matrices Σ̅ and Σ_n on the manifold ℳ can also be computed by a Euclidean distance in the tangent space around Σ̅, i.e. <cit.>,δ(Σ̅,Σ_n) =Log_Σ̅ (Σ_n)_Σ̅ =upper(Σ̅^-1/2Σ̂_n Σ̅^-1/2)_2=upper(logm(Σ̅^-1/2Σ_nΣ̅^-1/2))_2where the upper(·) operator keeps the upper triangular part of a symmetric matrix and vectorizes it by applying weight 1 for the diagonal elements and weight √(2) for the out-of-diagonal elements <cit.>.The RG mean <cit.>, or the intrinsic mean <cit.>, of N covariance matrices is defined as the matrix minimizing the sum of the squared Riemannian distances, i.e.,Σ̅≡𝔊(Σ_1,...,Σ_N) =min_Σ∑_n=1^Nδ^2(Σ,Σ_n)There is no closed-form expression for the RG mean, but an iterative gradient descent algorithm (see Algorithm <ref> <cit.>) can be used to find the solution. Note that Algorithm <ref> makes heavy use of the logarithmic and exponential maps. In this paper we used the implementation in the Matlab Covariance Toolbox[https://github.com/alexandrebarachant/covariancetoolbox.].§.§ Tangent Space Features for BCI Regression Problems To use the tangent space features for BCI regression problems, we first spatially filter each 𝐗_n to obtain 𝐗'_n in (<ref>), and then estimate its spatial covariance matrix Σ_n∈ℝ^KF× KF (note that each row of 𝐗'_n has zero mean):Σ_n=1/S𝐗_n' 𝐗_n'^T,n=1,..., NNext, we compute the Riemannian mean Σ̅ of all Σ_n by Algorithm <ref>, and take the KF(KF+1)/2 upper triangular part of logm(Σ̅^-1/2Σ_nΣ̅^-1/2) as our features. Note that we need to assign weight 1 to the diagonal elements of logm(Σ̅^-1/2Σ_nΣ̅^-1/2) and weight √(2) to the out-of-diagonal elements so that their Euclidean norm is equal to the Riemannian distance between Σ̅ and Σ_k. The weights do not have an effect when regression methods like LASSO are used, but are very important for distance based regression methods like kNN regression.The complete tangent space feature extraction procedure for BCI regression problems is summarized in Algorithm <ref>.§ EXPERIMENTS AND THE PERFORMANCE EVALUATION PROCESSThis section introduces a PVT experiment that was used to evaluate the performances of the proposed tangent space feature extraction method, and the corresponding RT and EEG data preprocessing procedures. §.§ Experiment SetupSeventeen university students (13 males; average age 22.4, standard deviation 1.6) from National Chiao Tung University (NCTU) in Taiwan volunteered to support the data-collection efforts over a 5-month period to study EEG correlates of attention and performance changes under specific conditions of real-world fatigue <cit.>, as determined by the percent effectiveness score of Readiband <cit.>. The Institutional Review Board of NCTU approved the experimental protocol.The customer-designed daily sampling system consists of a smartphone, actigraph, sleep diary, subjective scales of fatigue and stress, and software for recording, storing, transmitting, and analyzing data acquired from individuals in their natural environments on a daily basis. Each participant was provided a wrist-worn actigraph (Fatigue Science Readiband, Vancouver, BC), and was instructed to complete several subjective report scales and enter the percent effectiveness score from the actigraph approximately 30-60 minutes upon awakening each morning and to be available for experiment testing approximately once every 1-3 weeks over a 5-month period for a total of nine repeated sessions. Data recorded by the daily sampling system included electronically-adapted visual analog scales of fatigue and stress, the Karolinska Sleepiness Scale <cit.>, and the Pittsburgh Sleep Diary <cit.>. The daily sampling data were automatically uploaded from the smartphone to a designated secure server at NCTU on a daily basis. In this way we could track and identify periods when the participants were currently exhibiting low, normal, or high levels of fatigue based on the percent effectiveness score values (>90%, 70-90%, <70%, respectively). The goal was to examine the participants during experiment sessions three times within each of the three fatigue levels. Most participants finished all nine sessions.When the participants reported to the laboratory, we measured their fatigue level on site again right before the experiment to make sure it was close to the fatigue state reported via the smartphone. Upon completion of the related questionnaires and the informed consent form, subjects performed a PVT, a dynamic attention-shifting task, a lane-keeping task, and selected surveys preceding each condition. EEG data were recorded at 1000 Hz using a 64-channel NeuroScan Quik-Cap system (62 EEG channels and 1 electrocardiogram channel). The ground was between FPZ and FZ, and the reference channels were A1 and A2 at the mastoids.In this paper we focus on the PVT <cit.>, which is a sustained-attention task that uses RT to measure the speed with which a subject responds to a visual stimulus. It is widely used, particularly by NASA, for its ease of scoring, simple metrics, convergent validity, and free of learning effects. In our experiment, the PVT was presented on a smartphone with each trial initiated as an empty solid white circle centered on the touchscreen that began to fill in red displayed as a clockwise sweeping motion like the hand of a clock. The sweeping motion was programmed to turn solid red in one second or terminate upon a response by the participants, which required them to tap the touchscreen with the thumb of their dominant hand. The RT was computed as the elapsed time between the appearance of the empty solid white circle and the participant's response. Following completion of each trial, the circle went back to solid white until the next trial. Inter-trial intervals consisted of random intervals between 2-10 seconds.143 sessions of PVT data were collected from the 17 subjects, and each session lasted 10 minutes. Our goal is to predict the RT using a short EEG trial immediately before it. §.§ Performance Evaluation ProcessThe following procedure was used to evaluate the performances of different feature extraction methods: * RT data preprocessing to remove outliers.The number of trials and the mean RTs for the 17 subjects are shown in Table <ref>. Subject 17 may have data recording issues, because many of his RTs were longer than 5 seconds, which are highly unlikely in practice, and his mean RT was more than two times larger than the largest mean RT from other subjects. So we excluded him from consideration in this paper, and only used Subjects 1-16.The RTs were very noisy, and there were obvious outliers. It is very important to suppress the outliers and noise so that the performances of different algorithms can be more accurately compared. We employed the following 2-step procedure for RT data preprocessing: * Outlier removal, which aimed to remove abnormally large RTs. First, a threshold θ=m_y+3σ_y was computed for each subject, where m_y is the mean RT from all sessions of that subject, and σ_y is the corresponding standard deviation. Then, all RTs larger than θ were removed. Note that the threshold was different for different subjects.* Moving average smoothing, which replaced each RT by the average RT during a 60 seconds moving window centered at the onset of the corresponding PVT to suppress the noise.* EEG data preprocessing to remove or suppress artifacts and noise.Generally raw EEG data recorded from the scalp contain many artifacts (e.g., head motion, blinks, eye movements, etc.) and noise (e.g., power-line noise, noise caused by changes in electrode impedances, etc.) <cit.>, so it is very important to remove or suppress them to increase the signal-to-noise ratio before a machine learning algorithm is applied. This paper used the standardized early-stage EEG processing pipeline (PREP) <cit.>, which consists of three steps: a) remove line-noise, b) determine and remove a robust reference signal, and, c) interpolate the bad channels (channels with a low recording signal-to-noise ratio).The preprocessed EEG signals coming out of PREP were downsampled to 250 Hz. They were then epoched to 5-second trials according to the onset of the PVTs: if a PVT started at t, then the 62-channel EEG trial in [t-5, t] seconds was used to predict the RT, i.e., 𝐗_n∈ℝ^62× 1250. Each trial was then individually filtered by a [1, 20] Hz finite impulse response band-pass filter to make each channel zero-mean and to remove non-relevant high frequency components. * 5-fold cross-validation to compute the regression performance for each combination of feature set and regression method.We first randomly partitioned the trials into five folds; then, used four folds for supervised spatial filtering and regression model training, and the remaining fold for testing. We repeated this five times so that every fold was used in testing. Finally we computed the regression performances in terms of root mean square error (RMSE) and correlation coefficient (CC).We extracted the following three different feature sets for each preprocessed EEG trial: * Feature Set 1 (): Theta and Alpha powerband features from the band-pass filtered EEG trials. We computed the average power spectral density in the Theta band (4-8 Hz) and Alpha band (8-13 Hz) for each channel using Welch's method <cit.>, and converted these 62× 2=124 band powers to dBs as our features.* Feature Set 2 (): Theta and Alpha powerband features from EEG trials filtered by Algorithm <ref>. This procedure was almost identical to the above one, except that the band-pass filtered EEG trials were also spatially filtered by Algorithm <ref> before the powerband features were computed. We used 3 fuzzy sets for the RTs, and 10 spatial filters for each fuzzy class, so that the spatially filtered EEG trials had dimensionality 30× 1250, andhad 60 dimensions.* Feature Set 3 (): Riemannian tangent space features from EEG trials filtered by Algorithm <ref>. That is, we first band-pass filtered the raw EEG signals, then spatially filtered them by Algorithm 1 (K=10 and F=3), and further applied Algorithm <ref> to extract the tangent space features, which had 30× 31/2=465 dimensions. Two regression methods were used on each feature set: LASSO <cit.>, and kNN regression <cit.>.For labeled training data {𝐱_n, y_n}_n=1,...,N, LASSO solves the following minimization problem to find a sparse linear regression model:min_β_0,β[1/2N∑_n=1^N (y_n-β_0-β^T𝐱_n)^2+λβ_1]where λ> 0 is an adjustable parameter, which was optimized by an inner 5-fold cross-validation on the training dataset in this paper. Once β_0 and β are identified, the final LASSO regression model is:ŷ_n=β_0+β^T𝐱_n We used k=5 in kNN. Once the five nearest neighbors {𝐱_i, y_i}_i=1,...,5 to the new trial 𝐱_n are identified, the regression output is computed as a weighted average:ŷ_n=∑_i=1^5 w_iy_i/∑_i=1^5 w_iwhere the weights are the inverses of the feature distances:w_i=1/𝐱_n-𝐱_i_2* Repeat Step 3 10 times and compute the average regression performance.§ EXPERIMENTAL RESULTSThis section compares the informativeness of the features in ,and , and presents the regression performances. §.§ Informativeness of the Features Before studying the regression performance, it is important to check if the extracted features in ,andare indeed meaningful.In this first study, we computed the CC between the RT and powerband features inat different channel locations for each of the 16 subjects, and then averaged them. The corresponding topoplot is shown in Fig. <ref>. Both theta and alpha band powers show higher correlation at the central and central-frontal regions of the brain; however, generally the CC is small. This indicates thatfeatures are not very informative.In the second study, we picked a typical subject, partitioned his data randomly into 50% training and 50% testing, and extracted the powerband features . We then designed the spatial filters using Algorithm 1 on the training data, and extracted the corresponding powerband features , and the Riemannian tangent space featuresusing Algorithm 3. For each feature set, we identified the top three features that had the maximum CCs with the RT using the training data, and also computed the corresponding CCs for the testing data. The results are shown in Fig. <ref>, where in each panel the data on the left of the black dotted line were used for training, and the right for testing. The top thick curve is the RT, and the bottom three curves are the maximally correlated features identified from the training data. The training and testing CCs are shown on the left and right of the corresponding feature, respectively. For , we also show the corresponding channel labels and powerband names. For , we only show the powerband names of the top three features, as a channel here does not have a specific label (each channel inis a weighted combination of all 62 physical electrodes). Fig. <ref> shows thatgave much smoother features than , and also achieved much larger CCs to the RT, both in training and testing, suggesting that spatial filtering by Algorithm 1 can indeed increase the signal quality.further achieved larger training and testing CCs to the RT than , suggesting that the tangent space features are more informative than the powerband features.§.§ Estimation Performance Comparison The RMSEs and CCs of LASSO and kNN using three different feature sets are shown in Fig. <ref> for the 16 subjects. Recall that for each subject the feature extraction methods were run 10 times, each with randomly partitioned training and testing data, and the average regression performances are shown here. The average RMSEs and CCs across all subjects are also shown in the last group of each panel.Fig. <ref> shows that regardless of which regression method was used, generallyresulted in smaller RMSEs and larger CCs than , suggesting that the spatial filtering approach can indeed improve the regression performance. Fig. <ref> also shows thatfurther achieved better RMSEs and CCs than , suggesting that the tangent space features were more effective than the powerband features. Finally, LASSO had better performance than kNN on , but kNN became better onand . The RMSEs for Subjects 4, 9 and 11 in Fig. <ref> are much larger than others, because, as shown in Table <ref>, these three subjects have much larger RTs than others.To illustrate the performance differences among the three feature extraction methods from another viewpoint, Fig. <ref> shows the corresponding percentage performance improvements of LASSO and kNN using the three feature sets, where the legend “/" means the percentage performance improvement of LASSO onover LASSO on , and other legends should be understood in a similar manner. For LASSO, on averagehad 4.30% smaller RMSE than , and 6.59% larger CC. For kNN, on averagehad 8.30% smaller RMSE than , and 11.13% larger CC. These results again demonstrated that the tangent space features are more effective than the traditional powerband features.We also performed a two-way Analysis of Variance (ANOVA) for different regression algorithms to check if the raw RMSE and CC differences among the three feature sets (, , and ) were statistically significant, by setting the subjects as a random effect. The results are shown in Table <ref> as “p for raw values." Study results showed that there were statistically significant differences (at 5% level) in raw CCs among different feature sets for both LASSO and kNN, but not for raw RMSEs.However, because the RTs from different subjects had significantly different magnitudes, an ANOVA on the raw RMSEs and CCs may be unfair for those subjects with small RTs. So, we also performed a two-way ANOVA for different algorithms and feature sets on the ratios. For example, to compute the RMSE ratios for LASSO, we replaced all RMSEs forby 1, the RMSEs forby the ratios of the corresponding RMSEs fromover those from , and the RMSEs forby the ratios of the corresponding RMSEs fromover those from . In this way the RMSEs were normalized, and hence different subjects were treated equally. The corresponding ANOVA test results are shown in Table <ref> as “p for ratios." Observe that there were statistically significant differences (at 5% level) in both RMSE ratios and CC ratios among different feature sets for both LASSO and kNN.Then, non-parametric multiple comparison tests based on Dunn's procedure <cit.> were used to determine if the difference between any pair of algorithms was statistically significant, with a p-value correction using the False Discovery Rate method <cit.>. The p-values for the raw values are shown in Table <ref>, and the p-values for the ratios are shown in Table <ref>, where the statistically significant ones are marked in bold. Table <ref> shows that the raw RMSE difference betweenandwas statistically significant when kNN was used. Furthermore, the raw CC differences between all pairs of feature sets were statistically significant. Table <ref> shows that the ratio differences between all pairs of feature sets were statistically significant, for both LASSO and kNN.§ DISCUSSIONSThis section provides parameter sensitivity analysis and additional discussions. §.§ Parameter Sensitivity Analysis Tangent space feature extraction relies on the spatial filter in Algorithm 1, which has two adjustable parameters: K, the number of fuzzy classes for the RTs, and F, the number of spatial filters for each fuzzy class. The filtering performance is robust to K but changes noticeably when F changes <cit.>. As a result, the performance of the tangent space features also varies as F changes. In this subsection we study the sensitivity of the regression performance to F.The regression performances for F={5, 10, 15, 20} (K was fixed to be 3) are shown in Fig. <ref>. Algorithms 1 and 3 were repeated five times, each time with a random partition of training and testing data, and the average regression results are shown. Note that F cannot be too large because of three constraints: 1) F cannot exceed the number of channels (C) in the original EEG data, because Σ̅_kΣ̅^-1∈ℝ^C× C in (<ref>) has at most C eigenvectors; 2) the tangent space features have dimensionality KF(KF+1)/2, which increases rapidly with F; so, a large F can easily result in over-fitting; and, 3) there may be numerical difficulties in computing the RG mean when F is large, e.g., for Subjects 5, 8 and 15 in Fig. <ref> when F=20.Fig. <ref> shows that the regression performance increased when F increased from 5 to 15, but decreased when F further increased to 20. For the PVT experiment, F∈[10,15] seemed to achieve a good compromise between performance and computational cost.Additionally, in the previous subsection we used 5-second EEG trials to estimate the corresponding RT, and it is also interesting to study how the estimation performance changes with different trial lengths. The results are shown in Fig. <ref> for trial lengths of {1, 3, 5, 7, 9} seconds. In general, as trial length increased, the estimation performance improved. However, a longer trial means heavier computational cost and larger delay in estimation. Furthermore, a trial cannot be arbitrary long, as then it cannot capture the up-to-date RT. These effects should be taken into consideration when choosing the right trial length. §.§ Regression Performance versus the Number of Features Recall from Section <ref> thathas 124 features,has 60 features, andhas 465 features, i.e.,has much more features thanand . So, 's superior performance may be due to its increased number of features. In this subsection we investigate the relationship between the regression performance and the number of useful features.Because LASSO automatically selects the most useful features, whereas kNN always uses all the features, in this study we focus only on LASSO. For each subject and each feature set, we used all data in LASSO training, and recorded the number of selected features, as well as the corresponding training RMSEs and CCs. The results are shown in Fig. <ref>. On average LASSO selected 58.6 features from , 30.6 features from , and 69.1 features from . Although the selectedsubset was only about half the size of the selectedsubset, they resulted in similar overall training RMSEs and CCs. Connecting this observation with that in the previous subsection, i.e.,had much better testing RMSEs and CCs than , we can conclude that the CSPR-OVR spatial filter can aggregate the most useful information into just a small number of features, which reduces overfitting and improves the generalization performance. Fig. <ref> also shows that the selectedsubset was slightly larger than the selectedsubset, but thesubset resulted in much better training performance, and also much better testing performance, as presented in the previous subsection. These observations together suggest that the Riemannian geometry approach can indeed extract some novel informative features, which improve both the training and the testing performances. §.§ Computational cost The training of our feature extraction method () consists of three steps: 1) design the CSPR-OVR filter by Algorithm 1; 2) compute the RG mean Σ̅ by Algorithm 2; and, 3) map the spatially filtered EEG trials to the Riemannian tangent space by Algorithm 3. Once the training is done, feature extraction for a testing trial can be performed very efficiently: a matrix multiplication (<ref>) is first used to spatially filter it, and then another matrix multiplication (<ref>) is used to compute its spatial covariance matrix Σ_n; finally, compute logm(Σ̅^-1/2Σ_nΣ̅^-1/2) and take its upper triangular part as the features. Note that Σ̅ has been obtained in training, so Σ̅^-1/2 can be pre-computed, and hence Σ̅^-1/2Σ_nΣ̅^-1/2 is also a simple matrix multiplication. So, in this subsection we focus on the training computational cost only.Let N be the number of training samples. Then, the actual training time increased linearly with N, as shown in Fig. <ref>. The platform was a Dell XPS15 laptop (Intel i7-6700HQ CPU @2.60GHz, 16 GB memory) running Windows 10 Pro 64-bit and Matlab 2016b. A least squares curve fit shows that the training time is 0.0261+0.0030N seconds, which should not be a problem for a practical N. §.§ RT versus Fatigue State We also studied the relationship between the RT and the fatigue state. Our conjecture is that as the fatigue level goes up, the RT should be larger. Boxplots of the RT in different sessions for two typical subjects are shown in Fig. <ref>, where “L", “N" and “H" mean low, normal, and high fatigue, respectively. Fig. <ref> shows that the mean RT of a high fatigue sessions is generally larger than that of a low or normal fatigue session, and the former also has more extreme values and a larger variance. The difference between a low fatigue session and a normal fatigue session is not obvious. These observations suggest that although the fatigue state contains some useful information, it may be too coarse for accurate RT prediction. That's why it was not used in this paper.§ CONCLUSIONS AND FUTURE RESEARCHIn this paper, we have proposed a new feature extraction approach for EEG-based BCI regression problems: a spatial filter is first used to increase the EEG trial signal quality and also to reduce the dimensionality of the covariance matrix, and then Riemannian tangent space features are extracted. We validated the performance of the proposed approach in RT estimation from EEG signals measured in a large-scale sustained-attention PVT experiment, and showed that compared with the traditional powerband features, the tangent space features can reduce the estimation RMSE by 4.30-8.30%, and increase the estimation CC by 6.59-11.13%. To our knowledge, this is the first time that RG has been used in BCI regression problems.Our future research will focus on reducing the dimensionality of the tangent space features. As shown in Algorithm 3, the tangent space features have dimensionality KF(KF+1)/2, where K is the number of fuzzy classes for the RTs, and F is the number of spatial filters for each fuzzy class. So, the feature dimensionality increases quadratically with respect to both K and F, which quickly results in overwhelming computational cost, overfitting, and numerical problems. We will investigate effective dimensionality reduction approaches for the tangent space features to reduce the computational cost while maintaining or even improving the regression performance. § ACKNOWLEDGEMENT Research was sponsored by the U.S. Army Research Laboratory and was accomplished under Cooperative Agreement Numbers W911NF-10-2-0022 and W911NF-10-D-0002/TO 0023. 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"authors": [
"Dongrui Wu",
"Brent J. Lance",
"Vernon J. Lawhern",
"Stephen Gordon",
"Tzyy-Ping Jung",
"Chin-Teng Lin"
],
"categories": [
"cs.HC",
"cs.LG"
],
"primary_category": "cs.HC",
"published": "20170427123005",
"title": "EEG-Based User Reaction Time Estimation Using Riemannian Geometry Features"
} |
Dissecting Robotics — historical overview and future perspectives Irati Zamalloa, Risto Kojcev, Alejandro Hernández, Iñigo Muguruza, Lander Usategui, Asier Bilbao and Víctor MayoralAcutronic Robotics, April 2017 Received 1 March 2017 /Accepted 21 April 2017 =======================================================================================================================================================================plain plainRobotics is called to be the next technological revolution and estimations indicate that it will trigger the fourth industrial revolution. This article presents a review of some of the most relevant milestones that occurred in robotics over the last few decades and future perspectives. Despite the fact that, nowadays, robotics is an emerging field, the challenges in many technological aspects and more importantly bringing innovative solutions to the market still remain open. The need of reducing the integration time, costs and a common hardware infrastructure are discussed and further analysed in this work. We conclude with a discussion of the future perspectives of robotics as an engineering discipline and with suggestions for future research directions.robotics, review, hardware, H-ROS, artificial intelligence.§ INTRODUCTIONThe first use of the word "Robot" dates back in 1921 and it was introduced by Karel Čapek in his play Rossum's Universal Robots. The play describes mechanical men that are built to work on the factory assembly lines and that rebel against their human masters <cit.>. The etymological origin of the word Robot is from the Czech word robota, which means servitude or forced labor.The term "Robotics" was first mentioned by the Russian-born American science-fiction writer Isaac Asimov in 1942 in his short story Runabout <cit.>. Asimov had a much brighter and more optimistic opinion of the robot's role in human society compared to the view of Capek. In his short stories, he characterized the robots as helpful servants of man.Asimov defined robotics as the science that study robots. He created the Three Laws of Robotics, that state the following:* The First Law states that a robot should not harm a person or let a person suffer damage because of their inaction.* Second Law states that a robot must comply with all orders that a human dictates, with the proviso that occurs if these orders were in contradiction with the First Law.* The Third Law states that a robot must take care of its own integrity, except when this protection creates a conflict with the First or Second Law. Over the last decades, robotics evolved from fiction to reality becoming the science and technique that is involved in the design, manufacture and use of robots. Computer Science, Electrical Engineering, Mechanical Engineering and Artificial Intelligence (AI) are just some of the disciplines that are combined in robotics. The main objective of robotics is the construction of devices that perform user-defined tasks. The rapid growth of the field in scientific terms had led the development of different types of robots. Examples of different robotic systems are: industrial robots, manipulators, terrestrial, aerial, aquatic, research, didactic, entertainment robots or humanoids.Section <ref> covers the evolution of robotics over the last decades including current trends. In Section <ref>, we present the future perspectives of robotics based on the historical overview presented in <ref>. In Section <ref> we conclude this work and discuss our viewpoint for future development in robotics. § THE EVOLUTION OF ROBOTICSFigure <ref> illustrates a historical overview of the growth of robotics, divided into four columns: the first column presents the different robot generations, the second column (central) shows some of the most relevant milestones and our vision of the upcoming future of robotics,the third one the needs and social impact of robotics, and the fourth summarizes the different stages of development in robotics.The progress of robotics is influenced by the technological advances, for example, the creation of the transistor <cit.>, the digital computer <cit.>, the numerical control system <cit.> or the integrated circuits <cit.>. These technological advances further enhanced the properties of the robots, and helped them evolve from solely mechanical or hydraulic machines, to programmable systems, which can even be aware of their environment. Similar to other technological innovations, robotics has advanced and changed taking into account the needs of society.Based on their characteristics and the properties of the robots, we can classify the development of robotics into four generations:§.§ Generation 0: Pre-Robots (up to 1950) [colback=gray!10] Characteristics:* The first industrial robots were pneumatic orhydraulic.In 1495, the polymath Leonardo Da Vinci envisioned the desing of the first humanoid <cit.>. In the following years, different machines were manufactured using mechanical elements that helped the society and the industry. It was not until the first industrial revolution that factories began to think about automation as a way of improving manufacturing processes. The automated industrial machines of this generation were based on pneumatic or hydraulic mechanisms, lacking of any computing capacity and were managed by the workers. The first automation techniques were the punch cards <cit.>, used to enter information to different machinery (e.g. for controlling textile looms). The first electronic computers, for example the Colossus <cit.>, also used punched cards for programming. §.§ Generation 1: First Manipulators (1950-1967)[colback=gray!10] Characteristics:* Lack of information regarding the environment.* Simple control algorithms (point-to- point).Due to the rapid technological development and the efforts to improve industrial production, automated machines were designed to increase the productivity. The machining-tool manufacturers introduced the numerical controlled (NC) machines which enabled other manufacturers to produce better products <cit.>. The union between the NC capable machining tools and the manipulators paved the way to the first generation of robots.Robotics originated as a solution to improve output and satisfy the high quotas of the U.S. automotive industry. In parallel the technological growth led to the construction of the first digitally controlled mechanical arms which boosted the performance of repeatable, "simple" tasks such as pick and place. The first acknowledged robot is UNIMATE <cit.> (considered by many the first industrial robot), a programmable machine created by George Devol and Joe Engleberger that two years before funded the world's first robot company called Unimation (Universal Automation). In 1960, they secured a contract with General Motors to install the robotic arm in their factory located in Trenton (New Jersey). UNIMATE helped improve the production, which further motivated many companies and research centers to actively dedicate resources in robotics. §.§ Generation 2: Sensorized robots (1968-1977)[colback=gray!10] Characteristics:* More awareness of their surroundings.* Advanced sensory systems: for example, force, torque, vision.* Learning by demonstration. * These type of robots are used in the automotive industry and have large footprint. Starting from 1968, the integration of sensors marks the second generation of robots. These robots were able to react to the environment and offer responses that met different challenges. Shakey <cit.>, developed by Stanford Research Institute, was the first sensorized mobile robot, containing a diversity of sensors (for example tactile sensors) as well as a vision camera.During this period relevant investments were made in robotics. In the industrial environment, we have to highlight the PLC (Programmable Logic Controller) <cit.>, an industrial digital computer, which was designed and adapted for the control of manufacturing processes, such as assembly lines, robotic devices or any activity that requires high reliability. PLCs were at the considered to be easy to program. Due to these characteristics, PLCs became a commonly used device in the automation industry.In 1973, KUKA (one of the world's leading manufacturers of industrial robots) built the first industrial robot with 6 electromechanical-driven axes called Famulus <cit.>. One year later, the T3 robot <cit.> was introduced in the market by Cincinnati Milacron (acquired by ABB in 1990). The T3 robot was the first commercially available robot controlled by a microcomputer. §.§ Generation 3: Industrial robots (1978-1999)[colback=gray!10] Characteristics:* Robots now have dedicated controllers (computers).* New programming languages for robot control.* Reprogrammable robots.* Partial inclusion of artificial vision. Many consider that the Era of Robots started in 1980 <cit.>. Billions of dollars were invested by companies all around the world to automate basic tasks in their assembly lines. The investments in automation solutions increased the sales of industrial robots up to 80% compared to previous years. Robots populated many industrial sectors to automate a wide variety of activities such as painting, soldering, moving or assembly.Key technologies that still drive the development of robots appeared during these years: general Internet access was extended in 1980 <cit.>, Ethernet became a standard in 1983 <cit.> (IEEE 802.3), the Linux kernel was announced in 1991 <cit.> and soon after, real-time patches started appearing <cit.> to increase the determinism of Linux-based systems.The “robot programming languages” also became popular during this time. For example, Unimation started using VAL in 1979 <cit.> <cit.>, FANUC designed Karel in 1988 <cit.> and in 1994 ABB created Rapid <cit.> making robots re-programmable machines which also contained a dedicated controller. By the end of the 1990s, companies started thinking about robots outside industrial environments. Among the robots and kits created within this period, we highlight two that became an inspiration for hundreds of roboticists: * The first LEGO Mindstorms kit (1998) <cit.>, a set consisting of 717 pieces, including LEGO bricks, motors, gears, different sensors, and a RCX Brick with an embedded microprocessor to construct various robots using the same parts. The kit allowed to teach the principles of robotics. Creative projects have appeared over the years showing the potential of interchangeable hardware in robotics.* Sony’s AIBO (1999) <cit.>, the world’s first entertainment robot. Widely used for research and development. Sony brought robotics to everyone with a $1,500 robot that included a distributed hardware and software architecture. The OPEN-R architecture involved the use of modular hardware components —e.g. appendages that can be easily removed and replaced to change the shape and function of the robots—, and modular software components that can be interchanged to modify their behavior and movement patterns. OPEN-R represented an inspiration for future robotic frameworks and showed promise to minimize the need for programming individual movements or responses.Sony’s AIBO and LEGO’s Mindstorms were built upon the principle of modularity, both concepts were able to easily exchange components and both of them presented common infrastructures. Even though they came from the consumer side of robotics, one could argue that their success was strongly related to the fact that both products made use of interchangeable hardware and software modules. The use of a common infrastructure proved to be one of the key advantages of these technologies.§.§ Generation 4: Intelligent robots (2000-2017) [colback=gray!10] Characteristics:* Inclusion of advanced computing capabilities.* These computers not only work with data, they can also carry out logical reasoning and learn.* Artificial Intelligence begins to beincluded partially and experimentally.* More sophisticated sensors that send informa- tion to the controller and analyze it through complex control strategies. * The robot can base its actions on more solid and reliable information.* Collaborative robots are introduced. The fourth generation of robots, dating from the 2000, consisted of more intelligent robots that included advanced computers to reason and learn. These robots also contained more sophisticated sensors that helped them adapt more effectively to different circumstances. The robot Roomba <cit.> —the first vacuum cleaner domestic robot— introduced the robots in many homes. YuMi <cit.>, the first collaborative robot included many advances in the security systems beyond photoelectric barriers or interlocking devices, ensuring the coexistence of worker and robot in the same environment, improving the production process and the ergonomics of the operator. These advances both on the human-robot collaboration and improvements of the robot security systems, have allowed the robots to work together with humans in the same environment.Among the technologies that appeared in this period we highlight the Player Project <cit.> (2000, formerly the Player/Stage Project), the Gazebo simulator <cit.> (2004) and the Robot Operating System <cit.> (2007). Moreover, relevant hardware platforms appeared during these years. Single Board Computers (SBCs) like the Raspberry Pi <cit.> enabled millions of users all around the world to easily create robots. The following subsections will describe some relevantobservations our research obtained within this period:§.§.§ Decline in Industrial robot innovation Except for the appearance of the collaborative robots in 2015, the progress within the field of industrial robotics has significantly slowed down compared to previous decades. While industrial robots significantly improved their accuracy, speed or offer greater load capacity, industrial robots regarding innovation up till today are very stagnant.§.§.§ The boost of bio-inspired Artificial IntelligenceArtificial Intelligence and, particularly, of neural networks became relevant in this period as well. A lot of the important work on neural networks occurred in the 1980’s and in the 1990’s, however at that time computers did not have enough computational power. Data-sets were not big enough to be useful in practical applications. As a result, neural networks practically disappeared in the first decade of the 21st century. However, starting from 2009, neural networks gained popularity and started delivering good results in the fields of computer vision (2012) <cit.> or machine translation (2014) <cit.>. During the last years we have seen how these techniques have been translated to robotics for tasks such as robotic grasping (2016) <cit.>. In the coming years it is expected to see more innovations and these AI techniques will have high impact in robotics.§.§.§ Real-time communication solutionsAccording to <cit.>, since 2001 reputable industrial titans introduced different Industrial real-time Ethernet standards (EtherCAT, SERCOS, PROFINET, Ethernet/IP and Ethernet PowerLink) represented in Figure <ref>. These solutions have been widely used to criticize the deterministic and real-time capabilities of traditional Ethernet. However they do not serve to standardize the communications because the Industrial robots need to adapt and speak the factory language that have been chosen based on one of these technologies. This leads to incompatibility between different robotic or automation systems, therefore making integration task cumbersome. §.§.§ A common infrastructure for robotics An interesting approach would be to have manufacturers agree on a common infrastructure. Such an infrastructure could define a set of electrical and logical interfaces (leaving the mechanical ones aside due to the variability of robots) that would allow industrial robot companies to produce robots and components that could interoperate, be exchanged and eventually enter into new markets. This would also lead to a competing environment where manufacturers will need to demonstrate features rather than the typical obscured environment where only some are allowed to participate.Integration effort was identified as one of the main issues within robotics and particularly related to robots operating in industry. A common infrastructure typically reduces the integration effort by facilitating an environment where components can simply be connected and interoperate. Each of the infrastructure-supported components are optimized for such integration at their conception and the infrastructure handles the integration effort. At that point, components could come from different manufacturers, yet when supported by a common infrastructure, they will interoperate:A hardware/software standardization is needed"Currently, most of the time is spent dealing with the hardware/software interfaces and much less is put into behaviour development or real-world scenarios"For robots to enter new and different fields, it seems reasonable to accept that these robots will need to adapt to the environment itself. For industrial robotics, robots have to be fluent with factory languages (EtherCAT, SERCOS, PROFINET,Ethernet/IP and Ethernet PowerLink). One could argue that the principle is valid for service robots (e.g. households robots that will need to adapt to dish washers, washing machines, or media servers), medical robots and many other areas of robotics. Such reasoning led to the creation of the Hardware Robot Operating System (H-ROS), a vendor-agnostic hardware and software infrastructure for the creation of robot components that interoperate and can be exchanged between robots. H-ROS is built on top of ROS, which is the de-facto standard for robot software development <cit.>. In more detail, H-ROS utilizes the functionality of ROS 2; a redesigned version of the robot middleware that targets use cases not included in the initial design of ROS. The philosophy behind H-ROS is that creating robots should be about placing together components that are compliant with the standardized H-ROS interfaces, regardless of the manufacturer. H-ROS aims to facilitate a fast way of building robots choosing the best component for each use-case from a common robot marketplace. It complies with different environment requirements (industrial, professional, personal, and others) where variables such as time constraints are critical. Building or extending robots is simplified to the point of placing H-ROS compliant components together. The user simply needs to program the cognition part (in other words, the brain) of the robot and develop their own use-cases without facing the complexity of integrating different technologies and hardware interfaces.§ ENVISIONING THE FUTURE OF ROBOTICSIn order to predict the future of robotics we have analyzed the historical growth of robotics, divided into the following markets: industrial robots, professional robots and consumer robots. The presence of industrial robots led the growth of robotics since its beginning, as shown in Figure <ref>, however, since 2000, the use and aim of manyrobot companies and initiatives changed. A relevant amount of resources were invested into getting robots outside of industrial environments. We can distinguish two periods: * 1960-2000: boom of the automotive industry and increased interest in industrial robots. Many started including robots in their factories to increase productivity.* Since 2000: robot development and innovations pivot towards the consumer and professional markets which opens the door for faster innovation cycles.Figure <ref> illustrates the interest in robotics obtained from a joint review of publications, conferences and events, solutions and corporations:§.§ Generation 5: Collaborative and personal robots [colback=gray!10] Characteristics:* Robots and humans share same environment and collaborate.* Reconfigurable robots.* Robots help humans enhance every-day activities.* Modular robots and components.With latest developments in AI (e.g. AlphaGo beats the world-class player in Go <cit.>, an achievement that was not expected for many years) being translated to robotics and the recent investments in the field, there is a high expectation for the future development of robotics. Our team believes that the next generation robots will reflect all the technological advances that developers and researchers have made in recent years.According to the characteristics of the 5^th generation, robots will be able to coexist with the humans, enhance human capabilities, simplify and improve life. We foresee a boom in collaborative and personal robots:The list presented in the Figure <ref> presents our perception regarding the near future of robotics by taking into account its technical feasibility.§ DISCUSSION AND CONCLUSIONS Robotics did not grow as much as expected. Visionaries predicted that by 2020 robots would be in our every day lives helping with daily tasks. To the best of our knowledge, this fact is highly unlikely, however the results obtained through this research unveiled a promising future for the field. The importance of robotics and its potential is being intensely highlighted and a number of standardization bodies are taking steps towards creating a common set of rules to govern the interaction with these machines. The following subsections reflect some of the most relevant conclusions taken from our review: §.§ Lack of compatible systems * Hardware/software incompatibility: Building a new robot is a bumpy road. The incompatibility between components/elements/languages, take significant time and effort leading to the fact that during development of a new robotic solution most of the timeis dedicated in the integration and not to the behavior or developing innovative solutions. * Different programming languages and environments: In an attempt to dominate the market the main industrial robots manufacturers had created their own programming languages to keep their technology as closed as possible. As consequence there are many incompatible components/robots and integration between components from different manufactures is very cumbersome, time consuming process which sometimes is not even feasible. In order to "simplify" integration effort many users decide to bound to solutions provided from a single manufacturer which increases the overall costs. A common infrastructure for robot components is needed, H-ROS to lead the change. Making robotics hardware more affordable, versatile, and “standardized” is hugely important for the field, as Aaron Dollar, Francesco Mondada, Alberto Rodriguez, and Giorgio Metta, who guest edited the special issue <cit.>:"In the field of robotics, there has existed a relatively large void in terms of the availability of adequate hardware, particularly for research applications. The few systems that have been appropriate for advanced applications have been extremely costly and not very durable. For those and other reasons, innovation in commercially available hardware is extremely slow, with a historically small market and expensive and slow development cycles. Effective open source hardware that can be easily and inexpensively fabricated would not only substantially lower costs and increase accessibility to these systems, but would drastically improve innovation and customization of available hardware."* The Industrial robot industry —–will it remain only a supplier industry?— Forsome,theindustrialrobotindustryisasupplier industry. It supplies components and systems to largerindustries,mainly,themanufacturingindustry. Thesegroups argue that the manufacturing industry is dominated by the PLC, motion control and communication suppliers which together with the big customers are setting the standards to capitalize on the cost savings from Ethernet by extending their own standard to include Ethernet. In doing so, their customers receive some of the benefits from Ethernet but are still locked into the proprietary networks for the long term. Frequently forgotten in these discussions is the fact that it is not just technical properties such as performance and transfer rates that count, it is also the soft facts like ease of implementation, openness, vendor independence, risk avoidance, conformity, interoperability, long-term availability, and overall distribution that makes a standard gain acceptance and even thrive. The Industrial robots need to adapt and speak factory language (such as, PROFINET, ETHERCAT, Modbus TCP, Ethernet/IP, CANOPEN, DEVICENET) which for each factory, might be different. As a result, most robotic peripheral manufacturers suffer from supporting many different protocols which requires a lot of development time that does not add functionality to the product. * Competing by obscuring is slowing industryThe close attitude of most industrial robot companies is typically justified by the existing competition in this environment. Such attitude leads to a lack of understanding between different manufacturers and solutions but in exchange, some believe that it secures clients and favours competition. Our results indicate that this behavior is slowing progress, innovation and new solutions in the field of industrial robots. §.§ The hype cycle of robotics Robotics, like many other technologies, suffered from an inflated set of expectations which resulted in a decrease of the developments and results during the 1990s. Figure <ref> pictured the evolution of the general interest in robotics. Such graph displays a well known trajectory typically known as the hype cycle<cit.>. Figure <ref> pictures the hype cycle as defined by Linden and Fenn. The graph illustrates the five different states that a technology goes through before entering mass-adoption: a) technology trigger, b) peak of inflated expectations, c) trough of disillusionment, d) slope of enlightenment and e) plateau of productivity.Comparing Figures <ref> and <ref> weconclude that the state of robotics is past the "second-generation products, some services" and somewhere within the slope of enlightenment. Industrial robots on the rise Initiatives such as the U.S. Advanced Robotics for Manufacturing (ARM) Institute are pushing the boundaries of industrial robots once again <cit.>. Collaborative robots are increasingly improving the performance of the manufacturing processes while improving the interaction with humans. Robots everywhere Robots and automation are actively being introduced in many disciplines. With the growing popularity of such systems, we observe a transition that goes from mass manufacturing to a mass customization, particularly for industrial robots. Increasingly, personal (e.g. cleaning robots) and professional robots (e.g. service robots) are also demanding such customizations. We foresee that an infrastructure such as H-ROS will favor the creation of modular robots whose components could be easily exchanged or replaced meeting the growing needs for customization. §.§ Artificial Intelligence (AI) taking over:Current robot systems are designed by teams with multi-disciplinary skills. The design of the control mechanisms is one of the critical tasks. Typically, the traditional approach to design such systems require going from observations to final control commands through a) state estimation, b) modeling and prediction, c) planning and d) low level control (for example, inverse kinematics). This whole process requires fine tuning every step of the funnel incurring into a relevant complexity where optimization at every step have a direct impact in the final result.The work of Levine et al. <cit.> shows a promising path towards simplifying the construction of robot behaviors through the use of deep neural networks as a replacement of the whole funnel described above[The use of neural networks could also replace individual tasks within the funnel.]. Our team envisions that the use of deep neural networks will become of great relevance in the field of robotics in the coming years. These abstractions will empower roboticists to train robot models in specific applications (end-to-end) empowering engineers and robots to tackle more complex problems. Robots are going to change the society, in the same way computers changed our life. The introduction of robotics in our society will be disruptive. Work, communication, transportation and even social life will be affected by robotics. This will lead to change from mass customization to mass integration, so that robots and humans coexist helping each other.§ ACKNOWLEDGMENT This review was funded and supported by Acutronic Robotics[www.acutronicrobotics.com], a firm focused on the development of next-generation robot solutions for a range of clients.IEEEtran | http://arxiv.org/abs/1704.08617v1 | {
"authors": [
"Irati Zamalloa",
"Risto Kojcev",
"Alejandro Hernández",
"Iñigo Muguruza",
"Lander Usategui",
"Asier Bilbao",
"Víctor Mayoral"
],
"categories": [
"cs.RO"
],
"primary_category": "cs.RO",
"published": "20170427151118",
"title": "Dissecting Robotics - historical overview and future perspectives"
} |
Locality Preserving Projections for Grassmann manifold Boyue Wang^1, Yongli Hu^1Corresponding author: Yongli Hu ([email protected]). Yongli Hu, Yanfeng Sun and Baocai Yin are also with Beijing Advanced Innovation Center for Future Internet Technology., Junbin Gao^2, Yanfeng Sun^1, Haoran Chen^1 and Baocai Yin^3,1^1Beijing Key Laboratory of Multimedia and Intelligent Software Technology, Faculty of Information Technology, Beijing University of Technology, China^2Discipline of Business Analytics. The University of Sydney Business School, University of Sydney, Australia^3Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, China December 30, 2023 =============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== Learning on Grassmann manifold has become popular in many computer vision tasks, with the strong capability to extract discriminative information for imagesets and videos. However, such learning algorithms particularly on high-dimensional Grassmann manifold always involve with significantly high computational cost, which seriously limits the applicability of learning on Grassmann manifold in more wide areas. In this research, we propose an unsupervised dimensionality reduction algorithm on Grassmann manifold based on the Locality Preserving Projections (LPP) criterion. LPP is a commonly used dimensionality reduction algorithm for vector-valued data, aiming to preserve local structure of data in the dimension-reduced space. The strategy is to construct a mapping from higher dimensional Grassmann manifold into the one in a relative low-dimensional with more discriminative capability. The proposed method can be optimized as a basic eigenvalue problem. The performance of our proposed method is assessed on several classification and clustering tasks and the experimental results show its clear advantages over other Grassmann based algorithms. § INTRODUCTION Dimensionality reduction (DR), which extracts a small number of features from original data by removing redundant information and noise, can improve efficiency and accuracy in a wide range of applications, involving facial recognition <cit.>, feature extraction <cit.> and so on.The classic DR algorithms include Locality Preserving Projections (LPP) <cit.>, Principal Components Analysis (PCA) <cit.>, Canonical Correlation Analysis (CCA) <cit.> and Independent Component Analysis (ICA) <cit.>. Most existing DR algorithms are mainly designed to work with vector-valued data, which cannot be directly applied on multi-dimensional data or structured data (i.e., matrices, tensors).Simply vectorizing such structured data to fit vector-based DR algorithmsmay destroy valuable structural and/or spatial information hidden in data. Therefore, how to effectively and properly reduce the dimensionality of structured data becomes an urgent issue in the big data era.In practical application tasks such as those in computer vision, except for well-structured data like matrices or tensors, there exist data which are manifold-valued. For example, in computer vision, the movement of scattered keypoints in images can be described by subspaces, i.e., the points on the so-called Grassmann manifold <cit.>;and the covariance feature descriptors of images are SPD manifold-valued data <cit.>.How to design learning algorithms for these two types of special manifold-valued data has attracted great attention in the past two decades <cit.>.For our purpose in this paper, we will briefly review some recent progress about DR algorithms for structured and manifold-valued data.For a clear outline, we start with PCA. PCA is the most commonly used DR algorithm for vectorial data. The basic idea of PCA is to find a linear DR mapping such that as much variance in dataset as possible is retained. The classic PCA has been extended to process two dimensional data (matrices) directly with great success <cit.> (2DPCA). Wang et al. WangChenHuLuo2008consider the probabilistic 2DPCA algorithm including the algorithm for the mixture of local probabilistic 2DPCA. To identify outliers in structured data, Ju et al. JuSunGaoHuYin2015 introduce the Laplacian distribution into the probabilistic 2DPCA algorithm.Contrary to the global variance constraint in PCA-alike algorithms, LPP focuses on preserving the local structure of original data in the dimension-reduced space. The first work of extending LPP for 2D data was proposed in <cit.>, which is operated directly on image matrices. The experimental results show that 2DLPP performs better than 2DPCA and LPP. Xu et al. XuFengZhao2009 propose a supervised 2DLPP by constructing a discriminative graph of labeled data. To reduce the high computational cost of 2DLPP, Nyuyen et al. NguyenLiuVenkatesh2008 improve 2DLPP by using the ridge regression.However, the aforementioned DR algorithms for matrices are concerned in terms of Euclidean alike distance. Although the Riemannian structure has been shown to overcome the limitations of Euclidean geometry of data <cit.>,the computational cost of the resulting techniques increases substantially with the increasing dimensionality of manifolds (i.e., the dimension of its embedding space). To the best of our knowledge, few attention has been paid on DR for Riemannian manifold.Harandi et al. HarandiSalzmannHartley2014 extend PCA onto SPD manifold by employing its Riemannian metrics, and then incorporate a discriminative graph of the labeled manifold data to achieve a supervised DR algorithm for SPD manifold. Recent research has shown that the Grassmann manifold, another type of Riemannian matrix manifold, is a good tool to represent videos or imagesets <cit.>. In a newly proposed supervised metric learning on the Grassmann manifold <cit.>, an orthogonal matrix that maps the original Grassmann manifold into a more discriminative one is learned from data.In handling Grassmann-valued data, one usually employs one of three ways: embedding into a Hilbert feature space defined a Grassmann kernel function <cit.>; or embedding into the symmetric matrix manifold (a plain Euclidean space) <cit.>; or projecting data onto tangent spaces (extrinsic way) <cit.>. However the performance of all these ways can be hindered by the high dimensionality of given Grassmann manifold. It has become critical to reduce the dimensionality of Grassmann data. Motivated by <cit.>, we learn a projected matrix to reduce the dimensionality of Grassmann manifold in this paper. To fulfill the goal, we extend LPP local criterion onto Grassmann manifold through embedding Grassmann manifold into a symmetric matrices space <cit.> such that the local structure of original Grassmann data can be well preserved in the newly projected Grassmann manifold.Figure <ref> illustrates that a projected matrix 𝐀 is introduced to map the original high-dimensional Grassmann manifold into the one in a relative low-dimensional with more discriminative capability, which still preserves the structure of original Grassmann points. The contribution of this paper is summarized as follows, * A novel unsupervised DR algorithm in the context of Grassmann manifold is introduced. The DR is implemented by learning a mapping to a Grassmann manifold in a relative low-dimensional with more discriminative capability;* The proposed method generalizes the classic LPP framework to non-Euclidean Grassmann manifolds andonly involves the basic eigenvalue problem; andWe briefly review some necessary knowledge about LPP and Grassmann manifold in next section.§ BACKGROUNDS§.§ NotionsWe denote matrices by boldface capital letters (i.e., 𝐗, ...); vectors are denoted by boldface lowercase letters (i.e., 𝐚, 𝐱, ...) and scalars are denoted by italic letters (i.e., N, n, p, ...). The i-th matrices (i.e., 𝐗_i, ...), i-th vectors (i.e., 𝐱_i, ...) and the element (i.e., W_ij, ...) are also defined. As for transpose matrices and inverse matrices, we represent them as 𝐗^T and 𝐗^-1, respectively. Some regularization of matrix and vector are used. We represent Frobenius norm 𝐗_F= √(∑_ijx_ij^2). And tr(𝐗) is the trace of matrix 𝐗.Finally, 𝐈_p represents the identity matrix with p× p size.§.§ Locality Preserving Projections (LPP)LPP uses a penalty regularization to preserve the local structure of data in the new projected space.[Locality Preserving Projections] <cit.> Let 𝐗 = [𝐱_1, ..., 𝐱_N]∈ℝ^D× N be the data matrixwith N the number of samples and D the dimension of data. Given a local similarity 𝐖 = [w_ij] among data 𝐗, LPP seeks for the projection vector 𝐚 such that the projected value y_i = 𝐚^T𝐱_i (i=1, ..., N)fulfills the following objective,min_𝐚∑_i,j=1^N(𝐚^T𝐱_i - 𝐚^T𝐱_j)^2 · w_ij = ∑_i,j=1^N𝐚^T𝐗𝐋𝐗^T𝐚,with the constraint condition,𝐲𝐃𝐲^T=𝐚^T𝐗𝐃𝐗^T𝐚 = 1,where 𝐲 = [y_1, ..., y_N], 𝐋 = 𝐃 - 𝐖 is the graph Laplacian matrix and 𝐃 = diag[d_ii] with d_ii=∑_j=1^N w_ij. We generalized the constraint condition (<ref>) from the single projection to the multiple projection as,tr(𝐘𝐃𝐘^T) = 1⇒tr(𝐀^T𝐗𝐃𝐗^T 𝐀) = 1,where 𝐀 = [𝐚_1,...,𝐚_m]∈ R^n× m and 𝐘 = [𝐲_1,...,𝐲_N]∈ R^m× N, so that 𝐘 = 𝐀^T𝐗. A possible definition of 𝐖 is suggested as follows:w_ij =e^-𝐱_i - 𝐱_j^2/t,𝐱_i ∈𝒩(𝐱_j) or𝐱_j ∈𝒩(𝐱_i); 0, otherwise.where t ∈ℝ_+ and 𝒩(𝐱_i) denotes the k nearest neighbors of 𝐱_i. With the help of 𝐖, minimizing LPP objective function (<ref>) is to ensure if 𝐱_i and 𝐱_j are similar to each other, then the projected values y_i = 𝐚^T 𝐱_i and y_j=𝐚^T 𝐱_j are also similar.We can further find d more projection vectors so that the data dimension D can be reduced to d.§.§ Grassmann Manifold and its Distances(Grassmann Manifold) <cit.> The Grassmann manifold, denoted by 𝒢(p, d), consists ofall the p-dimensional subspaces embedded in d-dimensional Euclidean spaceℝ^d (0≤ p ≤ d). For example, when p=0, the Grassmann manifold becomes the Euclidean space itself. When p=1, the Grassmann manifold consists of all the lines passing through the origin in ℝ^d.As Grassmann manifold is abstract, there are a number of ways to realize it. One convenient way is to represent the manifold by the equivalent classes of all the thin-tall orthogonal matrices under the orthogonal group 𝒪(p) of order p. Hence we have the following matrix representation, 𝒢(p,d) = {𝐗∈ℝ^d× p: 𝐗^T𝐗 = 𝐈_p} / 𝒪(p).We refer a point on Grassmann manifold as to an equivalent class of all the thin-tall orthogonal matrices in ℝ^d× p, anyone in which can be converted to the other by a p× p orthogonal matrix. There are two popular methods to measure the distance on Grassmann manifold. One is to define consistent metrics in tangent spaces to make Grassmann manifold a Riemannian manifold. (Geodesic Distance) <cit.> For the Grassmann manifold, the geodesic distance between two Grassmann points 𝐗_1 and 𝐗_2∈𝒢(p,n) is given bydist_g(𝐗_1,𝐗_2)=Θ_2where Θ is the vector of principal angles between 𝐗_1 and 𝐗_2. Another is to embed the Grassmann manifold into symmetric matrices space where the Euclidean metric is available. The later one is easier and more effective in practice, therefore, we use the Embedding distance in this paper. (Embedding Distance) <cit.> Given Grassmann points 𝐗_1 and 𝐗_2, Grassmann manifold can be embedded into symmetric matrices space as,Π : 𝒢(p,d) →Sym(d),Π(𝐗)=𝐗𝐗^T,and the corresponding distance on Grassmann manifold can be defined as,dist^2_g(𝐗_1,𝐗_2) = 1/2Π(𝐗_1)-Π(𝐗_2)^2_F.§ THE PROPOSED METHOD In this section, we propose an unsupervised DR method for Grassmann manifold that maps a high-dimensional Grassmann point𝐗_i ∈𝒢(p,D) to a point in arelative low-dimensional Grassmann manifold 𝒢(p,d),D > d. The mapping 𝒢(p,D) →𝒢(p,d) to be learned is defined as,𝐘_i = 𝐀^T 𝐗_i,where 𝐀∈ℝ^D× d. To make sure that 𝐘_i ∈ℝ^d× p is well-defined as the representative of the mapped Grassmann point on lower dimension manifold, we need impose some conditions. Obviously, the projected data 𝐘_i is not an orthogonal matrix, disqualified as a representative of a Grassmann point. To solve this problem, we perform QR decomposition on matrix 𝐘_i as follows <cit.>,𝐘_i = 𝐀^T 𝐗_i = 𝐐_i 𝐑_i⇒ 𝐐_i = 𝐀^T (𝐗_i 𝐑_i^-1) = 𝐀^T 𝐗_i,where 𝐐_i ∈ℝ^d× p is an orthogonal matrix, 𝐑_i ∈ℝ^p× p is an invertible upper-triangular matrix, and 𝐗_i = 𝐗_i𝐑_i^-1∈ℝ^D× p denotes the normalized 𝐗_i. As both 𝐘_i and 𝐐_i generate the same (columns) subspace, the orthogonal matrix 𝐐_i (or 𝐀^T 𝐗_i) can be used as the representative of the low-dimensional Grassmann point mapped from 𝐗_i. §.§ LPP for Grassmann manifold (GLPP) The term ( 𝐚^T𝐱_i - 𝐚^T𝐱_j)^2 in LPP objective function (<ref>) means the distance between the projected data 𝐚^T 𝐱_i and 𝐚^T 𝐱_j; therefore, it is natural for us to reformulate the classic LPP objective function on Grassmann manifold as follows,min_𝐀∑_ij^Ndist_g^2(𝐐_i,𝐐_j)· w_ij =∑_ij^Ndist_g^2(𝐀^T 𝐗_i, 𝐀^T 𝐗_j)· w_ijwhere w_ij reflects the similarity between original Grassmann points 𝐗_i and 𝐗_j, andthe distance dist_g(·) is chosen as the Embedding distance (<ref>). Hence dist_g^2(𝐀^T 𝐗_i, 𝐀^T 𝐗_j)= 𝐀^T𝐗_i𝐗_i^T𝐀 - 𝐀^T𝐗_j𝐗_j^T𝐀_F^2 = 𝐀^T 𝐆_ij𝐀_F^2,where 𝐆_ij = 𝐗_i𝐗_i^T - 𝐗_j𝐗_j^T, which is a symmetric matrix of size D × D. Thus, the objective function (<ref>) can be re-written as, termed as GLPP, min_𝐀∑^N_i,j=1𝐀^T 𝐆_ij𝐀_F^2· w_ij. The next issue is how to construct the adjacency graph 𝐖 from the original Grassmann points. We extend the Euclidean graph 𝐖 onto Grassmann manifold as follows,[Graph 𝐖 on Grassmann manifold] Given a set of Grassmann points {𝐗_1, ..., 𝐗_N}, we define the graph asw_ij = e^-dist_g^2(𝐗_i,𝐗_j)where w_ij denotes the similarity of Grassmann points 𝐗_i and 𝐗_j. In this definition,we may set dist_g(𝐗_i,𝐗_j) to any one valid Grassmann distance. We select the Embedding distance in our experiments. §.§ GLPP with Normalized ConstraintWithout any constraints on 𝐀, we may have a trivial solution from problem (<ref>). To introduce an appropriate constraint, we have to firstly define some necessary notations. We split the normalized Grassmann point 𝐗_i ∈ R^D× p and the projected matrix 𝐐_i ∈ R^d× p in (<ref>) into their components𝐐_i = [𝐪_i1, ..., 𝐪_ip] = [𝐀^T 𝐱_i1, ..., 𝐀^T 𝐱_ip] = 𝐀^T 𝐗_iwhere 𝐪_ij∈ℝ^d and 𝐱_ij∈ℝ^D with j=1, 2, ..., p. For each j (1≤ j≤ p), define matrix𝐐^j = [𝐪_1j, 𝐪_2j, ..., 𝐪_Nj] ∈ℝ^d× Nand𝐗^j = [𝐱_1j, 𝐱_2j, ..., 𝐱_Nj] ∈ℝ^D× N. That is, from all N normalized Grassmann points 𝐐_i (or all N normalized Grassmann points 𝐗_i), we pick their j-th column and stack them together.Then, it is easy to check that𝐐^j = 𝐀^T 𝐗^j.For this particularly organized matrix 𝐐^j, considering the constraint condition similar to formula (<ref>), tr(𝐐^j 𝐃𝐐^jT) =tr(𝐃𝐐^jT𝐐^j) = tr(𝐃𝐗^jT𝐀𝐀^T𝐗^j).Hence, one possible overall constraint can be defined as∑^p_j=1tr(𝐃𝐗^jT𝐀𝐀^T𝐗^j) = 1.Rather than using the notation 𝐗^j, we can further simplify it into a form by using original normalized Grassmann points 𝐗_i.A long algebraic manipulation can prove that∑^p_j=1tr(𝐃𝐗^jT𝐀𝐀^T𝐗^j) =tr(𝐀^T (∑^N_i=1d_ii𝐗_i𝐗^T_i )𝐀).Hence, we add the following constraint conditiontr(𝐀^T (∑^N_i=1d_ii𝐗_i𝐗^T_i )𝐀) = 1.Define 𝐇 = ∑^N_i=1d_ii𝐗_i𝐗^T_i, then the final constraint condition can be written as,tr(𝐀^T 𝐇𝐀) = 1.Combining the objective function (<ref>) and constraint condition (<ref>), we get the overall GLPP model,min_𝐀∑_i,j=1^N𝐀^T𝐆_ij𝐀_F^2· w_ijs.t.tr(𝐀^T𝐇𝐀) = 1 In next section, we propose a simplified way to solve problem (<ref>) which is quite different from most Riemannian manifold based optimization algorithms such as in the Riemannian Conjugate Gradient (RCG) toolbox. § OPTIMIZATIONIn this section, we provide an iteration solution to solve the optimization problems (<ref>). First we write thecost functionas followsWe use an iterative algorithm starting with an initial 𝐀^(0). Suppose n>m, then you may choose𝐀^(0) = [I_m× m; random elements ]. In each step, you use last step 𝐀^(k-1) to calculate all 𝐗_i, then solve for 𝐀^(k) fromf(𝐀) = ∑^N_i,j=1tr( 𝐀^T 𝐆_ij𝐀𝐀^T𝐆_ij𝐀)· w_ij. For ease, we redefine a new objective function f_k in the k-th iteration by using the last step 𝐀^(k-1) as the following way,f_k(𝐀)= ∑^N_i,j=1w_ij·tr( 𝐀^T 𝐆_ij𝐀^(k-1)𝐀^(k-1)T𝐆_ij𝐀) = tr(𝐀^T ∑^N_i,j=1 w_ij𝐆_ij𝐀^(k-1)𝐀^(k-1)T𝐆_ij𝐀).Denoting𝐉 = ∑^N_i,j=1w_ij𝐆_ij𝐀^(k-1)𝐀^(k-1)T𝐆_ij,where 𝐆_ij is calculated according to 𝐀^(k-1) through both 𝐗_i and 𝐗_j. Then the simplified version of problem (<ref>) becomesmin_𝐀tr(𝐀^T𝐉𝐀), s.t. tr(𝐀^T𝐇𝐀) = 1.The Lagrangian function of (<ref>) is given by tr(𝐀^T 𝐉𝐀) + λ (1 - tr(𝐀^T𝐇𝐀)),which can be derived to solve and translated to a generalized eigenvalue problem,𝐉𝐚 = λ𝐇𝐚.Obviously, matrices 𝐇 and 𝐉 are symmetrical and positive semi-definite. By performing eigenvalue decomposition on 𝐇^-1𝐉, the transform matrix 𝐀 = [𝐚_1, ..., 𝐚_d]∈ℝ^D× d is given by the minimum d eigenvalue solutions to the generalized eigenvalue problem. We summarize the whole procedures as Algorithm <ref>. § EXPERIMENTS In this section, we evaluate our proposed method GLPP on several classification and clustering tasks, respectively.§.§ Experimental settings §.§.§ Datasets Extended Yale B dataset[http://vision.ucsd.edu/content/yale-face-database] is captured from 38 subjects and each subject has 64 front face images in different light directions and illumination conditions. All images are resized into 20× 20 pixels. Highway Traffic dataset[http://www.svcl.ucsd.edu/projects/traffic/]contains 253 video sequences of highway traffic. These sequences are labeled with three levels: 44 clips at heavy level, 45 clips at medium level and 164 clips at light level. Each video sequence has 42 to 52 frames. The video sequences are converted to gray images and each image is normalized to size24 × 24. UCF sport dataset[http://crcv.ucf.edu/data/]includes a total of 150 sequences. The collection has a natural pool of actions with a wide range of scenes and viewpoints.There are 13 actions in this dataset.Each sequence has 22 to 144 frames. We convert these video clips into gray images and each image is resized into 30× 30.Figure <ref> shows some samples from these three datasets. §.§.§ Parameters and Evaluation The reduced dimension d is the most important parameter for DR algorithms. Like PCA, we define d by the cumulative energy of the eigenvectors, i.e. given the remaining energy rate r (0 < r < 1), d is defined as follows, d = min{ d^*∈ℕ: ∑_i=1^d^*σ_i ≥ r ∑_i=1^Dσ_i},where σ_i is the i-th largest eigenvalue of 𝐏𝐏^T, in which we stack all the Grassmann points 𝐏 = [𝐗_1 ; ... ; 𝐗_N]. However, for different datasets and applications, it is difficult to set a proper r uniformly.For simplification and fairness, here we set r=0.95 in all our experiments.The performance of different algorithms is evaluated by Accuracy (ACC) and we also add Normalized Mutual Information (NMI) <cit.> as an additional evaluation method for clustering algorithms. ACC reflects the percentage of correctly labeled samples, while NMI calculates the mutual dependence of the predicted clustering and the ground-truth partitions.For the sake of saving space, we list all experimental parameters in Table <ref>. All the algorithms are coded in Matlab 2014a and implemented on an Intel Core i7-4600M 2.9GHz CPU machine with 8G RAM. §.§ Video/Imageset Classification We firstly evaluate the performance of GLPP on classification task, and we use K Nearest Neighbor on Grassmann manifold algorithm (GKNN) and Dictionary Learning on Grassmann manifold (GDL) <cit.> as baselines,* GKNN: KNN classifierbased on the Embedding distance on high-dimensional Grassmann manifold;* GKNN-GLPP: KNN classifier on low-dimensional Grassmann manifold obtained by the proposed method;* GDL: GDL on high-dimensional Grassmann manifold;* GDL-GLPP: GDL on low-dimensional Grassmann manifold. Human facial recognition is one of the hottest topics in computer vision and pattern recognize area. Affected by various factors, i.e., expression, illumination conditions and light directions, algorithms based on individual faces do not achieve great experimental performance. Therefore, we test our proposed method GLPP on classic Extended Yale B dataset. We wish to inspect the proposed method on a practical application in complex environment; therefore we pick the Highway Traffic video dataset which contains various weather conditions, such as sunny, cloudy and rainy. UCF sport dataset which contains more variations on scenes and viewpoints can be used to examine the robustness of the proposed methods in noised scenarios. In our experiments, each video clip is regarded as an imageset. To be fair, we set K=5 for GKNN algorithm in all three experiments, and the number of training and testing samples are listed in the first two columns in Table <ref>, while other parameters can be found in Table <ref> (i.e., D, d and r). Experimental results for classification tasks are shown in Table <ref>. Obviously, the experimental accuracy of GLPP-based algorithms is at least 5 percent higher than the corresponding compared methods in most cases. We distribute it to LPP is less sensitive to outliers since LPP is derived by preserving local information. The experimental results also demonstrate that the low-dimensional Grassmann points generated by our proposed method reflect more discrimination than on the original Grassmann manifold.How to infer the intrinsic dimensionality from high-dimensional data still is a challenging problem. The intrinsic dimensionality relies heavily on practical applications and datasets.In our method, the reduced dimensionality d is determined by the remaining energy rate r. Figure <ref> shows that there exist different optimal r or d for different datasets. Extended Yale B dataset contains much rich information (e.g., face contour, texture, expression, illustration conditions and light directions) which has strong impacts on face recognition accuracy. When the reduced dimensionality d is less than the intrinsic dimensionality, the data in reduced dimensionality may lose some useful discriminative information. Therefore, the accuracy increases with larger reduced dimensionality in a certain range, e.g., the remaining energy rate r from 0.6 to 0.95 for Extended Yale B dataset. We find the optimal value is r=0.96. For the Traffic and UCF datasets, the simple or static backgrounds occupy main area of images. The foreground, e.g. car action and human action, is more valuable information for classification. When the optimal reduced dimensionality d is achieved at relative small r, here 0.7 and 0.8, the data in reduced dimensionality actually contain the ¡°right¡± information for Traffic and UCF data. In other words, when the reduced dimensionality d is getting larger (> the intrinsic dimensionality), the information from the background may lead to negative influence for the accuracy. §.§ Video/Imageset ClusteringTo further verify the performance of GLPP, we apply it on clustering tasks, and select K-means on Grassmann manifold (GKM) <cit.> as the compared mathod,* GKM : K-means based on the Embedding distance on high-dimensional Grassmann manifold.* GKM-GLPP: K-means on low-dimensional Grassmann manifold obtained by our proposed method. Table <ref> shows ACC and NMI values for all algorithms. Clearly, after drastically reducing dimensionality from D to d (see Table <ref>)by our proposed method, the new low-dimensional Grassmann manifold still maintain fairly higher accuracy than the original high-dimensional Grassmann manifold for all algorithms, which attests that our proposed DR scheme significantly boosts the performance of GKM.§ CONCLUSIONIn this paper, we extended the unsupervised LPP algorithm onto Grassmann manifold by learning a projection from the high-dimensional Grassmann manifold into the one in a relative low-dimensional with more discriminative capability, based on the strategy of embedding Grassamnn manifolds onto the space of symmetric matrices. The basic idea of LPP is to preserve the local structure of original data in the projected space.Our proposed model can besimplified as a basic eigenvalue problem for an easy solution. Compared with directly using the high-dimensional Grassmann manifold, the experimental results illustrate the effectiveness and superiority of the proposed GLPP on video/imageset classification and clustering tasks.§ ACKNOWLEDGEMENTSThe research project is supported by the Australian Research Council (ARC) through the grant DP140102270 and also partially supported by National Natural Science Foundation of China under Grant No. 61390510, 61672071, 61632006, 61370119, Beijing Natural Science Foundation No. 4172003, 4162010, 4152009, Beijing Municipal Science & Technology Commission No. Z171100000517003, Project of Beijing Municipal Education Commission No. KM201610005033, Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality No.IDHT20150504 and Beijing Transportation Industry Science and Technology Project. named[Absil et al.2008]AbsilMahonySepulchre2008 P. Absil, R. Mahony, and R. Sepulchre. 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"authors": [
"Boyue Wang",
"Yongli Hu",
"Junbin Gao",
"Yanfeng Sun",
"Haoran Chen",
"Baocai Yin"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170427072435",
"title": "Locality Preserving Projections for Grassmann manifold"
} |
Institut für Astronomie und Astrophysik, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany{daniel.thun@, wilhelm.kley@}uni-tuebingen.deUniversitäts-Sternwarte, Ludwig-Maximilians-Universität München, Scheinerstr. 1, D-81679 München, [email protected] around a central binary system play an important role in star and planet formation and in the evolution of galactic discs. These circumbinary discs are strongly disturbed by the time varying potential of the binary system and display a complex dynamical evolution that is not well understood. Our goal is to investigate the impact of disc and binary parameters on the dynamical aspects of the disc.We study the evolution of circumbinary discs under the gravitational influence of the binary using two-dimensional hydrodynamical simulations. To distinguish between physical and numerical effects we apply three hydrodynamical codes. First we analyse in detail numerical issues concerning the conditions at the boundaries and grid resolution.We then perform a series of simulations with different binary parameters (eccentricity, mass ratio) and disc parameters (viscosity, aspect ratio) starting from a reference model with Kepler-16 parameters.Concerning the numerical aspects we find that the length of the inner grid radius and the binary semi-major axis must be comparable, with free outflow conditions applied such that mass can flow onto the central binary. A closed inner boundary leads to unstable evolutions. We find that the inner disc turns eccentric andprecesses for all investigated physical parameters. The precession rate is slow with periods (T_prec) starting at around 500 binary orbits (T_bin) for high viscosity and a high aspect ratio H/R where the inner hole is smaller and more circular. Reducing α and H/R increases the gap size and T_prec reaches 2500 T_bin.For varying binary mass ratios q_bin the gap size remains constant, whereas T_prec decreases with increasing q_bin. For varying binary eccentricities e_bin we find two separate branches in the gap size and eccentricity diagram. The bifurcation occurs at arounde_crit≈ 0.18 where the gap is smallest with the shortest T_prec. For e_bin lower and higher than e_crit, the gap size and T_prec increase. Circular binaries create the most eccentric discs.Circumbinary discs: Numerical and physical behaviour Daniel Thun 1 Wilhelm Kley 1 Giovanni Picogna 2====================================================§ INTRODUCTIONCircumbinary discs are accretion discs that orbit a binary system that consists for example of a binary star or a binary black hole system.A very prominent example of a circumbinary disc orbiting two stars is the system GG Tau.Due to the large star separation, the whole system (stars and disc) can be directly imaged by interferometric methods <cit.>.Later data yielded constraints on the size of the dust in the system and have indicated that the central binary may consist of multiple stellar systems (see <cit.>, <cit.> and references therein). More recent observations have pointed to possible planet formation <cit.> and streamers from the disc onto the stars <cit.>.A summary of the properties of the GG Tau system is given in <cit.>.An additional important clue for the existence and importance of circumbinary discs is given by the observed circumbinary planets.These are planetary systems where the planets do not orbit one star, but instead orbit a binary star system. These systems have been detected in recent years by the Kepler Space Mission and have inspired an intense research activity.The first such circumbinary planet, Kepler-16b, orbits its host system consisting of a K-type main-sequence star and an M-type red dwarf, in what is known as a P-type orbit with a semi-major axis of 0.7048au and a period of 228.7 days <cit.>. Presently there are ten Kepler circumbinary planets known and a summary of the properties of the first five systems discovered is given in <cit.> [Additional known main-sequence binaries with circumbinary planets besides Kepler-16 are Kepler-34 and -35 <cit.>, Kepler-38 <cit.>, Kepler-47 <cit.>, Kepler-64 <cit.>, Kepler-413 <cit.>, Kepler-453 <cit.>, and Kepler-1647 <cit.>]. The observations have shown that in these systems the stars are mutually eclipsing each other, as well as the planets.The simultaneous eclipses of several objects is a clear indication of the flatness of the system because the orbital plane of the binary star has to coincide nearly exactly with the orbital plane of the planet. In this respect these systems are flatter than our own solar system. Because planets form in discs, the original protoplanetary disc in these systems had to orbit around both stars, hence it must have been a circumbinary disc that was also coplanar with respect to the orbital plane of the central binary.In addition to their occurrence around younger stars, circumbinary discs are believed to orbit supermassive black hole binaries in the centres of galaxies <cit.>, where they may play an important role in the evolution of the host galaxies <cit.>.In the early phase of the Universe there were many more close encounters between the young galaxies that alread had black holes in their centres.The gravitational tidal forces between them often resulted in new merged objects that consisted of a central black hole binary surrounded by a large circumbinary disc <cit.>.The dynamical behaviour of such a system is very similar to the one described above, where we have two stars instead of black holes.To understand the dynamics of discs around young binary stars such as GG Tau or the planet formation process in circumbinary discs or the dynamical evolution of a central black hole binary in a galaxy, it is important to understand in detail the behaviour of a disc orbiting a central binary. The strong gravitational perturbation by the binary generates spiral waves in the disc; these waves transport energy and angular momentum from the binary to the disc <cit.> and subseqently alter the evolution of the binary <cit.>. The most important impact of the angular momentum transfer to the disc is the formation of an inner cavity in the disc with a very low density whose size depends on disc parameters (such as viscosity or scale height) and on the binary properties (mass ratio, eccentricity) <cit.>. The numerical simulation of these circumbinary discs is not trivial and over the years several attempts have been made. The first simulations of circumbinary discs used the smoothed particle hydrodynamics (SPH) method, a Lagrangian method where the fluid is modelled as individual particles <cit.>.On the one hand, this method has the advantage that the whole system can be included in the computational domain and hence the accretion onto the single binary components can be studied. On the other hand, because of the finite mass of the SPH-particles, the maximum density contrast that can be resolved is limited.Later <cit.> simulated circumbinary discs with the help of a finite difference method on a polar grid. The grid allowed for a much larger range in mass resolution than the SPH codes used to date. A polar grid is well suited for the outer parts of a disc, but suffers from the fact that there will always be an inner hole in the computational domain since the minimum radius R_min cannot be made arbitrarily small. Decreasing the minimum radius will also strongly reduce the required numerical time step, rendering long simulations unfeasible. But since circumbinary disc simulations have to be simulated for tens of thousands of binary orbits to reach a quasi-steady state, the minimum radius is often chosen such that the motion of the central stars is not covered by the computational domain, and the mass transfer from the disc to the binary cannot be studied.<cit.> used a special dual-grid technique to use the best of the two worlds, the high resolution of the grid codes and the coverage of the whole domain of the particle codes. For the outer disc they used a polar grid and they overlaid the central hole with a Cartesian grid. In this way it was possible to study the complex interaction between the binary and the disc in greater detail. All these simulations indicated that despite the formation of an inner gap, material can enter the inner regions around the stars crossing the gap through stream like features to eventually become accreted onto the stars and influence their evolution <cit.>.Another interesting feature of circumbinary discs concerns their eccentricity. As shown in hydrodynamical simulations for planetary mass companions, discs develop a global eccentric mode even for circular binaries <cit.>. The eccentricity is confined to the inner disc region and shows a very slow precession <cit.>. These results were confirmed for equal mass binaries on circular orbits <cit.>.The disc eccentricity is excited by the 1:3 outer eccentric Lindblad resonance <cit.>, which is operative for a sufficiently cleared out gap.For circular binaries a transition, in the disc structure from more circular to eccentric is thought to occur for a mass ratio of secondary to primary above 1/25 <cit.>. However, this appears to contradict the mentioned results by <cit.> and <cit.> who show that this transition already occurs for a secondary in the planet mass regime with details depending on disc parameters such as pressure and viscosity.Driven by the observation of planets orbiting eccentric binary stars and the possibility of studying eccentric binary black holes, a large number of numerical simulations have been performed over the last few years dealing with discs around eccentric central binaries <cit.>. The recent simulations of circumbinary discs by <cit.>, <cit.> and <cit.> used grid-based numerical methods on a polar grid with an inner hole. This raises questions about the location and imposed boundary conditions at the innermost grid radius R_min.In particular, the value ofR_min must be small enough to capture the development of the disc eccentricity through gravitational interaction between the binary and the disc, especially through the 1:3 Lindblad resonance <cit.>.Therefore R_min has to be chosen in a way that all important resonances lie inside the computational domain.In addition to the position of the inner boundary, there is no common agreement on which numerical boundary condition better describes the disc. In simulations concerning circumbinary planets mainly two types of boundary conditions are used: closed inner boundaries <cit.> that do not allow for mass flow onto the central binary, and outflow boundaries <cit.> that do allow for accretion onto the binary. While <cit.> do not find a significant difference between these two cases, <cit.> see a clear impact on the surface density profile, and they were also able to construct discs with (on average) constant mass flow through the disc, which is not possible for closed boundaries.Driven by these discussions in the literature we decided to perform dedicated numerical studies to test the impact of the location of R_min, the chosen boundary condition, and other numerical aspects in more detail.During our work we became aware of other simulations that were tackling similar problems. In October 2016 alone three publications appeared on astro-ph that described numerical simulations of circumbinary discs <cit.>. While the first paper considered SPH-simulations, in the latter two papers grid-codes were used and in both the boundary condition at R_min is discussed. In our paper we start out with numerical considerations and investigate the necessary conditions at the inner boundary in detail, and we discuss aspects of the other recent results in the presentation of our findings below. Additionally, we present a detailed parameter study of the dynamical behaviour of circumbinary discs as a function of binary and disc properties.We organised this paper in the following way. In Sect. <ref> we describe the numerical and physical setup of our circumbinary simulations. In Sect. <ref> we briefly describe the numerical methods of the different codes we used for our simulations. In Sect. <ref> we examine the disc structure by varying the inner boundary condition and its location. The influence of different binary parameters are examined in Sect. <ref>. Different disc parameters and their influence are studied in Sect. <ref>. Finally we summarise and discuss our results in Sect. <ref>.§ MODEL SETUPTo study the evolution of circumbinary discs we perform locally isothermal hydrodynamical simulations. As a reference system we consider a binary star having the properties of Kepler-16, whose dynamical parameters are presented in Table <ref>.This system has a typical mass ratio and a moderate eccentricity of the orbit, and it has been studied frequently in the literature. In our parameter study we start from this reference system and vary the different aspects systematically.§.§ Physics and equationsInspired by the flatness of the observed circumbinary planetary systems, for example in the Kepler-16 system the motion takes place in single plane to within 0.5^∘ <cit.>, we make the following two assumptions: * The vertical thickness H of the disc is small compared to the distance from the centre R;* There is no vertical motion.It is therefore acceptable to reduce the problem to two dimensions by vertically averaging the hydrodynamical equations. In this case it is meaningful to work in cylindrical coordinates (R, φ, z)[With r⃗ we denote the three-dimensional positional vector r⃗ = Rê⃗_⃗R⃗ + zê⃗_⃗z⃗ and with R⃗ the two-dimensional positional vector in the R-φ-plane R⃗ = R ê⃗_⃗R⃗] and the averaging process is done in the z-direction. Furthermore, we choose the centre of mass of the binary as the origin of our coordinate system with the vertical axis aligned with the rotation axis of the binary. The averaged hydrodynamical equations are then given bytΣ + ·(Σu⃗) = 0,t(Σu⃗) + ·(Σu⃗⊗u⃗ - Π⃗) = - P -ΣΦ,where Σ = ∫_-∞^∞ϱz is the surface density, P = ∫_-∞^∞ p z the vertically integrated pressure, and u⃗ = (u_R, u_φ)^T the two-dimensional velocity vector. We close this system of equations with a locally isothermal equation of stateP = c_s^2(R) Σ,with the local sound speed c_s(R).The gravitational potential Φ of both stars is given byΦ = - ∑_k=1^2 G M_k/[(R⃗ - R⃗_k)^2 + (ε H )^2 ]^1/2.Here M_k denotes the mass of the k-th star, G is the gravitational constant, and R⃗ - R⃗_k a vector from a point in the disc to the primary or secondary star. From a numerical point of view, the smoothing factor ε H is not necessary if the orbit of the binary is not inside the computational domain. We use this smoothing factor to account for the vertically extended three-dimensional disc in our two-dimensional case <cit.>. For all simulations we use a value of ε = 0.6.In our simulations with no bulk viscosity the viscous stress tensor Π⃗ is given in tensor notation byΠ⃗_ij = 2η[ 1/2( u_j;i + u_i;j)- 1/3δ_ij (·u⃗)],where η = Σν is the vertically integrated dynamical viscosity coefficient. To model the viscosity in the disc we use the α-disc model by <cit.>, where the kinematic viscosity is given by ν = α c_s H. The parameter α is less than one, and for our reference model we use α = 0.01.To calculate the aspect ratio h = HR we assume a vertical hydrostatic equilibrium1/ϱpz = - Φz,with the density ϱ and pressure p. To solve this equation we use the full three-dimensional potential of the binary Φ = -∑_k GM_k/|r⃗ - r⃗_k| and the isothermal equation of state p = c_s^2(R)ϱ (we also assume an isothermal disc in the z-direction). Integration over z yieldsϱ = ϱ_0 exp{ -1/2z^2/H^2},with the disc scale height <cit.>H = [∑_k 1/c_s^2GM_k/|R⃗ - R⃗_k|^3]^-1/2and the midplane densityϱ_0 = Σ/√(2π) H,which can be calculated by using the definition of the surface density.If we concentrate the mass of the binary M_bin = M_A + M_B in its centre of mass (R⃗_k → 0), this reduces toH = c_s/u_K R,with the Keplerian velocity u_K = √(GM_bin/R). Test calculations with the simpler disc height (<ref>) showed no significant difference compared to calculations with the more sophisticated disc height (<ref>). Therefore, we use the simpler disc height in all our calculation to save some computation costs.For our locally isothermal simulations we use a temperature profile of T ∝ R^-1, which corresponds to a disc with constant aspect ratio. The local sound speed is then given by c_s(R) = h u_K∝ R^-12. If not stated otherwise we use a disc aspect ratio of h = 0.05. §.§ Initial disc parametersFor the initial disc setup we follow <cit.>. The initial surface density in all our models is given byΣ(t=0) = f_gapΣ_ref R^-α_Σ,with the reference surface density Σ_ref = 10^-4 M_ au^-2.The initial slope is α_Σ = 1.5 and the gap function, which models the expected cavity created by the binary-disc interaction, is given byf_gap = [1 + exp(- R-R_gap/Δ R) ]^-1<cit.>, with the transition width Δ R = 0.1 R_gap and the estimated size of the gap R_gap = 2.5 a_bin <cit.>.The initial radial velocity is set to zero u_R(t=0) = 0, and for the initial azimuthal velocity we choose the local Keplerian velocity u_φ(t=0) = u_K. §.§ Initial binary parametersFor models run with Pluto or Fargo we start the binary atperiastron at time t=0 (upper signs in equation (<ref>) and (<ref>)). Models carried out with Rh2d start the binary atapastron (lower signs in equation (<ref>) and (<ref>)).R⃗_A= [ K_1 a_bin(1∓ e_bin); 0 ],v⃗_A= [0; K_1 2π/T_bin a_bin√(1 ± e_bin/1 ∓ e_bin) ],R⃗_B= [ -K_2 a_bin(1∓ e_bin);0 ], v⃗_B= [0; -K_22π/T_bina_bin√(1 ± e_bin/1∓ e_bin) ],with K_1 = M_B/M_bin, K_2 = M_A/M_bin, and the period of the binary T_bin=2π[a^3_bin/(G M_bin)]^1/2. §.§ NumericsIn our simulations we use a logarithmically increasing grid in the R-direction and a uniform grid in the φ-direction.The physical and numerical parameter for our reference system used in our extensive parameter studies in Sect. <ref> and Sect. <ref> are quoted in Table <ref> below. In the radial direction the computational domain ranges from R_min = 0.25au to R_max = 15.4au and in the azimuthal direction from 0 to 2π, with a resolution of 762× 582 grid cells.For our numerical studies we also use also different configurations as explained below.Because of the development of an inner cavity where the surface density drops significantly, we use a density floor Σ_floor = 10^-9 (in code units) to avoid numerical difficulties with too low densities. Testsimulations using lower floor densities did not show any differences.At the outer radial boundary we implement a closed boundary condition where the azimuthal velocity is set to the local Keplerian velocity. At the inner radial boundary the standard condition is the zero-gradient outflow condition as in <cit.> or <cit.>, but we also test different possibilities such as closed boundaries or the viscous outflow condition <cit.>. The open boundary is implemented in such a way that gas can leave the computational domain but cannot reenter it. This is done by using zero-gradient boundary conditions (∂/∂ R = 0) for Σ and negative u_R. For positive u_R we use a reflecting boundary to prevent mass in-flow. Since there is no well-defined Keplerian velocity at the inner boundary, due to the strong binary-disc interaction, we also use a zero-gradient boundary condition for the angular velocity Ω_φ = u_φ/R. By using the zero-gradient condition for the angular velocity instead of the azimuthal velocity we ensure a zero-torque boundary.In the φ-direction we use periodic boundary conditions.In all our simulations we use dimensionless units. The unit of length is R_0 = 1au, the unit of mass is the sum of the primary and secondary mass M_0 = M_A + M_B and the unit of time is t_0 = √(R_0^3/(G M_0)) so that the gravitational constant G is equal to one. The unit density is then given by Σ_0 = M_0 / R_0^2. §.§ Monitored parametersSince our goal is to study the evolution of the disc under the influence of the binary we calculated the disc eccentricity e_disc and the argument of the disc periastron ϖ_disc ten times per binary orbit. To calculate these quantities we treat each cell as a particle with the cells mass and velocity on an orbit around the centre of mass of the binary. Thus, the eccentricity vector of a cell is given bye⃗_cell = u⃗×j⃗/G M_bin - R⃗/|R⃗|with the specific angular momentum j⃗ = R⃗×u⃗ of that cell. We have a flat system, thus the angular momentum vector only has a z-component. The eccentricity e_cell and longitude of periastron ϖ_cell of the cell's orbit are therefore given bye_cell = |e⃗_cell|,ϖ_cell = atan2(e_y, e_x).The global disc values are then calculated through a mass-weighted average of each cell's eccentricity and longitude of periastron <cit.>e_disc = [∫_R_1^R_2∫_0^2πΣ e_cell R φR] / [∫_R_1^R_2∫_0^2πΣ R φR],ϖ_disc = [∫_R_1^R_2∫_0^2πΣϖ_cell R φR] / [∫_R_1^R_2∫_0^2πΣ R φR].The integrals are simply evaluated by summing over all grid cells. The lower bound is always R_1 = R_min. For the disc eccentricity we integrate over the whole disc (R_2 = R_max) if not stated otherwise, whereas for the disc's longitude of periastron it is suitable to integrate only over the inner disc (R_2 = 1.0au) to obtain a well-defined value of ϖ_disc since animations clearly show a precession of the inner disc <cit.>. The radial eccentricity distribution of the disc is given bye_ring(R) = [∫_R^R+R∫_0^2πΣ e_cell R' φR'] / [∫_R^R+R∫_0^2πΣ R' φR']. § HYDRODYNAMIC CODESSince the system under analysis is, in particular near the central binary, very dynamical we decided to compare the results from three different hydrodynamic codes to make sure that the observed features are physical and not numerical artefacts. We use codes with very different numerical approaches. Pluto solves the hydrodynamical equations in conservation form with a Godunov-type shock-capturing scheme, whereas Rh2d and Fargo are second-order upwind methods on a staggered mesh. In the following sections we describe each code and its features briefly. §.§ PlutoWe use an in-house developed GPU version of the Pluto 4.2 code <cit.>. Pluto solves the hydrodynamic equations using the finite-volume method which evolves volume averages in time. To evolve the solution by one time step, three substeps are required. First, the cell averages are interpolated to the cell interfaces, and then in the second step a Riemann problem is solved at each interface. In the last step the averages are evolved in time using the interface fluxes. For all three substeps Pluto offers many different numerical options. We found that the circumbinary disc model is very sensitive to the combination of these options. Third-order interpolation and time evolution methods lead very quickly to a negative density from which the code cannot recover, even though we set a density floor.We therefore use a second-order reconstruction of states and a second-order Runge-Kutta scheme for the time evolution. Another important parameter is the limiter, which is used during the reconstruction step to avoid strong oscillations. For the most diffusive limiter, the minmod limiter, no convergence is reached for higher grid resolutions. For the least diffusive limiter, the mc limiter, the code again produces negative densities and aborts the calculation. Thus, we use the van Leer limiter which, in kind of diffusion, lies between the minmod and the mc limiter.For the binary position we solve Kepler's equation using the Newton–Raphson method at each Runge-Kutta substep. §.§ Rh2dThe Rh2d code is a two-dimensional radiation hydrodynamics code originally designed to study boundary layers in accretion discs <cit.>, but later extended to perform flat disc simulations with embedded objects <cit.>.It is based on the second-order upwind method described in <cit.> and <cit.>. It uses a staggered grid with second-order spatial derivatives and through operator-splitting the time integration is semi-second order.Viscosity can be treated explicitly or implicitly, artificial viscosity can be applied, and the Fargo algorithm has also been added.The motion of the binary stars is integrated using a fourth-order Runge-Kutta algorithm. §.§ FargoWe adopted the adsg version of Fargo <cit.> updated by <cit.>.This code uses a staggered mesh finite difference method to solve the hydrodynamic equations. Conceptually, the Fargo code uses the same methods as Rh2d or the Zeus code, but employs the special Fargo algorithm that avoids the time step limitations that are due to the rotating shear flow <cit.>. In our situation the application of the Fargo algorithm is not always beneficial because of the larger deviations from pure Keplerian flow near the binary. The position of the binary stars is calculated by a fifth-order Runge-Kutta algorithm.§ NUMERICAL CONSIDERATIONSBefore describing our results on the disc dynamics, we presenttwo important numerical issues that can have a dramatic influence on the outcome of the simulations: the inner boundary condition (open or closed) and the location of the inner radius of the disc. We show that unfortunate choices can lead to incorrect results.In this section we use a numerical setup different from that in Table <ref>. Specifically, the base model has a radial extent from R_min = 0.25au to R_max = 4.0au, which is covered with 448×512 gridcells for simulations with Rh2d and FARGO, and 512×580 with Pluto (see Appendix <ref> for an explanation of why we use different resolutions for different codes). For simulations with varying inner radii, the number of gridcells in the radial direction is adjusted to always give the same resolution in the overlapping domain. §.§ Inner boundary conditionAll simulations using a polar-coordinate grid expericence the same problem: there is a hole in the computational domain because R_min cannot be zero.Usually this hole exceeds the binary orbit. Therefore, the area where gas flows from the disc onto the binary and where circumstellar discs around the binary components form is not part of the simulation. This complex gas flow around the binary has been shown by e.g. <cit.> with a special dual-grid technique that covers thewhole inner cavity. Their code is no longer in use andmodern efficient codes (Fargo and Pluto) that run in parallel do not have the option of a dual-grid.To reproduce nevertheless some results of <cit.>, we carried out a simulation on a polar grid with an inner radius of R_min = 0.02au so that both orbits of the primary and secondary lie well inside the computational domain.A snapshot of this simulation is plotted in Fig. <ref>. The logarithm of the surface density is colour-coded, while the orbits of the primary and secondary stars are shown in white and green. The red cross marks the centre of mass of the binary, which lies outside the computational domain. Figure <ref> shows circumstellar discs around the binary components, as well as a complex gas flow through the gap onto the binary. Although including the binary orbit inside the computational would be desirable, it is at the moment not feasible for long-term simulations because of the strong time step restriction resulting from the small cell size at the inner boundary. This means that for simulations where the binary orbit is not included in the computational domain, the boundary condition at the inner radius has to allow for flow into the inner cavity, at least approximately. Two boundary conditions are usually used in circumbinary disc simulations: closed boundaries <cit.> and outflow boundaries <cit.>. Given the importance of the boundary condition at R_min, we carried out dedicated simulations for a model adapted from <cit.> and used an inner radius of R_min= 0.345au, but otherwise it was identical to our standard model. We applied closed boundaries and open ones in order to examine their influence on the disc structure. We also varied different numerical parameters (resolution, radial grid spacing, integrator) for this simulation series. The top panel of Fig. <ref> shows the azimuthally averaged density profiles for different grid resolutions and spacings after 16000 binary orbits. All these simulations were performed with Pluto and a closed inner boundary. The strong dependence of the density distribution and inner gap size on the numerical setup stands out.This dependence on numerics is very surprising and not at all expected since the physical setup was identical in all these simulations and therefore the density profiles should all be similar and converge upon increasing resolution.Not only does the gap size show this strong dependence, but also the radial eccentricity distribution (bottom panel of Fig. <ref>). To examine this dependence further, which could in principle be the result of a bug in our code, we reran the identical physical setup with a different code, Rh2d.The results of these Rh2d simulations show the same strong numerical dependence as the Pluto results. In Fig. <ref> the total disc eccentricity time evolution for the Rh2d simulations is plotted. Again, one would expect that different numerical parameters should produce approximately the same disc eccentricity, but our simulations show a radical change in the disc eccentricity if the numerical methods are slightly different.We note that using a viscous outflow condition where the radial velocity at R_min is fixed to - β 3 ν / (2 R) with β=5 (as suggested by <cit.>) resulted in the same dynamical behaviour as the closed boundary condition. The reason for this lies in the fact that the viscous speed is very low in comparison to the radial velocity induced by the perturbations of the central binary and hence there is little difference between a closed and a viscous boundary.In contrast to the closed inner boundary simulations, simulations with a zero-gradient outflow inner boundary reach a quasi-steady state.For the open boundaries Pluto simulations with different numerical parameters produce very similar surface density and radial disc eccentricity profiles (Fig. <ref>) for different resolutions and grid spacings.Only the values for the uniform radial grid with a resolution of 395×512 (purple line in Fig. <ref>) deviate. Here the resolution in the inner computational domain is not high enough, because a simulation with a higher resolution uniform grid (790×512, orange line in Fig. <ref>) produces approximately the same results as the simulations with a logarithmic radial grid spacing. We do not have a full explanation for this strong variability and non-convergence of the flow when using the closed inner boundary, but this result seems to imply that this problem is ill-posed.A closed inner boundary creates a closed cavity, which apparently implies in this case that the details of the flow depend sensitively on numerical diffusion as introduced by different spatial resolutions, time integrators, or codes.As a consequence, for circumbinary disc simulations a closed inner boundary is not recommended for two reasons. Firstly, it produces the described numerical instabilities and secondly, for physical reasons the inner boundary should be open because otherwise no material can flow into the inner region and be accreted by the stars.This is also indicated in Fig. <ref> which shows mass flow through the inner gap, which cannot be modelled with a closed inner boundary.§.§ Location of the inner radiusAfter determining that a zero gradient outflow condition is necessary, we use it now in all the following simulations and investigate in a second step the optimal location of R_min. It should be placed in such a way that it simultaneously insures reliable results and computational efficiency.For the excitation of the disc eccentricity through the binary-disc interaction, non-linear mode coupling, and the 3:1 Lindblad resonance are important <cit.>.Therefore, the location of the inner boundary is an important parameter and R_min should be chosen such that all major mean-motion resonances between the disc and the binary lie inside the computational domain. To investigate the influence of the inner boundary position, we set up simulations with an inner radius from R_min = 0.12au to R_min = 0.5au for the standard system parameter as noted in Table <ref>. Fig. <ref> shows the results of these simulations. Displayed are the azimuthally averaged surface density and the radial disc eccentricity distribution after 16000 binary orbits. In agreement with <cit.>, we find that the radial disc eccentricity does not depend much on the location of the inner radius and remains low in the outer disc (R > 2au). Only for a large inner radius of R_min = 0.5au do we observe almost no disc eccentricity growth (brown line in Fig. <ref>).In this case the 3:1 Lindblad resonance (R_3:1 =0.46au) lies outside the computational domain, confirming the conclusion from <cit.> about the importance of the 3:1 Lindblad resonance for the disc eccentricity growth.The variation of the radial disc eccentricity for inner radii smaller than 0.3au occurs because the precessing discs are in different phases after 16000 binary orbits. Contrary to the eccentricity distribution, the azimuthally averaged surface density shows a stronger dependence on the inner radius. As seen in the top panel of Fig. <ref>, the maximum of the surface density increases monotonically as the location of the inner boundary decreases because a smaller inner radius means also that the area where material can leave the computational domain decreases. For inner radii smaller than R_min≤0.25au the change in the maximum surface density becomes very low upon further reduction of R_min.This implies that for too large R_min there is too much mass leaving the domain and it has to chosen small enough to capture all dynamics.Although the maximum value of the surface density increases, the position of the maximum does not depend on the inner radius and remains at approximately R = 1.1au as long as the 3:1 Lindblad resonance lies inside the computational domain. Furthermore, we find that the slope of the surface density profile at the gap's edge does not depend on the location of the inner boundary.These results are in contrast to <cit.> who find a stronger dependence of the density profile on the location of the inner radius.However, their results may be a consequence of the viscous outflow condition used. As is discussed below, the disc becomes eccentric with a slow precession that depends on the binary eccentricity, e_bin.To explore how the location of the inner radius affects the disc dynamics, we ran additional simulations with varying R_min for different e_bin. We chose e_bin = 0.08 and e_bin = 0.32 in addition to the Kepler-16 value of e_bin = 0.16, which lies near the bifurcation point, e_crit. Fig. <ref> shows the precession period of the inner gap for varying inner radii and three different binary eccentricities, where the blue curve refers to the model shown above with e_bin = 0.16.For a higher binary eccentricity of e_bin = 0.32 (green curve in Fig. <ref> and on the upper branch of the bifurcation diagram) the disc dynamics seems to be captured well even for higher values of R_min. Only for values lower than R_min = 0.22au are deviations seen.The simulation with R_min = 0.20au was more unstable, probably because the secondary moved in and out of the computational domain on its orbit.On the lower branch of the bifurcation diagram the convergence with decreasing R_min is slower as indicated by the case e_bin = 0.08 (red curve in Fig. <ref>).Here, an inner radius of R_bin = a_bin or even slightly lower may be needed.One explanation for this behaviour is that on the lower branch (for low e_bin) the inner gap is smaller but nevertheless quite eccentric such that the disc is influenced more by the location of the inner boundary.The convergence of the results for smaller inner radii is also visible in the azimuthally averaged surface density and radial eccentricity profiles for e_bin= 0.08 and e_bin = 0.32 displayed in Fig. <ref>. In summary, from the results shown in Figs. <ref>, <ref>, and <ref> we can conclude that to model the circumbinary disc properly, the inner radius should correspond to R_min≈ a_bin.This should be used in combination with a zero-gradient outflow (hereafter just outflow) condition.Since a smaller inner boundary increases the number of radial cells and imposes a stricter condition on the time step, long-term simulations with a small inner radius are very expensive. As a compromise to make long-term simulations possible, we have chosen in all simulations below an inner radius of R_min = 0.25au. §.§ Location of the outer radiusWhile investigating the numerical conditions at the inner radius, we used an outer radius of R_max = 4.0au and assumed that this value is high enough, so that the outer boundary would not interfere with the dynamical behaviour of the inner disc. During our study of different binary and disc parameters, we found that in some cases this assumption is not true.Especially for binary eccentricities greater than 0.32, reflections from the outer boundary interfered with the complex inner disc structure. Therefore, we increased the outer radius of all the following simulations to R_max = 15.4au = 70 a_bin, a value used by <cit.>. In order to keep the same spatial resolution as before we also increased the resolution to 762×582 cells. This ensures that in the radial direction we still have 512 grid cells between 0.25au and 4.0au. To save computational time it is also possible to use a smaller outer radius (R_max≈ 40 a_bin) with a damping zone where the density and radial velocity are relaxed to their initial value. A detailed description of the implementation of such a damping zone can be found in <cit.>.Hence, for our subsequent simulations we use from now on the numerical setup quoted in Table <ref>, unless otherwise stated. To study the dependence on different physical parameter we start from the standard values of the Kepler-16 system in Table <ref> and vary individual parameter separately. § VARIATION OF BINARY PARAMETERSIn this section we study the influence of the central binary parameter specifically its orbital eccentricity and mass ratio on the disc structure. Throughout this section we use a disc aspect ratio of h=0.05 and a viscosity α = 0.01. For the inner radius and the inner boundary condition we use the results from the previous section and apply the outflow condition at R_min (see also Table <ref>). All the resultsin this section were obtained with Pluto, but comparison simulations using Rh2d show very similar behaviour (see Fig. <ref>). §.§ Dynamics of the inner cavityBefore discussing in detail the impact of varying e_bin and q_bin, we comment first on the general dynamical behaviour of the inner disc.Fig. <ref> shows snapshots of the inner disc after 16000 binary orbits for different binary eccentricities.As seen in several other studies (as mentioned in the introduction) we find that the inner disc becomes eccentric and shows a coherent slow precession.In the figure we overplot ellipses (white dashed lines) that are approximate fits to the central inner cavity, where the semi-major axis (a_gap) and eccentricity (e_gap) are indicated. To calculate these parameters we assumed first that the focus of the gap-ellipse is the centre of mass of the binary. We then calculated from our data the coordinates (R_Σmax, φ_Σmax) of the cell with the highest surface density value, which defines the direction to the gap's apocentre.We defined the apastron of the gap R_apa as the minimum radius along the line (R, φ_Σmax) which fulfils the conditionΣ(R, φ_Σmax) ≥ 0.1 ·Σ(R_Σmax, φ_Σmax).The periastron of the gap R_peri is defined analogously as the minimum radius along the line (R, φ_Σmax + π) in the opposite direction which fulfilsΣ(R, φ_Σmax+π) ≥ 0.1 ·Σ(R_Σmax, φ_Σmax).Using the apastron and periastron of the gap as defined above, the eccentricity and semi-major axis of the gap are given bya_gap = 0.5 (R_apa + R_peri), e_gap= R_apa / a_gap - 1.As shown in Fig. <ref> this purely geometrical construction matches the shape of the central cavity very well.Even though the overall disc behaviour shows a rather smooth slow precession, the dynamical action of the central binary is visible as spiral waves near the gap's edges, most clearly seen in the first and last panel. We prepared some online videos to visualise the dynamical behaviour of the inner disc.An alternative way to characterise the disc gap dynamically is by using the orbital elements (e_max, a_max) of the cell with the maximum surface density. These orbital elements can be calculated with equation (<ref>) and the vis-viva equationa = ( 2/R - u⃗^2/GM_bin)^-1 .The time evolution of these gap characteristics are plotted in Fig. <ref>.The gap's size and eccentricity show in phase oscillatory behaviour with a larger amplitude in the eccentricity variations. The period of the oscillation is identical to the precession period of the disc. Clearly, the extension of the gap always lies inside of the location of maximum density a_gap < a_max, but for the eccentricities the ordering is not so clear. The radial variations of the disc eccentricity as shown in Figs. <ref> and <ref> indicate that the inner regions of the disc typically have a higher eccentricity. In Fig. <ref> e_disc is lowest because it is weighted with the density which is very low in the inner disc regions. The eccentricity for the gap e_gap stems from a geometric fit to the very inner disc regions and is the highest. Inside the maximum density and even slightly beyond that radius the disc precesses coherently in the sense that the pericentres estimated at different radii are aligned.The data show further that when the disc eccentricity is highest the disc is fully aligned with the orbit of the secondary star, i.e.the pericentre of disc and binary lie in the same direction (see also Appendix <ref>).§.§ Binary eccentricityFor simulations in this section we fixed q_bin = 0.29 and varied e_bin from 0.0 to 0.64. The binary eccentricity strongly influences the size of the gap as well as the disc precession period of the inner disc. In Fig. <ref> the azimuthally averaged surface density profiles are shown for the various e_bin at 16000 binary orbits.In all cases there is a pronounced density maximum visible which is the strongest for the circular binary with e_bin=0(red curve). The position of the peak varies systematically with binary eccentricity.Increasing e_bin from zero to higher values the peak shifts inward. Not only does the gap size decrease, but the maximum surface density also drops with increasing e_bin until a minimum at e_bin = 0.18 is reached. Increasing e_bin further causes the density peak to move outward again, whereas the maximum surface density stays roughly constant for higher binary eccentricities.As mentioned above, the eccentric disc precesses slowly and in Fig. <ref> the longitude of the inner disc's pericentre is shown versus time over 16000 binary orbits. The disc displays a slow prograde precession with inferred period of several thousand binary orbits.For the measurement of the precession period we discarded the first 6000 binary orbits since plots of the longitude of periastron show that during this time span the precession period is not constant yet (see Fig. <ref>). The period is then calculated by averaging over at least two full periods. To be able to average over two periods, models with a very long T_prec (e.g. e_bin = 0.64) were simulated for more than 16000 binary orbits.Fig. <ref> summarises the results from simulations with varying binary eccentricities.The top panel of Fig. <ref> shows different measures for the gap size for varying binary eccentricities. In addition to the radial position where the azimuthally averaged surface density reaches its maximum (R_peak), we plot the positions where the density drops to 50 and 10 percent of its maximum value (R_0.5 and R_0.1). The value R_0.5 was used by <cit.> as a measure for the gap size, whereas <cit.> used R_0.1 and <cit.> R_peak. Starting from a non-eccentric binary the curves for R_peak and R_0.5 decrease with increasing binary eccentricity. For e_bin≈ 0.18 the gap size reaches a minimum and then increases again for higher binary eccentricities. In agreement with <cit.> we see an almost monotonic increase of R_0.1 for increasing binary eccentricities. The gap size, a_gap, correlates well with R_0.5 and isalways about 14% smaller. The middle panel of Fig. <ref> shows the eccentricity of the inner cavity, calculated with the method described in Sec. <ref>. The disc eccentricity also changes systematically with e_bin. For circular binaries it reaches e_gap≈ 0.44, and then it drops down to about 0.25 for the turning point e_bin = 0.18, and increases again reaching e_gap≈ 0.4 for the highest e_bin = 0.64. Hence, for nearly circular binaries e_gap can be higher than for more eccentric binaries.A similar variation of the disc's eccentricity and precession rate with binary eccentricity has been noticed by <cit.> who attribute this to the possible excitation of higher order resonances for higher disc eccentricities.The bottom panel of Fig. <ref> shows the precession period of the inner disc (R < 1au) for different binary eccentricities. The curve for the precession period shows a similar behaviour to the upper curvesfor the gap size. To investigate further the correlation of the gap size (here represented by R_0.5) and the precession period, we show in Fig. <ref> the two quantities plotted against each other.Two branches can be seen as indicated by the dashed curves.One starts at e_bin = 0.0 and goes to e_bin = 0.18 where the gap size and precession period decrease with increasing binary eccentricities, and the other branch starts at the minimum at e_bin = 0.18 where both gap properties increase again with increasing binary eccentricities. These two branches may indicate two different physical processes that are responsible for the creation of the eccentric inner gap and show the direct correlation between precession period and gap size.In our simulations all discs became eccentric and showed a prograde precession,and we did not find any indication for stand still discs that areeccentric but without any precession.In contrast, <cit.> find for their disc setup (h=0.1, α=0.1) that the disc does not precess for binary eccentricities between 0.2 and 0.4. We do not find this behaviour for our disc models, but note that we use a disc with a lower aspect ratio and a lower α-value. To investigate this further we set up a simulation with the same numerical parameters as <cit.> and used a physical setup where they did not observe a precession of the disc (q_bin =1.0, e_bin = 0.4, h=0.1, and α = 0.1). We carried out two simulations, one with an inner radius of R_min = (1 + e_bin) a_bin and one with an inner radius of R_min = 1.136 a_bin. The first radius was used by <cit.> and in this case we also observe no precession of the disc. The second radius corresponds to R_min = 0.25au, which is the radius we established in Sect. <ref> as the optimum location of the inner boundary. In the case with the smaller inner radius we observe a clear precession of the inner disc (Fig. <ref>) with a relatively short precession period T_prec as expected for high viscosity and high h (see below).This is another indication of the necessity of choosing R_min sufficiently small to capture all effects properly.§.§ Binary mass ratioIn this section we study the influence of the binary mass ratio. We carried out simulations with a mass ratio from q_bin = 0.1 to q_bin=1.0; all other parameters were set according toTable <ref>. Fig. <ref> shows the azimuthally averaged density profiles for varying binary mass ratios. The density profiles do not show such a strong variation as in simulations with varying binary eccentricities. Only the density profiles of the two lowestsimulated mass ratios q_bin=0.1 and q_bin=0.2 differsignificantly from the other profiles. For these models the position of the peak surface density is closer to the binary and the maximum surface density roughly 20 percent lower. The different gap size measures, introduced in the previous section, are shown in the top panel of Fig. <ref>.In general, the variations of all three gap-size indicators with mass ratio q_bin are relatively weak. All three gap size measurements remain nearly constant for mass ratios greater than 0.3. Since the density profiles do not show a significant variation this is not surprising. At the same time the size and eccentricity of the gap do not vary significantly and lie in the range a_gap≈ 4 a_bin and e_gap≈ 0.27. Overall, compared to simulations with varying binary eccentricity the binary mass ratio does not have such a strong influence on the inner disc structure for q_bin≳ 0.3. In contrast, the precession period of the inner disc shows a far stronger dependence on the binary mass ratio than the gap size (bottom panel of Fig. <ref>).Discs around binaries with a higher mass ratio have a lower precession period than discs around binaries with a low mass ratio. This dependence can be understood in terms of free particle orbits around a binary star that also display a reduction of the precession period with increasing q, as we discuss in more detail in Appendix <ref>. § VARIATION OF DISC PARAMETERSIn this section we explore the influence of discs parameters, namely the aspect ratio and the alpha viscosity, on the inner disc structure. We use the Kepler-16 values for the binary and our standard numerical setup (Table <ref>). While performing the simulations for this paper we observed that models with low pressure (low H/R) and high viscosity (high α) seem to be very challenging for the Riemann-Code Pluto.As a consequence we also present in this section results using the Upwind-Code Rh2d.As discussed in Appendix <ref>, the two codes produce results in very good agreement with each other for our standard model. For other parameters they do not necessarily produce results with such good agreement, but the simulation results from both codes always show the same trend.Therefore, we decided to mix simulation results from Pluto and Rh2d to cover a broader parameter range. §.§ Disc aspect ratioFirst we varied the disc aspect ratio h from 0.03 to 0.1. Our simulation results show a decreasing gap size for higher aspect ratios (Fig. <ref>). The gaps precession period is again directly correlated to the gap size, an observation we have already seen for different binary eccentricities. In general, a drop in T_prec with higher h is expected because an increase in pressure will tend to reduce the gap size, which will in turn lead to a faster precession. This trend is indeed observed in our simulations for higher h.For simulations with h < 0.05 we observed a decrease in gap size and precession period. The drop in T_prec with lower values of h may be partly due to the lack of numerical resolution and partly to the lack of pressure support, which may no longer allows for coherent disc precession. §.§ Alpha viscosityIn this section we discuss the influence of the magnitude of viscosity on the disc structure and therefore set up simulations with different viscosity coefficients ranging from α = 0.001 to α = 0.1. We then analysed the structure and behaviour of the inner cavity as before. The results are summarised in Fig. <ref>. A clear trend is visible for the gap size and for the precession period of the gap. For higher viscosities the gap size shrinks and the precession period decreases. Again, we see the direct correlation of gap size and precession rate, as already observed for aspect ratio and binary eccentricity variations. The decrease in the disc's gap size can be explained: for higher α values the viscous spreading of the disc increases. This viscous spreading counteracts the gravitational torques of the binary, which are responsible for the gap creation.§ SUMMARY AND DISCUSSIONUsing two-dimensional hydrodynamical simulations, we studied the structure of circumbinary discs for different numerical and physical parameters. Since these simulations are, from a numerical point of view, not trivial and often it is not easy to distinguish physical features and numerical artefacts, we checked our results with three different numerical codes.In the following we summarise the most important results from our simulations with different numerical and physical parameters.Concerning the numerical treatment the most crucial issue with simulations using grid codes in cylindrical coordinates is the treatment of the inner boundary condition.As there has been some discussion in the literature about this issue<cit.>, we decided to perform a careful study of the location of the inner boundary and the conditions on the hydrodynamical variables at R_min. Our results can be summarised as follows: 1.0em* Inner boundary condition. Since there was no agreement on which type of inner boundary condition should be used for circumbinary disc simulations, especially if closed or open boundary conditions should be used, we investigated the influence of the inner boundary condition through dedicated simulations with two different codes. We observed thatclosed boundaries can lead to numerical instabilities for all codesused in our comparison and no convergence to a unique solution could be found. The use of a viscous outflow condition where the velocity is set to the mean viscous disc speed produced very similar results to the closed boundary because the viscous speed is very low in comparison to the velocities induced by the dynamical action of the binary stars.Open inner boundaries, on the other hand, lead to numerically stable results and are also, from a physical point of view, more logical because material can flow onto the binary and be accreted by one of the stars. Hence, open boundaries with free outflow are the preferred conditions at R_min. * Location of inner boundary. The location of the inner boundary is also an important numerical parameter.Since an inner hole in the computational domain cannot be avoided in polar coordinates, the inner radius has to be sufficiently small to capture all relevant physical effects. In particular, all important mean motion resonances, which may be responsible for the development of the eccentric inner cavity, should lie inside the computational domain. Through a parameter study we were able to determine that the radius of the inner boundary R_min has to be of the order of the binary separation a_bin to capture all physical effects in a proper way.After having determined the best values for the numerical issues we performed,in the second part of the paper a careful study to determine the physical aspects that determine the dynamics of circumbinary discs. Starting from a reference model, where we chose the binary parameter of Kepler-16, we varied individual parameters of the binary and the disc and studied their impact on the disc dynamics. First of all, for all choices of our parametersthe models produce an eccentric inner disc that shows a coherent prograde precession. However, the size of the gap, its eccentricity, and its precession rate depend on the physical parameters of binary and disc. Our results can be summarised as follows: 1.0em* Binary parameters To study the influence of the binary star on the disc we systematically varied the eccentricity (e_bin) and the mass ratio (q_bin) of the binary.The parameter study showed that e_bin has a strong influence on the gap size and on the precession period of the gap. We found that two regimes exist where the disc behaves differently to an increasing binary eccentricity (see Fig. <ref>).The two branches bifurcate at a critical binary eccentricity, e_crit = 0.18 from each other. From e_bin = 0.0 to e_crit the gap size and precession period decrease, and from e_bin = 0.18 onward both gap parameters become larger again, as displayed in Fig. <ref>. The bifurcation of the two branches near e_crit = 0.18 strongly suggests that different physical mechanisms, responsible for the creation of the eccentric inner cavity, operate in the two regimes. For the lower branch theexcitation at the 3:1 outer Lindblad resonance may be responsible, which is supported by a simulation where the inner boundary was outside the 3:1 radius and no disc eccentricity was found. For the upper branch (high e_bin) non-linear effects may be present, as suggested by <cit.> who found similar behaviour. As second binary parameter we varied the mass ratio of the secondary to the primary star, q_bin, between [0.1,1.0] and studied its impact on the gap size and precession period.Overall the variation of R_0.5 with q_bin is weak.For low q_bin the gap size increases until it becomes nearly constant for q_bin≳ 0.3 with R_0.5≈ 4.6 a_bin, as shown in Fig. <ref>.The precession period decreases on average with increasing T_prec and for large q_bin the behaviour is equivalent to a single particle at a separation of 4.9au, as we discuss in more detail in Appendix <ref>. * Disc parameters We also varied different disc parameters, namely the pressure (through the aspect ratio h=H/R) and the viscosity (through α).The results are displayed in Figs. <ref> and Fig. <ref>.For changes in the aspect ratio no clear trend was visible; Both gap size and precession period first increase with increasing h until they reach a maximum at h=0.05; from there both gap properties decrease again with increasing aspect ratio. For high h the behaviour can be understood in term of the gap closing tendency of higher disc pressure. On the other hand, for very low pressure it may be more difficult to sustain a coherent precession of a large inner hole in the disc due to the reduced sound speed. Additionally, the damping action of the viscosity becomes more important for discs with lower sound speed. A clear monotonic trend was visible for the viscosity variation, where higher α leads to smaller gaps and shorter gap precession periods. This can be directly attributed to the gap closing tendency of viscosity. To allow for an alternative view of the impact of different binary and disc parameters on the dynamical structure of the disc we display in Fig. <ref> the eccentricity (top) and the precession period (bottom) of the gap plotted against the semi-major axis of the gap for all our simulations where different colours stand for different model series. Pluto simulations are represented by dots whereas squares stand for Rh2d simulations.The reference model is marked with a star. Interestingly the majority of the models lie on a main-sequence branch, which has the trend of increasing eccentricity and precession period with increasing gap width.The smaller branch that bifurcates near the reference model represents the low e_bin sequence, which has higher e_gap but smaller T_prec.By coincidence, our reference model with the Kepler-16 parameters lies near the bifurcation point.How the location of the bifurcation point depends of the disc and binary parameter needs to be explored in subsequent numerical studies. A first simulation series with q_bin = 0.6 suggest that this critical eccentricity does not depend on the mass ratio of the binary. As outlined in the Appendix, for parameters very near to the critical e_bin the models can have the tendency to switch between the two states during a single simulation. The physical cause of the bifurcation needs to be analysed in subsequent simulations.In addition, there are a more possibilities to further improve our model. First, the use of a non-isothermal equation should be considered to study the effects of viscous heating and radiative cooling. Especially for simulations, which also evolve planets inside the disc, <cit.> showed that radiative effects should been taken into account.Self-gravity effects have been studied by <cit.> and <cit.> and are important for galactic discs around a central binary black hole.The backreaction of the disc onto the binary could also be considered in more detail. Angular momentum exchange with the binary occurs through gravitational torques from the disc and direct accretion of angular momentum from the material that enters the central cavity.The first part always leads a decrease in the binary semi-major axis and an increase in eccentricity <cit.>, while <cit.> pointed out that angular momentum advection might even be dominant leading to a binary separation. Another interesting question is the evolution of planets inside the circumbinary disc and their influence on the disc structure. Since an in situ formation of the observed planets seems very unlikely, the migration and final parking position of planets is an interesting topic that requires further investigations.In view of our extensive parameter studies there is the indication that in several past studies <cit.> an inner radius was considered that was too large and that did not allow for full disc dynamics.Through fully three-dimensional simulations it will be possible to study the dynamical evolution of inclined discs. Daniel Thun was funded by grant KL 650/26 of the German Research Foundation (DFG), and Giovanni Picogna acknowledges DFG support grant KL 650/21 within the collaborative research program The first 10 Million Years of the Solar System.Most of the numerical simulations were performed on the bwForCLuster BinAC, supported by the state of Baden-Württemberg through bwHPC, and the German Research Foundation (DFG) through grant INST 39/963-1 FUGG. All plots in this paper were made with the Python library matplotlib <cit.>.aa§ CONVERGENCE STUDYThe strong gravitational influence of the binary onto the disc, and the complex dynamics of the inner disc presents a big challenge for numerical schemes of grid codes. We therefore study in this appendix the dependence of the disc structure on numerical parameters, such as the numerical scheme or the grid resolution. First, we simulated our reference system, using an open inner boundary and an inner radius of R_min = 0.25au, with the different codes described in Sec. <ref>.In Fig. <ref> the long-time evolution of the disc eccentricity (top) and of the longitude of periastron (bottom) are shown. All three codes produce comparable results, although we should note that to get this agreement simulations performed with Pluto need a slightly higher resolution. Pluto simulations with a resolution of 448×512 grid cells produce a slightly smaller precession period of the inner disc (red line in Fig. <ref>, see also Table <ref>).One possible explanation for this could be the size of the numerical stencil. Fargo and Rh2d have a very compact stencil because they use a staggered grid. In contrast, Pluto has a wider stencil because a collocated grid is used, where all variables are defined at the cell centres.In particular the viscosity routine of Pluto has a very wide stencil to achieve the correct centring of the viscous stress tensor components. This could explain why for Pluto a slightly higher resolution is needed. The π-shift of the longitude of periastron calculated with Rh2d compared to the values calculated with Pluto and Fargo (bottom Panel of Fig. <ref>) can be explained in the following way. Simulations performed with Pluto and Fargo started the binary at periastron with the pericentre of the secondary at φ =π (see red ellipses in Fig. <ref>), whereas Rh2d started the binary at apastron with the pericentre of the secondaries orbit at φ = 0.The data show that when the disc eccentricity is at its maximum the disc is aligned with the orbit of the secondary star.Since the pericentre of the secondary in Rh2d simulations is shifted by π we can also see this π-shift in the disc's longitude of periastron.Fig. <ref> shows the azimuthally averaged density profiles at t=16000 T_bin. Even after such a long time span all three codes agree very closely. All codes produce roughly the same density maximum and the same density slopes. To check the influence of the numerical resolution on the disc structure we run Pluto simulations with different resolutions, summarised in Table <ref>. The results of this resolution test are shown in Fig. <ref>.The disc eccentricity converges to the same value, whereas it seems that the disc precession rates increases with higher resolution.A closer look at the data shows that after around 6000 orbits the precession rate for the higher resolution converged. Table <ref> summarises the measured precession rate for different resolutions. Since there is no large variation between 512 × 580 and 600 × 1360 grid cells, we used a resolution of 512 × 580 grid cells for our Pluto simulations. As already discussed in Sect. <ref> in some cases (for example high binary eccentricities) an outer radius ofR_max = 4.0au can lead to wave reflections which interfere with the inner disc. Therefore we increased the outer boundary to R_max = 15.4au = 70 a_bin (and to ensure the same resolution between 0.25au and 4.0au we also increased the number of grid points to 762 × 582). Figure <ref> compares the effect of different outer radii for the case of varying binary eccentricities. For binary eccentricities greater than 0.32, a larger outer radius makes adifference. Also, around the bifurcation point, at e_bin = 0.18, a larger outer radius changes the gap size and precession rate of the disc. We also added results from Rh2d simulations with an outer radius of R_max = 18.18 a_bin.These Rh2d results follow the same trend as the Pluto simulations, although for most binary eccentricities we could not reproduce the very good agreement of the e_bin = 0.16 case. Around e_bin = 0.18, which is the critical binary eccentricity where the two branches bifurcate, simulations tend to switch between two states. This jump happens after roughly 3000 binary orbits and does not depend on physical parameters. Changes in numerical parameters, like the Courant-Friedrichs-Lewy condition or R_max, can trigger the jump. This can be seen in Fig. <ref> for e_bin = 0.16 where simulations with R_max = 18.18 a_bin (red curve) and R_max = 70.0 a_bin (blue curve) produce different values for the gap size and the precession period. An outer radius of R_max = 18.18 a_bin is in the case of e_bin = 0.16 not too small, since a simulation with R_max = 100 a_bin again produced the same values for the gap size and the precession period. For simulations with e_bin far away from e_crit, we did not observe these jumps between states. So far we have only observed these jumps in simulations carried out with Pluto.§ COMPARISON WITH MASSLESS TEST PARTICLE In our circumbinary disc evolution we found that the inner disc becomes eccentric with a constant precession rate. In this section we compare this to test particle trajectories around the binary and investigate how a massless particle with orbital elements similar to the disc gap (Fig. <ref>) would behave under the influence of the binary potential. In particular, we would like to know at what distance from the binary a test particle has to be positioned such that its precession period agrees approximately with that of the disc. Using secular perturbation theory for the coplanar motion of a massless test particle around an eccentric binary star <cit.> derived the following formula for the particle's precession period for low binary eccentricitiesT_prec = 4/3(q_bin + 1)^2/q_bin(a_p/a_bin)^7/2(1 +3/2 e_bin^2)^-1T_bin ,where a_p is the semi-major axis of the particle. The same relation was quoted by <cit.> based on results by <cit.>. A similar relation was given by <cit.> without the e_bin term.In order to verify the applicability of expression (<ref>) when varying of binary and planet parameters, we performed series of three-body simulations with very low planet mass (10^-6 M_⊙), where we varied individually q_bin, e_bin, and a_p. In addition we varied the planet eccentricity e_p. For the simulations we used the parameter of the Kepler-16 systems as a reference (see Table <ref>) and integrated the system for several 10000 binary orbits. Our results of these three-body simulations are shown in Fig. <ref>. We find that the precession rate scales with q_bin, a_p, and e_bin exactly as expected from relation (<ref>).The fourth panel of Fig. <ref> indicates that the precession period also depends on the particle eccentricity as T_prec∼ (1 - e_p^2)^2, where we plotted the average eccentricity of the orbit which is equivalent to the particle's free eccentricity. The agreement holds up to about e_p^2 ∼ 0.5 for the used a_p. Clearly, for higher values of e_p the particle's orbit will be unstable as this leads to close encounters with the binary. The value of e_p where this happens depends on the distance from the binary star. For lower a_p the range will be more limited. In summary, we find for the precession period of a particle around a binary starT_prec = 4/3(q_bin + 1)^2/q_bin(a_p/a_bin)^7/2(1 - e_p^2)^2/(1 + 3/2 e_bin^2)T_bin .This relation is plotted in Fig. <ref> as the solid black line.In addition to the scaling behaviour we find that the numerical results are about 3-5%lower than the theoretical estimate. The agreement of the numerical results with the theoretical prediction becomes better for larger a_p because eq. (<ref>) is derived from an approximation for large a_p/a_bin. The full relation (<ref>) can be inferred directly from eq. (11) in <cit.>.Now we compare the particle precession rate to our disc simulations.The best option for this comparison is to check the scaling of the precession rate for models where q_bin has been varied becausefor binary mass ratio greater than q_bin = 0.3 the gap size and eccentricity do not change much (purple points inFig. <ref>). This is supported by the small variation of the gap radii in Fig. <ref>. For higher mass ratios the gap size is approximately a_gap = 4.0 a_bin and e_gap = 0.27.As seen from eq. (<ref>), a test particle with these orbit elements would have a precession period that is far too short. However, if we assume that the test particle has a semi-major axis of a_p = 4.9 a_bin (roughly 20 percent more than a_gap), the precession period of the gap and the particle match very well for different q_bin (Fig. <ref>), at least for the higher mass ratios.In general, however, due to the strong sensitivity of the precession period with a_p,an exact agreement will be difficult to obtain. | http://arxiv.org/abs/1704.08130v2 | {
"authors": [
"Daniel Thun",
"Wilhelm Kley",
"Giovanni Picogna"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170426140717",
"title": "Circumbinary discs: Numerical and physical behaviour"
} |
Counter-propagating solitons in microresonators Qi-Fan Yang^∗, Xu Yi^∗, Ki Youl Yang, and Kerry Vahala^†T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA.^∗These authors contributed equally to this work.^†Corresponding author: [email protected] December 30, 2023 =============================================================================================================================================================================================================================================================================Solitons occur in many physical systems when a nonlinearity compensates wave dispersion. Their recent formation in microresonators opens a new research direction for nonlinear optical physics and provides a platform for miniaturization of spectroscopy and frequency metrology systems. These microresonator solitons orbit around a closed waveguide path and produce a repetitive output pulse stream at a rate set by the round-trip time. In this work counter-propagating solitons that simultaneously orbit in an opposing sense (clockwise/counter-clockwise) are studied. Despite sharing the same spatial mode family, their round-trip times can be precisely and independently controlled. Furthermore, a state is possible in which both the relative optical phase and relative repetition rates of the distinct soliton streams are locked. This state allows a single resonator to produce dual-soliton frequency-comb streams having different repetition rates, but with high relative coherence useful in both spectroscopy and laser ranging systems. The recent demonstration of optical solitons in microresonators has opened a new chapter in nonlinear optical phenomena <cit.>. These dissipative solitons <cit.> use the Kerr nonlinearity to balance wave dispersion and to compensate cavity loss <cit.>. The resulting dissipative Kerr solitons (DKSs) exhibit Raman-related phenomena <cit.>, optical Cherenkov radiation <cit.> and can form ordered arrays called soliton crystals <cit.>.Soliton mode locking also creates a new and very stable frequency microcomb with distinct advantages over earlier microcombs<cit.>. For example, internal broadening of these combs by dispersive-wave generation <cit.> enables offset frequency measurement for comb self referencing<cit.>. Also, the soliton repetition rate has an excellent phase noise stability <cit.> and its spectral envelope is stable and reproducible so that the resulting microcombs are suitable for dual comb spectroscopy <cit.>. All whispering-gallery microresonators feature clockwise (CW) and counter-clockwise (CCW) optical whispering gallery modes, and this degree of freedom has not been explored for generation of soliton pulse trains. In this work counter-propagating (CP) solitons are generated by counter-pumping on a single microcavity resonance (fig.1a). Because DKSs are phase coherent with their respective optical pump, the tuning of two counter-propagating pumps causes an offset in the optical frequency of the two soliton pulse streams. Also, on account of the Raman-induced soliton self-frequency shift (SSFS), the repetition rate of a soliton pulse stream depends on the detuning of the pump frequency relative to the resonant frequency being pumped <cit.>. As a result, the pulse rate of each CP soliton pulse stream can be independently controlled. Besides independent repetition rate control there are two soliton phase locking effects that are observed. In both of these locked states, the soliton streams are also optically phase locked even though the soliton frequencies reside on a distinctly different grid of optical frequencies. In the first, the CW and CCW solitons are observed to phase lock with identical repetition rates. In the second locking effect the CP solitons experience relative rate locking at different repetition rates.As a result, the microresonator produces two, soliton streams having different repetition rates but with high relative coherence. This form of locking is potentially useful in dual comb spectroscopy and in laser ranging systems <cit.> (LIDAR) where it would eliminate the need for independent and mutually-locked frequency combs.The counter-propagating solitons are typically several-hundred femtoseconds in duration and the microcavity round-trip time is 46 ps. To produce the solitons, a continuous-wave fiber laser is amplified and split using a directional coupler so as to pump CW and CCW modes of a microcavity resonance using a fiber taper coupler (see experimental setup in fig. 1b). Two acousto-optic modulators (AOM) are used to control the pump power and frequencies in each pumping direction. The residual transmitted pump power is filtered by a fiber Bragg grating filter (FBG).The CP solitons are stabilized indefinitely using the active capture technique <cit.>. It is found that application of this locking technique to only one of the soliton pulse streams automatically locks the other pulse stream. In figure 1c the measured optical spectra and the autocorrelation traces (insets) for typical CW and CCW soliton streams are shown. The system can be controllably triggered and locked with a single or a specified numberof solitons in each propagation direction. The microresonator, a high-Q silica wedge design <cit.> with 3 mm diameter, has anomalous dispersion at the pumping wavelength near 1.55 microns and is engineered to produce minimal avoided mode crossings over the optical band of the solitons <cit.>. A feature of the dissipative Kerr soliton when viewed as a mode-locked frequency comb, is that the pump provides one of the comb frequencies and is therefore coherent with the soliton. Non-degenerate counter-pumping therefore introduces a controlled frequency offset between CW and CCW solitons. Because the counter-propagating pumps are derived from a single laser source, the mutual optical coherence of CW and CCW pump comb teeth is excellent and determined primarily by the stability of the radio-frequency signals used to drive the AOMs shown in fig. 1b. A key parameter that controls the soliton properties is the cavity-pump detuning frequency δω_ cw, ccw = ω_0 - ω_ cw, ccw where ω_0 is the cavity resonant frequency which is pumped and ω_ cw, ccw are the CW and CCW pump frequencies. Soliton pulse width <cit.>, average power <cit.>, and self-Raman-shift<cit.> depend upon this detuning. In cases where the self-Raman shift is strong, the soliton repetition rate also depends upon the cavity-pump detuning <cit.> and the CW and CCW soliton repetition rates (f_cw and f_ccw) can be separately controlled by tuning of the respective pump frequencies.To measure the CW and CCW soliton repetition rates, their pulse streams are combined and simultaneously photodetected. The electrical spectrum of the photocurrent is shown in fig. 2a when the difference in pumping frequencies is set to Δν≡ (ω_ccw-ω_cw) / 2 π = 3.9 MHz and δω_cw∼ 20 MHz. A zoom-in of the spectrum in the upper panel of fig. 2b shows that two strong central spectral peaks differ by 60 kHz. These peaks are the fundamental repetition rates associated with the CW and CCW soliton streams. The weaker, non-central beats appearing in fig. 2a and the upper panel in fig. 2b are inter-soliton beat frequencies between comb teeth belonging to different soliton combs. These beat frequencies are equally separated by the difference in the repetition rates (60 kHz). As an aside, the maxima at the extreme wings of the spectrum are caused by the mode crossing distortion in the comb spectra seen in fig. 1c near 1542 nm. An interferogram showing the electrical time trace of the co-detected dual-soliton pulse streams is shown in fig. 2c. This time trace can be understood as a stroboscopic interference of the respective soliton pulses on the detector. The strobing occurs at the rate difference (Δ f ≡ (f_ccw-f_cw)) of the two soliton streams giving the repetitive signal a period of 16.5 μs. By varying the pump detuning, Δν, it is possible to observe tuning of the repetition rate difference, Δ f, as shown in fig. 2d. A theoretical fit discussed in Methods is provided in the figure. Near Δν = 0 locking of the repetition rates is observed over a range of Δν around 150 kHz. The associated electrical zoom-in spectrum under this locked condition is shown in the lower panel of fig. 2b. Importantly, nearly all of the weaker peaks that appear in the unlocked spectrum shown in fig. 2a disappear as a result of locking. This can be understood to result from the high relative temporal stability of the two pulse streams. In particular, under the locking condition, inter-soliton pulse mixing on the photo-detector, which is guaranteed under conditions of unequal repetition rates, now requires strict spatial-temporal alignment of the two pulse streams at the detector.Consistent with this physical picture, the interferogram trace is observed to show no periodic strobing behavior. This locking behavior is believed to occur when pump light from a given pumping direction is backscattered into the opposing direction where the two pump signals can mix by the Kerr-effect.This induces four-wave mixing sidebands on the soliton comb lines that subsequently induce locking. In addition to locking at identical repetition rates (degenerate locking), the soliton pulse streams are observed to lock when their repetition rates are different. Fig. 3a illustrates the principle of this locking mechanism. Therein, soliton spectra for CW and CCW directions are presented. A zoom-in of the higher frequency portion of the spectra is shown in which the respective soliton spectral lines are superimposed next to shaded areas representing the cavity resonances. The mode index μ = 0, which is by convention the optical pump, is also indicated. As required for DKS generation, this pump frequency and the other soliton comb teeth are red-detuned in frequency relative to their respective cavity resonances.At μ = 0, the two pump lines are separated by the pump frequency difference, Δν. Under conditions where these pump frequencies are well separated so that degenerate rate locking does not occur (see fig. 2d), the soliton having the more strongly red-detuned pump will feature a slightly lower repetition rate on account of the self-Raman-effect discussed above and in Methods. Accordingly, the CW and CCW comb lines will shift in frequency so as to become more closely spaced as μ decreases. For a certain negative value of μ the CW and CCW comb lines will achieve closest spectral separation. In the illustration, this occurs at comb tooth μ = r where CW and CCW comb lines have frequency separation δ = Δν + r Δ f. Backscattering within the resonator will couple power between these nearly resonant lines. This power coupling is shown in Methods to induce locking with a corresponding bandwidth.Because the original comb teeth at μ = 0 are derived from the same laser, the additional locking at μ=r causes the CW and CCW solitons to be mutually phase locked. Moreover, the difference in the soliton repetition rates must be an integer fraction (1/|r|) of pump frequency difference, Δ f = -Δν / r,This result shows that pulse rates have a relative stability completely determined by the radio frequency signal used to set the pump frequency offset. Accordingly, the beat signal between the CCW and CW solitons exhibits very high stability when the system is locked in this way. The above relation also shows that the locked CP solitons play the role of a frequency divider of the pump frequency difference into the pulse-rate difference frequency. The phase noise of the rate difference is therefore r^2 lower than the phase noise of the relative pump signal,S_Δ f=1/r^2S_Δν.where S_Δ f and S_Δν are the phase noise spectral density functions of the inter-soliton fundamental beat signal and the pump difference signal. Fig. 3b illustrates the effect of the locking condition on the electrical spectrum produced by photodetection of combined CCW and CW soliton streams.Under unlocked conditions, the electrical spectrum will feature two distinct spectra with spacing Δ f. However, under locked conditions the difference in the frequency of the comb teeth at μ = 0 (i.e., optical pumps) is an integer multiple of the difference in the repetition rates. As result, the two electrical spectra merge to form a single spectrum. Fig. 3c shows a typical measured RF spectrum in the locked state. It is obtained by Fourier transforming an interferogram recorded over 1s. A set of equidistant spectral lines is observed with a 50 dB signal-to-noise ratio (SNR) at the 1 Hz resolution bandwidth (RBW). In this measurement, Δν is set to be 1.5 MHz which is 60 times Δ f = 25 kHz.Meanwhile, the unlocked state shown in fig. 3d features relatively noisier spectral lines and lower SNR. This noise results from fluctuations of the absolute pump frequencies which induce fluctuations in the two Raman-shifted repetition rates <cit.>. The resulting noise is multiplied with each comb tooth index relative to the pump comb tooth.Fig. 3e plots the spacing between the RF comb lines. It shows collapse to a sub-Hz stability under the locked condition.In strong contrast to the unlocked case, the spectral line beatnotes in fig. 3c actually improve in stability with decreasing order relative to the pump line. This a consequence of the frequency division noted in eq. (1) and eq. (2). In particular, the lowest frequency inter-soliton beatnote features the minimum linewidth as shown in fig. 3f. Furthermore, to confirm the scaling of phase noise with the frequency division given in eq. (2), fig. 3g shows the phase noise spectral density versus the spectral beatnote number measured at two phase-noise offset frequencies (1 Hz and 10 Hz). The dependence follows the predicted quadratic form typical of a frequency divider. It is noted that theinferred linewidth for the lowest order beatnote is 40 μHz ( assuming that it is limited by white frequency noise).Counter-propagating solitons have been demonstrated in a high-Q optical microresonator. Both the repetition rates and the spectral location for the clockwise and counter-clockwise directions are independently tuned by tuning of the corresponding optical pumping frequencies. Two distinctly different locking phenomena have been observed while tuning the soliton repetition frequencies. In the first, the repetition rates lock to the same value. The pumping frequencies are different when this locking occurs so that the two soliton comb spectra are offset slightly in the optical frequency, but have identical comb line spacings. The interferogram of the two pulse trains has no baseband time dependence when this locking occurs. In the second form of locking, the pumps are typically tuned apart to larger difference frequencies and the solitons are observed to lock at different repetition rates with a difference that divides into the pump-frequency difference. The origin of this locking is associated with optical locking of two comb teeth, one from each soliton. Since the two pumps are derived from the same laser, this additional comb tooth locking effectively results in the two comb spectra being locked at two different positions in their spectra. The resulting high level of mutual soliton coherence is observable in the base-band inter-soliton beat spectra which features very narrow spectral lines spaced by the difference in the locked soliton repetition rates. In effect, this second form of locking creates two frequency combs in the same device with distinct repetition rates and optical frequencies, but that are optically locked. It is potentially useful in dual comb spectroscopy and dual comb LIDAR applications where it would obviate the need for two separate frequency combs and the associated inter-comb locking hardware. Finally, it is noted that while single clockwise and counter-clock-wise solitons have been generated, it is also possible to create states containing multiple solitons.Methods Repetition rate control of CP solitons. The Raman SSFS, Ω_R, is dependent on the pump-cavity detuning, δω, by <cit.>Ω_R=-32D_1^2 τ_R/15κ D_2δω^2where τ_R is the Raman shock time, κ is the cavity decay rate, D_1 (D_2) is the free-spectral-range (second-order dispersion) at mode μ = 0 (the pumping mode). The soliton repetition rate, f, is coupled to the SSFS as2π f=D_1+Ω_R D_2/D_1Therefore the interferogram between the counter-propagating solitons with cavity-pump detuning δω_cw and δω_ccw has a repetition rate differencef_ccw-f_cw=-16 D_1 τ_R/15πκ(δω_ccw^2-δω_cw^2) = -16 D_1 τ_R/15πκ (2 δω_ccwΔω - Δω^2)The second form of this equation uses Δω = ω_ccw - ω_cw = 2 πΔν and is applied for the theoretical plot in fig. 2d.Locking of CP solitons The dissipative Kerr solitons are governed by the Lugiato-Lefever equation augmented by the Raman term <cit.>. The presence of scattering centers can induce coupling between the CP solitons as follows,∂ A(ϕ, t)/∂ t= -(κ/2 +iδω_A)A+iD_2/2∂^2 A/∂ϕ^2+F +ig|A|^2A+igτ_RD_1A∂ |A|^2/∂ϕ+i∫_0^2πΓ(θ)B(ϕ-2θ,t)e^-iΔω tdθ ∂ B(ϕ, t)/∂ t= -(κ/2 +iδω_B)B+iD_2/2∂^2 B/∂ϕ^2+F +ig|B|^2B+igτ_RD_1B∂ |B|^2/∂ϕ+i∫_0^2πΓ(θ) A(ϕ+2θ,t)e^iΔω tdθHere A and B denote the slowly varying field envelopes of the CW and CCW solitons, respectively. ϕ is the angular coordinate in the rotational frame <cit.>. g is the normalized Kerr nonlinear coefficient <cit.>, F denotes the normalized continuous-wave pump term and Γ(θ) represents the backscattering coefficient in the lab frame θ.Considering the spectral misalignment of CP soliton comb lines presented in fig. 3a, it is assumed that only the r-th comb lines will induce inter-soliton coupling.Accordingly, the equation of motion for the soliton field amplitude A, eq. <ref>, is reduced to the following,∂ A(ϕ, t)/∂ t= -(κ/2 +iδω_A)A+iD_2/2∂^2 A/∂ϕ^2+F +ig|A|^2A+igτ_RD_1A∂ |A|^2/∂ϕ+iGb_r e^irϕwhere the expansion B(ϕ,t)e^iΔω t=∑_μ b_μ e^iμϕ is used to extract the r-th comb line from soliton field B. A similar equation of motion to eq.(<ref>) holds for the amplitude B (with corresponding expansion A(ϕ,t)=∑_μ a_μ e^iμϕ). The coupling coefficient G=∫Γ(θ)exp(-2irθ)dθ.The soliton field amplitude in the presence of the soliton self-frequency shift can be expressed as, <cit.>A=B_s sech[(ϕ-ϕ_Ac)/D_1τ_s]e^iμ_A(ϕ-ϕ_Ac)+iψ_Awhere B_s and τ_s are the pulse amplitude and duration, respectively. μ_A is the mode number of the soliton spectral maximum (μ=0 is the mode number of the pump mode). This mode number is related to the soliton self-frequency shift by Ω_R = μ_A D_1.ψ_A is a constant phase determined by the pump <cit.>. ϕ_Ac is the peak position of the CW soliton, which is coupled to μ_A by <cit.>∂ϕ_Ac/∂ t=μ_AD_2. The soliton energy E_A and the spectral maximum mode number μ_A are given byE_A = ∑_μ|a_μ|^2 = 1/2π∫_-π^+π|A|^2dϕ = B_s^2τ_sD_1/π μ_A = ∑_μμ|a_μ|^2/E_A = -i/4π E_A∫_-π^+π(A^*∂ A/∂ϕ-A∂ A^*/∂ϕ)dϕ Taking the time derivative of eq.(<ref>) and substituting ∂ A/ ∂ t using eq.(<ref>), the equation of motion for μ_A is obtained as∂μ_A/∂ t = -κμ_A -gτ_RD_1/2π E_A∫_-π^+π(∂ |A|^2/∂ϕ)^2 dϕ-1/2π E_A∫_-π^+π (G^* b_r^* e^-irϕ∂ A/∂ϕ-irGA^*b_r e^irϕ)dϕThe second term on the right-hand-side corresponds to the steady-state Raman-induced center shift <cit.> and is denoted by κ R_A. The third term is the soliton spectral shift caused by coupling to the opposing CP soliton through its comb tooth b_r. By using A=∑_μ a_μ e^iμϕ, eq. (<ref>) yields,∂μ_A/∂ t =-κμ_A+κ R_A-i r/E_A(a_r b_r^*G^*-c.c.)=-κμ_A+κ R_A+2 r/E_A|a_r b_r G|sinΘ.where Θ=(ψ_rA-ψ_rB-ψ_G) with the phases, ψ_rA and ψ_rB, of the comb lines a_r and b_r given by the following expression,ψ_rA=ψ_A-rϕ_Ac. ψ_rB=ψ_B-rϕ_Bc +Δω t.Also, Ψ_G is the phase of the backscatter coefficient G. The time dependence of ψ_rA can be derived from eq. <ref> as,∂ψ_rA/∂ t=-r∂ϕ_Ac/∂ t=-rμ_AD_2.Similarly, the derivative of the phase of b_r is given by,∂ψ_rB/∂ t=-rμ_BD_2+Δω.Therefore the time derivative of the phase term Θ=(ψ_rA-ψ_rB-ψ_G) is given by,∂Θ/∂ t=Δω+rD_2(μ_B-μ_A) = 2 π(Δν+ r Δ f ) = 2 πδ.Similar to eq.(<ref>), a parallel equation exists for the soliton B and is given by, ∂μ_B/∂ t=-κμ_B+κ R_B-2 r/E_B|a_r b_r G|sinΘ.Taking a time derivative of eq.(<ref>) and using eq.(<ref>) and eq.(<ref>) gives the following equation of motion for the relative phase Θ,∂ ^2 Θ/∂ t^2 + κ∂Θ/∂ t = -2r^2 D_2 (1/E_A+1/E_B)|a_r b_r G|sinΘ + 2πκδ',where 2πδ' =Δω + rD_2(R_B-R_A) is the frequency difference between the r_th comb lines induced by the shifted pumps and Raman SSFS when the CP solitons have no interaction. The above equation is similar to the Alder equation of injection locking <cit.>, only with an additional second order time-derivative term. Setting the time derivatives of Θ equal to zero gives the locking bandwidth, ω_L, of δ' asω_L=4π|δ'_max|=4r^2D_2/κ(1/E_A+1/E_B)|a_r b_r G|.Moreoever, eq.(<ref>) gives δ = 0 so that the pump frequency difference Δν is divided by the repetition rate difference as follows,Δ f=-Δν/r,which is eq. (1) in the main text.Parameters. In the measurement, the loss rate is κ/2π=1.5 MHz. D_2/2π=16 kHz and r=-60. For a soliton with τ_s=150 fs, the mode number of the Raman SSFS is μ_R∼ -20 and the ratio |a_r|^2/E_A=D_1τ_ssech^2[π (r-μ_R) D_1τ_s/2]/8∼ 7× 10^-4. As the CP solitons have similar powers, the locking bandwidth is estimated as ω_L∼ |G|/4. In this case a backscattering rate of 4 kHz can provide a 1 kHz locking bandwidth.AcknowledgmentThe authors gratefully acknowledge the Defense Advanced Research Projects Agency under the PULSE and DODOS programs, NASA, the Kavli Nanoscience Institute. | http://arxiv.org/abs/1704.08409v2 | {
"authors": [
"Qi-Fan Yang",
"Xu Yi",
"Ki Youl Yang",
"Kerry Vahala"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170427015450",
"title": "Counter-propagating solitons in microresonators"
} |
2.0cm -1.0cm 3.0cm -1.5cm | http://arxiv.org/abs/1704.08017v3 | {
"authors": [
"Roderich Tumulka"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170426090119",
"title": "Bohmian Mechanics"
} |
[email protected] [email protected] [email protected] [email protected]^1Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo. Edificio C-3, Cd. Universitaria. C. P. 58040 Morelia, Michoacán, México.^2Unidad Académica de Física, Universidad Autónoma de Zacatecas, Calzada Solidaridad esquina con Paseo a la Bufa S/N C.P. 98060, Zacatecas, México^3Centro Universitario de la Ciénega, Ave. Universidad 1115, C.P. 47820 Ocotlán, Jalisco, México^4Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582. Colonia Crédito Constructor, Del. Benito Juárez C.P. 03940, Ciudad de México, México.The presence of phantom dark energy in brane world cosmology generates important new effects, causing a premature Big Rip singularity when we increase the presence of extra dimension andconsiderably competing with the other components of our Universe. The idea is based first, in only considering a field with the characteristic equation ω<-1 and after that, consider the explicit form of the scalar field with a potential with a maximum (with the aim of avoid a Big Rip singularity). In both cases we study the dynamic in a robust form through the dynamical analysis theory, detailing in parameters like the deceleration q and the vector field associated to the dynamical system. Results are discussed with the purpose of deeming the cosmology with a phantom field as dark energy in a Randall-Sundrum scenario.The presence of a phantom field in a Randall-Sundrum scenario J. C. López-Domínguez^2 December 30, 2023 ============================================================= § INTRODUCTION Recent observations at high redshift with supernovae of the Type Ia <cit.>, along with observations of anisotropies of cosmic microwave background radiation (CMB) <cit.>, among others <cit.>, show evidence of the current accelerated expansion of the Universe; suggesting the existence of a repulsive energy with the capability of accelerating the Universe, known as dark energy (DE). These same observations also confirm that DE comprises the ∼67% of the total components and only has played role in the recent history of the Universe evolution.In addition, from a theoretical treatment of Raychaudhuri equation, it is possible to see that in order to obtain an accelerated expansion it is necessary that the dark fluid fulfills the equation of state (EoS) ω<-1/3. In this vein and with the aim to explain DE, the less expensive candidate is the well known cosmological constant (CC), originally introduced by Einstein, but with a modern point of view regarding to its origin, expressing the EoS as ω=-1 to obtain an accelerated expansion. Despite the excellent agreement of CC with observations <cit.>, CC has a fundamental problem since we assume that it comes from the contributions of quantum vacuum fluctuations <cit.>, having ∼120 orders of magnitude of difference between the theoretical expectation valueand the observational value <cit.>. In this sense, the theoretical community has been exploring many alternatives to control this problem, without a clear resolution so far <cit.>. However, this fundamental problem has encouraged the scientific community to propose alternative candidates for DE like the quintessence, phantom field, Chaplygin gas and extra dimensions models, among others (for a thorough review of all these alternative models see <cit.>); however until now this problem remains open, with important ongoing theoretical and observational efforts with the aim of finally understand the elusive nature of DE.As we previously mentioned, extra dimensions models are some of the most accepted candidates to understand the accelerated expansion; being a natural solution due to the straightforward way of confronting the origin of the problem. Extra dimensional models like the one proposed by Dvali, Gabadadze and Porrati (DGP) <cit.> is one of the most promising models to solve the DE problem, because it is possible to obtain a natural threshold between four and five-dimensional physics, explaining how gravity could leak to the bulk and vice versa, imitating the actual accelerated expansion. Other highly successful models are the Randall-Sundrum models <cit.>,originally created to solve the hierarchy problem between the standard model of particles (SM) and gravity. One of them is Randall-Sundrum I (RSI), which is characterized by the introduction of a five dimensional AdS compactified extra dimension between two Minkowski branes. The second one, is the Randall-Sundrum II (RSII) which has a non compactified extra dimension with the same capability of solving the hierarchy problem in a more economical way. Furthermore, in the cosmological context, RSII has been the most successful model due that it provides the capability of modifying the structure of Einstein's field equations. Additionally, it's important to notice that RSII leads to three new tensors: The first one is associated with the second order corrections to the energy-momentum tensor; the second one, is a tensor associated with the existence of matter in the bulk and finally a tensor that contains non-local effects associated with the Weyl's tensor <cit.>. In this sense, we also emphasize that the disadvantage of this model, is the necessity to introduce the DE fluid on hand, because the geometrical characteristics are not enough to obtain a natural accelerated period.Therefore, the RSII model provides a new paradigm for the study the Universe evolution with different components. According to this idea, we propose a dynamical analysis of the modified Friedmann equations with the addition of a matter fluid (dark and baryonic) and phantom DEin order to study the Big Rip singularity in this context. As we know from traditional literature <cit.>, phantom DE produces a Big Rip singularity at 22 Gyrs which can be avoided if the potential has a maximum <cit.>. Another important characteristic is that phantom field minimally coupled to gravity has the sign of the kinetic term, in contrast to the ordinary scalar fields (see also <cit.> as a complementary literature of the phantom field). Moreover, the presence of a phantom field itself in a RSII scenario will generate a more abrupt Big Rip coupled with the brane tension λ which is the free parameter of the theory. In this sense, we establish two limits enunciated as: ρ≫λ which is the high energy limit (early times) and ρ≪λ, being the low energy limit (late times). In this vein, there are several reported attempts to constraint the brane tension parameter, through Table-Top experiments <cit.>, astrophysics observations <cit.> and cosmological analysis like BBN <cit.> and CMB <cit.>; indeed the brane tension lies on λ_CMB>3.44×10^6 eV^4 in the first one and λ_TT≳138.59×10^48 eV^4 in the latter one[The first one is related with Cosmic Microwave Background radiation (CMB) and the latter one with Table Top experiments (TT).], showing an enormous difference between the results, but constraining the region of possible lambda values. Setare et. al. <cit.> made a braneworld model with a non-minimally coupled panthom field where they observed that this non-minimally coupling provides a mechanism for an indirect bulk-brane gravity interaction, for late-time cosmological evolution they achieved the -1-crossing of its equation of state parameter. From here, we are in position to organize the paper in the following way. Sec. <ref> is dedicated to construct the modified Friedmann equation from the modified Einstein's equation on the brane, as well as set the necessary condition to obtain an accelerated Universe in this theory. Following these ideas, we construct the Subsec. <ref> in order to generate a numerical analysis taking into account the baryonic and the dark matter (DM) components as a dust fluid and the phantom DE, there we also discuss the possibility of an earlier Big Rip compared with GR predictions. In Sec. <ref>, we revisit the dynamical system theory, in order to apply in the following sections. Sec. <ref>, is dedicated to study phantom DE through the dynamical systems theory, focusing our attention in the evolution, the vector field and the deceleration parameter, always just considering ω<-1, as the main characteristic of the phantom field. In order to extend our study, we develop Sec. <ref> with the aim of generate a detailed study of the phantom scalar field, using a scalar potential with a maximum; similarly, we focus our attention in the evolution, vector field and deceleration parameter in this case. Finally, in Sec. <ref>, we discuss our results and we draw important conclusions.We will henceforth use units in which c=ħ=k_B=1. § FROM MODIFIED EINSTEIN'S FIELD EQUATION TO BRANE COSMOLOGY We start this analysis writing the Einstein's field equation projected onto the brane as:G_μν+ξ_μν=κ^2_(4)T_μν + κ^4_(5)Π_μν + κ^2_(5)F_μν,where T_μν is the four-dimensional energy-momentum tensor of the matter trapped inside the brane, G_μν is the classical Einstein's tensor and the rest of terms on the right and left sides of the equation are explicitly given by: κ^2_(4)=8π G_N=κ^4_(5)/6λ,Π_μν=-1/4T_μαT_ν^α+TT_μν/12+g_μν/24(3T_αβT^αβ-T^2),F_μν=2T_ABg_μ^Ag_ν^B/3+2g_μν/3(T_ABn^An^B-^(5)T/4),ξ_μν=^(5)C^E_AFBn_En^Fg^A_μg^B_ν, where λ is related with the brane tension, κ_(4) and κ_(5) are the four and five-dimensional coupling constants of gravity, G_N is Newton's gravitational constant, Π_μν represents the quadratic corrections of the energy-momentum tensor on the brane and F_μν gives the contributions of the energy-momentum tensor in the bulk projected onto the brane through the unit normal vector n_A, having always in mind that latin capital letters take the values 0,1,2,3,4. In addition ξ_μν gives the contributions of the five-dimensional Weyl's tensor, also projected onto the brane manifold <cit.>.We start the cosmological analysis proposing the traditional homogeneous and isotropic line element as:ds^2=-dt^2+a(t)^2(dr^2+r^2(dθ^2+sin^2θ dφ^2)),where a(t) represents the scale factor and we have assumed a flat geometry, as recent observations indicate <cit.> i.e. k=0. Using Eq. (<ref>), with matter in the brane in the form of perfect fluids and assuming no matter in the bulk, it is possible to write the modified Friedman equation and the covariant Raychaudhuri equation in the following way <cit.>: H^2 = κ^2∑_iρ_i(1+ρ_i /2λ),Ḣ = -3κ^2/2∑_i(ρ_i+p_i)(1+ρ_i/λ), where H=ȧ/a is the Hubble parameter, κ^2=8π G_N/3=κ^2_(4)/3 is the renamed gravitational coupling constant and ρ_i is the energy density of the different components of the Universe. Notice that λ is the free parameter of the theory, giving the threshold between low and high energy regimes of the Universe evolution. Is it important to notice how the regime of low energy is recovered when the following ratio is applied: ρ_i/2λ→0, recovering the traditional cosmological behavior.In addition, the EoS necessary to accelerate the Universe satisfies the constraintw_p<-1/3[1+2ρ_p/λ/1+ρ_p/λ],where in this case, the equation corresponds to a DE fluid. The previous equation can be easily calculated assuming ä/a>0 in Eq. (<ref>), to obtain an accelerated Universe <cit.>. If we are also considering phantom DE we additionally impose the condition ω_p<-1 <cit.>, which implies that ρ_p/λ>-2 for the phantom field.§.§ First Integrals First of all, we specify two fundamental quantities that are dominant in the actual stage of the Universe evolution: matter (dark and baryonic) and phantom DE, i.e. ω_p<-1 <cit.>; as we previously mentioned, we assume that the other components are negligible for late times and we also consider non interaction between the different components i.e. non crossed terms.Under these assumptions, the Friedmann equation can be written as: H^2=κ^2[ρ_0m/a^3(1+ρ̅_0m/a^3)+ρ_0p/a^3(1+ω_p)(1+ρ̅_0p/ a^3(1+ω_p)) ],where we define ρ̅_0m≡ρ_0m/2λ, ρ̅_0p≡ρ_0p/2λ,which depend on the free parameter of the theory. From here, it is possible to write the equations in terms of quadratures with the aim of integrating numerically. Thus, the previous equation can be written as:∫_a(τ_0)^a(τ_f)da/√(Ω_0m(a^-1+ρ̅_0ma^-4)+Ω_0p(a^7/2+ρ̅_0pa^5))=Δτ,where it is convenient to define the following dimensionless variables: Ω_0m≡κ^2ρ_0m/H_0^2, Ω_0p≡κ^2ρ_0p/H_0^2 and τ≡ H_0t.As we can see in Fig. <ref>, the Big Rip singularity occurs earlier than predicted by GR, provided that we increase the presence of extra dimensions mediated by the brane tension. In this sense, high energy in early times in the Universe evolution could have caused totally different dynamics in contrast to the expected under the presence of a phantom field. Moreover, the reader can verify that we reproduce the results obtained by Ref. <cit.> for the Big Rip singularity at 22 Gyrs, using the observational value of the Hubble constant <cit.>. Thus, we notice that observations can constraint the brane tension parameter to bound the presence of extra dimensions in the case where DE is modeled by a phantom field. § REVISITING A DYNAMICAL SYSTEM ANALYSIS The dynamical systems play an important role in Cosmology <cit.> particularly in understanding the evolution of the Universe via the solutions of either the numerical or the analytical equations. Many of the tools given by this area have been widely applied in some models offering the possibility of studying different epochs of the Universe in this process (see for instance <cit.>). As we mention above, in the following section we present a model with phantom dark energy in a brane world to be analyzed comprehensively. However, previous to our analysis we do a theoreticalrevision of dynamical systems.First, consider the non-linear three dimensional systemx^'= xf(x,y,z)+α x, y^'= yf(x,y,z)-β y,z^'= zf(x,y,z)+γ z, wheref(x,y,z)=-α x^k+β y^k-γ z^k,k ≥ 1,and α,β,γ∈ℝ^+∖{0}. For each selection of these parameters we have the associated critical pointss_i=(δ_i1,δ_i2,δ_i3),i=0,1,2,3,with δ_ij the Kronecker delta. The Jacobian matrix of the system is 𝒥= ( [ -α(k+1) x^k+β y^k- γ z^k+α-α k yx^k-1-α k zx^k-1;-β k xy^k-1 -α x^k+β(k+1) y^k- γ z^k-β β k zy^k-1;-γ k xz^k-1-γ k yz^k-1 -α x^k+β y^k- γ (k+1)z^k+γ;]). If x=(x,y,z) and considering a small perturbation x→ s_i+δx, we obtain the associated system δx^'=𝒥_s_iδx, where 𝒥_s_iis the Jacobian at the point s_i and the Hartman-Grobman theorem guarantees the existence of a neighborhood for a critical point on which the flow of the system (<ref>) is topologically equivalent to the linearized one. Thus the eigenvalues areλ^i_1 = α-(k+1)αδ_i1+βδ_i2- γδ_i3, λ^i_2 = -β-αδ_i1+(k+1)βδ_i2 -γδ_i3 , λ^i_3 = γ-αδ_i1+βδ_i2- (k+1)γδ_i3,then, fixing the parameters β=3/2^k-1 y γ=6/2^k-1 we define the functionsg(i,k,α)(j)=λ^i_j,where it is possible to identify the values of α for which g<0, g>0, g=0; and from the Table <ref>, we are able to identify the kind of point of the nonlinear system.From here, we observe that for every k and α≠ 0 the critic point s_0 is always a saddle point, and if α decays at the rate 0<b < 1/2^k-1, we find stability at the point s_3 since we have an hyperbolic system.Now, for the system x^'= f_1(x,y,z)+xF(x,y,z), y^'= f_2(x,y,z)+yF(x,y,z),z^'= f_3(x,y,z)+zF(x,y,z), with f_1,f_2,f_3,P,Q ∈ℝ[x,y,z] (the ring of polynomials in three variables with real coefficients) that satisfies f_1(0)=f_2(0)=f_3(0)=P(0)=0 and F=P/Q an element of the set of rational function in three variables ℝ(x,y,z), with the possibility that it is not defined at the origin, also f_1, f_2, f_3, P are not irreducible polynomials. Taking in mind the nonempty set𝒵(𝔞)={𝐱∈ℝ^3 :f_1(𝐱)=f_2(𝐱)=f_3(𝐱)=P(𝐱)=0},with 𝔞=(f_1,f_2,f_3,P) ℝ[x,y,z] being an ideal associated to the system. For the idealℐ(𝒵(𝔞))={p ∈ℝ[x,y,z] :p(𝐱)=0, ∀𝐱∈𝒵(𝔞)},if f^n∈ℐ(𝒵(𝔞)) for some n ∈ℕ then, f ∈ℐ(𝒵(𝔞)), because ℝ[x,y,z] is an integer domain, showing the inclusion √(𝔞)⊂ℐ(𝒵(𝔞)), where √(𝔞) denotes the radical of 𝔞.Now, let f ∈ℐ(𝒵(𝔞))∖{0} and consider the ideal 𝔟=(f_1,f_2,f_3,P,(wf-1)) ℝ[x,y,z,w],in the ring ℝ[x,y,z,w]. We notice that 𝒵(𝔟)=∅, hence 𝔟=ℝ[x,y,z,w] so, there are polynomials p_1,p_2,p_3,p_4,p_5 such that1=p_1f_1+p_2f_2+p_3f_3+p_4P+p_5(wf-1).Now, considering the ring ℝ(x,y,z)[w] with w=1/f, we have1=p_1(𝐱,1/f)f_1+p_2(𝐱,1/f)f_2+p_3(𝐱,1/f)f_3+p_4(𝐱,1/f)P,and for some k ∈ℕf^k=q_1f_1+q_2f_2+q_3f_3+q_4P,where q_1,q_2,q_3,q_4 ∈ℝ[x,y,z], obtaining the inclusion ℐ(𝒵(𝔞))⊂√(𝔞) and the equality ℐ(𝒵(𝔞))=√(𝔞). Using this result, a set of critical points for (<ref>) is 𝒵(𝔞)∖{0}, being a possibility the points in Eq. (<ref>) as * s_1if f_1(x,0,0)Q(x,0,0)+xP(x,0,0)∼_a f_i(x,0,0) i=2,3.* s_2 if f_2(0,y,0)Q(0,y,0)+yP(0,y,0)∼_b f_i(0,y,0)i=1,3.* s_3 if f_3(0,0,z)Q(0,0,z)+zP(0,0,z)∼_cf_i(0,0,z)i=1,2,with a,b,c ≠ 0 and ∼_r denotes the equivalence relation for polynomial functions in one variable that vanish at the point r. § PHANTOM DARK ENERGY IN A RS SCENARIO We now start rearranging the Eq. (<ref>) in the form:H^2=8π G_N/3∑_i(ρ_i+ρ_i^2/2λ),where, redefining the expression ρ̅_i≡ρ_i^2/2λ, it is possible to write:H^2=8π G_N/3∑_i(ρ_i+ρ̅_i).In order to visualize the dynamic equations we propose two different methods. Both methods will generate the same dynamical information but in some cases we will choose one over the other, due to the different information we will be able to obtain from each of them. * Method 1. When the dimensionless variables are constrained into a four dimensional sphere, with Friedman constraint 1=∑_i(x_i^2+y_i^2) and dimensionless variables:x_i^2≡8π G_N/3H^2ρ_i, y_i^2≡8π G_N/3H^2ρ̅_i.Applying the Friedmann constraint, the dynamical equations reduces to:2x_2^'/3x_2= -3/2x_2^2+y_1^2-2y_2^2+3/2, 2y_1^'/3y_1= -3/2x_2^2+y_1^2-2y_2^2-1, 2y_2^'/3y_2= -3/2x_2^2+y_1^2-2y_2^2+2.We notice that this is the dynamical system (<ref>) with k=2 and α=9/4. * Method 2. When the dimensionless variables are constrained into a four dimensional plane, with Friedman constraint 1=∑_i(x_i+y_i) and dimensionless variables: x_i≡8π G_N/3H^2ρ_i, y_i≡8π G_N/3H^2ρ̅_i. In the same way as in method 1, the Friedmann constraint helps us to reduce the dynamical equations, as in the the case k=1 in (<ref>) with α=9/2: x_2^'/3x_2= -3/2x_2+y_1-2y_2+3/2, y_1^'/3y_1= -3/2x_2+y_1-2y_2-1, y_2^'/3y_2= -3/2x_2+y_1-2y_2+2. In both cases, the primes denote an e-folding derivative N=ln(a), where we also made use of the fact that ω_m,(DM)=ω_1=0 and ω_p=ω_2=-3/2, due that we had assumed that the only components of the Universe are phantom DE and matter (baryonic and DM). Notice that the choice of the phantom EoS is based on Planck satellite observations <cit.>.For instance, the dynamical system represented by Eqs. (<ref>) can be solved numerically, which graphical solutions are shown in Figs. <ref>, establishing the initial conditions for Ω_0m and Ω_0p through the Planck satellite constraints <cit.>, while the other initial conditions for Ω̅_0m and Ω̅_0p, can be also constrained with <cit.> having a permitted region to manipulate the density parameters coupled by the brane tension. Here we separate the different components with the aim of visualizing the behavior. As the reader can observe, the phantom DE coupled with branes dominates in later stages of Universe evolution while in similar conditions matter coupled with branes dominates in earlier stages of Universe; in this sense, it is important to give a more restrictive constriction on the brane tension parameter based on observations, in order to elucidate the effects of extra dimensions.In addition, the deceleration parameter q=-ä/aH^2 can be written in terms of Eqs. (<ref>) as:q(N)=1/2(1-3/2x_2-7/3y_1-10/3y_2),where we have used the Friedmann constraint to eliminate the x_1 variable. The corresponding plot can be seen in Fig. <ref>, assuming the following initial conditions: x_2(0)≡Ω_p=0.6 for the four plots and y_1(0)≡Ω̅_m=0, y_2(0)≡Ω̅_p=0 (Dashed plot), Ω̅_m=10^-4, Ω̅_p=4×10^-4 (Red plot), Ω̅_m=10^-3, Ω̅_p=4×10^-3 (Blue plot), Ω̅_m=10^-2, Ω̅_p=4×10^-2 (Green plot) which are permitted small values according to the observations <cit.>. Notice how brane terms, even if they are minimal, can cause an accelerated expansion process. Indeed, phantom dynamics in a non brane theory has a region where the Universe does not present an acceleration epoch, however, we can see that the more the brane effects are present the non-accelerated stages are less pronounced, which is clearly a contradiction to observations.Another complementary analysis is shown in Figs. (<ref>) where we present a vectorial dynamical analysis, showing only the region of interest, ergo, the region given by the Friedmann constriction. We start the analysis finding the equilibrium points and eigenvalues associated with Eq. (<ref>), defining the critical points as (dx_i,y_i/dN)_x_0=0. In this case as can be seen in Tab. <ref>, the critical points are associated with matter domination, phantom DE domination, matter coupled with branes domination and phantom coupled with branes domination respectively.In addition, we define the vector x=(x_2,y_1,y_2) and consider a linear perturbation of the form (<ref>) and the Jacobian matrix 𝒥_s_i associated with the linearized system. The table <ref> shows the eigenvalues and eigenvectors associated with the critical points of 𝒥_s_i Then, as seen in the previous section, the critical points can be classified according to the eigenvalues of the Jacobian of the linearized vector field at a specific point and also for the values below. Thus (0,0,1) is an attractor and (0,1,0) is a source since the eigenvalues associated with these points are all positive, while the origin (0,0,0) and (1,0,0) are saddle points of the non-linear system since their eigenvalues have opposite signs. The region of interest is formed by families of solutions or dynamical fluxes, providing a qualitative description of the evolution of the system as a whole. The dynamics of a particular solution is governed by the initial conditions, x_2(0)≡Ω_0p, y_1(0)≡Ω̅_0m, y_2(0)≡Ω̅_0p, which are called solution curves. In Figs. <ref> it is shown the vector field and some numerical solutions (solid lines) for different initial conditions, all of them satisfying the Friedmann constriction. § PHANTOM DARK ENERGY: A REFINED ANALYSIS In the previous sections we only considered the phantom field under the constriction ω_p<-1, however we now propose a deeper analysis through the explicit form of the phantom field.The coupling with gravity is given by the action <cit.>S[g,ϕ]=∫ d^4x√(-g)[1/2(∇ϕ)^2-V(ϕ)],with an opposite sign in the kinetic term, where ϕ is the phantom scalar field. Therefore, density and pressure are written as ρ=-ϕ̇^2/2+V(ϕ) and p=-ϕ̇^2-V(ϕ).Furthermore, as we previously mentioned, it is possible to avoid the Big Rip singularity if the potential:V(ϕ)=V_0[cosh(√(G_N)βϕ)]^-1,has a maximum, where β is a constant <cit.>.Thus, the Friedmann equation can be written as: H^2=8π G_N/3{ρ_m(1+ρ_m/2λ)-1/2ϕ̇^2(1-ϕ̇^2/4λ)+V_0/cosh(√(G_N)βϕ)(1+1/2λ[V_0/cosh(√(G_N)βϕ)-ϕ̇^2])}, along with the following equationsϕ̈+3Hϕ̇+∂_ϕV=0,ρ̇_m+3Hρ_m=0.Now, defining the appropriate dimensionless equations:x^2≡8π G_Nρ_m/3H^2,y^2≡4π G_Nϕ̇^2/3H^2,k^2≡3H^2/16π G_Nλ, u^2≡8π G_NV_0/3H^2cosh(√(G_N)βϕ),l^2≡√(3/4π)βtanh(√(G_N)βϕ), it is possible to reduce the modified Friedmann Eq. (<ref>) to:1=x^2+(u^2-y^2)+k^2[x^4+(u^2-y^2)^2],recovering the traditional Friedmann equation when k→0. Then, the dynamical system can be written as:x^'=-(3/2+H^'/H)x,y^'=-(3+H^'/H)y+1/2u^2l^2, u^'=-(1/2l^2y+H^'/H)u, altogether withl=Ω_l[σtanh(2σ∫ y^2dN)]^1/2,where σ≡√(3/4π)β is another free parameter, which must be assigned in order to solve the previous equations.Notice that we also made use of the Friedmann constriction. Additionally we have:H^'/H = -3/2x^2-1/2u^2l^2y-1-x^2-(u^2-y^2)/x^4+(u^2-y^2)^2(6y^4-2u^2l^2y^3 -6u^2y^2+3/2u^4l^2y).Therefore the critical points for the system (<ref>) are (±1,0,0),(0,3±√(9-l^4)/l^2 ,1),(0,3±√(9-l^4)/l^2 ,-1) and it is possible to plot the dynamical system as we show in Fig. <ref> with the initial conditions: Ω_m=0.33, Ω_ϕ̇=0.33, Ω_V=0.72 and Ω_l=10^-1, obtaining the expected behavior for a matter domination at earlier times and a posterior domination of the phantom DE, mainly in the potential of the field (see Fig. <ref>); immediately we recognize a state where V≫ϕ̇^2 for large values of N, producing an accelerated state.Another conclusive study can be performed through the deceleration parameter q(N) which can be written in terms of the dimensionless variables (<ref>) as:q(N) = 1/2x^2-2y^2-u^2+1-x^2-(u^2-y^2)/x^4+(u^2-y^2)^2× (2x^4+5y^4-4y^2u^2-u^4),where its behavior is shown in Fig. <ref>. From here, it is possible to observe a transition phase between an unaccelerated and accelerated state at N∼-0.4 as would be expected in the traditional Universe behavior. However our results show a sudden phase transition in a short region of N, remaining stable for the value of q≃-1 and a Universe in continuous state of acceleration.Additionally, some extra information comes from the k parameter which has a dynamical equation k^'=kH^'/H, related with the brane tension. The numerical solution can be observed in Fig. <ref>, where we see a domination of the brane tension component in the earlier times of the Universe evolution, along with an abrupt peak related to the transition between an unaccelerated and accelerated Universe, and finally a subdominant epoch at later times.This evolution is always constrained with the brane tension which is bounded by current observations <cit.>.Finally, we explore the vector field of the system (<ref>), fixing the variables l=k=10^-1, which is shown in Fig. <ref>; our results present two repulsers (the baryonic matter and the kinetic part of the phantom) and one attractor associated to the phantom DE potential. § CONCLUSIONS AND DISCUSSION The results presented in Figs. <ref> show that branes generates a premature Big Rip singularity while the brane tension is accentuated. We emphasize, that this important result can not be obtained in the traditional cosmological analysis with phantom dark energy. In this vein, the analysis developed in this paper, unmasks the dominant components in the beginning and in the end of the Universe (see Figs. <ref>), showing that the density parameters in function of the brane tension will dominates in the future. The previous results are corroborates by Figs. <ref> where it is shown the repulsers associated with the components in the beginning of the Universe and the attractors related with the presence of extra dimensions, in future epochs (see also Table <ref>).Moreover, some important extra information can be obtained from the deceleration parameter q, showing the differences with the standard cosmological model (ΛCDM). Here it is possible to observe an ever-accelerating Universe, as the presence of the extra dimension increases. These results from the deceleration parameter agree with those shown in previous figures. In addition, the TT experiments or others, could restrict the dynamics of the presence of extra dimensions, mimicking to a large extent, the standard cosmological model. As a complement, we develop an analysis, when the form of the phantom DE is explicitly written as a scalar field. Indeed, we assume the same potential used in Ref. <cit.>, with the aim of avoiding a Big Rip singularity, but now in the brane-world context. Thus, this scenario generates a matter dominant era and a posterior domination of the phantom field through the variable Ω_V, which depends on the SF potential, implying an accelerated Universe, for values ω∼-1. In addition to this, the deceleration parameter (see Fig. <ref> ) also give us information about the Universe passing from a non accelerated to an accelerated state (which is an expected result), having an abrupt change of phase (decelerated→accelerated) between N=-0.8 and N=0.2; coinciding with the region where the brane tension presents an anomalous behavior (see Fig. <ref>). It is also possible to discuss that in Fig. <ref> the density parameter related with brane tension presents the expected behavior with a dominant brane tension in early epochs and subdominant brane tension in late epochs. Finally, the vector field presented in Fig. <ref> corroborates our results, presenting the expected attractor related to phantom dark energy potential and a repulser in the early times of the Universe evolution.As a final comments, phantom field in brane-world scenario generates a premature Big Rip by the presence of brane tension, which can be stopped with a the presence of a potential with a local maximum. This last analysis show us a more adequate and congruent behavior as expected by observations, except for the transition region. Further analysis with observations will help us to constrict the brane tension, however this is work that will be presented elsewhere.We would like to thank the referees for thoughtful comments which helped to improve the manuscript. ROA-C and JAA-M acknowledge support from PhD CONACYT fellowship, MAG-A acknowledge support from SNI-México and CONACyT research fellow. We acknowledge as well the invaluable support of Cesar Martinez, due to his careful reading of the first draft and suggestions. We also want to thank Instituto Avanzado de Cosmología (IAC) collaborations. | http://arxiv.org/abs/1704.08611v2 | {
"authors": [
"Rubén O. Acuña-Cárdenas",
"J. A. Astorga-Moreno",
"Miguel A. Garcia-Aspeitia",
"J. C. López-Domínguez"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170427145914",
"title": "The presence of a Phantom field in a Randall-Sundrum scenario"
} |
[The Loss Surface of Deep and Wide Neural Networks equal* Quynh Nguyento Matthias Heinto toDepartment of Mathematics and Computer Science, Saarland University, GermanyQuynh [email protected] local minima, global optimality, loss surface, deep neural networks, wide neural networks0.3in ]While the optimization problem behind deep neural networks is highly non-convex, it is frequently observed in practicethat training deep networks seems possible without getting stuck in suboptimal points. It has been argued that this is the case as all local minima are close to being globally optimal. We show that this is (almost) true, in fact almost all local minima are globally optimal, for a fully connected network with squared loss and analytic activation function given that the number of hidden units of one layer of the network is larger than the number of training points and the network structure from this layer on is pyramidal.§ INTRODUCTIONThe application of deep learning <cit.> has in recent years lead to a dramatic boost in performance in many areas such as computer vision, speech recognition or natural language processing. Despite this huge empirical success, the theoretical understanding of deep learning is still limited.In this paper we address the non-convex optimizationproblem of training a feedforward neural network.This problem turns out to be very difficult as there can be exponentially many distinct local minima <cit.>. It has been shown that the training of a network with a single neuron with a variety of activation functionsturns out to be NP-hard <cit.>. In practice local search techniques like stochastic gradient descent or variants are used for training deep neural networks.Surprisingly, it has been observed <cit.> that in the training of state-of-the-artfeedforward neural networks with sparse connectivity like convolutional neural networks <cit.>or fully connected ones one does not encounter problems with suboptimal local minima.However, as the authors admit themselves in <cit.>,the reason for this might be that there is a connection between the fact that these networks have good performanceand that they are easy to train. On the theoretical side there have been several interesting developments recently, see <cit.>.For some class of networks one can show that one can train them globally optimal efficiently.However, it turns out that these approaches are either not practical <cit.>as they require e.g. knowledge about the data generating measure,or they modify the neural network structure and objective <cit.>.One class of networks which are simpler to analyze are deep linear networks for which it has been shown thatevery local minimum is a global minimum <cit.>.While this is a highly non-trivial result as the optimization problem is non-convex, deep linear networks are not interesting in practice as one efficiently just learns a linear function. In order to characterize the loss surface for general networks, an interesting approach has been taken by <cit.>.By randomizing the nonlinear part of a feedforward network with ReLU activation function and making some additional simplifying assumptions,they can relate it to a certain spin glass model which one can analyze. In this model the objective of local minima is close to the global optimum and the number of bad local minima decreases quickly with the distanceto the global optimum. This is a very interesting result but is based on a number of unrealistic assumptions <cit.>.It has recently been shown <cit.> that if some of these assumptions are dropped one basically recovers the result of the linear case, but the model is still unrealistic. In this paper we analyze the case of overspecified neural networks, that is the network is larger thanwhat is required to achieve minimum training error. Under overspecification <cit.> have recently analyzed under which conditionsit is possible to generate an initialization so that it is in principle possible to reach the global optimumwith descent methods.However, they can only deal with one hidden layer networks and have to make strong assumptions on the datasuch as linear independence or cluster structure. In this paper overspecification means that there exists a very wide layer,where the number of hidden units is larger than the number of training points. For this case, we can show that a large class of local minima is globally optimal.In fact, we will argue that almost every critical point is globally optimal. Our results generalize previous work of <cit.>, who have analyzed a similar setting for one hidden layer networks, to networks of arbitrary depth.Moreover, it extends results of <cit.> who have shown that for certain deep feedforward neural networksalmost all local minima are globally optimal whenever the training data is linearly independent.While it is clear that our assumption on the number of hidden units is quite strong,there are several recent neural network structures which contain a quite wide hidden layerrelative to the number of training points e.g. in <cit.> they have 50,000 training samples and the network has one hidden layer with 10,000 hidden units and <cit.> have 1.1 million training samples and a layer with 400,000 hidden units. We refer to <cit.> for other examples where the number of hidden units of one layer is on the order of the number of training samples.We conjecture that for these kind of wide networks it still holds that almost all local minima are globally optimal.The reason is that one can expect linear separability of the training data in the wide layer.We provide supporting evidence for this conjecture by showing that basically every critical point for which the training data is linearly separable in the wide layer is globally optimal.Moreover, we want to emphasize that all of our results hold for neural networks used in practice. There are no simplifying assumptions as in previous work.§ FEEDFORWARD NEURAL NETWORKS AND BACKPROPAGATIONWe are mainly concerned with multi-class problems but our results also apply to multivariate regression problems. Let N be the number of training samples anddenote by X=[x_1,…,x_N]^T∈^N× d, Y=[y_1,…,y_N]^T∈^N× m the input resp. output matrix for the training data (x_i,y_i)_i=1^N, where d is the input dimension and m the number of classes. We consider fully-connected feedforward networks with L layers, indexed from 0,1,2,…,L, which correspond to the input layer, 1st hidden layer, etc, and output layer. The network structure is determined by the weight matrices (W_k)_k=1^L∈:=^d× n_1×…×^n_k-1× n_k×…×^n_L-1× m;where n_k is the number of hidden units of layer k (for consistency, we set n_0=d,n_L=m),and the bias vectors (b_k)_k=1^L ∈:=^n_1×…×^n_L.We denote by =× the space of all possible parameters of the network. In this paper, [a] denotes the set of integers 1,2,…,aand [a,b] the set of integers from a to b. The activation function σ:→ is assumed at least to be continuously differentiable, that is σ∈ C^1(). In this paper, we assume that all the functions are applied componentwise. Let f_k,g_k:^d→^n_k be the mappings from the input space to the feature space at layer k, which are defined as f_0(x)=x,f_k(x) = σ(g_k(x)),g_k(x)=W_k^T f_k-1(x) + b_kfor every k∈[L], x∈^d.In the following, let F_k=[f_k(x_1), f_k(x_2), …, f_k(x_N)]^T∈^N× n_k and G_k=[g_k(x_1), g_k(x_2), …, g_k(x_N)]^T∈^N× n_kbe the matrices that store the feature vectors of layer k after and before applying the activation function. One can easily check thatF_1=σ(XW_1+_N b_1^T) , F_k=σ(F_k-1W_k+_N b_k^T), fork∈[2,L].In this paper we analyze the behavior of the loss of the network without any form of regularization,that is the final objective Φ:→ of the network is defined asΦ( (W_k,b_k)_k=1^L) = ∑_i=1^N ∑_j=1^m l(f_Lj(x_i) - y_ij)where l:→ is assumed to be a continuously differentiable loss function, that is l ∈ C^1(). The prototype loss which we consider in this paper is the squared loss, l(α)=α^2, which is one of the standard loss functions in the neural network literature. We assume throughout this paper that the minimum of (<ref>) is attained. The idea of backpropagation is the core of our theoretical analysis. Lemma <ref> below shows well-known relations for feed-forward neural networks, which are used throughout the paper.The derivative of the loss w.r.t. the value of unit j at layer k evaluated at a single training sample x_iis denoted as δ_kj(x_i)=∂Φ/∂ g_kj(x_i). We arrange these vectors for all training samples into a single matrix Δ_k, defined asΔ_k=[δ_k:(x_1), …, δ_k:(x_N)]^T∈^N× n_k.In the following we use the Hadamard product ∘,which for A,B ∈^m × n is defined as A ∘ B ∈^m × n with (A ∘ B)_ij=A_ijB_ij. Let σ,l∈ C^1(). Then it holds * Δ_k= l'(F_L-Y)∘σ'(G_L),k=L (Δ_k+1W_k+1^T) ∘σ'(G_k),k∈[L-1] * ∇_W_kΦ= X^T Δ_1,k=1 F_k-1^T Δ_k, k∈[2,L] * ∇_b_kΦ=Δ_k^T_N ∀ k∈[L] * By definition, it holds for every i∈[N],j∈[n_L] that (Δ_L)_ij =δ_Lj(x_i)=∂Φ/∂ g_Lj(x_i)=l'(f_Lj(x_i)-y_ij) σ'(g_Lj(x_i))=l'((F_L)_ij-Y_ij) σ'((G_L)_ij) and hence, Δ_L=l'(F_L-Y)∘σ'(G_L). For every k∈[L-1], the chain rule yields for every i∈[N],j∈[n_k] that (Δ_k)_ij =δ_kj(x_i)=∂Φ/∂ g_kj(x_i)=∑_l=1^n_k+1∂Φ/∂ g_(k+1)l(x_i)∂ g_(k+1)l(x_i)/∂ g_kj(x_i)=∑_l=1^n_k+1δ_(k+1)l (x_i) (W_k+1)_jlσ'(g_kj(x_i)) =∑_l=1^n_k+1(Δ_(k+1))_il(W_k+1)^T_ljσ'((G_k)_ij) and hence Δ_k=(Δ_k+1W_k+1^T) ∘σ'(G_k). * For every r∈[d],s∈[n_1] it holds ∂Φ/∂ (W_1)_rs = ∑_i=1^N ∂Φ/∂ g_1s(x_i)∂ g_1s(x_i)/∂ (W_1)_rs=∑_i=1^N δ_1s(x_i) x_ir =∑_i=1^N (X^T)_ri(Δ_1)_is=(X^TΔ_1)_rs and hence ∇_W_1Φ=X^TΔ_1. For every k∈[2,L],r∈[n_k-1],s∈[n_k], one obtains ∂Φ/∂ (W_k)_rs = ∑_i=1^N ∂Φ/∂ g_ks(x_i)∂ g_ks(x_i)/∂ (W_k)_rs=∑_i=1^N δ_ks(x_i) f_(k-1)r(x_i) =∑_i=1^N (F_k-1^T)_ri(Δ_k)_is=( F_k-1^T Δ_k)_rs and hence ∇_W_kΦ=F_k-1^TΔ_k. * For every k∈[1,L], s∈[n_k] it holds ∂Φ/∂ (b_k)_s = ∑_i=1^N ∂Φ/∂ g_ks(x_i)∂ g_ks(x_i)/∂ (b_k)_s=∑_i=1^N δ_ks(x_i) = (Δ_k^T_N)_s and hence ∇_b_kΦ=Δ_k^T_N.Note that Lemma <ref> does not apply to non-differentiable activation functions like the ReLU function, σ_ReLU(x)=max{0,x}. However, it is known that one can approximate this activation function arbitrarily well bya smooth function e.g. σ_α(x)=1/αlog(1+e^α x) (a.k.a. softplus) satisfies lim_α→∞σ_α(x)=σ_ReLU(x) for any x ∈.§ MAIN RESULTWe first discuss some prior work and present then our main result together with extensive discussion.For improved readability we postpone the proof of the main result to the next section which contains several intermediate results which are of independent interest. §.§ Previous WorkOur work can be seen as a generalization of the work of <cit.>. While <cit.> has shown thatfor a one-hidden layer network, that if n_1=N-1, then every local minimum is a global minimum, the work of <cit.> considered also multi-layer networks. For the convenience of the reader, we first restate Theorem 1 of <cit.> using our previously introduced notation. The critical points of a continuously differentiable function f:^d → are the points where the gradient vanishes, that is ∇ f(x)=0. Note that this is a necessary condition for a local minimum. <cit.> Let Φ:→ be defined as in (<ref>) with least squares loss l(a)=a^2. Assume σ:→[d,d̅] to be continuously differentiable with strictly positive derivative and[ lim_a→∞σ'(a)/d̅-σ(a)>0, lim_a→∞-σ”(a)/d̅-σ(a)>0; lim_a→-∞σ'(a)/σ(a)-d>0,lim_a→-∞σ”(a)/σ(a)-d>0 ]Then every critical point (W_l,b_l)_l=1^L of Φ which satisfies the conditions * (W_l)=n_l for all l∈[2,L], * [X,_N]^TΔ_1=0 implies Δ_1=0is a global minimum.While this result is already for general multi-layer networks,the condition “[X,_N]^TΔ_1=0 implies Δ_1=0” is the main caveat. It is already noted in <cit.>, that “it is quite hard to understand its practical meaning”as it requires prior knowledge of Δ_1 at every critical point. Note that this is almost impossible as Δ_1 depends on all the weights of the network. For a particular case, when the training samples (biases added) are linearly independent, ([X,_N])=N, the condition holds automatically.This case is discussed in the following Theorem <ref>, where we consider a more general class of loss and activation functions.§.§ First Main Result and DiscussionA function f:^d → is real analytic if the corresponding multivariate Taylor series converges to f(x) on an open subset of ^d <cit.>. All results in this section are proven under the following assumptions on the loss/activation function and training data.* There are no identical training samples, x_i ≠ x_j for all i≠ j,* σ is analytic on , strictly monotonically increasing and * σ is boundedor* there are positive ρ_1,ρ_2,ρ_3,ρ_4, s.t. |σ(t)|≤ρ_1 e^ρ_2 t for t< 0 and |σ(t)|≤ρ_3t + ρ_4 for t≥ 0* l∈ C^2() and if l'(a)=0 then a is a global minimum These conditions are not always necessary to prove some of the intermediate results presented below, but we decided to provide the proof under the above strong assumptions for better readability. For instance, all of our results also hold for strictly monotonically decreasing activation functions. Note that the above conditions are not restrictive as many standard activation functions satisfy them. The sigmoid activation function σ_1(t)=1/1+e^-t,the tangent hyperbolic σ_2(t)=tanh(t)and the softplus function σ_3(t)=1/αlog(1+e^α t) for α>0satisfy Assumption <ref>. Note that σ_2(t)=2/1+e^-2t-1.Moreover, it is well known that ϕ(t)=1/1+t is real-analytic on_+={t ∈ | t≥ 0}. The exponential function is analytic with values in (0,∞). As composition of real-analytic function is real-analytic (see Prop 1.4.2 in <cit.>),we get that σ_1 and σ_2 are real-analytic. Similarly, since log(1+t) is real-analytic on (-1,∞) and the composition with the exponential function is real-analytic, we get that σ_3 is a real-analytic function.Finally, we note that σ_1,σ_2,σ_3 are strictly monotonically increasing.Since σ_1,σ_2 are bounded, they both satisfy Assumption <ref>. For σ_3, we note that 1+e^α t≤ 2e^α t for t≥ 0, and thus it holds for every t≥ 0 that0 ≤σ_3(t) = 1/αlog(1+e^α t) ≤1/αlog(2 e^α t) = log(2)/α + t,and with log(1+x)≤ x for x>-1 it holds log(1+e^α t) ≤ e^α t for every t∈.In particular0 ≤σ_3(t) ≤ e^α t/α∀ t<0which implies that σ_3 satisfies Assumption <ref>for ρ_1=1/α,ρ_2=α,ρ_3=1,ρ_4=log(2)/α.The conditions on l are satisfied for any twice continuously differentiable convex loss function. A typical example is the squared loss l(a)=a^2or the Pseudo-Huber loss <cit.> given as l_δ(a)=2δ^2(√(1+a^2/δ^2)-1)which approximates a^2 for small a and is linear with slope 2δ for large a. But also non-convex loss functions satisfy this requirement, for instance: * Blake-Zisserman: l(a)=-log(exp(-a^2)+δ) for δ>0. For small a, this curve approximates a^2, whereas for large a the asymptotic value is -log(δ).* Corrupted-Gaussian:l(a)=-log(αexp(-a^2)+(1-α)exp(-a^2/w^2)/w) for α∈[0,1], w>0. This function computes the negative log-likehood of a gaussian mixture model.* Cauchy: l(a)=δ^2log(1+a^2/δ^2)for δ≠ 0. This curve approximates a^2 for small a and the value of δ determines for what range of a this approximation is close.We refer to <cit.> (p.617-p.619) for more examples and discussion on robust loss functions. As a motivation for our main result, we first analyze the casewhen the training samples are linearly independent, which requires N≤ d+1.It can be seen as a generalization of Corollary 1 in <cit.>. Let Φ:→ be defined as in (<ref>) and let the Assumptions <ref> hold.If the training samples are linearly independent, that is ([X,_N])=N, then every critical point (W_l,b_l)_l=1^L of Φfor which the weight matrices (W_l)_l=2^L have full column rank, that is(W_l)=n_l for l∈[2,L],is a global minimum. The proof is based on induction. At a critical point it holds ∇_W_1Φ=X^TΔ_1=0 and ∇_b_1Φ=Δ_1^T_N=0 thus [X,_N]^TΔ_1=0. By assumption, the data matrix [X,_N]^T∈^(d+1)× N has full column rank, this implies Δ_1=0. Using induction, let us assume that Δ_k=0 for some 1 ≤ k ≤ L-1, then by Lemma <ref>,we have Δ_k=(Δ_k+1W_k+1^T) ∘σ'(G_k)=0.As by assumption σ' is strictly positive, this is equivalent to Δ_k+1W_k+1^T=0 resp. W_k+1Δ_k+1^T=0.As by assumption W_k+1 has full column rank, it follows Δ_k+1=0. Finally, we get Δ_L=0.With Lemma <ref> we thus get l'(F_L-Y)∘σ'(G_L)=0which implies with the same argument as above l'(F_L-Y)=0. From our Assumption <ref>, it holds that if l'(a)=0 then a is a global minimum of l. Thus each individual entry of (F_L-Y) must represent a global minimum of l. This combined with (<ref>) implies that the critical point must be a global minimum of Φ.Theorem <ref> implies that the weight matrices of potential saddle pointsor suboptimal local minima need to have low rank for one particular layer. Note however that the set of low rank weight matrices inhas measure zero.At the moment we cannot prove that suboptimal low rank local minima cannot exist. However, itseems implausible that such suboptimal low rank local minima exist as every neighborhood of such points containsfull rank matrices which increase the expressiveness of the network. Thus it should be possible to use this degree of freedom to further reduce the loss,which contradicts the definition of a local minimum.Thus we conjecture that all local minima are indeed globally optimal.The main restriction in the assumptions of Theorem <ref>is the linear independence of the training samples as it requires N≤ d+1, which is very restrictive in practice.We prove in this section a similar guaranteein our main Theorem <ref> by implicitly transporting this condition to some higher layer.A similar guarantee has been proven by <cit.> for a single hidden layer network,whereas we consider general multi-layer networks. The main ingredient of the proof of our main result is the observation in the following lemma.Let Φ:→ be defined as in (<ref>) and let the Assumptions <ref> hold.Let (W_l,b_l)_l=1^L ∈ be given. Assume there is some k∈[L-1] s.t. the following holds * ([F_k,_N])=N * (W_l)=n_l, l∈[k+2,L] * ∇_W_k+1Φ( (W_l,b_l)_l=1^L )=0 ∇_b_k+1Φ( (W_l,b_l)_l=1^L )=0 then (W_l,b_l)_l=1^L is a global minimum. By Lemma <ref> it holds that ∇_W_k+1Φ =F_k^TΔ_k+1 =0, ∇_b_k+1Φ =Δ_k+1^T_N =0,which implies [F_k,_N]^T Δ_k+1=0. By our assumption, ([F_k,_N])=N it holds that Δ_k+1=0. Since (W_l)=n_l, l∈[k+2,L], we can apply a similar induction argument as in the proof of Theorem <ref>,to arrive at Δ_L=0 and thus a global minimum.The first condition of Lemma <ref> can be seen as a generalization of the requirementof linearly independent training inputs in Theorem <ref> to a condition of linear independence of the feature vectors at a hidden layer.Lemma <ref> suggests that if we want to make statements about the global optimality of critical points,it is sufficient to know when and which critical points fulfill these conditions. The third condition is trivially satisfied by a critical pointand the requirement of full column rank of the weight matrices is similar toTheorem <ref>. However, the first one may not be fulfilled since ([F_k,_N]) is dependent not only on the weights but also on the architecture.The main difficulty of the proof of our following main theorem is to prove that this first condition holds under the rather simple requirement that n_k≥ N-1 for a subset of all critical points.But before we state the theorem we have to discuss a particular notion of non-degenerate critical point.Let f:D→ be a twice-continuously differentiable function defined on some open domain D⊆^n. The Hessian w.r.t. a subset of variables S⊆x_1,…,x_nis denoted as ∇^2_S f(x)∈^|S|×|S|. When |S|=n, we write ∇^2 f(x)∈^n× n to denote the full Hessian matrix.We use this to introduce a slightly more general notion of non-degenerate critical point.Let f:D→ be a twice-continuously differentiable function defined on some open domain D⊆^n. Let x∈ D be a critical point, ∇ f(x)=0, then * x is non-degenerate for a subset of variables S⊆x_1,…,x_n if ∇^2_S f(x) is non-singular. * x is non-degenerate if ∇^2 f(x) is non-singular. Note that a non-degenerate critical point might not be non-degenerate for a subset of variables,and vice versa, if it is non-degenerate on a subset of variables it does not necessarily imply non-degeneracy on the whole set.For instance,∇^2f(x)= [ 1 0 0 0; 0 1 0 0; 0 0 0 0; 0 0 0 0; ],∇^2f(y)= [ 1 0 1 0; 0 1 0 1; 1 0 0 0; 0 1 0 0; ]Clearly, ∇^2f(x)=0 but ∇^2_x_1,x_2f(x)≠ 0, and ∇^2f(y)≠ 0 but ∇^2_y_3,y_4f(y)= 0. The concept of non-degeneracy on a subset of variables is crucial for the following statement of our main result.Let Φ:→ be defined as in (<ref>) and let the Assumptions <ref> hold.Suppose n_k≥ N-1 for some k∈[L-1]. Then every critical point (W^*_l,b^*_l)_l=1^L of Φ which satisfies the following conditions * (W^*_l,b^*_l)_l=1^L is non-degenerate on (W_l,b_l)l∈ℐ, for some subset ℐ⊆k+1,…,L satisfying k+1∈ℐ, * (W^*_l)_l=k+2^L has full column rank, that is, (W^*_l)=n_l for l∈[k+2,L], is a global minimum of Φ.First of all we note that the full column rank condition of (W_l)_l=k+2^L in Theorem <ref>,and <ref> implicitly requires that n_k+1≥ n_k+2≥…≥ n_L. This means the network needs to have a pyramidal structure from layer k+2 to L.It is interesting to note that most modern neural network architectures have a pyramidal structure from some layer,typically the first hidden layer, on.Thus this is not a restrictive requirement.Indeed, one can even argue that Theorem <ref> gives an implicit justification as it hints on the fact that such networks are easy to train if one layer is sufficiently wide.Note that Theorem <ref> does not require fully non-degenerate critical points but non-degeneracy is only needed for some subset of variables that includes layer k+1.As a consequence of Theorem <ref>, we get directly a stronger result for non-degenerate local minima.Let Φ:→ be defined as in (<ref>) and let the Assumptions <ref> hold.Suppose n_k≥ N-1 for some k∈[L-1].Then every non-degenerate local minimum (W^*_l,b^*_l)_l=1^L of Φ for which (W^*_l)_l=k+2^L has full column rank, that is (W^*_l)=n_l,is a global minimum of Φ. The Hessian at a non-degenerate local minimum is positive definite andevery principal submatrix of a positive definite matrix is again positive definite,in particular for the subset of variables (W_l,b_l)_l=k+1^L.Then application of Theorem <ref> yields the result.Let us discuss the implications of these results. First, note that Theorem <ref> is slightly weaker than Theorem <ref> as it requires also non-degeneracy wrt to a set of variables including layer k+1. Moreover, similar to Theorem <ref>it does not exclude the possibility of suboptimal local minima of low rank in the layers “above” layer k+1.On the other hand it makes also very strong statements.In fact, if n_k ≥ N-1 for some k ∈ [L-1] theneven degenerate saddle points/local maxima are excluded as long as they are non-degenerate with respectto any subset of parameters of upper layers that include layer k+1 and the rank condition holds.Thus given thatthe weight matrices of the upper layers have full column rank ,there is not much room left for degenerate saddle points/local maxima.Moreover, for a one-hidden-layer network for which n_1≥ N-1,every non-degenerate critical point with respect to the output layer parameters is a global minimum,as the full rank condition is not active for one-hidden layer networks. Concerning the non-degeneracy condition of main Theorem <ref>,one might ask how likely it is to encounter degenerate points of a smooth function.This is answered by an application of Sard's/Morse theorem in <cit.>.If f:U ⊂^d → is twice continuously differentiable. Then for almost all w ∈^d with respect to the Lebesgue measure it holds that f' defined as f'(x)=f(x)+w,x has only non-degenerate critical points.Note that the theorem would still hold if one would draw w uniformly at random from the set {z ∈^d | z_2≤ϵ} for any ϵ>0.Thus almost every linear perturbation f' of a function f will lead to the fact all of its critical points are non-degenerate.Thus, this result indicates that exact degenerate points might be rare.Note however that in practice the Hessian at critical points can be close to singular (at least up to numerical precision),which might affect the training of neural networks negatively <cit.>.As we argued for Theorem<ref> our main Theorem <ref>does not exclude the possibility of suboptimal degenerate local minima or suboptimal local minima of low rank.However, we conjecture that the second case cannot happen as every neighborhood of the local minima contains full rank matrices which increase the expressiveness of the network and this additional flexibilitycan be used to reduce the loss which contradicts the definition of a local minimum.As mentioned in the introduction the condition n_k ≥ N-1 looks at first sight very strong. However, as mentioned in the introduction, in practice often networks are used where one hidden layer is rather wide, that is n_k is on the order of N (typically it is the first layer of the network). As the condition of Theorem <ref> is sufficient and not necessary, one can expect out of continuity reasons that the loss surface of networks where the condition is approximately true, is still rather well behaved, in the sense that still most local minima are indeed globally optimal and the suboptimal ones are not far away from the globally optimal ones.§ PROOF OF MAIN RESULT For better readability, we first prove our main Theorem <ref> for a special case where ℐ is the whole set of upper layers, ℐ=k+1,…,L, and then show how to extend the proof to the general case where ℐ⊆k+1,…,L. Our proof strategy is as follows.We first show that the output of each layer are real analytic functions of network parameters. Then we prove that there exists a set of parameters such that ([F_k,_N])=N. Using properties of real analytic functions, we conclude that the set of parameters where ([F_k,_N])<N has measure zero. Then with the non-degeneracy condition, we can apply the implicit-function theorem to conclude thateven if ([F_k,_N])=N is not true at a critical point,then still in any neighborhood of it there exists a point where the conditions of Lemma <ref> are true and the loss is minimal.By continuity of Φ, this implies that the loss must also be minimal at the critical point.We introduce some notation frequently used in the proofs. Let B(x,r)={ z ∈^d | x-z_2 < r} be the open ball in ^d of radius r around x. If the Assumptions <ref> hold, then the output of each layer f_l for every l∈[L] are real analytic functions of the network parameters on . Any linear function is real analytic andthe set of real analytic functions is closed under addition, multiplication and composition,see e.g. Prop. 2.2.2 and Prop. 2.2.8 in <cit.>. As we assume that the activation function is real analytic,we get that all the output functions of the neural network f_kare real analytic functions of the parameters as compositions of real analytic functions.The concept of real analytic functions is important in our proofs as these functions can never be “constant” in a set of the parameter space which has positive measure unless they are constant everywhere. This is captured by the following lemma. <cit.> If f:^n→ is a real analytic function which is not identically zero then the set x∈^nf(x)=0 has Lebesgue measure zero.In the next lemma we show that there exist network parameters such that ([F_k,_N])=N holds if n_k ≥ N-1.Note that this is only possible due to the fact that one uses non-linear activation functions.For deep linear networks, it is not possible for F_k to achieve maximum rank if the layers below it are not sufficiently wide. To see this, one considers F_k=F_k-1W_k + _N b_k^T for a linear network,then (F_k)≤min{(F_k-1), (W_k)}+1since the addition of a rank-one term does not increase the rank of a matrix by more than one. By using induction, one gets (F_k)≤(W_l)+k-l+1 for every l∈[k].The existence of network parameters where ([F_k,_N])=Ntogetherwith the previous lemma will then be used to show that the set of network parameterswhere([F_k,_N])<N has measure zero.If the Assumptions <ref> hold and n_k≥ N-1 for some k∈[L-1],then there exists at least one set of parameters (W_l,b_l)_l=1^k such that ([F_k,_N])=N.We first show by induction that there always exists a set of parameters (W_l,b_l)_l=1^k-1 s.t. F_k-1 has distinct rows. Indeed, we have F_1=σ(XW_1+_Nb_1^T). The set of (W_1,b_1) that makes F_1 to have distinct rows is characterized by σ(W_1^T x_i+b_1)≠σ(W_1^T x_j+b_1), ∀ i≠ j.Note, that σ is strictly monotonic and thus bijective on its domain. Thus this is equivalent toW_1^T (x_i-x_j) ≠ 0, ∀ i ≠ j.Let us denote the first column of W_1 by a, then the existence of a for whicha^T(x_i-x_j) ≠ 0, ∀ i ≠ j,would imply the result. Note that by assumption x_i≠ x_j for all i≠ j. Then the set { a ∈^d|a^T(x_i-x_j)=0} is a hyperplane, which has measure zero and thus the set where condition (<ref>) fails corresponds to the union of N(N-1)/2 hyperplanes which again has measure zero. Thus there always exists a vector a such that condition (<ref>) is satisfied and thus there exists (W_1,b_1) such that the rows of F_1 are distinct. Now, assume that F_p-1 has distinct rows for some p≥ 1,then by the same argument as above we need to construct W_p such thatW_p^T (f_p-1(x_i)-f_p-1(x_j))≠ 0, ∀ i ≠ j.By construction f_p-1(x_i)≠ f_p-1(x_j) and thus with the same argument as above we can choose W_p such that this condition holds.As a result, there exists a set of parameters (W_l,b_l)_l=1^k-1 so that F_k-1 has distinct rows.Now, given that F_k-1 has distinct rows, we show how to construct (W_k,b_k) in such a way that [F_k,_N]∈^N× (n_k+1) has full row rank. Since n_k≥ N-1, it is sufficient to make the first N-1 columns of F_k together with the all-ones vector become linearly independent. In particular, let F_k=[A,B] where A∈^N× (N-1) and B∈^N×(n_k-N+1) be the matrices containing outputs of the first (N-1) hidden units and last (n_k-N+1) hidden units of layer k respectively. Let W_k=[w_1,…,w_N-1,w_N,…,w_n_k]∈^n_k-1× n_k and b_k=[v_1,…,v_N-1,v_N,…,v_n_k]∈^n_k. Let Z=F_k-1=[z_1,…,z_N]^T∈^N× n_k-1 with z_i≠ z_j for every i≠ j. By definition of F_k, it holds A_ij = σ(z_i^T w_j+v_j) for i∈[N],j∈[N-1]. As mentioned above, we just need to show there exists (w_j,v_j)_j=1^N-1so that ([_N,A])=N because then it will follow immediately that ([F_k,_N])=N. Pick any a∈^n_k-1 satisfying potentially after reordering w.l.o.g.az_1<az_2<…<az_N. By the discussion above such a vector always exists since the complementary set is contained in ⋃_i≠ ja∈^n_k-1z_i-z_ja=0 which has measure zero.We first prove the result for the case where σ is bounded. Since σ is bounded and strictly monotonically increasing,there exist two finite values γ,μ∈ with μ < γ s.t. lim_α→-∞σ(α)=μ andlim_α→+∞σ(α)=γ.Moreover, since σ is strictly monotonically increasing it holds for every β∈, σ(β)>μ. Pick some β∈. For α∈, we define w_j=-α a, v_j=α z_j^T a + β for every j∈[N-1]. Note that the matrix A changes as we vary α. Thus, we consider a family of matrices A(α) defined as A(α)_ij=σ(z_i^T w_j+v_j)=σ(α (z_j-z_i)^T a + β). Then it holds for every i∈[N],j∈[N-1] lim_α→+∞ A(α)_ij = γ j>i σ(β) j=i μ j<i Let E(α)=[_N, A(α)] then it holds lim_α→+∞ E(α)_ij = 1 j=1 μ j∈[2,N],i≥ j σ(β) j=i+1 γ else Let Ê(α) be a modified matrix where one subtracts every row i by row (i-1) of E(α), in particular, let Ê(α)_ij = E(α)_ij i=1,j∈[N]E(α)_ij-E(α)_i-1,j i>1,j∈[N] then it holds lim_α→+∞Ê(α)_ij = 1 i=j=1 μ-σ(β)< 0 i=j>10 i>j We do not show the values of other entries as what matters is that the limit, lim_α→+∞Ê(α), is an upper triangular matrix. Thus, the determinant is equal to the product of its diagonal entries which is non-zero. Note that the determinant of Ê(α) is the same as that of E(α) as subtraction of some row from some other row does not change the determinant, and thus we get that lim_α→+∞E(α) has full rank N. As the determinant of E(α) is a polynomial of its entries and thus continuous in α,there exists α_0∈ s.t. for every α≥α_0 it holds (E(α))=([_N, A(α)])=N. Moreover, since A is chosen as the first (N-1) columns of F_k, one can always choose the weights of the first (N-1) hidden units of layer k so that ([F_k,_N])=N. In the case where theactivation function fulfills |σ(t)|≤ρ_1 e^ρ_2 t for t< 0 and |σ(t)|≤ρ_3 t + ρ_4 for t≥ 0 we consider directly the determinant of the matrix E(α).In particular, let us pick some β∈ such that σ(β)≠ 0.We consider the family of matrices A(α) defined asA(α)_ij=σ(z_i^T w_j+v_j)=σ(α(z_j-z_i)^T a+β) where w_j=-α a, v_j=α z_j^T a + β for every j∈[N-1]. Let E(α)=[A(α), _N]. Note that the all-ones vector is now situated at the last column of E(α) instead of first column as before.This column re-ordering does not change the rank of E(α).By the Leibniz-formula one has (E(α)) = ∑_π∈ S_Nsign(π) ∏_j=1^N-1 E(α)_π(j) j, where S_N is the set of all N! permutations of the set {1,…,N} and we used the fact that the last column of E(α) is equal to the all ones vector.Define the permutation γ as γ(j)=j for j∈[N]. Then we have (E(α))= (γ) σ(β)^N-1 + ∑_π∈ S_N \{γ}sign(π) ∏_j=1^N-1 E(α)_π(j) j. The idea now is to show that ∏_j=1^N-1 E(α)_π(j) j goes to zero for every permutation π≠γ as α goes to infinity. And since the whole summation goes to zero while σ(β)≠ 0, the determinant would be non-zero as desired.With that, we first note that for any permutation π≠γ there has to be at least one component π(j) where π(j)>j, in which case, δ_j=(z_j-z_π(j))^T a<0 and thus for sufficiently large α, it holds αδ_j + β<0. Thus|E(α)_π(j) j|=|σ(α (z_j-z_π(j))^T a + β)| ≤ρ_1 e^ρ_2β e^-αρ_2 |δ_j|. If π(j)=j then E(α)_π(j) j=σ(β).In cases where π(j)<j (j≠ N) it holds that δ_j=(z_j-z_π(j))^T a>0 and thus for sufficiently large α, it holds αδ_j + β>0 and we have|E(α)_π(j) j|=|σ(α (z_j-z_π(j))^T a + β)|≤ρ_3 δ_j α + ρ_3 β +ρ_4.So far, we have shown that |E(α)_π(j) j| can always be upper-boundedby an exponential function resp. affine function of α when π(j)>j resp. π(j)<j orit is just a constant when π(j)=j. The above observations imply that there exist positive constants P,Q,R,S,Tsuch that it holds for every π∈ S_N∖γ,|∏_j=1^N-1 E(α)_π(j) j|≤R (P α + Q)^S e^-α T.As α→∞ the upper bound goes to zero. As there are only finitely many such terms, we getlim_α→∞(E(α)) = (γ) σ(β)^N-1≠ 0,and thus with the same argument as before we can argue that there exists a finite α_0for which E(α) has full rank. Now we combine the previous lemma with Lemma <ref> to conclude the following.If the Assumptions <ref> hold and n_k≥ N-1 for some k∈[L-1] thenthe set S(W_l,b_l)_l=1^k([F_k,_N])<N has Lebesgue measure zero. Let E_k=[F_k,_N]∈^N× (n_k+1).Note that with Lemma <ref> the output F_k of layer k is an analytic function ofthe network parameters on . The set of low rank matrices E_k can be characterized by a system of equationssuch that the n_k+1N determinants of all N × N submatrices of E_k are zero.As the determinant is a polynomial in the entries of the matrix and thus an analytic function of the entries and composition of analytic functions are again analytic, we conclude that each determinant is an analytic function of the network parameters of the first k layers.By Lemma <ref> there exists at least one set of network parametersof the first k layerssuch that one of these determinant functions is not identically zero and thus by Lemma <ref> the set of network parameters where this determinant is zero has measure zero. But as all submatrices need to have low rank in order that ([F_k,_N])<N,it follows that the set of network parameters where ([F_k,_N])<N has measure zero.We conclude that for n_k≥ N-1 even if there are network parameters such that ([F_k,_N])<N, then every neighborhood of these parameters contains network parameters such that ([F_k,_N])=N. If the Assumptions <ref> hold and n_k≥ N-1 for some k∈[L-1], then for any given (W^0_l,b^0_l)_l=1^k andfor every ϵ>0,there exists at least one (W_l,b_l)_l=1^k∈ B((W^0_l,b^0_l)_l=1^k,ϵ)s.t. ([F_k,_N])=N. Let S(W_l,b_l)_l=1^k([F_k,_N])<N. The ball B((W_l,b_l)_l=1^k,ϵ) has positive Lebesgue measurewhile S has measure zero due to Lemma <ref>.Thus, for every (W_l,b_l)_l=1^k ∈ B((W^0_l,b^0_l)_l=1^k,ϵ) ∖ Sit holds ([F_k,_N])=N.The final proof of our main Theorem <ref> is heavily based on the implicit function theorem, see <cit.>. Let Ψ:^s×^t→^t be a continuously differentiable function. Suppose (u_0,v_0)∈^s×^t and Ψ(u_0,v_0)=0. If the Jacobian matrix w.r.t. v, J_vΨ(u_0,v_0) = [ ∂Ψ_1/∂ v_1⋯ ∂Ψ_1/∂ v_t;⋮⋮; ∂Ψ_t/∂ v_1⋯ ∂Ψ_t/∂ v_t ]∈^t× t is non-singular at (u_0,v_0), then there is an open ball B(u_0,ϵ) for some ϵ>0 and a unique function α:B(u_0,ϵ)→^t such that Ψ(u,α(u))=0 for all u∈ B(u_0,ϵ). Furthermore, α is continuously differentiable. With all the intermediate results proven above, we are finally ready for the proof of the main result.Proof of Theorem <ref> for case ℐ=k+1,…,L Let us divide the set of all parameters of the network into two subsets where one corresponds to all parameters of all layers up to k, for that we denote u=[(⃗W_1)^T,b_1^T,…,(⃗W_k)^T,b_k^T]^T, and the other corresponds to the remaining parameters, for that we denote v=[(⃗W_k+1)^T,b_k+1^T,…,(⃗W_L)^T,b_L^T]^T. By abuse of notation, we write Φ(u,v) to denote Φ( (W_l,b_l)_l=1^L ). Let s=(u), t=(v) and (u^*,v^*)∈^s×^t be the corresponding vectors for the critical point (W^*_l,b^*_l)_l=1^L. Let Ψ:^s×^t→^t be a map defined as Ψ(u,v)=∇_v Φ(u,v) ∈^t, which is the gradient mapping of Φ w.r.t. all parameters of the upper layers from (k+1) to L. Since the gradient vanishes at a critical point, it holds that Ψ(u^*,v^*)=∇_v Φ(u^*,v^*) =0. The Jacobian of Ψ w.r.t. v is the principal submatrix of the Hessian of Φ w.r.t. v, that is, J_vΨ(u,v)=∇^2_vΦ(u,v) ∈^t× t. As the critical point is assumed to be non-degenerate with respect to v, it holds that J_vΨ(u^*,v^*)=∇^2_vΦ(u^*,v^*) is non-singular. Moreover, Ψ is continuously differentiable since Φ∈ C^2() due to Assumption <ref>. Therefore, Ψ and (u^*,v^*) satisfy the conditions of the implicit function theorem <ref>. Thus there exists an open ball B(u^*, δ_1)⊂^s for some δ_1>0 and a continuously differentiable function α:B(u^*,δ_1)→^t such that Ψ(u,α(u)) = 0,∀u∈ B(u^*, δ_1) α(u^*) = v^* By assumption we have (W^*_l)=n_l, l∈[k+2,L], that is the weight matrices of the “upper” layers have full column rank. Note that (W^*_l)_l=k+2^L corresponds to the weight matrix part of v^* where one leaves out W^*_k+1. Thus there exists a sufficiently small ϵ such that for any v ∈ B(v^*,ϵ), the weight matrix part (W_l)_l=k+2^L of v has full column rank. In particular, this, combined with the continuity of α, implies that for a potentially smaller 0<δ_2 ≤δ_1, it holds for all u ∈ B(u^*,δ_2) that Ψ(u,α(u))=0,α(u^*)=v^*, and that the weight matrix part (W_l)_l=k+2^L of α(u) ∈^t has full column rank. Now, by Corollary <ref> for any 0<δ_3 ≤δ_2 there exists a ũ∈ B(u^*,δ_3) such that the generated output matrix F̃_k at layer k of the corresponding network parameters of ũ satisfies ([F̃_̃k̃, _N])=N. Moreover, it holds for ṽ=α(ũ) that Ψ(ũ,ṽ)=0 and the weight matrix part (W̃_l)_l=k+2^L of ṽ has full column rank. Assume (ũ, ṽ) corresponds to the following representation ũ= [(⃗W̃_1)^T,b̃_1^T,…,(⃗W̃_k)^T,b̃_k^T]^T ∈^s ṽ= [(⃗W̃_k+1)^T,b̃_k+1^T,…,(⃗W̃_L)^T,b̃_L^T]^T∈^t We obtain the following Ψ(ũ,ṽ)=0 ⇒∇_W_k+1Φ( (W̃_̃l̃,b̃_̃l̃)_l=1^k)=0Ψ(ũ,ṽ)=0 ⇒∇_b_k+1Φ( (W̃_̃l̃,b̃_̃l̃)_l=1^k)=0(W̃_̃l̃)=n_l, ∀ l∈[k+2,L]([F̃_̃k̃, _N]) = N Thus, Lemma <ref> implies that (W̃_̃l̃,b̃_̃l̃)_l=1^L is a global minimum of Φ. Let p^* = Φ( (W̃_l,b̃_l)_l=1^L ) = Φ(ũ,ṽ). Note that this construction can be done for any δ_3∈(0,δ_2]. In particular, let (γ_r)_r=1^∞ be a strictly monotonically decreasing sequence such that γ_1=δ_3 and lim_r→∞γ_r=0. By Corollary <ref> and the previous argument, we can choose for any γ_r>0 a point ũ_̃r̃∈ B(u^*,γ_r) such that ṽ_r=α(ũ_r) has full rank and Φ(ũ_r,ṽ_r)=p^*. Moreover, as lim_r→∞γ_r=0, it follows that lim_r→∞ũ_r=u^* and as α is a continuous function, it holds with ṽ_r=α(ũ_r) that lim_r→∞ṽ_r=lim_r→∞α(ũ_r)=α(lim_r→∞ũ_r)=α(u^*)=v^*. Thus we get lim_r →∞ (ũ_r,ṽ_r) = (u^*,v^*) and as Φ is a continuous function it holds lim_r→∞Φ( (ũ_r,ṽ_r))=Φ(u^*,v^*) = p^*, as Φ attains the global minimum for the whole sequence (ũ_r,ṽ_r). Proof of Theorem <ref> for general case In the general case ℐ⊆k+1,…,L,the previous proof can be easily adapted. The idea is that we fix all layers in k+1,…,L∖ℐ. In particular, letu=[(⃗W_1)^T,b_1^T,…,(⃗W_k)^T,b_k^T]^T v=[(⃗W_ℐ(1))^T,b_ℐ(1)^T,…,(⃗W_ℐ(|ℐ|))^T,b_ℐ(|ℐ|)^T]^T.Let s=(u), t=(v) and (u^*,v^*)∈^s×^tbe the corresponding vectors at (W^*_l,b^*_l)_l=1^L. Let Ψ:^s×^t→^t be a map defined asΨ(u,v)=∇_v Φ((W_l,b_l)_l=1^L) with Ψ(u^*,v^*)=∇_v Φ( (W^*_l,b^*_l)_l=1^L)=0.The only difference is that all the layers fromk+1,…,L∖ℐ are hold fixed. They are not contained in the arguments of Ψ, thus will not be involved in our perturbation analysis. In this way, the full rank property of the weight matrices of these layers are preserved,which is needed to obtain the global minimum.§ RELAXING THE CONDITION ON THE NUMBER OF HIDDEN UNITS We have seen that n_k≥ N-1 is a sufficient condition which leads to a rather simple structure of the critical points, in the sense that all local minima which have full rank in the layers k+2 to Land for which the Hessian is non-degenerate on any subset of upper layers that includes layer k+1are automatically globally optimal. This suggests that suboptimal locally optimal points are either completely absent or relatively rare. We have motivated before that networks with a certain wide layer are used in practice, which shows that the condition n_k≥ N-1 is not completely unrealistic. On the other hand we want to discuss in this section how it could be potentially relaxed. The following result will providesome intuition about the case n_k<N-1, but will not be as strong as our main result <ref> which makes statements about a large class of critical points.The main idea is that with the condition n_k≥ N-1 the data is linearly separable at layer k. As modern neural networks are expressive enough to represent any function, see <cit.> for an interesting discussion on this, one can expect that in some layer the training data becomes linearly separable. We prove that any critical point, for which the “learned” network outputs at any layer are linearly separable(see Definition <ref>) is a global minimum of the training error.A set of vectors (x_i)_i=1^N∈^d from m classes (C_j)_j=1^mis called linearly separable if there exist m vectors (a_j)_j=1^m∈^d and m scalars (b_j)_j=1^m∈ so thata_j^T x_i+b_j >0 for x_i∈ C_j and a_j^T x_i+b_j <0 for x_i∉ C_j for every i∈[N],j∈[m]. In this section, we use a slightly different loss function than in the previous section. The reason is that the standard least squares loss is not necessarily small when the data is linearly separable. Let C_1,…,C_m denote m classes. We consider the objective function Φ:→ from (<ref>)Φ( (W_l,b_l)_l=1^L) = ∑_i=1^N ∑_j=1^m l(f_Lj(x_i) - y_ij)where the loss function now takes the new forml(f_Lj(x_i) - y_ij) = l_1(f_Lj(x_i) - y_ij) x_i∈ C_j l_2(f_Lj(x_i) - y_ij) x_i∉ C_jwhere l_1,l_2 penalize the deviation from the label encoding for the true class resp. wrong classes.We assume that the minimum of Φ is attained over . Note that Φ is bounded from below by zero as l_1 and l_2 are non-negative loss functions. The results of this section are made under the following assumptions on the activation and loss function.* σ∈ C^1() and strictly monotonically increasing.* l_1:→_+, l_1∈ C^1, l_1(a)=0⇔ a≥0, l_1'(a)=0⇔ a≥0 and l_1'(a)<0∀ a<0 * l_2:→_+, l_2∈ C^1, l_2(a)=0⇔ a≤0, l_2'(a)=0⇔ a≤0 and l_2'(a)>0∀ a>0In classification tasks, this loss function encourages higher values for the true class and lower values for wrong classes. An example of the loss function that satisfies Assumption <ref> is given as (see Figure <ref>): l_1(a)= a^2 a≤0 0 a≥0l_2(a)= 0 a≤0 a^2 a≥0Note that for a {+1,-1}-label encoding, +1 for the true class and -1 for all wrong classes,one can rewrite (<ref>) asΦ( (W_l,b_l)_l=1^L) = ∑_i=1^N ∑_j=1^m max{0,1-y_ij f_Lj(x_i)}^2,which is similar to the truncated squared loss (also called squared hinge loss) used in the SVM for binary classification.Since σ and l are continuously differentiable,all the results from Lemma <ref> still hold.Our main result in this section is stated as follows.Let Φ:→_+ be defined as in (<ref>) and let the Assumptions <ref> hold. Then it follows: * Every critical point of Φ for which the feature vectors contained in the rows of F_k are linearly separable and all the weight matrices (W_l)_l=k+2^L have full column rank is a global minimum. * If the training inputs are linearly separable then every critical point of Φ for which all the weight matrices (W_l)_l=2^L have full column rank is a global minimum. * Let F̃_̃k̃=[F_k,_N]. Since F_k contains linearly separable feature vectors, there exists m vectors h_1,…,h_m∈^n_k+1 s.t. h_j(F̃_̃k̃)_i:>0 for x_i∈ C_j and h_j(F̃_̃k̃)_i:<0 for x_i∉ C_j. Let H=[h_1,…,h_m]∈^(n_k+1)× m, one obtains (H^T F̃_̃k̃^T)_ji = h_j(F̃_̃k̃)_i:>0 x_i∈ C_j <0 x_i∉ C_j. On the other hand, (Δ_L)_ij = δ_Lj(x_i)= ∂Φ/∂ g_Lj(x_i)= l_1'(f_Lj(x_i)-y_ij)σ'(g_Lj(x_i)) x_i∈ C_j l_2'(f_Lj(x_i)-y_ij)σ'(g_Lj(x_i)) x_i∉ C_j We show that H^T F̃_̃k̃^T Δ_L=0 if and only if Δ_L=0. Indeed, if Δ_L=0 the implication is trivial. For the other direction, assume that H^T F̃_̃k̃^T Δ_L=0. Then it holds for every j∈[m] that 0=(H^T F̃_̃k̃^T Δ_L)_jj=∑_i=1^N (H^T F̃_̃k̃^T)_ji (Δ_L)_ij. In particular, ∑_i=1^N (H^T F̃_̃k̃^T)_ji (Δ_L)_ij =∑_ix_i ∈ C_j(F̃_̃k̃)_i,:,h_jl_1'(f_Lj(x_i)-y_ij)σ'(g_Lj(x_i)) + ∑_ix_i ∉ C_j(F̃_̃k̃)_i,:,h_jl_2'(f_Lj(x_i)-y_ij)σ'(g_Lj(x_i))≤ 0 Note that under the assumptions on the loss and activation function and since the features are separable, the terms in both sums are non-positive and thus the sum can only vanish if all terms vanish which implies l_1'(f_Lj(x_i) - y_ij)=0 x_i∈ C_j l_2'(f_Lj(x_i) - y_ij)=0 x_i∉ C_j ∀ i∈[N],j∈[m] which yields Δ_L=0. Back to the main proof, the idea is to prove that Δ_L=0 at the given critical point. Let us assume for the sake of contradiction that Δ_L≠ 0. For every l∈[L], i_l∈[n_l] define Σ_i_l^l=diag(σ'(g_l i_l(x_1)), …, σ'(g_l i_l(x_N))). Since the cumulative product ∏_l=k+1^L-1Σ_i_l^l is a N× N diagonal matrix which contains only positive entries in its diagonal, it does not change the sign pattern of Δ_L, and thus it holds with, H^T F̃_̃k̃^T Δ_L=0 if and only if Δ_L=0, for every (i_k+1,…,i_L-1)∈[n_k+1]×…×[n_L-1] that 0 ≠ H^T F̃_̃k̃^T (∏_l=k+1^L-1Σ_i_l^l) Δ_L 0 ≠ H^T F̃_̃k̃^T (∏_l=k+1^L-1Σ_i_l^l) Δ_L W_L^T ((W_L)=n_L) , where the last inequality is implied by (<ref>) as W_L has full column rank n_L=m. Since the above product of matrices is a non-zero matrix, there must exist a non-zero column, say p∈[n_L-1], then 0≠ H^T F̃_̃k̃^T (∏_l=k+1^L-1Σ_i_l^l) (Δ_L W_L^T)_:p Since i_L-1 is arbitrary, pick i_L-1=p one obtains 0 ≠ H^T F̃_̃k̃^T (∏_l=k+1^L-2Σ_i_l^l) Σ_p^L-1(Δ_L W_L^T)_:p_(Δ_L-1)_:p Moreover, it holds for every i∈[N] ( Σ_p^L-1(Δ_L W_L^T)_:p)_i =σ'(g_(L-1)p(x_i)) ∑_j=1^n_Lδ_Lj(x_i) (W_L)_pj=δ_(L-1)p(x_i) =(Δ_L-1)_ip and thus from (<ref>), 0 ≠ H^T F̃_̃k̃^T (∏_l=k+1^L-2Σ_i_l^l) (Δ_L-1)_:p ⇒0 ≠ H^T F̃_̃k̃^T (∏_l=k+1^L-2Σ_i_l^l) Δ_L-1 Compared to (<ref>), we have reduced the product from ∏_l=k+1^L-1 to ∏_l=k+1^L-2, By induction, one can easily show that 0 ≠ H^T F̃_̃k̃^T Σ_i_k+1^k+1Δ_k+2 and hence 0≠ H^T F̃_̃k̃^T Δ_k+1,which implies 0≠F̃_̃k̃^T Δ_k+1=[(∇_W_k+1)^T Φ,∇_b_k+1Φ]^T. However, this is a contradiction to the fact that we assumed that (W_l,b_l)_l=1^L is a critical point. Thus it follows that it has to hold Δ_L=0. As Δ_L=0 it holds (<ref>) which implies f_Lj(x_i)≥ y_ijx_i∈ C_j f_Lj(x_i)≤ y_ijx_i∉ C_j∀ i∈[N],j∈[m]. This in turn implies Φ((W_l,b_l)_l=1^L)=0. Thus the critical point (W_l,b_l)_l=1^L is a global minimum. * This can be seen as a special case of the first statement. In particular, assume one has a zero-layer which coincides with the training inputs, namely F_0=X, then the result follows immediately. Note that the second statement of Theorem <ref> can be considered as a special case of the first statement. In the case where L=2 and training inputs are linearly separable, the second statement of our Theorem <ref> recovers the similar result of <cit.> for one-hidden layer networks.Even though the assumptions of Theorem <ref> and Theorem <ref> are different in terms of class of activation and loss functions, their results are related. In fact, it is well known that if a set of vectors is linearly independent then they are linearly separable, see e.g. p.340 <cit.>. Thus Theorem <ref> can be seen as a direct generalization of Theorem <ref>.The caveat, which is also the main difference to Theorem <ref>, is that Theorem <ref> makes only statements for all the critical pointsfor which the problem has become separable at some layer, whereas there is no such condition in Theorem <ref>.However, we still think that the result is of practical relevance, as one can expect for a sufficiently large network that stochastic gradient descent will lead to a network structure where the data becomes separable at a particular layer. When this happens all the associated critical points are globally optimal.It is an interesting question for further research if one can show directly under some architecture condition that the network outputs become linearly separable at some layer for any local minimum and thus every local minimum is a global minimum.§ DISCUSSIONOur results show that the loss surface becomes well-behaved when there is a wide layer in the network. Implicitly, such a wide layer is often present in convolutional neural networks used in computer vision. It is thus an interesting future research question how and if our result can be generalized to neural networks with sparse connectivity. We think that the results presented in this paper are a significant addition to the recent understanding why deep learning works so efficiently. In particular, since in this paper we aredirectly working with the neural networks used in practice without any modifications or simplifications.§ ACKNOWLEDGMENTThe authors acknowledge support by the ERC starting grant NOLEPRO 307793. icml2017 | http://arxiv.org/abs/1704.08045v2 | {
"authors": [
"Quynh Nguyen",
"Matthias Hein"
],
"categories": [
"cs.LG",
"cs.AI",
"cs.CV",
"cs.NE",
"stat.ML"
],
"primary_category": "cs.LG",
"published": "20170426102454",
"title": "The loss surface of deep and wide neural networks"
} |
Quasistationary solutions of scalar fields around collapsing self-interacting boson stars José A. Font December 30, 2023 ========================================================================================= Language models are typically applied at the sentence level, without access to the broader document context.We present a neural language model that incorporates document context in the form of a topicmodel-like architecture, thus providing a succinct representation ofthe broader document context outside of the current sentence.Experiments over a range of datasets demonstrate that our model outperforms a pure sentence-based model in terms of language model perplexity, and leads to topics that are potentially more coherent than those produced by astandard LDA topic model.Our model also has the ability to generaterelated sentences for a topic, providing another way to interprettopics.§ INTRODUCTION Topic models provide a powerful tool for extracting the macro-level content structure of a document collection in the form of the latent topics (usually in the form of multinomial distributions over terms), with a plethora of applications in NLP <cit.>. A myriad of variants of the classical LDA method <cit.> have been proposed, including recent work on neural topic models <cit.>.Separately, language models have long been a foundational component of any NLP task involving generation or textual normalisation of a noisy input (including speech, OCR and the processing of social media text). The primary purpose of a language model is to predict the probability of a span of text, traditionally at the sentence level, under the assumption that sentences are independent of one another, although recent work has started using broader local context such as the preceding sentences <cit.>. In this paper, we combine the benefits of a topic model and language model in proposing a topically-driven language model, whereby we jointly learn topics and word sequence information. This allows us to both sensitise the predictions of the language model to the larger document narrative using topics, and to generate topics which are better sensitised to local context and are hence more coherent and interpretable.Our model has two components: a language model and a topic model. We implement both components using neural networks, and train them jointly by treating each component as a sub-task in a multi-task learning setting.We show that our model is superior to other language models that leverage additional context, and that the generated topics are potentially more coherent than LDA topics.Thearchitecture of the model provides an extra dimensionality of topicinterpretability, in supporting the generation of sentences from a topic(or mix of topics). It is also highly flexible, in its ability to be supervised and incorporate side information, which we show to further improve language model performance. An open source implementation of our model is available at:<https://github.com/jhlau/topically-driven-language-model>.§ RELATED WORK <cit.> propose a model that learns topics and worddependencies using a Bayesian framework. Word generation is driven byeither LDA or an HMM. For LDA, a word is generated based on a sampled topicin the document.For the HMM, a word is conditioned on previous words. Akey difference over our model is that their language model is driven by an HMM, which uses a fixed window and is therefore unable to track long-range dependencies.<cit.> relate the topic model view of documents and words — documents having a multinomial distribution over topics and topics having a multinomial distributional over words— from a neural network perspective by embedding these relationships in differentiable functions. With that, the model lost the stochasticity and Bayesian inference of LDA but gained non-linear complex representations. The authors further propose extensions to the model to do supervised learning where document labels are given. <cit.> and <cit.> relax the sentence independence assumption inlanguage modelling, and use preceeding sentences as additional context. By treating words in preceeding sentences as a bag of words,<cit.> use an attentional mechanism to focus on these wordswhen predicting the next word. The authors show that the incorporationof additional context helps language models.§ ARCHITECTUREThe architecture of the proposed topically-driven language model (henceforth “”) is illustrated in model.There are two components in : a language model and a topic model. The language model is designed to capture word relations in sentences, while the topic model learns topical information in documents. The topic model works like an auto-encoder, where it is given the document words as input and optimised to predict them.The topic model takes in word embeddings of a document and generates adocument vector using a convolutional network.Given the documentvector, we associate it with the topics via an attention scheme to compute a weighted mean of topic vectors, which is then used topredict a word in the document.The language model is a standard LSTM language model<cit.>, but it incorporates the weightedtopic vector generated by the topic model to predict succeeding words.Marrying the language and topic models allows the language model to betopically driven, i.e. it models not just word contexts but alsothe document context where the sentence occurs, in the form of topics.§.§ Topic Model ComponentLet 𝐱_i ∈ℝ^e be the e-dimensional word vectorfor the i-th word in the document. A document of n words isrepresented as a concatenation of its word vectors:𝐱_1:n = 𝐱_1 ⊕𝐱_2 ⊕ ...⊕𝐱_nwhere ⊕ denotes the concatenation operator. We use a number ofconvolutional filters to process the word vectors, but for clarity wewill explain the network with one filter.Let 𝐰_v ∈ℝ^eh be a convolutional filter whichwe apply to a window of h words to generate a feature.A featurec_i for a window of words 𝐱_i:i+h-1 is given as follows:c_i = I(𝐰^⊺_v𝐱_i:i+h-1 + b_v)where b_v is a bias term and I is the identity function.[A non-linear function is typically used here, but preliminary experiments suggest that the identity function works best for .] A feature map 𝐜 is a collection of features computed from all windows of words:𝐜 = [c_1, c_2, ..., c_n-h+1]where 𝐜∈ℝ^n-h+1. To capture the most salientfeatures in 𝐜, we apply a max-over-time pooling operation <cit.>, yielding a scalar:d = max_i c_i In the case where we use a filters, we have 𝐝∈ℝ^a, and this constitutes the vector representation of thedocument generated by the convolutional and max-over-time poolingnetwork.The topic vectors are stored in two lookup tables 𝐀∈ℝ^k × a (input vector) and 𝐁∈ℝ^k × b (output vector), where k is the number oftopics, and a and b are the dimensions of the topic vectors.To align the document vector 𝐝 with the topics, we compute an attention vector which is used to compute a document-topic representation:[The attention mechanism was inspired by memory networks <cit.>.We explored various attention styles (including traditional schemes which use one vector for a topic), but found this approach to work best.]𝐩 = softmax(𝐀𝐝)𝐬 = 𝐁^⊺𝐩where 𝐩∈ℝ^k and 𝐬∈ℝ^b. Intuitively, 𝐬 is a weighted mean of topic vectors, with theweighting given by the attention 𝐩. This is inspired by thegenerative process of LDA, whereby documents are defined as having amultinomial distribution over topics.Finally 𝐬 is connected to a dense layer with softmax outputto predict each word in the document, where each word is generated independentlyas a unigram bag-of-words, and the model is optimised using categoricalcross-entropy loss.In practice, to improve efficiency we compute lossfor predicting a sequence of m_1 words in the document, where m_1 isa hyper-parameter. §.§ Language Model Component The language model is implemented using LSTM units<cit.>:𝐢_t= σ(𝐖_i 𝐯_t + 𝐔_i𝐡_t-1 + 𝐛_i)𝐟_t= σ(𝐖_f 𝐯_t + 𝐔_f𝐡_t-1 + 𝐛_f)𝐨_t= σ(𝐖_o 𝐯_t + 𝐔_o𝐡_t-1 + 𝐛_i)𝐜̂_t= tanh(𝐖_c 𝐯_t +𝐔_c 𝐡_t-1 + 𝐛_c)𝐜_t= 𝐟_t ⊙𝐜_t-1 + 𝐢_t⊙𝐜̂_t𝐡_t= 𝐨_t ⊙tanh(𝐜_t)where ⊙ denotes element-wise product; 𝐢_t,𝐟_t, 𝐨_t are the input, forget and outputactivations respectively at time step t; and 𝐯_t,𝐡_t and 𝐜_t are the input word embedding, LSTMhidden state, and cell state, respectively. Hereinafter 𝐖,𝐔 and 𝐛 are used to refer to the model parameters.Traditionally, a language model operates at the sentence level,predicting the next word given its history of words in the sentence. The language model of incorporates topical information byassimilating the document-topic representation (𝐬) with thehidden output of the LSTM (𝐡_t) at each time step t.Toprevent from memorising the next word via the topic model network,we exclude the current sentence from the document context.We use a gating unit similar to a GRU <cit.> to allow to learn the degree of influence of topical information on the language model:𝐳_t= σ(𝐖_z 𝐬 + 𝐔_z𝐡_t + 𝐛_z)𝐫_t= σ(𝐖_r 𝐬 + 𝐔_r𝐡_t + 𝐛_r)𝐡̂_t= tanh(𝐖_h 𝐬 +𝐔_h (𝐫_t ⊙𝐡_t) + 𝐛_h)𝐡'_t= (1 - 𝐳_t) ⊙𝐡_t +𝐳_t ⊙𝐡̂_twhere 𝐳_t and 𝐫_t are the update and reset gateactivations respectively at timestep t. The new hidden state𝐡'_t is connected to a dense layer with linear transformationand softmax output to predict the next word, and the model is optimisedusing standard categorical cross-entropy loss.§.§ Training and Regularisation is trained using minibatches and SGD.[We use Adam as theoptimiser <cit.>.] For the language model, a minibatchconsists of a batch of sentences, while for the topic model it is abatch of documents (each predicting a sequence of m_1 words).We treat the language and topic models as sub-tasks in a multi-tasklearning setting, and train them jointly using categorical cross-entropyloss. Most parameters in the topic model are shared by the languagemodel, as illustrated by their scopes (dotted lines) in model.Hyper-parameters of are detailed in hyperparameters. Wordembeddings for the topic model and language model components are notshared, although their dimensions are the same (e).[Wordembeddings are updated during training.]For m_1, m_2 and m_3,sequences/documents shorter than these thresholds are padded.Sentenceslonger than m_2 are broken into multiple sequences, and documentslonger than m_3 are truncated.Optimal hyper-parameter settings aretuned using the development set; the presented values are used forexperiments in [s]lm and <ref>.To regularise , we use dropout regularisation<cit.>.We apply dropout to 𝐝 and𝐬 in the topic model, and to the input word embedding andhidden output of the LSTM in the language model<cit.>. § LANGUAGE MODEL EVALUATIONWe use standard language model perplexity as the evaluation metric. In terms of dataset, we use document collections from 3 sources: , and . is a collection of Associated Press[<https://www.ap.org/en-gb/>.] news articles from 2009 to 2016.is a set of movie reviews collected by <cit.>. is the written portion of the British National Corpus <cit.>, which contains excerpts from journals, books, letters, essays, memoranda, news and other types of text. For and , we randomly sub-sample a set of documents for our experiments.For preprocessing, we tokenise words and sentences using Stanford CoreNLP <cit.>. We lowercase all word tokens, filter word types that occur less than 10 times, and exclude the top 0.1% most frequent word types.[For the topic model, we remove word tokens that correspond to these filtered word types; for the language model we represent them astokens (as for unseen words in test).] We additionally remove stopwords for the topic model document context.[We use Mallet's stopword list: <https://github.com/mimno/Mallet/tree/master/stoplists>.]All datasets are partitioned into training, development and test sets; preprocessed dataset statistics are presented in dataset.We tune hyper-parameters of based on development set languagemodel perplexity. In general, we find that optimal settings are fairlyrobust across collections, with the exception of m_3, as document length iscollection dependent; optimal hyper-parameter values are given inhyperparameters.In terms of LSTM size, we explore2 settings: a small model with 1 LSTM layer and 600 hiddenunits, and a large model with 2 layers and 900 hiddenunits.[Multi-layer LSTMs are vanilla stacked LSTMs without skipconnections <cit.> or depth-gating <cit.>.] Forthe topic number, we experiment with 50, 100 and 150 topics. Wordembeddings are pre-trained 300-dimension Google Newsvectors.[<https://code.google.com/archive/p/word2vec/>.] For comparison, we compare with:[Note that all models use the same pre-trained vectors.]: A standard LSTM language model, using the same hyper-parameterswhere applicable. This is the baseline model. : A larger context language model that incorporates context from preceding sentences <cit.>, by treating the preceding sentence as a bag of words, and using an attentional mechanism when predicting the next word. An additional hyper-parameter in is the number of preceeding sentences to incorporate, which we tune based on a development set (to 4 sentences in each case). All other hyper-parameters (such as n_𝑏𝑎𝑡𝑐ℎ, e, n_𝑒𝑝𝑜𝑐ℎ, k_2) are the same as . : A standard LSTM language model that incorporates LDA topic information. We first train an LDA model <cit.> to learn 50/100/150 topics for , and .[Based on Gibbs sampling; α = 0.1, β = 0.01.] For a document, the LSTM incorporates the LDA topic distribution (𝐪) by concatenating it with the output hidden state (𝐡_t) to predict the next word (i.e. 𝐡'_t = 𝐡_t ⊕𝐪).That is, it incorporates topical information into the language model, but unlike the language model and topic model are trained separately.We present language model perplexity performance in lm. All models outperform the baseline , with performing the best across all collections. is competitive over the , although the superiority of for the other collections is substantial. performs relatively well over and , but very poorly over .The strong performance of over suggests that compressing document context into topics benefits language modelling more than using extra context words directly.[The context size of (4 sentences) is technically smaller than (full document), however, note that increasing the context size does not benefit , as the context size of 4 gives the best performance.] Overall, our results show that topical information can help language modelling and that joint inference of topic and language model produces the best results.§ TOPIC MODEL EVALUATIONWe saw that performs well as a language model, but it is also a topic model, and like LDA it produces: (1) a probability distribution over topics for each document (attn); and (2) a probability distribution over word types for each topic.Recall that 𝐬 is a weighted mean of topic vectors for adocument (hiddenreps).Generating the vocabulary distributionfor a particular topic is therefore trivial: we can do so by treating𝐬 as having maximum weight (1.0) for the topic of interest,and no weight (0.0) for all other topics. Let 𝐁_t denote thetopic output vector for thetopic. To generate themultinomial distribution over word types for the t-th topic, wereplace 𝐬 with 𝐁_t before computing the softmaxover the vocabulary.Topic models are traditionally evaluated using model perplexity.There are various ways to estimate test perplexity <cit.>, but <cit.> show that perplexity does not correlate with the coherence of the generated topics. <cit.> propose automatic approaches to computing topic coherence, and <cit.> summarises these methods to understand their differences. We propose using automatic topic coherence as a means to evaluate the topic model aspect of .Following <cit.>, we compute topic coherence using normalised PMI (“NPMI”) scores.Given the top-n words of a topic, coherence iscomputed based on the sum of pairwise NPMI scores between topic words,where the word probabilities used in the NPMI calculation are based onco-occurrence statistics mined from English Wikipedia with a slidingwindow <cit.>.[We use this toolkit tocompute topic coherence:<https://github.com/jhlau/topic_interpretability>.]Based on the findings of <cit.>, we average topic coherence over the top-5/10/15/20 topic words.To aggregate topic coherence scores for a model, we calculate the mean coherence over topics.In terms of datasets, we use the same document collections (,and ) as the language model experiments (lm).We usethe same hyper-parameter settings for and do not tune them.For comparison, we use the following topic models: : We use a LDA model as a baseline topic model. We use the same LDA models as were used to learn topic distributions for (lm). : is a neural topic model proposed by <cit.>. Thedocument-topic and topic-word multinomials are expressed from a neuralnetwork perspective using differentiable functions. Modelhyper-parameters are tuned using development loss.Topic model performance is presented in tm. There are two models of (and ), which specify the size of its LSTM model (1 layer+600 hidden vs. 2 layers+900 hidden; see lm).achieves encouraging results: it has the best performance over , and is competitive over ., however, produces more coherent topics over .Interestingly, coherence appears to increase as the topic number increases for , but the trend is less pronounced for .performs the worst of the 3 topic models, and manual inspection reveals that topics are in generalnot very interpretable.Overall, the results suggest that topicsare competitive: at best they are more coherent than topics, and atworst they are as good as topics.To better understand the spread of coherence scores and impact ofoutliers, we present box plots for all models (number of topics =100)over the 3 domains in boxplot. Across all domains, haspoor performance and larger spread of scores. The difference betweenand is small (> in , but < in ), which is consistent with our previous observation that topics arecompetitive with topics.§ EXTENSIONS One strength of is its flexibility, owing to it taking the form of a neural network. To showcase this flexibility, we explore two simple extensions of , where we: (1) build a supervised model using document labels (supervised); and (2) incorporate additional document metadata (metadata). §.§ Supervised Model In datasets where document labels are known, supervised topic modelextensions are designed to leverage the additional information toimprove modelling quality. The supervised setting also has an additionaladvantage in that model evaluation is simpler, since models can bequantitatively assessed via classification accuracy.To incorporate supervised document labels, we treat documentclassification as another sub-task in .Given a document and itslabel, we feed the document through the topic model network to generatethe document vector 𝐝 and document-topic representation𝐬, and then we concatenate both and connect it to anotherdense layer with softmax output to generate the probability distributionover classes.During training, we have additional minibatches for the documents. Westart the document classification training after the topic and languagemodels have completed training in each epoch.We use in this experiment, which is a popular dataset for text classification.is a collection of forum-like messages from 20 newsgroups categories. We use the “bydate” version of the dataset, where the train and test partition is separated by a specific date.We sample 2K documents from the training set to create the development set. For preprocessing we tokenise words and sentence using Stanford CoreNLP <cit.>, and lowercase all words. As with previous experiments (lm) we additionally filter low/high frequency word types and stopwords.Preprocessed dataset statistics are presented in 20news.For comparison, we use the same two topic models as in tm: and .Both and have natural supervised extensions <cit.> for incorporating document labels.For this task, we tune the model hyper-parameters based on development accuracy.[Most hyper-parameter values for are similar to those used in the language and topic model experiments; the only exceptions are: a = 80, b= 100, n_𝑒𝑝𝑜𝑐ℎ=20, m_3=150. The increase in parameters is unsurprising, as the additional supervision provides more constraint to the model.] Classification accuracy for all models is presented in classification. We present results using only the small setting of LSTM (1 layer + 600 hidden), as we found there is little gain when using a larger LSTM.performs very strongly, outperforming both and by a substantial margin.Comparing and , achieves better performance, especially when there is a smaller number of topics.Upon inspection of the topics we found that topics are much less coherent than those of and , consistent with our observations from tm. §.§ Incorporating Document MetadataIn , each news article contains additional document metadata, including subject classification tags, such as “General News”, “Accidents and Disasters”, and “Military and Defense”.We present an extension to incorporate document metadata in to demonstrate its flexibility in integrating this additional information.As some of the documents in our original sample were missing tags, we re-sampled a set of articles of the same size as our original, all of which have tags. In total, approximately 1500 unique tags can be found among the training articles.To incorporate these tags, we represent each of them as a learnablevector and concatenate it with the document vector before computing theattention distribution. Let 𝐳_i ∈ℝ^f denote thef-dimension vector for the i-th tag. For the j-th document, we sumup all tags associated with it:𝐞 = ∑^n_tags_i=1𝕀(i,j) 𝐳_iwhere n_tags is the total number of unique tags, and function𝕀(i,j) returns 1 is the i-th tag is in the j-th documentor 0 otherwise. We compute 𝐝 as before (tmc), andconcatenate it with the summed tag vector: 𝐝' = 𝐝⊕𝐞.We train two versions of on the new dataset: (1) the vanilla version that ignores the tag information; and (2) the extended version which incorporates tag information.[Model hyper-parameters are the same as the ones used in the language (lm) and topic model (tm) experiments.] We experimented with a few values for the tag vector size (f) and find that a small value works well; in the following experiments we use f=5. We evaluate the models based on language model perplexity and topic model coherence, and present the results in tag.[As the vanilla is trained on the new dataset, the numbers are slightly different to those in [s]lm and <ref>.]In terms of language model perplexity, we see a consistent improvementover different topic settings, suggesting that the incorporation of tagsimproves modelling.In terms of topic coherence, there is a small butencouraging improvement (with one exception).To investigate whether the vectors learnt for these tags are meaningful, we plot the top-14 most frequent tags in tagemb.[The 5-dimensional vectors are compressed using PCA.] The plot seems reasonable: there are a few related tags that are close to each other, e.g. “State government” and “Government and politics”; “Crime” and “Violent Crime”; and “Social issues” and “Social affairs”.§ DISCUSSION Topics generated by topic models are typically interpreted by way of their top-N highest probability words. In , we can additionally generate sentences related to the topic, providing another way to understand the topics. To do this, we can constrain the topic vector for the language model to be the topic output vector of a particular topic (incorporatetm).We present 4 topics from a model (k=100; LSTM size = “large”) and 3 randomly generated sentences conditioned on each topic in sample.[Words are sampled with temperature = 0.75. Generation is terminated when a special end symbol is generated or when sentence length is greater than 40 words.] The generated sentences highlight the content of the topics, providing another interpretable aspect for the topics. These results alsoreinforce that the language model is driven by topics. § CONCLUSION We propose , a topically driven neural language model. hastwo components: a language model and a topic model, which are jointly trained using a neural network. We demonstrate that outperforms a state-of-the-art language model that incorporates larger context, and that its topics are potentially more coherent than LDA topics. Weadditionally propose simple extensions of to incorporateinformation such as document labels and metadata, and achievedencouraging results. § ACKNOWLEDGMENTS We thank Shraey Bhatia for providing an open source implementation of, and the anonymous reviewers for their insightful comments andvaluable suggestions. This work was funded in part by the Australian Research Council.acl_natbib | http://arxiv.org/abs/1704.08012v2 | {
"authors": [
"Jey Han Lau",
"Timothy Baldwin",
"Trevor Cohn"
],
"categories": [
"cs.CL"
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"primary_category": "cs.CL",
"published": "20170426083314",
"title": "Topically Driven Neural Language Model"
} |
[fitpaper=true,pages=-]opda | http://arxiv.org/abs/1704.07987v3 | {
"authors": [
"Jianqiao Wangni"
],
"categories": [
"cs.LG",
"stat.ML"
],
"primary_category": "cs.LG",
"published": "20170426070713",
"title": "Training L1-Regularized Models with Orthant-Wise Passive Descent Algorithms"
} |
Chemical Enhancements in Shock-accelerated Particles: Ab-initio Simulations Anatoly Spitkovsky December 30, 2023 =========================================================================== Robust network flows are a concept for dealing with uncertainty and unforeseen failures in the network infrastructure.One of the most basic models is theproblem: Given a network and an integer k, the task is to find a path flow of maximum robust value, i.e., the guaranteed value of surviving flow after removal of any k arcs in the network. The complexity of this problem appeared to have been settled a decade ago: Aneja et al. <cit.> showed that the problem can be solved efficiently when k = 1, while an article by Du and Chandrasekaran <cit.> established that the problem is -hard for any constant value of k larger than 1.We point to a flaw in the proof of the latter result, leaving the complexity for constant k open once again. For the case that k is not bounded by a constant, we present a new hardness proof, establishing -hardness even for instances where the number of paths is polynomial in the size of the network. We further show that computing optimal integral solutions is already -hard for k = 2 (whereas for k=1, an efficient algorithm is known) and give a positive result for the case that capacities are in {1, 2}.§ INTRODUCTIONNetwork flows are an important tool for modeling vital network services, such as transportation, communication, or energy transmission. In many of these applications, the flow is subjected to uncertainties such as failures of links in the network infrastructure.This motivates the study of robust optimization versions of network flows, which offer a concept to anticipate and counteract such failures.A fundamental optimization problem within this framework is to find a flow that maximizes the amount of surviving flow after it is affected by a worst-case failure of k links in the network for some fixed number k. This problem is also known as theproblem. In this paper, we discuss the complexity of . We point out an error in an earlier result on this problem, which claimed -hardness for the case that k is fixed to a constant value larger than 1. In its stead, we give a new hardness proof which, however, requires the number k to be a non-constant part of the input.Before we discuss these results in detail, we give a formal definition of the problem and discuss related literature. Problem definition We are given a directed graph G = (V, E) with source s, sink t, capacities u ∈ℤ_+^E, and an integer k, specifying the number of possible link failures. Let 𝒫 denote the set of s-t-paths in G and let 𝒮 := {S ⊆ E|S| = k}. An s-t-flow is a vector x ∈ℝ_+^𝒫 respecting the capacity constraints ∑_P : e ∈ P x(P) ≤ u(e) for all e ∈ E.The goal is to find an s-t-flow x that maximizes the robust flow value(x) := ∑_P ∈𝒫 x(P) - max_S ∈𝒮 P ∈𝒫 : P ∩ S ≠∅ x(P), i.e., the amount of remaining flow after failure of any set of k arcs. Related workAneja, Chandrasekaran, and Nair <cit.> were the first to investigate . They showed that if k = 1, the problem can be solved in polynomial time by solving a parametric linear program. In fact, their LP yields a flow x that simultaneously maximizes (x) and the nominal flow value (x) := ∑_P ∈𝒫 x(P). They also show that a maximum integral robust flow can be found in polynomial time for k=1, even though its value might be strictly lower than that of the optimal fractional solution.Following up on this work, Du and Chandrasekaran <cit.> investigated the problem for values of k larger than 1. They presented a hardness proof forwith k=2. Unfortunately, however, this proof is incorrect. We explain this error in detail in <ref>.Because of the presumed hardness of the problem, later work focused on approximation algorithms. Bertsimas, Nasrabadi, and Stiller <cit.> use a variation of the parametric LP to obtain an approximation algorithm forwhose factor depends on the fraction of flow lost through the failure. More recently, Bertsimas, Nasrabadi, and Orlin <cit.> gave an alternative analysis of the same algorithm, establishing an approximation factor of 1 + (k/2)^2/(k+1). Another related concept are k-route flows introduced by Aggarwal and Orlin <cit.>. A k-route flow is a conic combination of elementary flows, each sending flow uniformly along k disjoint paths. This structure ensures that the failure of any arc can only destroy a 1/k fraction of the total flow. Baffier et al. <cit.> observed that computing a maximum (k+1)-route flow yields a (k+1)-approximation for .Several alternative robustness models for flows have been proposed in different application contexts.Taking a less conservative approach, Bertsimas, Nasrabadi, and Stiller <cit.> and Matuschke, McCormick, and Oriolo <cit.> proposed different models of flows that can be rerouted after failures occur. Matuschke et al. <cit.> investigated variants of robust flows in which an adversary can target individual flow paths and the network can be fortified against such attacks. Gottschalk et al. <cit.> devised a robust variant of flows over time in which transit times are uncertain.Robust flows can be seen as a dual version of network flow interdiction, where the task is to find a subset of k arcs whose removal minimizes the maximum flow value in the remaining network. Wood <cit.> proved that this problem is strongly -hard. The reduction presented in <ref> also exploits the fact that interdiction is -hard, but the construction is considerably more involved in order to couple network flow and interdiction decisions in the correct way. For an overview of results on network flow interdiction, see the recent article by Chestnut and Zenklusen <cit.> on the approximability of the problem. Results and structure of this paper In <ref>, we give the background necessary to understand Du and Chandrasekaran's reduction <cit.> and the reason why it does not imply hardness forwith k=2.In <ref>, we then give a new reduction that establishes -hardness forwhen k is an arbitrarily large number given in the input. Our reduction even works whenthe number of paths in the graph is polynomial in the size of the network and only two different capacity values occur (capacity 1 and a capacity that is large but polynomial in the size of the network). We also point out that the problem becomes easy for the case that all capacities are equal.In <ref>, we show that it is -hard to compute an optimal integral solution for k=2. Note that this is in contrast to the case k=1, where the optimal integral solution can be computed efficiently <cit.>. While-hardness for the integral case even holds when capacities are bounded by 3, we show that the problem can be solved efficiently when capacities are bounded by 2, even for arbitrary values of k. § BACKGROUNDThe hardness result in <cit.> is based on an LP formulation ofand the equivalence of optimization and separation, which we shortly recapitulate in this section. §.§ LP formulation For our further discussion of , the following linear programming formulation of the problem will be useful: max P ∈𝒫 x(P) - λ s.t.P : e∈P x(P) ≤u(e)∀e ∈EP : P ∩S ≠∅ x(P) - λ≤0∀S ∈𝒮x(P) ≥0∀P ∈𝒫Note that λ = max_S ∈𝒮∑_P ∈𝒫 : P ∩ S ≠∅ x(P) in any optimal solution to , i.e., λ represents the amount of flow lost in a worst-case failure scenario for flow x. We also consider the dual of : min ∑_e ∈E u(e) y(e) s.t.∑_e∈P y(e)+S : P ∩S ≠∅ z(S)≥1∀P ∈𝒫∑_S ∈𝒮 z(S)=1y(e) ≥0∀e ∈Ez(S) ≥0∀S ∈𝒮Note that the number of s-t-paths in G and hence the number of variables ofcan be exponential in |E|. On the other hand, the number of variables ofis |E| + |E|k, which is polynomial in |E| for constant values of k. In such a situation, a standard approach is to solve the dual via its separation problem, which is described in the next section. §.§ Equivalence of Optimization and Separation Let Q ⊆ℝ^n be a rational polyhedron. By a classic result of Grötschel, Lovasz, and Schrijver <cit.>, optimizing arbitrary linear objectives over Q is polynomially equivalent to finding out wether a given point is in Q and finding a hyperplane separating the point from Q if not. We give a formal statement of this result below.Separation(Q)Input: a vector y ∈ℝ^n Task: Assert that y ∈ Q, or find a separating hyperplane, i.e., a vector d ∈ℝ^n such that d^Tx < d^Ty for all x ∈ Q. Optimization(Q) Input: a vector c ∈ℝ^n Task: Either assert that Q = ∅, or find x, d ∈ℝ^n such that c^Td > 0 and x + α d ∈ Q for all α≥ 0, or find x ∈ Q maximizing c^Tx.The optimization problem for Q can be solved in oracle-polynomial time given an oracle for the separation problem for Q, and vice versa. §.§ Dual Separation forLet Q be the set of feasible solutions of the dual program , i.e.,Q := {(y, z) ∈ℝ^E ×𝒮∑_S ∈𝒮 z(S) = 1, ∑_e ∈ P y(e) + S : P ∩ S ≠∅ z(S) ≥ 1∀ P ∈𝒫}. In the separation problem for Q, we are given (y, z) ∈ℝ^E ×𝒮 and have to decide whether (y, z) ∈ Q. Since checking whether ∑_S ∈𝒮 z(S) = 1 can be done in polynomial time for constant values of k, the separation problem is polynomial-time equivalent to finding a path P such that ∑_e ∈ P y(e) + ∑_S : P ∩ S ≠∅ z(S) < 1 or deciding that no such path is exists. Du and Chandrasekaran <cit.> showed that Separation(Q) is -hard, even when k = 2. They concluded that by the equivalence of optimization and separation, solvingand hence solvingis NP-hard. However, this claim is not correct. It is true that the hardness of Separation(Q) implies that also Optimization(Q) is -hard. However,is only a special case of Optimization(Q): The objective function ofis not an arbitrary vector in ℝ^E ×𝒮, but it is restricted to those objective functions where all coefficients corresponding to the z-variables are 0. Indeed, it turns out that the instances ofwith k = 2 constructed in the reducton of <cit.> contain an s-t-cut of cardinality 2—implying every s-t-flow has robust value 0 in these instances.§ ROBUST FLOWS WITH LARGE NUMBER OF FAILING ARCS is strongly NP-hard, even when restricted to instances where the number of paths is polynomial in the size of the graph.We show this by a reduction from Clique: Given a graph G' = (V', E') and k' ∈ℤ_+, is there a clique of size k' in G'? We will construct an instance ofconsisting of a graph G = (V, E), source s, sink t, capacities u E →ℚ_+∪{∞}, and k ∈ℤ_+ from the Clique instance (at the end of the proof, we show how to obtain an equivalent instance with finite and integral capacities). Letℓ := |V'| + 2|E'|, k := k' ℓ + (|V'| - k') + 2|E'|,ε := 1/ℓ, M := (1+ε)k.For every vertex v ∈ V' we introduce a node a_v and two additional groups of ℓ nodes each, A_v = {a_v, 1, …, a_v, ℓ} and B_v = {b_v, 1, …, b_v, ℓ}. We connect a_v to every node in B_v by an arc of capacity M, and we also connect each node a_v, i to b_v, i by an arc of capacity 1. For every edge e = {u, v}∈ E' we introduce two nodes a'_e, a”_e and arcs (a'_e, b_u, i), (a”_e, b_u, i), (a'_e, b_v, i), (a”_e, b_v, i) for i ∈{1, …, ℓ}, each of capacity M. We denote A := ⋃_v ∈ V' ({a_v}∪ A_v) ∪⋃_e ∈ E'{a'_e, a”_e} andB := ⋃_v ∈ V' B_v. We also introduce a source s and a sink t and arcs (s, a) for every a ∈ A and (b, t) for every b ∈ B, all of infinite capacity. We then add k parallel s-t-arcs e_1, …, e_k. Defining h := 2 ·k'2 - 2, we set the capacity of e_1, …, e_h to 1 + ε and the capacity of e_h+1, …, e_k to 1.We finally add two additional nodes v', v”, together with two s-v'-arcs e'_1, e'_2, two v”-t-arcs e”_1, e”_2, and arcs (s, v”), (v', t), (v', v”). We set the capacities u(e'_1) = u(e”_1) = 1, u(e'_2) = u(e”_2) = u(v', v”) = ε and u(s, v”) = u(v', t) = 1 + ε. We let E_H denote the arcs in the subgraph H induced by the node set {s, v', v”, t}.We now prove the following lemma, which implies Theorem <ref>. For convenience we will use the notation x(e) := ∑_P : e ∈ P x(P) for the total flow through an arc e. Let (x^*, λ^*) be an optimal solution to . Then there is a clique of size k' in G' if and only if x^*(v', v”) > 0.In order to prove Lemma <ref> we first observe that, without loss of generality, we can assume that all arcs in E ∩ (A × B) and the arcs e_1, …, e_k are saturated by x^*: If any of these arcs is not saturated, we can increase the flow along the unique path containing that arc and increase λ^* by the same value, not decreasing the value of the solution and not changing the flow on (v', v”). Consider the set F := {e_1, …, e_k}∪ E_H and definef_x^*(r) := max{∑_e ∈ F' x^*(e) :F' ⊆ F,|F'| ≤ r}for r ∈ℕ. We derive the following lemma. Let h^* := max{|E'[U]|:U ⊆ V', |U| ≤ k' }. Then,λ^* = (|V'| + 4|E'|)ℓ M + k'ℓ + f_x^*(2h^*). Let S ∈𝒮 be such that ∑_P ∈𝒫 : S ∩ P ≠∅ x^*(P) = λ^*. Without loss of generality, we can assume that S ∩ (A × B) = ∅: If S contains an arc (a, b) ∈ A × B, we can replace it by either of the arcs (s, a) or (b, t), each of which intersect the unique s-t-path containing (a, b).Now defineU := {v ∈ V':(b, t) ∈ Sfor allb ∈ B_v }.Note that |U| ≤⌊ k/ℓ⌋≤ k' by choice of k and ℓ.Furthermore, note that x^*(P) = M for exactly (|V'| + 4|E'|)ℓ paths P ∈𝒫 by our earlier assumption that arcs in E ∩ (A × B) are fully saturated. Also, by choice of M and since every other path carries at most 1+ε units of flow, the only possibility to destroy at least (|V'| + 4|E'|)ℓ M units of flow is for S to intersect all these paths, and by maximality of λ^*, this must indeed be the case. Therefore, we can assume that for every v ∈ V', either v ∈ U or {(s, a_v)}∪{(s, a'_e), (s, a”_e):e ∈δ(v)}⊆ S. This implies that U already determines a subset S_U ofk_U := ℓ |U| + |V'| - |U| + 2(|E'| - |E'[U]|)arcs in S, destroying a flow of (|V'| + 4|E'|)ℓ M + |U| ℓ units. The remaining k - k_U arcs in S can destroy an additional flow of at most f_x^*(k - k_U), as no arc in E ∖ F carries more than 1 unit of flow after destruction of the flow paths of value M and there are at least k arcs in F with flow value at least 1. Furthermore observe that f_x^*(r' + r”) ≤ f_x^*(r') + (1+ε)r” as none of the arcs in F carries more than 1 + ε units of flow. We deduce thatλ^*≤(|V'| + 4|E'|)ℓ M + |U| ℓ + f_x^*(k - k_U) = (|V'| + 4|E'|)ℓ M + |U| ℓ + f_x^*((k' - |U|)(ℓ - 1) + 2|E'[U]|)≤(|V'| + 4|E'|)ℓ M + |U|ℓ + f_x^*(2|E'[U]|) + (1 + ε)(k' - |U|)(ℓ - 1) = (|V'| + 4|E'|)ℓ M + k'ℓ + (k' - |U|)(ε(ℓ - 1) - 1_≤ 0) + f_x^*(2|E'[U]|_≤ h^*)≤ (|V'| + 4|E'|)ℓ M + k'ℓ + f_x^*(2h^*). Now let U^* ⊆ V' be such that |U^*| = k' and |E'[U^*]| = h^* and let F^* ⊆ F be such that |F^*| = 2h^* and ∑_e ∈ F^* x^*(e) = f_x^*(2h^*) (recall that we may assume arcs in F to be saturated). Consider the set S^* := ⋃_v ∈ U^* B_v∪ {a_v:v ∈ V' ∖ U^*} ∪ {a'_e, a”_e:e ∉ E'[U^*]} ∪F^*and observe that ∑_P : P ∩ S^* ≠∅ x^*(P) = (|V'| + 4|E'|)ℓ M + k'ℓ + f_x^*(2h^*). This proves Lemma <ref>.We use Lemma <ref> to prove Lemma <ref> as follows. Observe that (x^*, λ^*) maximizes the quantity ∑_P ∈𝒫 x^*(P) - λ^*. As we already fixed the flow value on all paths outside of the subgraph H, we know that∑_P ∈𝒫 x^*(P) = (|V'| + 4|E'|)ℓ M + ℓ|V'| + ∑_i = 1^k u(e_i)_C_1 := + ∑_P ∈𝒫 : P ⊆ E_H x^*(P) = C_1 + x^*(v', t) + x^*(v', v”) + x^*(s, v”), where the last three summands together determine the total nominal flow through H. Defining C_2 := (|V'| + 4|E'|)ℓ M + k'ℓ, Lemma <ref> states that λ^* = C_2 + f_x^*(2h^*). As C_1 and C_2 do not depend on the flow in E_H, we deduce that the flow x^* in E_H maximizes the quantityx^*(v', t) + x^*(v', v”) + x^*(s, v”) - f_x^*(2h^*).First, assume G' has no clique of size k', i.e., h^* ≤k'2 - 1. In this case, 2h^* ≤ h and hence f_x^*(2h^*) = 2h^*(1+ε), independent of the flow values in the subgraph H, as no arc in E_H can carry more than 1 + ε units of flow and there are already h arcs with flow value 1 + ε in F ∖ E_H. Therefore, x^* maximizes x^*(v', t) + x^*(v', v”) + x^*(s, v”), which implies it is the unique maximum flow in H fulfilling ∑_P : (v', v”) ∈ P x^*(P) = 0.Now assume G' has a clique of size k' and thus h^* = k'2. In this case 2h^* = h + 2 and hencef_x^*(2h^*) = 2h · (1+ε) + max{1,x^*(v', t)} + max{1,x^*(s, v”)}, as (v', t) and (s, v”) are the only two arcs in F outside {e_1, …, e_h} that can carry more than 1 unit of flow. Thus x^* maximizes x^*(v', t) + x^*(v', v”) + x^*(s, v”) - max{1,x^*(v', t)} - max{1,x^*(s, v”)}.This term is maximized for x^*(v', t) = x^*(s, v”) = 1 and x^*(v', v”) = ε.The above two observations conclude the proof of Lemma <ref>.Note that the size of the graph G = (V, E) constructed in the reduction is polynomial in the size of G'. Furthermore observe that |𝒫| ≤ |E| and that all capacities are polynomial in the size of G (note that the capacity ∞ can be replaced by |E|M, and multiplying all capacities with ℓ yields integral capacities). This concludes the proof of Theorem <ref>.Reduction to two capacity values We now observe that there is a pseudopolynomial transformation that converts general instances ofto instances with where the capacity of each arc is one of two different values: 1 or u_max, where u_max is the maximum capacity value occurring in the original instance. There is an algorithm that given an instance ofI = ((V, E), s, t, u, k) computes in time 𝒪(|E| u_max) an instance ofI' = ((V', E', s', t', u', k) with u'(e) ∈{1, u_max} for all e ∈ E' such that the maximum robust flow value of I and I' is identical, where u_max := max_e ∈ E u(e). Moreover, given an (integral) flow x' in I' one can compute in time polynomial in |E| and |x'|an (integral) flow x in I with (x) = (x').To obtain an equivalent instance in which only the capacities 1 and u_max occur, observe that we can replace any arc of capacity u by a the concatenation of an arc of capacity u_max andu parallel arcs of capacity 1. Failure of the original arc corresponds to failure of the infinite capacity arc in the modified instance. See Figure <ref> for an illustration. In particular, this implies that our hardness result still holds in the more restricted setting of capacities 1 and ∞ (where ∞ can also be replaced by a number that is polynomially bounded in the network size).is NP-hard, even when restricted to instances where u(e) ∈{1, ∞} for all e ∈ E and where the number of paths is polynomial in the size of the graph.Unit capacities On the other hand, it is not hard to see that the problem becomes easy in the unit capacity case. If u ≡ 1 then any maximum flow also is a maximum robust flow.Let C be a minimum s-t-cut in G.If |C| ≤ k, then every flow has robust value 0. Thus assume |C| > k. Let x be any s-t-flow. Clearly, (x) ≤ |C| - k, as after removal of any k arcs from C, the remaining flow must traverse the |C| - k remaining arcs in the cut. Now assume x is a maximum flow, i.e., (x) = |C|. Since every arc carries at most 1 unit of flow, the removal of any k arcs from G can only decrease the flow value by k, thus x is an optimal solution to .§ INTEGRAL ROBUST FLOWS In this section, we show that finding an maximum integral robust flow is -hard already for instances with k = 2. This is in contrast to the case k = 1, for which it is possible to compute the best integral solution. In fact, our reduction implies that it is hard to distinguish instances with optimal value 2 or 3, resulting in hardness of approximation for the integral problem. Interestingly, the fractional version of the problem admits a 4/3-approximation algorithm for k = 2 <cit.>, indicating that the integral problem is indeed harder. Unless P =, there is no (3/2 - ε)-approximation algorithm for , even when restricted to instances where k=2 and u(e) ≤ 3 for all e ∈ E.We reduce from Arc-disjoint Paths, which is well-known to be -hard <cit.>.As input of Arc-disjoint Paths, we are given a directed graph G' = (V', E') and two pairs of nodes (s_1, t_1) and (s_2, t_2). The task is to decide whether there is an s_1-t_1-path P'_1 and and s_2-t_2-path P'_2 in G' with P'_1 ∩ P'_2 = ∅.From the input graph G' = (V', E'), we construct an instance ofby adding 6 new nodes and 13 new arcs, obtaining a new directed graph G = (V, E) withV =V' ∪{s,t,v,v',v”,w}E =E' ∪{(s, v),(s, v'),(s, v”),(v, s_1),(v, v'),(v, v”),(v', t),(v”, t),(s, w),(t_1, w),(w, t),(s, s_2),(t_2, t)}.We set u(s, v) = 3 and u(v', t) = u(v”, t) = u(w, t) = 2. All other arcs have capacity 1. The whole construction is depicted in <ref>.We show that there is an integral flow x with (x) ≥ 3 if an only if there is an s_1-t_1-path P'_1 and and s_2-t_2-path P'_2 in G' with P'_1 ∩ P'_2 = ∅. It is thus -hard to distinguish instances ofwith optimal value at least 3 from those with optimal value at most 2.First assume there is an integral flow x with (x) = 3. Consider the arc set S' = {(s, v),(w, t)}. As ∑_P : P ∩ S' = ∅ x(P) ≥ 3, there must be three s-t-pathscarrying 1 unit of flow each and not intersecting with S'. These paths must thus start with the arcs (s, v'), (s, v”), and (s, s_2), respectively.In particular, the latter path must end with (t_2, t), as t_2 cannot be reached from v' or v”. Let P_2 be this unique flow-carrying path starting with (s, s_2) and ending with (t_2, t). Note that all arcs of P_2 have unit capacity and thus no other flow-carrying path can intersect P_2. Now consider the arc set S” = {(v', t),(v”, t)}.Because the arc (v, s_1) is part of an s-t-cut with capacity 3 in the network (V, E ∖ S”), there must be a flow path P_1 containing (v, s_1). As (t_2, t) is already saturated by the flow on P_2, the path P_1 must use (t_1, w). In particular, P_1 contains an s_1-t_1-path P'_1 and P_2 contains an s_2-t_2-path P'_2, and P_1 ∩ P_2 = ∅.Conversely, assume there is an s_1-t_1-path P'_1 and and s_2-t_2-path P'_2 in G' with P'_1 ∩ P'_2 = ∅. Let P_1 = {(s, v),(v, s_1)}∪ P'_1 ∪{(t_1, w),(w, t)} and let P_2 = {(s, s_1)}∪ P'_2 ∪{(t_2, t)}. Send 1 unit of flow along each of the paths P_1, P_2 and the five remaining paths s-v'-t, s-v”-t, s-v-v'-t, s-v-v”-t, and s-w-t, obtaining a flow x. Now assume by contradiction that (x) < 3. Because (x) = 7, there must be S ∈𝒮 with ∑_P : P ∩ S ≠∅ x(P) > 4. In particular, the arc (s, v) must be contained in S, as it is the only arc carrying more than 2 units of flow. The other arc in S must be one of the arcs with capacity 2, i.e., (v, v'), (v, v”), or (w, t). However, each of these three arcs is contained in one of the three flow paths using (s, v). Thus ∑_P : P ∩ S ≠∅ x(P) = 4, a contradiction. For the reduction given above to work, it is sufficient to have arcs of capacity at most 3. We now argue that the problem can be solved efficiently for arbitrary values of k when capacities are bounded by 2.restricted to instances with u(e) ≤ 2 for all e ∈ E can be solved in polynomial time.Let x^* be an optimal solution to . Let x_1 be a maximum flow in G with respect to unit capacities and let x_2 be a maximum flow in G with respect to capacities u. As capacities are integral, we can assume without loss of generality that x_1 and x_2 are integral. We will show that(x^*) = max{0, (x_1) - k, (x_2) - 2k}.To prove this claim, consider a minimum cardinality s-t-cut C in G.We greedily construct a set S ⊆ C with |S| ≤ k as follows:Starting with S = ∅, iteratively add an arc e ∈ C ∖ S to S that maximizesΔ(S, e) := ∑_P ∈𝒫 e ∈ P,P ∩ S = ∅ x^*(P)until |S| = k or S = C. In other words, we greedily add an arc from C to S that removes the most flow from x^*. Note that throughout this selection process, the value Δ(S, e) ∈{0, 1, 2} is non-increasing for each e ∈ C. After construction of S, let Δ := 0 if S = C and Δ := max_e ∈ C ∖ SΔ(S, e) otherwise. * If Δ = 0, then ∑_P ∈𝒫 : P ∩ S ≠∅ x^*(P) = (x^*) and therefore (x^*) = 0. In this case, any integral flow is an optimal solution.* If Δ = 1, then ∑_P ∈𝒫∈𝒫: e ∈ P, P ∩ S = ∅ x^*(P) ≤ 1 for every arc e ∈ C ∖ S. Therefore (x^*) ≤∑_P : P ∩ S = ∅ x^*(P) ≤ |C ∖ S| = (x_1) - k ≤(x_1), where the equality follows from (x_1) = |C| and the last inequality follows from the fact that every arc carries at most 1 unit of flow in x_1. We conclude that x_1 is an optimal solution in this case.* If Δ = 2, then ∑_P ∈𝒫 : P ∩ S ≠∅ x^*(P) = 2k, because in every iteration an arc e with Δ(S, e) = 2 was added to S.Thus (x^*) ≤(x^*) - 2k ≤(x_2) - 2k ≤(x_2), where the last inequality follows from the fact that every arc carries at most 2 units of flow in x_2. We conclude that x_2 is an optimal solution in this case.Computing x_1 and x_2 and identifying whether (x_1) - k ≥(x_2) - 2k can be done in polynomial time. § CONCLUSIONIn this article, we point out that the computational complexity of theproblem, which was believed to be settled 10 years ago, is indeed open. We show that the problem is -hard when the number of failing arcs is part of the input. However, it remains a challenging open research question to determine the complexity when this number is bounded by a constant. Our hardness result also does not give bounds on the approximability of the problem (other than ruling out fully polynomial-time approximation schemes) and it would be interesting to see whether the O(k)-approximation by Bertsimas, Nasrabadi, and Orlin <cit.> can be improved to a constant factor.Acknowledgements We thank Tom McCormick and Gianpaolo Oriolo for numerous helpful discussions that led to the discovery of the flaw in <cit.>. This work was supported by the `Excellence Initiative' of the German Federal and State Governments, the Graduate School CE at TU Darmstadt, and by the Alexander von Humboldt Foundation with funds of the German Federal Ministry of Education and Research (BMBF). | http://arxiv.org/abs/1704.08241v2 | {
"authors": [
"Yann Disser",
"Jannik Matuschke"
],
"categories": [
"cs.DM"
],
"primary_category": "cs.DM",
"published": "20170426175614",
"title": "The Complexity of Computing a Robust Flow"
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EEG-Based User Reaction Time EstimationUsing Riemannian Geometry Features Dongrui Wu1, Senior Member, IEEE, Brent J. Lance2, Senior Member, IEEE, Vernon J. Lawhern23, Member, IEEE, Stephen Gordon4,Tzyy-Ping Jung5, Fellow, IEEE, Chin-Teng Lin6, Fellow, IEEE 1DataNova, NY USA2Human Research and Engineering Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, MD USA3Department of Computer Science, University of Texas at San Antonio, San Antonio, TX USA4DCS Corp, Alexandria, VA USA5Swartz Center for Computational Neuroscience & Center for Advanced Neurological Engineering,University of California San Diego, La Jolla, CA6Centre for Artificial Intelligence, Faculty of Engineering and Information Technology, University of Technology Sydney, AustraliaE-mail: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]=====================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================This paper aims to catalyze research discussions about text feature extraction techniques using neural network architectures. The research questions discussed here focus on the state-of-the-art neural network techniques that have proven to be useful tools for language processing, language generation, text classification and other computational linguistics tasks. § MOTIVATION A majority of the methods currently in use for text-based feature extraction rely on relatively simple statistical techniques. For instance, a word co-occurrence model like n-grams or a bag-of-words model like TF-IDF.The motivation of this research project is to identify and survey the techniques that use neural networks and study them in juxtaposition with the traditional text feature extraction models to show their differences in approach.Feature extraction of text can be used for a multitude of applications including - but not limited to - unsupervised semantic similarity detection, article classification and sentiment analysis.The goal of this project is to document of the differences, advantages and drawbacks in the domain of feature extraction from text data using neural networks. It also sketches the evolution of such techniques over time.This report could serve as a quick cheat-sheet for engineers looking to build a text classification or regression pipeline, as the discussion (Section <ref>) would serve to map a use-cases to feature extraction implementation specifics.§ RESEARCH QUESTIONS RQ1 What are the relatively simple statistical techniques to extract features from text?RQ2 Is there any inherent benefit to using neural networks as opposed to the simple methods?RQ3 What are the trade-offs that neural networks incur as opposed to the simple methods?RQ4 How do the different techniques compare to each other in terms of performance and accuracy?RQ5 In what use-cases do the trade-offs outweigh the benefits of neural networks? § METHODOLOGY The research questions listed in Section <ref> will be tackled by surveying a few of the important overview papers on the topic<cit.><cit.><cit.>. A few of the groundbreaking research papers in this area will also be studied, including word embeddings<cit.><cit.><cit.>.In addition to this, other less-obvious methods of features extraction will be surveyed, including tasks like part-of-speech tagging, chunking, named entity recognition, and semantic role labeling<cit.><cit.><cit.><cit.><cit.><cit.>. § BACKGROUND This section provides a high level background of the tasks within Computational Linguistics. §.§ Part-of-Speech Tagging * POS tagging aims to label each word with a unique tag that indicates its syntactic role, like noun, verb, adjective etc.* The best POS taggers are based on classifiers trained on windows of text, which are then fed to a bidirectional decoding algorithm during inference.* In general, models resemble a bi-directional dependency network, and can be trained using a variety of methods including support vector machines and bi-directional Viterbi decoders.§.§ Chunking * Chunking aims to label segments of a sentence with syntactic constituents such as noun or verb phrases. It is also called shallow parsing and can be viewed as a generalization of part-of-speech tagging to phrases instead of words.* Implementations of chunking usually require an underlying POS implementation, after which the words are compounded or chunked by concatenation.§.§ Named Entity Recognition * NER labels atomic elements in a sentence into categories such as “PERSON” or “LOCATION”.* Features to train NER classifiers include POS tags, CHUNK tags, prefixes and suffixes, and large lexicons of the labeled entities.§.§ Semantic Role Labeling * SRL aims to assign a semantic role to a syntactic constituent of a sentence.* State-of-the-art SRL systems consist of several stages: producing a parse tree, identifying which parse tree nodes represent the arguments of a given verb, and finally classifying these nodes to compute the corresponding SRL tags.* SRL systems usually entail numerous features like the parts of speech and syntactic labels of words and nodes in the tree, the syntactic path to the verb in the parse tree, whether a node in the parse tree is part of a noun or verb phrase etc. § DOCUMENT VECTORIZATION Document vectorization is needed to convert text content into a numeric vector representation that can be utilized as features, which can then be used to train a machine learning model on. This section talks about a few different statistical methods for computing this feature vector<cit.>. §.§ N-gram ModelN-grams are contiguous sequences of `n' items from a given sequence of text or speech. Given a complete corpus of documents, each tuple of `n' grams, either characters or words are represented by a unique bit in a bit vector, which, when aggregated for a body of text, form a sparse vectorized representation of the text in the form of n-gram occurrences. §.§ TF-IDF ModelTerm frequency - inverse document frequency (TF-IDF), is a numerical statistic that is intended to reflect how important a word is to a document in a collection or corpus <cit.>. The TF-IDF value increases proportionally to the number of times a word appears in the document, but is offset by the frequency of the word in the corpus, which helps to adjust for the fact that some words appear more frequently in general. It is a bag-of-words model, and doesn't preserve word ordering. §.§ Paragraph Vector Model A Paragraph Vector model is comprised of an unsupervised learning algorithm that learns fixed-size vector representations for variable-length pieces of texts such as sentences and documents <cit.>. The vector representations are learned to predict the surrounding words in contexts sampled from the paragraph. Two distinct implementations have gained prominence in the community. * Doc2Vec: A Python library implementation in Gensim. [https://radimrehurek.com/gensim/models/doc2vec.html].* FastText: A standalone implementation in C++. <cit.> <cit.>.§ A PRIMER OF NEURAL NET MODELS FOR NLP<CIT.> * Fully connected feed-forward neural networks are non-linear learners that can be used as a drop-in replacement wherever a linear learner is used.* The high accuracy observed in experimental results is a consequence of this non-linearity along with the availability of pre-trained word embeddings.* Multi-layer feed-forward networks can provide competitive results on sentiment classification and factoid question answering* Convolutional and pooling architecture show promising results on many tasks, including document classification, short-text categorization, sentiment classification, relation type classification between entities, event detection, paraphrase identification, semantic role labeling, question answering, predicting box-office revenues of movies based on critic reviews, modeling text interestingness, and modeling the relation between character-sequences and part-of-speech tags.* Convolutional and pooling architectures allow us to encode arbitrarily large items as fixed size vectors capturing their most salient features, but, they do so by sacrificing most of the structural information.* Recurrent and recursive networks allows using sequences and trees and preserve the structural information.* Recurrent models have been shown to produce very strong results for language modeling as well as for sequence tagging, machine translation, dependency parsing, sentiment analysis, noisy text normalization, dialog state tracking, response generation, and modeling the relation between character sequences and part-of-speech tags.* Recursive models were shown to produce state-of-the-art or near state-of-the-art results for constituency and dependency parse re-ranking, discourse parsing, semantic relation classification, political ideology detection based on parse trees, sentiment classification, target-dependent sentiment classification and question answering.* Convolutional nets are observed to to work well for summarization related tasks, just as recurrent/recursive nets work well for language modeling tasks.§ A NEURAL PROBABILISTIC LANGUAGE MODEL Goal: Knowing the basic structure of a sentence, one should be able to create a new sentence by replacing parts of the old sentence with interchangeable entities<cit.>. Challenge: The main bottleneck is computing the activations of the output layer, since it is a fully-connected softmax activation layer.Description: * One of the major contributions of this paper in terms of optimizations was data parallel processing (different processors working on a different subsets of data) and asynchronous processor usage of shared memory.* The authors propose to fight the curse of dimensionality by learning a distributed representation for words which allows each training sentence to inform the model about an exponential number of semantically neighboring sentences.* A fundamental problem that makes language modeling and other learning problems difficult is the curse of dimensionality. It is particularly obvious in the case when one wants to model the joint distribution between many discrete random variables (such as words in a sentence, or discrete attributes in a data-mining task).* State-of-the art results are typically obtained using trigrams.* Language generation via substitution of semantically similar language constructs of existing sentences can be done via shared-parameter multi-layer neural networks.* The objective of this paper is to obtain real-valued vector sequences of words and learn a joint probability function for those sequences of words alongside the feature vector, and hence, jointly learn both the real-valued vector representation and the parameters of the probability distribution.* This probability function can be tuned in order to maximize log-likelihood of the training data, while penalizing the cost function, similar to the penalty term one used in Ridge regression.* This will ensure that semantically similar words end up with an almost equivalent feature vectors, called learned distributed feature vectors.* A challenge with modeling discrete variables like a sentence structure as opposed to a continuous value is that the continuous valued function can be assumed to have some form of locality, but the same assumption cannot be made in case of discrete functions.* N-gram models try to achieve a statistical modeling of languages by calculating the conditional probabilities of each possible word that can follow a set of n preceding words.* New sequences of words can be generated by effectively gluing together the popular combinations i.e. n-grams with very high frequency counts.§ HIERARCHICAL PROBABILISTIC NEURAL NETWORK LANGUAGE MODEL Goal: Implementing ahierarchical decomposition of the conditional probabilities that yields a speed-up of about 200 both during training and recognition. The hierarchical decomposition is a binary hierarchical clustering constrained by the prior knowledge extracted from the WordNet[https://wordnet.princeton.edu/] semantic hierarchy<cit.>. Description: * Similar to the previous paper, attempts to tackle the `curse of dimensionality' (Section <ref>) and attempts to produce a much faster variant.* Back-off n-grams are used to learn a real-valued vector representation of each word.* The word embeddings learned are shared across all the participating nodes in the distributed architecture.* A very important component of the whole model is the choice of the words binary encoding, i.e. of the hierarchical word clustering. In this paper the authors combine empirical statistics with prior knowledge from the WordNet resource.§ A HIERARCHICAL NEURAL AUTOENCODER FOR PARAGRAPHS AND DOCUMENTS Goal: Attempts to build a paragraph embedding from the underlying word and sentence embeddings, and then proceeds to encode the paragraph embedding in an attempt to reconstruct the original paragraph<cit.>. Description: * The implementation uses an LSTM layer to convert words into a vector representation of a sentence. A subsequent LSTM layer converts multiple sentences into a paragraph.* For this to happen, we need to preserve, syntactic, semantic and discourse related properties while creating the embedded representation.* Hierarchical LSTM utilized to preserve sentence structure.* Parameters are estimated by maximizing likelihood of outputs given inputs, similar to standard sequence-to-sequence models.* Estimates are calculated using softmax functions to maximize the likelihood of the constituent words.* Attention models using the hierarchical autoencoder could be utilized for dialog systems, since it explicitly models for discourse.§ LINGUISTIC REGULARITIES IN CONTINUOUS SPACE WORD REPRESENTATIONSGoal: In this paper, the authors examine the vector-space word representations that are implicitly learned by the input-layer weights. These representations are surprisingly good at capturing syntactic and semantic regularities in language, and that each relationship is characterized by a relation-specific vector offset. This allows vector-oriented reasoning based on the offsets between words<cit.>. This is one of the seminal papers that led to the creation of Word2Vec, which is a state-of-the-art word embedding tool<cit.>. Description: * A defining feature of neural network language models is their representation of words as high dimensional real-valued vectors.* In this model, words are converted via a learned lookup-table into real valued vectors which are used as the inputs to a neural network.* One of the main advantages of these models is that the distributed representation achieves a level of generalization that is not possible with classical n-gram language models.* The word representations in this paper are learned by a recurrent neural network language model.* The input vector w(t) represents input word at time t encoded using 1-of-N coding, and the output layer y(t) produces a probability distribution over words. The hidden layer s(t) maintains a representation of the sentence history. The input vector w(t) and the output vector y(t) have dimensionality of the vocabulary.* The values in the hidden and output layers are computed as follows:s(t) = f(Uw(t) + Ws(t-1)) y(t) = g(Vs(t))where f(z) = 1/1 + e^-z and g(z_m) = e^z_m/∑_k e^z_k * One of the biggest features of having real-valued feature representations is the ability to compute the answer to an analogy question a:b; c:d where d is unknown. With continuous space word representations, this becomes as simple as calculatingy = x_b - x_a + x_cy is the best estimate of d that the model could compute. If there is no vector amongst the trained words such that y == x_w, the nearest vector representation can be estimated using cosine similarity.w^* = argmax_w x_w y/||x_w|| ||y||§ BETTER WORD REPRESENTATIONS WITH RECURSIVE NEURAL NETWORKS FOR MORPHOLOGY Goal: The paper aims to address the inaccuracy in vector representations of complex and rare words, supposedly caused by the lack of relation between morphologically related words<cit.>. Description: * The authors treat each morpheme as a basic unit in the RNNs and construct representations for morphologically complex words on the fly from their morphemes. By training a neural language model (NLM) and integrating RNN structures for complex words, they utilize contextual information to learn morphemic semantics and their compositional properties.* Discusses a problem that the Word2Vec syntactic relations likex_apples - x_apple≈ x_cars - x_carmight not hold true if the vector representation of a rare word is inaccurate to begin with.*operates at the morpheme level rather than the word level. An example of the this is illustrated in Figure <ref>. * Parent words are created by combining a stem vector and an affix vector, as shown in Equation <ref>.p = f (W_m [x_stem ; x_affix] + b_m) * The cost function is expression in terms of the squared Euclidean loss between the newly constructed representation p_c(x_i) and the reference representation p_r(x_i). The cost function is given in Equation <ref>.J(θ) = ∑_i=1^N (|| p_r(x_i) - p_c(x_i) ||^2_2) + λ/2 ||θ||^2_2 * The paper describes both context sensitive and insensitive versions of the Morphological RNN.* Similar to a typical RNN, the network is trained by computing the activation functions and propagating the errors backward in a forward-backward pass architecture.* This RNN model performs better than most of the other neural language models, and could be used to supplement word vectors.§ EFFICIENT ESTIMATION OF WORD REPRESENTATIONS IN VECTOR SPACE Goal: The main goal of this paper is to introduce techniques that can be used for learning high-quality word vectors from huge data sets with billions of words, and with millions of words in the vocabulary<cit.>. Challenge: The complexity that arises at the fully-connected output layer of the neural network is the dominant part of the computation. A couple of methods suggested to mitigate this is to use hierarchical versions of the softmax output activation units, or to refrain from performing normalization at the final layer altogether. Description: * The ideas presented in this paper build on the previous ideas presented by <cit.>.* The objective was to obtain high-quality word embeddings that capture the syntactic and semantic characteristics of words in a manner that allows algebraic operations to proxy the distances in vector space.man - woman = king - queenortell - told = walk - walked * The training time here scales with the dimensionality of the learned feature vectors and not on the volume of training data.* The approach attempts to find a distributed vector representation of values as opposed to a continuous representation of values as computed by methods like LSA and LDA.* The models are trained using stochastic gradient descent and backpropagation.* The RNN models are touted to have an inherently better representation of sentence structure for complex patterns, without the need to specify context length.* To allow for the distributed training of the data, the framework DistBelief was used with multiple replicas of the model. Adagrad was utilized for asynchronous gradient descent.* Two distinct models were conceptualized for the training of the word vectors based on context, both of which are continuous and distributed representations of words. These are illustrated in Figure <ref>.* Continuous Bag-of-Words model: This model uses the context of a word i.e. the words that precede and follow it, to predict the current word.* Skip-gram model: This model uses the current word to predict the context it appeared in.The experimental results show that the CBOW and skip-gram models consistently out-perform the then state-of-the-art models. It was also observed that after a point, increasing the dimensions and the size of the data began providing diminishing returns. § DISTRIBUTED REPRESENTATIONS OF WORDS AND PHRASES AND THEIR COMPOSITIONALITY Goal: This paper builds upon the idea of the Word2Vec skip-gram model, and presents optimizations in terms of quality of the word embeddings as well as speed-ups while training. It also proposes an alternative to the hierarchical softmax final layer, called negative sampling<cit.>. Description: * One of the optimizations suggested is to sub-sample the training set words to achieve a speed-up in model training.* Given a sequence of training words [w_1 , w_2 , w_3 , ... , w_T], the objective of the skip-gram model is to maximize the average log probability shown in Equation <ref>1/T∑_t=1^T ∑_-c ≤ j ≤ c; j ≠ 0log P(w_t+j, w_t)where c is the window or context surrounding the current word being trained on. * As introduced by <cit.>, a computationally efficient approximation of the full softmax is the hierarchical softmax. The hierarchical softmax uses a binary tree representation of the output layer with the W words as its leaves and, for each node, explicitly represents the relative probabilities of its child nodes. These define a random walk that assigns probabilities to words.* The authors use a binary Huffman tree, as it assigns short codes to the frequent words which results in fast training. It has been observed before that grouping words together by their frequency works well as a very simple speedup technique for the neural network based language models.* Noise Contrastive Estimation (NCE), which is an alternative to hierarchical softmax,posits that a good model should be able to differentiate data from noise by means of logistic regression.* To counter the imbalance between the rare and frequent words, we used a simple sub-sampling approach: each word within the training set is discarded with probability computed by the below formula.P(w_i) = 1 - √(t/f(w_i))This is similar to a dropout of neurons from the network, except that it is statistically more likely that frequent words are removed from the corpus by virtue of this method.* Discarding the frequently occurring words allows for a reduction in computational and memory cost.* The individual words can easily be coalesced into phrases using unigram and bigram frequency counts, as shown below.score(w_i, w_j) = count(w_i w_j) - δ/count(w_i) * count(w_j) * Another interesting property of learning these distributed representations is that the word and phrase representations learned by the skip-gram model exhibit a linear structure that makes it possible to perform precise analogical reasoning using simple vector arithmetic. § GLOVE: GLOBAL VECTORS FOR WORD REPRESENTATION Goal: This paper proposes a global log-bilinear regression model that combines the advantages of the two major model families in the literature: global matrix factorization and local context window methods<cit.>. Description: * While methods like LSA efficiently leverage statistical information, they do relatively poorly on word analogy tasks, indicating a sub-optimal vector space structure. Methods like skip-gram may do better on analogy tasks, but they poorly utilize the statistics of the corpus since they train on separate local context windows instead of on global co-occurrence counts.* The relationship between any arbitrary words can be examined by studying the ratio of their co-occurrence probabilities with various probe words.* The authors suggest that the appropriate starting point for word vector learning should be with ratios of co-occurrence probabilities rather than the probabilities themselves.* We can express this co-occurrence relation as shown belowF((w_i - w_j)^T w_k) = P_ik/P_jkThis makes the feature matrix interchangeable with its transpose.* An additive shift is included in the logarithm,log(X_ik) ⇒ log(1 + X_ik)which maintains the sparsity of X while avoiding the divergences while computing the co-occurrences matrix.* The model obtained in the paper could be compared to a global skip-gram model as opposed to a fixed window-size skip-gram model as proposed by <cit.>.* The performance seems to increase monotonically with an increase in training data.§ DISCUSSION Following the literature survey, this section re-visits the original research questions and provides a succinct summary that can be inferred from the experimental results and conclusions drawn from the original papers. RQ1 What are the relatively simple statistical techniques to extract features from text? Word count frequency models like n-gram and simple bag-of-words models such as TF-IDF are still the easiest tools to obtain an numeric vector representation of text.RQ2 Is there any inherent benefit to using neural networks as opposed to the simple methods? The benefit of using neural nets primarily is their ability to identify obscure patterns, and remain flexible enough for a varied set of application areas from topic classification to syntax parse-tree generation.RQ3 What are the trade-offs that neural networks incur as opposed to the simple methods? The trade-offs are typically expressed in terms of computational cost and memory usage, although model complexity is a factor too, given that neural nets can be trained to learn arbitrarily complex generative models.RQ4 How do the different techniques compare to each other in terms of performance and accuracy? This question can only be answered subjectively as it varies from application to application. Typically, document similarity can be tackled with a simple statistical approach like TF-IDF. CNNs inherently model input data in a manner that iteratively reduces the dimensionality, making it a great fit for topic classification and document summarization. RNNs are great at modeling sequences of text, which make them apt for language syntax modeling. Amongst the frameworks, GloVe's pre-trained word-embeddings perform better than vanilla Word2Vec, which is considered state-of-the-art.RQ5 In what use-cases do the trade-offs outweigh the benefits of neural networks? As explained for the previous question, for a simple information retrieval use case such as document ranking, models such as TF-IDF, and word PMI (pointwise mutual information) are sufficient, and neural networks would be overkill in such use-cases.§ CONCLUSION This paper has summarized the important aspects of the state-of-the-art neural network techniques that have emerged in recent years. The field of machine translation, natural language understanding and natural language generation are important areas of research when it comes to developing a range of applications from a simple chatbot, to the conceptualization of a general AI entity.The discussion section aggregates the results of the surveyed papers and offers a ready reference for new-comers to the field. For future work, it is intended to experimentally compare different word-embedding approaches to act as a bootstrapping method to iteratively build high quality datasets for future machine learning model usage. § ACKNOWLEDGMENTS The author would like to thank Dr. Pascal Poupart for his constructive feedback on the survey proposal. acl_natbib | http://arxiv.org/abs/1704.08531v1 | {
"authors": [
"Vineet John"
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"categories": [
"cs.CL",
"68T50"
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"primary_category": "cs.CL",
"published": "20170427122725",
"title": "A Survey of Neural Network Techniques for Feature Extraction from Text"
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^1Department of Physics, University of Guilan, Rasht 41635-1914, Iran^2Department of Physics, Shahid Beheshti University, Velenjak, Tehran 19839, Iran^3 School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran Corresponding author email: [email protected] Corresponding author email: [email protected] We introduce a pipeline including multifractal detrended cross-correlation analysis (MF-DXA) modified by either singular value decomposition or the adaptive method to examine the statistical properties of the pulsar timing residual (PTR) induced by a gravitational wave (GW) signal. We propose a new algorithm, the so-called irregular-MF-DXA, to deal with irregular data sampling. Inspired by the quadrupolar nature of the spatial cross-correlation function of a gravitational wave background, a new cross-correlation function, σ̅_×, derived from irregular-MF-DXA will be introduced. We show that, this measure reveals the quadrupolar signature in the PTRs induced by stochastic GWB.We propose four strategies based on the y-intercept of fluctuation functions, the generalized Hurst exponent, and the width of the singularity spectrum to determine the dimensionless amplitude and power-law exponent of the characteristic strain spectrum as ℋ_c(f)∼𝒜_yr(f/f_yr)^ζ for stochastic GWB. Using the value of Hurst exponent, one can clarify the type of GWs. We apply our pipeline to explore 20 millisecond pulsars observed by Parkes Pulsar Timing Array. The computed scaling exponents confirm that all data are classified into a nonstationary class implying the universality feature. The value of the Hurst exponent is in the rangeH∈ [0.56,0.87]. The q-dependency of the generalized Hurst exponent demonstrates that the observed PTRs have multifractal behavior, and the source of this multifractality is mainly attributed to the correlation of data which is another universality of the observed datasets. Multifractal analysis of available PTRs datasets reveals an upper bound on the dimensionless amplitude of the GWB, 𝒜_yr< 2.0× 10^-15.§ INTRODUCTION Pulsar timing has received extensive attention for astrophysical interests due to possessing a stable rotational mechanism <cit.>. The pulsar timing residual (PTR)which is an important observable, is defined by the difference between the measured time of arrival (TOA) and those anticipated by a timing model <cit.>.The observed PTR is a precise indicator to elucidate some interesting physical properties of pulsars and other cosmological and astrophysical foreground processes <cit.>. The influence of unknown physical phenomena on the variation of the pulsar's spin and spin-down and the presence of foreground effects impose randomness on the PTR. Therefore, the PTR is categorized in a (1+1)-dimensional stochastic process (where only one of the parameters is independent, while the other parameter is represented as a function of the independent parameter). Therefore, the stochastic nature of the data requires implying robust methods to extract reliable information from the PTR.Millisecond pulsars (MSPs) were first suggested as detectors of gravitational waves (GWs) by <cit.> and<cit.>because of the high stability and predictability of their rotational behavior (see also <cit.>). Indeed, GWs can be produced by different mechanisms ranging from the early epoch to the present era. Continuous wave sources <cit.>,burst sources <cit.> and stochastic backgrounds <cit.> are the most well-known classes among the GW sources. As an illustration,we refer to relic GWs, including GWs by cosmic strings and primordial perturbations <cit.>. The GWs are also produced in the formation of supermassive black holes and binary black hole mergers <cit.>. Recently, the GWs of black hole mergers have been detected by LIGO instruments, which can be an evidence of dark matter in the early universe or can correspond to the binary black hole of stellar origin <cit.>. Other classes of GWs include the continuous, inspiral, burst, and stochastic types of GWs <cit.>.Many approaches have beenproposed and utilized during past decades to detect mentioned types of GWs <cit.>. The very low amplitude of GWs, different sources and mechanisms for GW production on one hand, andthe extended range of frequency on the other hand lead to introducing various indirect approaches such as predictions of energy loss due to GW emission <cit.> and direct approaches such as detecting the effect of GWs on pulsar timing residuals <cit.>. The two main methods for detection of GWs are known as interferometers (such as LISA and LIGO) and pulsar timing arrays <cit.>.For the frequency interval ν∈ [10^-8,10^-6], there are several pulsar timing array projects that observe the imprint of GWs using pulsar timing detectors <cit.>. In the context of pulsar timing array approach, some famous projects have been proposed, namely the Parkes Pulsar Timing Array (PPTA) <cit.>, the European Pulsar Timing Array (EPTA) <cit.>, the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) <cit.> and International Pulsar Timing Array (IPTA) <cit.>. The Square Kilometer Array (SKA) <cit.> radio telescope would further improve the sensitivity of pulsar timing measurements to detect GWs. For a recent and more complete discussions on various experiments and methods to detect GWs, see<cit.> . A pulsar timing array utilizing the Parkes radio telescope in Australia is an experiment to detect GWs by observing 20 bright MSPs <cit.>. The PPTA observations must be continued more than 5 yr in order to achieve a precision of 100 ns. Since pulsar timing residuals are induced by GWs <cit.>, therefore some authors used the statistical correlation of pulsars timing residuals to evaluate their capability of detecting GWs <cit.>.In order to elaborate the benchmark of different types of GWs based on PTR datasets,we should consider four aspects. First of all, various sources of GWs encourage us to find deep insight regarding the properties of GWs emitted by different sources. Second, we should examine the performance of different statistical methods and their sensitivities. Third, we should take into account the quadrupolar signature of GWs. Finally, the upper limit on the amplitude of the gravitational wave background (GWB) should be computed. The main method to detect a stochastic GW employing pulsar timing arrays is to search for a correlation between PTRs and compare it with quadrupole spatial cross-correlation calculated by Hellings and Downs <cit.>. If such a signature is not detected, one can set an upper bound on the amplitude of GWs using frequentist or Bayesian approaches. The upper limits provided by <cit.> and <cit.> for a stochastic background produced by supermassive black hole binaries is 𝒜_yr≤ 6×10^-15 and 𝒜_yr≤ 1×10^-15, respectively, where the latter is the lowest claimed upper limit so far. In addition, <cit.> have used the high-precision pulsar timing data recorded as part of the NANOGrav project and finally providedan upperlimitonthe power-spectrum amplitude of the nHz-frequency stochastic supermassive black hole GWB.Pulsar datasets are manipulated by trends and noises. Statistical models for noises, trends, and signals play crucial roles in any parametric GW detection approaches. Subsequently, it is necessary to implement robust and novel methods for removing destructive effects from desired parts of signals.Our work in this paper has the following advantages and novelties. i) Inspired by theproperties of a self-similar process characterized by a scaling exponent called "Hurst exponent" <cit.>, for the first time, we have used Multifractal Detrended Fluctuation Analysis (MF-DFA) <cit.>, Multi-Fractal Detrended Moving Average Analysis (MF-DMA) <cit.>and Multifractal Detrended Cross-correlation Analysis (MF-DXA) <cit.> methods to analyze the observed (including 20 MSPs inferred from <cit.>) and simulated pulsar timing residuals induced by GW signals (simulated by the TEMPO2 software package <cit.>).We will evaluate the multiscaling behavior of the underlying data from statistical point of view.ii) We modify MF-DFA, MF-DMA and MF-DXA by additional denoising algorithms, namely Adaptive Detrending (AD) <cit.> or Singular Value Decomposition (SVD) <cit.> methods to exclude or at least to reduce the contribution of unknown trends and noises as much as possible.These methods are used as precomplementary denoising procedures.iii) The standard version of multifractal analysis is a reliable algorithm when the input is a regular sampling series. Observed PTRs are unevenly sampled data sets; we therefore modify parts of the MF-DXA algorithm and call it the irregular MF-DXA method. In addition, noise modeling can be revealed by multifractal analysis.iv) We check the multifractal nature ofpulsar timing residuals. We also determine the sources of multifractality based on our statistical approaches.v) According to the quadrupolar signature on the spatial cross-correlation function of PTRs, the detectability of the stochastic GWB is evaluated according to the MF-DXA of PTRs. The cross-correlation exponent will be determined. We also give a new spatial cross-correlation functionfor pulsar timing residues.vi) We introduce some criteria not only for discrimination of the stochastic GWB footprint and single sources GWs on pulsar timing residuals but also for determining the dimensionless amplitude of GWB. Then, an upper bound on the amplitude of stochastic GWs will be computed. Finally, this view provides a new insight to use pulsar timing residuals for further astrophysical studies.The rest of this paper is organized as follows. In section <ref>,we will explain MF-DFA, MF-DMA, and MF-DXA, dealing with irregular sampled data, AD, and SVD in detail. A newmeasure for the spatial cross-correlation of PTR is presented in this section. Noise and trend modeling andposterior analysis to obtain scaling exponents are also discussed in section <ref>. Section <ref> is devoted to the theoretical notions of the GWB and data description for observed as well as synthetic datasets. We will implement the multifractal methods on simulatedtiming residuals series in section <ref>. We will also study a new spatial cross-correlation function derived by the MF-DXA method in the search of the footprint of the stochastic GWB in the sensitive range of pulsar timing residual series.Four strategies to reveal the imprint of GWs on the residual time series in a noiseless observation will alsobe explained in section <ref>. We will examine the multifractality of the observed pulsar timing residuals in section <ref>. In that section, we will also look for an upper bound on the amplitude of stochastic GWs using some observed PTRs. Section <ref> is devoted to summary and conclusion.§ METHODOLOGY: MULTIFRACTAL ANALYSISNonstationary sources such astrends and artificial noises usually influence the observed time series. To infer reliable results, these spurious effects should be well characterized and distinguished from the intrinsic fluctuations. Concerning trends, <cit.> stated that, in principle, there is no universal definition for trends, and any proper algorithm for evoking trends from underlyingseries should remove the contribution of trends withoutdestroying fluctuations. One of the well-studied methodsfor this purpose is MF-DFA <cit.>, used in various areas, such as economical time series <cit.>, river flow <cit.>, sunspot fluctuations <cit.>, cosmic microwave background radiations <cit.>, music <cit.>, plasma fluctuations <cit.>, identification of a defective single layer in two-Dimensional material <cit.>, traffic jamming <cit.>, image processing, medical measurements <cit.>, and astronomy <cit.>. Cross-correlation has also been introduced and applied in some cases <cit.>. The MF-DXA examining higher-order detrended covariance was introduced by <cit.>. Although the approaches in multifractal detrended analysis, such as the MF-DFA and MF-DXA methods, diminish polynomial trends, previous researche demonstrated that sinusoidal and power-law trends are not completely removed <cit.>. Mentioned trends make some crossovers in fluctuation functions <cit.>. Several robust methods have been proposed to eliminate cross-overs produced by sinusoidal and power-law trends: Fourier Detrended Fluctuations Analysis (F-DFA) <cit.>,Singular Value Decomposition (SVD) <cit.>, Adaptive Detrending method (AD) <cit.>, and Empirical Mode Decomposition (EMD)<cit.>. In this work, we implement the AD and SVD methods to reduce the contribution of noise and magnify the effect of GWs in our results for further cleaning preprocessors.§.§ Multifractal-based analysisFinding scaling exponents in the context of auto-correlation and cross-correlation analysis has many inaccuracies due to nonstationarity,noises, and undesired trends. To resolve the mentioned difficulties, a well-known method based on decomposing the original signal into its positive and negative fluctuation components has been proposed by <cit.>. Motivated by such adecomposition method, Podobnik et al. introduced the cross-correlation between two non-stationary fluctuations by means of the DFA method <cit.>. A modification of detrended cross-correlation analysis (DCCA) is known as MF-DXA was invented by <cit.>. The pipeline of MF-DXA is considered as follows <cit.>[If both signals are identical, we have the MF-DFA/MF-DMA method.].(1): We consider two typical PTR series named byPTR_a and PTR_b, located at n̂_a and n̂_b with respect to the line of sight, respectively, as the input data sets to study their mutual multifractal property:PTR_a(i),PTR_b(i),i=1,...,NThe pulsar timing observations are almost unevenly sampled datasets.We need equidistant sampling series. A trivial but not essentially optimum way is to interpolate between two successive data.Different methods to reconstruct regular serieswill be explained in subsection <ref>. Therefore, here we assume that the input data are regular and ready for further tasks. Moreover, the observed data have variable error bars, and, to take into account heteroskedasticity, we use error-propagation formalism in all statistical analysis, such as averaging, fitting, and computing fluctuation functionsthroughout this paper.(2): To magnify the hidden self-similarity property, we make profile series according to:X_♢(j) = ∑_i=1^j [ PTR_♢(i)-⟨ PTR_♢⟩],j=1,...,NHere the subscript ♢ can be replaced by "a" or "b".(3-a): The above profile series must be divided into N_s=int(N/s) nonoverlapping segments of length s. The range of nonoverlapping window values is N_s∈[N_s^ min,N_s^ max]. To take into account the remaining unused part of the data from the opposite end of the data, the enumeration must to be repeated from the mentioned part. In this case, we will have 2N_s segments. In the framework of the MF-DCCA method, we should compute the following fluctuation function in each segment as follows:ℰ_×(s,ν)=1/s∑^s_i=1[ X_a(i+(ν-1)s)-X̃_a^(ν)(i)] ×[ X_b(i+(ν-1)s)-X̃_b^(ν)(i)]for segments ν=1,...,N_s. For the opposite end, we have:ℰ_×(s,ν)=1/s∑^s_i=1[ X_a(i+N-(ν-N_s)s)-X̃_a^(ν)(i)] ×[ X_b(i+N-(ν-N_s)s)-X̃_b^(ν)(i)]where ν=N_s+1,···,2 N_s and X̃_♢^(ν)(i) is a weighted fitting polynomial function in the νth segment with an arbitrary order describing the local trend for data with variable error bars. Usually a linear function for modeling local trends is considered <cit.>. The MF-DCCAm denotes that the order of the polynomial function used in the MF-DCCA is "m”. Throughout this paper, we take m=1 unless stated otherwise. To reduce the statistical uncertainties in the computed fluctuation functions, we set s>m+2 <cit.>. On the other hand, this method becomes unreliable for very large window sizes, i.e. s>N/4. There is a discontinuity for fitting a polynomial at the boundary of each partition in the MF-DCCA method; to resolve this discrepancy, MF-DMA has been introduced <cit.>. Accordingly, instead of doing item (3-a), we carry out the following procedure:(3-b):For each moving window with size s, we calculate the moving average function:X_♢(j)=1/s∑_k=-s_1^s_2X_♢(j-k)where s_1=⌊(s-1)θ⌋ and s_2=⌈(s-1)(1-θ)⌉. The symbol ⌊ a ⌋ represents the largest integer value not greater than a and ⌈ a⌉ is devoted to the smallest integer value not smaller than a. In the above equation, θ plays a crucial role. The θ=0 refersto the backward moving average, while θ=1 is the so-called forward moving average; finally θ=0.5 is related to the centered moving average <cit.>. Therefore, detrended data are constructed by subtracting the calculated moving average function from the cumulative series, X_♢ as:ε_X_♢(i)= X_♢(i)-X_♢(i)where s-s_1 ≤ i≤ N-s_1.Now ε_X_♢(i) values are divided into N_s= int[N/s] nonoverlapping windows with the same size of s and we calculate the fluctuation function:ℰ_×(s,ν) = 1/s∑_i=1^sε_X_a(i+(ν-1)s)×ε_X_b(i+(ν-1)s) (4): Using Eqs. (<ref>) and (<ref>) for the MF-DCCA (MF-DFA) based method <cit.> and Eq. (<ref>) for the MF-DMA algorithm, the corresponding qth-order fluctuation function can be computed by:ℱ_×(q,s)=( 1/ 2N_s∑^ 2N_s_ν=1 |ℰ_×(s,ν)|^q/2)^1/qFor q=0, we have:ℱ_×(0,s)=exp( 1/4 N_s∑^ 2N_s_ν=1ln |ℰ_×(s,ν)| ) (5): The scaling behavior of the fluctuation function according to:ℱ_×(q,s)∼ s^h_×(q)gives the cross-correlation exponent h_×(q). The q-parameter enables us to quantify the contribution of different values of fluctuation functions in Eqs. (<ref>) and (<ref>).The small fluctuations play a major role in summation for q<1, while large fluctuations become dominant for q≥1.We emphasize that for heteroskedastic data, the summation in Eqs. (<ref>) and (<ref>) should incorporate variable errorbars, and weighted fitting polynomials must be considered. It turns out that for a=b, the usual generalized Hurst exponent, h(q), is retrieved. In this case we have:ℱ_q(s)= 𝒢_h(q)s^h(q)for q=2, the 𝒢 is𝒢 = σ^2/2H+1-4σ^2/2H+2+3σ^2 ( 2/H+1-1/2H+1 ) -3σ^2/H+1 (1-1/(H+1)(2H+1))and σ^2=⟨ PTR^2⟩ for zero mean data. Any q-dependency of h(q), confirms that the underlying data set is a multifractal process. For the class of the nonstationary series (corresponding to a fractional Brownian motion; fBm) the exponent derived by using MF-DFA is h(q=2)>1. Therefore, in this case, the Hurst exponent is given by H=h(q=2)-1. In the stationary case, h(q=2)<1 (corresponding to a fractional Gaussian noise; fGn) and H=h(q=2). For completely stationary random data, H=0.5, while for a persistent data set,0.5<H<1.0. For an anticorrelated data set, H<0.5 <cit.>. When the Hurst exponent is determined, the scaling exponents of autocorrelation for an fGn process read as 𝒞(τ)=⟨ x(t)x(t+τ)⟩∼τ^-γ for τ≫0 with γ=2-2H, while for a fBm signal, we have 𝒞(t_i,t_j)=⟨ x(t_i)x(t_j)⟩∼ t_i^-γ+t_j^-γ-|t_i-t_j|^-γ for |t_i-t_j|≫0 with γ=-2H. The associated power spectrum is S(f)∼ f^-β with β = 2H-1 and β =2H+1 for the fGn and fBm processes, respectively. The relation between the generalized Hurst exponent and the scaling exponent of the partition function known as the multifractal scaling exponent based on the standard multifractal formalism becomes<cit.>:ξ(q)=qh(q)-1For a monofractal data set, ξ(q) is a linear function <cit.>. The generalized multifractal dimension is also given by:D(q)=ξ(q)/q-1=qh(q)-1/q-1where D(q = 0) = D_f is the fractal dimension of the time series and D(q = 1) is related to the so-called entropy of the underlying system <cit.>. A more complete quantitative measure of multifractality is the singularity spectrum and indicates how the box probability ofstandard multifractal formalism behaves at small scales. It is defined by the Legendre transformation of ξ(q) as <cit.>:f(α)=α q-ξ(q)and the Hölder exponent is α≡ dξ(q)/dq. In the case of multifractality, a spectrum ofthe Hölder exponentis obtained instead of a single exponent. The domain of the Hölder spectrum, α∈ [α_ min,α_ max], becomes <cit.>:α_ min= lim_q→ +∞∂ξ(q)/∂ q,α_ max= lim_q→ -∞∂ξ(q)/∂ qSubsequently, the width Δα≡α_ max-α_ min is a reliable measure for quantifying the multifractal nature of the underlying data. The higher value of Δα is associated with the higher multifractal nature reflecting the complexity of the signal. As other complexity measures, one can point to the q-orderLyapunov exponent <cit.>, and the Lempel-Ziv complexity <cit.>. Inspired by the common cross-correlation definition, relying on Eq. (<ref>), we define the new cross-correlation function <cit.>:σ_×(Θ_ab)≡∑_s(∑^ 2N_s_ν=1ℰ_×(s,ν)/√([∑^ 2N_s_ν=1ℰ_a(s,ν)][∑^ 2N_s_ν=1ℰ_b(s,ν)]))here Θ_ab=arccos|n̂_a.n̂_b|. Averaging on all available pairs separated by Θ leads to:σ̅_×(Θ )=1/4π∫ dΩσ_×(Θ_ab)The σ̅_× introduced by Eq. (<ref>) based on fluctuation functions computed in the context of detrended cross-correlation containsthe quadrupolar signatureif PTRs are modified by the GWB signal. Therefore, this is a new criterion that enables us to assess the footprint of GWs more precisely.Now we turn to the spatial cross-correlation function for PTRs taking into account stationarity as: 𝒞_×(Θ_ab) = ⟨ PTR_a(t,n̂_a)PTR_b(t,n̂_b)⟩_tIn the presence of an isotropic GWB, by averaging the cross-correlation on all available pairs separated by Θ leads to:𝒞_×(Θ) = ⟨ 𝒞_×(Θ_ab)⟩_ pairs∼Γ(Θ)The Γ(Θ) is given by the Hellings and Downs equation <cit.>:Γ(Θ)=3/2ψln(ψ)-ψ/4+1/2where ψ≡[1-cos(Θ)]/2. We should notice that the Hellings and Downs curve is only a function of the angular separation between pulsar pairs separated by Θ, and it is independent of the frequency <cit.>.The new cross-correlation coefficient defined by Eq. (<ref>) is related to the traditional cross-correlation 𝒞_× in a complex way, and the relation is not analytically tractable without any approximation, and we will evaluate it numerically in the next section. However, according to Eq. (<ref>), the mapping between 𝒞_× and σ_× does not change the sign of σ_×. Thus, the quadrupolar signature of the Hellings and Downs function is preserved. It is worth mentioning that, besides probable GW signal superimposed in the PTRs, the following fluctuations can be existed in the recorded data: the correlated red (fractal) noise; clock errors, which are the same in all pulsars (i.e., monopolar); and ephemeris errors (which are dipolar).There are no known noise sources other than GWs that are quadrupolar <cit.>. Applying MF-DXA on PTRs determines the value of the temporal scaling exponent, h_×.We expect to find constant h_×(q) with respect to different separation angles (Θ) for an isotropic GWB, while for the other local source of GWs, the h_×(q) depends on Θ_ab in an arbitrary manner.§.§ Dealing with irregularly sampled dataThe pulsar timing observations are unevenly sampled; i.e. they are not a set of equidistant sampling values, and the underlying series is nonuniform, requiring some sort of interpolation technique. The Lomb-Scargle periodogram proposed a least-squares pipeline to resolve this problem<cit.>. Radon transformations have also been used for irregular sampling analysis <cit.>; see also <cit.> and references therein. Extrapolation of irregularly recorded data onto a regular grid was introduced by <cit.>. For constructing Fourier expansion,nonuniform discrete Fourier transform was introduced by <cit.>. A trivial but not necessarily optimum method with less computational burden is to interpolate between two successive data pointsin recorded series. A more robust method is to apply kernel functions on the irregular data, as (see also <cit.>):PTR_reg(t)=∫ dt' PTR_irre(t')𝒲(t-t')where PTR_reg and PTR_irre are regular and nonuniform sampled data, respectively. Here 𝒲 is a normalized window function. A typical functional form for this window function can be Gaussian.In general, the choice of the window function, 𝒲, depends on the smoothness, accuracy requirements, and computation efficiency <cit.>.Here we propose a new approach to find robust scaling properties for irregular sampled data. If there is no a priori information for the smoothing procedure, we suggest applying a gaussian kernel to the data followed by a linear interpolation to regularize datasets. Subsequently, we can construct the profile using such regular data (Eq.(<ref>)).To reduce the contribution of artificial data points produced in this interpolation, we introduce tge irregular MF-DXA method. In this new algorithm, we modify the fluctuation function procedure given by Eqs. (<ref>) and (<ref>)for identical PTRs as:ℰ^2(s,ν)=1/s_ν'(s)∑^s_ν'(s)_i=1[ X(i+(ν-1)s')-X̃_ν(i)]^2In the above equation, only the data points recorded during observation in each segment with size s will be considered for further computations. Therefore, the number of data in the νth window with size s is representedby s_ν'(s) which in general is not equal to s. Now Eq. (<ref>) becomes a weighted average:ℱ_q(s)=(∑^2 N_s_ν=1[ℰ^2(s,ν)]^q/2/σ_ℰ^2(s,ν,q)/∑^2 N_s_ν=11/σ_ℰ^2(s,ν,q))^1/qwhere σ_ℰ^2(s,ν,q) is the variance of [ℰ^2(s,ν)]^q/2. We similarly replace the averaging procedure in any relevant parts with the weighted averaging.Recently, <cit.> showed that the global scaling exponents of long-correlated signals remain unchanged for up to 90% of data loss, while for anticorrelated series, even less than 10%of data loss creates a significant modification in the original scaling exponents. This research shows that one can compute the scaling exponents for long-range correlated irregularly sampled data points if one regularizes the data set through linear interpolation and then applies DFA. But for an anticorrelated signal, the DFA method does not lead to reasonable results. Our new proposal demonstratesthatfor synthetic series with known Hurst exponents, our modification leads to more reliable estimations for scaling exponents, not only for correlated series but also for anticorrelated datasets. Our simulations show that the PTR can be considered as long-range correlated fluctuation. Therefore, our results are almost are not affected by the type of regularization.§.§ SVDIt is important to find trends and noise sectors in data analysis, especially in the astronomical data. When we use MF-DFA, MF-DMA, and MF-DXA, an essential demand corresponding to presenting a scaling behavior must be satisfied, as represented by Eqs. (<ref>) and (<ref>).In some cases, there exist one or more crossovers corresponding to different correlation behaviors of the pattern in various scales <cit.>. The MF-DFA and MF-DXA methods cannot remove the effect of all undesired parts of the underlying signal; therefore, we implement complementary tasks to properly recover the scaling behavior of fluctuation functions properly and to obtain the reliable scaling exponents. There are some preprocessing methods for denoising in the literature; for instance, The EMD method <cit.>, the Fourier-detrended (Fourier-based filtering) method <cit.>, the SVD method <cit.> and the AD algorithm <cit.>. In this paper, we utilize the SVD method and AD algorithm. The main part of the SVD method can be described in the following steps <cit.>:(I): Construct a matrix whose elements are PTRs in the following order:Γ≡( [PTR_1PTR_1+τ... PTR_1+N-(d-1)τ-1;⋮⋮⋮⋮;PTR_iPTR_i+τ... PTR_i+N-(d-1)τ-1;⋮⋮⋮⋮;PTR_dPTR_d+τ... PTR_d+N-(d-1)τ-1;])whered is the embedding dimension, τ is the time delay, and 1 ≤ i≤ d. Considering a time series of size N, the maximum value of the embedding dimension d is equal to d≤ N-(d-1)τ+1 <cit.>.(II): Decompose the matrix Γ to left (𝐔_d× d) and right (𝐕_(N-(d-1)τ)×(N-(d-1)τ)) orthogonal matrices:Γ=𝐔𝐒𝐕^†where 𝐒_d× (N-(d-1)τ) is a diagonal matrix and its elements are the desired singular values. If we are interested in examining the fluctuations with high frequency, we should remove dominant wavelengths. In this case, for removing trends containing p dominant wavelengths, we set 2p+1 largest eigenvalues of matrix 𝐒 to zero; therefore, long periods or short frequencies are eliminated. In other words, p dominant eigenvalues and associated eigenvectors correspond to long wavelength (short-frequency part) subspace, while d-p eigenvalues and the corresponding eigen-decomposed vectors represent short-wavelength (high-frequency part) subspace.In this paper, we look for the footprint of GWs superimposed on the PTRs signals. As shown in Fig. <ref>, the GW part behaves as a dominant trend in PTRs; consequently, we essentially need to do denoising using the SVD method to magnify the contribution of superimposed GWs. To this end, we should remove small eigenvalues corresponding to a low-pass filter. In this paper, we eliminate the high-frequency part of the signal by keeping the 2p+1 largest eigenvalues of the matrix 𝐒. Finally, the new eigenvalues matrix, 𝐒̃, is determined. According to the filtered matrix, Γ̃=𝐔𝐒̃𝐕^†, the cleaned time series is constructed by:PTR_i+j-1=Γ̃_ij.Here 1≤ i≤ d and 1≤ j≤ N-(d-1)τ. Now the cleaned PTR datasets will be used as input for the MF-DFA or MF-DXA discussed in previous subsections.§.§ AD algorithm Another robust algorithm to examine trends is the AD method introduced by <cit.>. The implementation of the AD algorithm is a complementary method for determining local and global trends. Therefore, after applying the AD method on observed pulsar timing series, the corresponding dominant trend output data will be used as an input for the MF-DFA or MF-DXA methods. The AD method includes the following steps <cit.>. A discrete series, PTR(i) with i=1,⋯,N is partitioned with overlapping windows of length 2n+1 and, accordingly,each neighboring segment has n+1 overlapping points. An arbitrary polynomial Y is constructed in each window of length 2n+1.In order to have the continuous trend function avoid a typical sharp jump in it, the following weighted function for the overlapping part of the νth segment is considered <cit.>:Y_ν^ overlap(j)=(1-j-1/n) Y_ν(j+n)+j-1/n Y_ν+1(j)where j=1,2,⋯,n+1. The two free parameters, namely n and the order of the fitting polynomial, should be determined properly <cit.>.The size of each segment was calculated by 2n+1= 2× int[(N-1)/(w_ adaptive+1)]+1. It turns out that by increasing the value of w_ adaptive and the order of the fitting polynomial, the fluctuations disappear, and, consequently,the fluctuations are suppressed. For the nonoverlapping segments, the AD data are given by PTR(i)- Y_ν (i), while for the overlap part it isPTR(i)- Y_ν^ overlap(i). Since the GW, as the dominant part of the signal, is our desired part of the signal, we instead use PTR(i)= Y_ν (i), while for the overlap part, we consider PTR(i)= Y_ν^ overlap(i). Now PTR(i) is used for further analysis in MF-DFA or MF-DXA.§.§ Trend and noise modeling In real observational data to carry out parametric detection, reliable statistical models of the noise and signal should be well established. A proposal for noise modeling is based on the denoiseing procedure carried out by the SVD or AD algorithms. Previously, we were interested in removing the contribution of undesired noise modulated on real data. Now weconcentrate on the PTR given by Eq. (<ref>) in the context of SVD analysis as a model of trends andPTR- PTR for noise. Also, if we use the AD approach, the global variation part of the signal corresponds to both 𝒴 and 𝒴^ overlapp (Eq. (<ref>)). For the noise part, we should consider PTR(i)- Y_ν (i), while for the overlap part, it is PTR(i)- Y_ν^ overlap(i). Therefore, SVD or AD, as well as the internal part of the MF-DFA and MF-DXA algorithms, are able to give a robust model for trends and noise. Also, extracting intrinsic functions based on EMD can be a good proposal for this purpose <cit.>. §.§ Posterior Analysis In this paper, we turn to Bayesian statistics <cit.> to compute the reliable value of the generalized Hurst exponent (Eqs. (<ref>) and (<ref>)). Let {𝒟}:{ℱ_q(s)} and {Υ} : {h(q)} represent the measurements and model parameters, respectively. The posterior function is defined by:𝒫(Υ |𝒟)=ℒ(𝒟|Υ)𝒫(Υ)/∫ℒ(𝒟|Υ)𝒫(Υ)dΥwhere ℒ is the likelihood and 𝒫(Υ) is the prior probability function including all information concerning model parameters. Here we adopt the top-hat function for 𝒫(h(q)) in the interval h(q)∈[0,4]. According to the central limit theorem, the functional form of likelihood becomes multivariate Gaussian, i.e. ℒ(𝒟|Υ)∼exp(-χ^2/2). The χ^2 for determining the best-fit value for the scaling exponent coordinated by multifractal formalism reads as:χ^2(Υ)≡Δ^†.C^-1.Δwhere Δ≡ [ℱ_q^ obs.-ℱ_q^ the.] and C is the covariance matrix. The ℱ_q^ obs.(s) and ℱ_q^ the.(s;h(q)) are fluctuation functions computed directly from the data and determined by Eqs.(<ref>) or (<ref>), respectively. In the case of the diagonal covariance matrix, the χ^2 becomes:χ^2(h(q))=∑_s=s_ min^s=s_ max[ℱ_q^ obs.(s)-ℱ_q^ the.(s;h(q))]^2/σ_ obs.^2(s)Here σ_ obs.(s)=⟨[δℱ_q^ obs.(s)]^2⟩, which is related to the diagonal elements of C and can be computed using a standard statistical error propagator from primary uncertainties on PTR datasets (Eq. (<ref>) to Eqs. (<ref>) and (<ref>)). The 1σ error bar of h(q) is determined by:68.3%=∫_-σ^-_h(q)^+σ^+_h(q)ℒ(ℱ_q(s)|h(q))dh(q)Subsequently, we report the best value of the scaling exponent at a 1σ confidence interval as h(q)_-σ^-_h(q)^+σ^+_h(q). § DATA DESCRIPTIONIn this section, we will describe theoretical models for GW signals. The observational datasets, synthetic series for pure timing residuals, and GWs,in order to examine the multiscaling behavior ofPTRs as an indicator of GWs, will be described in this section. §.§ Theoretical notions of the GWB on PTRsThe potential sources of GWs could be massive accelerated objects <cit.>, burst sources <cit.> or stochastic background sources <cit.>. Isotropic stochastic GWB produced by coalescing supermassive binary black hole mergers is the strongest potentially detectable signal of GWs <cit.>. Therefore, we use the GWB model to produce synthetic data. The characteristic strain spectrum, ℋ_c(f), for a stochastic GWB can be described by the power-law relation <cit.>:ℋ_c(f) = 𝒜_yr(f/f_1yr)^ζwhere f is the frequency of GWs, f_1yr≡1/1yr; 𝒜_yr is the dimensionless amplitude of the GWB; and ζ is a scaling exponent and for almost all expected GWsis ζ <0. The corresponding ζ exponent takes the following values for different mechanisms:ζ=-2/3 for coalescing black hole binaries, ζ=-1 for cosmic strings, and ζ=-7/6 for primordial GWs from the Big Bang <cit.>. We should mention that the power-law relation obtained in Equation (<ref>) is not unique and there is another framework represented by <cit.>. The dimensionless amplitude of GWs has been predicted by most authors in the range of 𝒜_yr∈[10^-15,10^-14]; however, according to Refs. <cit.> the expected range of 𝒜_yr for a stochastic GWB is 𝒜_yr∈ [10^-16 , 3×10^-15]. §.§ Synthetic Data Sets for GWBTo simulate synthetic series, we use the TEMPO2 software package that carries out the fitting procedure of TOA <cit.>. This package is used to simulate pure timing residuals <cit.>. To simulate the GWB, the "GWbkgrd" plug-in of TEMPO2 will be used <cit.>. In the absence of GW signal, we have pure pulsar timing residual represented byPTR_ pure, while signal induced by GWB is indicated by PTR(t). In order to test the effect of GWs on the PTRs, we simulate 100 timing residuals with 1076 data points that are separated by 13 days with an rms of 100 ns. Then we add the effect of GWB on the simulated pure PTR using different seeds for a given 𝒜_yr. The chosen accuracy for simulation has been used in other work as a level at which a GWB might be detected <cit.>; however, it should be noted that only two of the PPTA pulsars (J0437-4715 and J1909-3744) have rms noise of this order (Table <ref>). The GWB introduces two terms for each polarization, one set of which is referred to as the Earth terms. These Earth terms are correlated. However, the other set, referred to as the pulsar terms, has equal amplitude but a long and unknown time delay, so these terms are effectively uncorrelated noise with the same red spectrum as the Earth terms. Our simulations include both the Earth and the pulsar terms. We simulate 20 pure PTRs for pulsars separated in the sky according to the ephemeris of 20 MSPs observed in the PPTA project (Table <ref>). An isotropic GWB induces a particular spatial cross-correlation in PTRs leading to a quadrupolar signature (Hellings and Downs curve) <cit.>. Subsequently, to examine the GWB, we will examine the cross-correlation property of the simulated data. The upper panel of Fig. <ref> indicates a typical pure timing residual simulated by TEMPO2 with zero mean uncorrelated series.We also depict the superposition of pure timing residuals with the GW model introduced in <cit.>, in the middle panel of Fig.<ref>. §.§ Observed DataWe use the timing residual data of 20 MSPs observed by the PPTA project at three bandwidths, namely10, 20, and 50cm, by implementing the Parkes 64 m radio telescope (PTA) <cit.>. The PTA telescope is located in Australia at an altitude of -33^∘ and can observe all of the inner Galaxy. Due to the higher stability of the short-period MSPs,the observed pulsars have short periods and are selected from bright ones. Also, these MSPs have narrow pulse widths in order to reduce uncertainties in the corresponding TOA. Finally, isolated wide-binary MSPs have been selected to avoid the effects of the companion star.The PTR series for these MSPs as observed datasets are publicly available[ <https://datanet.csiro.au/dap/>]. We have used the TEMPO2 software to extract post-fitted PTRs from timing model data presented by <cit.>. The spectralModel[<http://www.atnf.csiro.au/research/pulsar/tempo2>] plug-in is utilized for temporal smoothing and making an equally spaced grid of observed data <cit.>. Then, we applied our analysis on post-fitted data. The names of 20 MSPs with the corresponding rms and total time span are reported in Table <ref>. It is worth noting that several phenomena, such as atmospheric delays, vacuum retardation due to observatory motion, Einstein delay, and Shapiro delay, can affectthe TOA <cit.> and they should be dismissed to have apost-fitted timing residual that is called PTR. The lower panel of Fig. <ref> illustrates a typical post-fit pulsar timing residual of PSR J0437-4715 observed by the PPTA project <cit.>. The fitting procedure has been done with the TEMPO2 software.§ MULTIFRACTAL ANALYSIS OF SYNTHETIC PTR SERIESIn this section we will evaluate the multifractal nature of synthetic datasets. The capability of our analysis as a detector of gravitational waves and a pipeline for determining the type of GWB will be explained in this section. §.§ Multifractal nature of synthetic DataAt first, we examine the multifractal nature of synthetic PTR_ pure and its superposition with simulated GWB. Since, in simulation, our data are regular, therefore we apply common assessment algorithms. Fig. <ref> illustrates the fluctuation functions versus scale computed by DMA for PTR_ pure. The results derived by the DFA method are in agreement with the DMA algorithm.The average value of the Hurst exponent for all simulated pure PTRs is ⟨ H⟩=0.51±0.02 at a 1σ level of confidence, confirming that PTR_ pure is an uncorrelated data set <cit.>.Now we superimpose the syntheticPTR_ pure(t) with simulated GWB with a given set of free parameters. We apply DFA and DMA on simulated PTR(t) for various GWB amplitudes. Fig. <ref> illustrates ℱ_2(s) as a function of s for the simulated series. These results confirm that there is at least one crossover in fluctuation function versus s. We should eliminate the crossover in fluctuation function to determine the generalized Hurst exponent. To this end, we apply either SVD or AD to the datasets, and the clean series are used for further analysis by either the DFA or DMA methods. For SVD, we consider p=1 and d=40; therefore, the three largest eigenvalues are set to zero, and the new eigenvalues matrix (𝐒̃), filtered matrix (Γ̃), and cleaned data(PTR) are constructed.Fig. <ref> indicatesℱ_2(s) computed by the DFA and DMA algorithms after applying either the SVD or AD method. <cit.> demonstrated that DMA with θ=0 (backward) has the best performance; therefore, we use the backward DMA method throughout this paper.We deduce that applying an SVD preprocess can efficiently remove the crossover, and we are able to assign a scaling exponent for fluctuation function versus s. The situation forAD preprocessing is somehow different, but it is consistent with the SVD results. The generalized Hurst exponent and ξ versus q for three types of PTRs superimposed by different values of GWB amplitudes are depicted in Fig. <ref>. The upper panels of Fig. <ref> illustrate the h(q) and ξ(q) for synthetic PTRs affected by GWB with different amplitudes with the same ζ. As we expect, the value of h(q=2) that is related to ζ for all samples is almost same. The lower panel shows h(q) and ξ(q) for simulated PTRs with different ζ.§.§ Irregular MF-DXA of simulated PTRsThe quadrupolar signature of the spatial cross-correlation function of PTRs is considered as a particular measure for detecting the imprint of the GWB <cit.>. Previously, the Hellings and Downs curve has been examined for detection of stochastic GWB<cit.>.Implementation of Irregular MF-DXA on PTRs provides a reliable cross-correlation exponent and coefficient in the presence of nuisance trends and noises. Irregular MF-DXA is indeed a crucial part of our pipeline for searching the significance of GWB. Here, due to the regularity of the simulated data, we consider the usual MF-DXA. To show the validity of this idea, we simulate 20 pure PTRs for pulsars separated in the sky according to the ephemeris of 20 MSPs observed in the PPTA project given in Table <ref>. Then, we add the effect of GWB to each pure PTR. In Fig. <ref>, we show σ̅_×(Θ) for simulated PTRs. Here we have simulated 50 realizations for 20 pulsars. The points plotted in Fig. <ref> are the average of these 50 realizations. As indicated in this figure, when synthetic PTRs are affected by GWB with 𝒜_yr=50× 10^-17 and ζ=-2/3, we can recognize a quadrupolar feature in σ̅_×(Θ) which is a benchmark for existing GWB. This behavior is similar to the Hellings and Downs curve indicated by the dashed line in Fig. <ref>. One of the advantages of this new measure is that, when undesired parts exist in the observed series, we are able to infer the contribution of the GWB signal robustly.Eqs. (<ref>) and (<ref>) also confirm that σ̅_× is almost insensitive to the value of 𝒜_yr. To make a more conservative pipeline for assessing the GWB signal, it is necessary to compute the cross-correlation coefficient, σ̅_×(Θ), in addition to the usual spatial cross-correlation function known as the Hellings and Downs curve. After obtainingthe feature, we carry out the rest part of the MF-DMA analysis to determine the type and amplitude of GWB signal.§.§ Strategies for Searching GWsAccording to the results presented in the previous sections, the randomness of pure PTRs exhibits that deviations from uncorrelated behavior can be considered as additional features presented in the recorded data. Unfortunately, the observed PTRs may include intrinsic fractal noise, interstellar plasma, uncertainties in the Earth's motion, master clocks, and receiver signals. It has been demonstrated that the noise from some of these sources is wavelength dependent andhas spatial correlation, either monopole or dipole in nature. Subsequently, relying on multifractal analysis modified by preprocessing algorithms such as the AD or SVD methods of individual PTRs probably gives rise spurious results in the framework of GW searching. To get rid of the effect of undesired components, we rely on the quadrupole structure of the GWB and carry out the irregular MF-DXAapproach. Therefore, we begin withIrregular-MF-DXA on all available PTRs distributed over all directions and then compute σ̅_×(Θ) as a function of separation angle, Θ. The existence of a feature similar to Fig. <ref> in observed PTRs would imply detection of a GWB. Note that Fig. <ref> is the average of 50 realizations. One observation with these parameters would have error bars almost 7 times larger, so the GWB would be detected but the significance would be much less. Thereafter,we will turn to the multifractal behavior of thePTR series to determine the type and amplitude of the GWB. In order to determine the type of stochastic GWB with a strain spectrum modeled by Eq. (<ref>), after preprocessing to remove noise and foreground, we apply multifractal methods to compute a reliable Hurst exponent. This exponent is related to the power-spectrum exponent. Finally, the best-fit value of ζ and its associated error bar are determined <cit.>. However, there are many complications in the real data sets, making the inference procedure less straightforward to assess GWs. We therefore introduce four criteria as follows: I) According to Eqs. (<ref>) and (<ref>), the intercept of fluctuation function forPTRs contains the intensity ofsuperimposed GWs. Therefore, after recognizing a quadrupolar signature in analyzing pairs of PTRs, the following quantity is able to indicate the intensity of GWB: Δ h_1(𝒜_yr,ζ)≡∑_q=q_min^q=q_max | 𝒢_h(q)(𝒜_yr,ζ)-𝒢_h(q)(𝒜_yr=0)|.In practice, we find a robust mathematical relation between Δ h_1(𝒜_yr,ζ) and 𝒜_yr for any given ζ (or, equivalently, H) and rms of white noise, as follows. We do many simulations for a given value of ζ with different 𝒜_yr values. Then, we apply either SVD or AD to make clean data. The clean data are used for further analysis. According to our simulation for ζ=-2/3 and rms=100 ns,the mathematical relation between 𝒜_yr and Δ h_1in the range of 𝒜_yr∈[10^-17,10^-15]reads as:(𝒜_yr/10^-17)=a Δ h_1^2 + bΔ h_1 + cwhere a =(-1.15±0.40)× 10^12, b = (2.84±0.54)× 10^7 and c = -74.45±16.88. This fitting function is not unique, and here we select one with a high goodness of fit before going further. Also, for any other rms dictated by experiment, the above analysis should be repeated again to find the corresponding fitting function.II) For pure PTRs, we found that the Hurst exponent is almost 0.5, while there will be deviations in the generalized Hurst exponent for PTR signals affected by GWs (Eq. (<ref>)) for a given amplitude 𝒜_yr, and ζ.Therefore, another powerful measure to quantify the intensity of the GWB would beΔ h_2(𝒜_yr,ζ)≡∑_q=q_min^q_max |h(q;𝒜_yr,ζ)-h_ shuf(q;𝒜_yr,ζ)|. Where h_ shuf(q;𝒜_yr,ζ) is for completely randomized PTR and "shuf" refers to shuffled. In practice, we find a robust mathematical relation between Δ h_2(𝒜_yr,ζ) and 𝒜_yr for any given ζ (or, equivalently, H) and rms of white noise.The corresponding shuffled series are produced using original series. Now by calculating the generalized Hurst exponent for original and shuffled data, one can compute Δ h_2. We find that the following function is a good fit to our simulations for 𝒜_yr in the range of 𝒜_yr∈[10^-17,10^-15] versus Δ h_2 forζ=-2/3 and rms=100 ns:(𝒜_yr/10^-17) = aΔ h_2^3+bΔ h_2^2+cΔ h_2where a=0.19±0.06, b=-1.57±0.92, and c= 7.40±3.30. This fitting function is not unique, and here we select a high goodness of fit. Before going further, it is worth noting thatthe whitened noise generation is serious in many simulations. An optimal algorithm to evaluate noise quality in many simulations, especially in data generation by the TEMPO2 software, can be carried outby the shuffling procedure explained here. Subsequently, our proposal in this regard can be straightforwardly implemented as a new plug-in. III) Since GWs may induce non-Gaussianity in PTR, it is interesting totake into account Δ h_3(𝒜_yr,ζ)≡∑_q=q_min^q_max |h(q;𝒜_yr,ζ)-h_ sur(q,𝒜_yr,ζ)|. In the mentioned criterion, h_ sur(q;𝒜_yr,ζ) is the generalized Hurst exponents computed for Gaussian datasets with the same correlation function as the original series. Here "sur" represents surrogated data or phase-randomized surrogated series, including the multiplication of Fourier-transform data by a random phase with a uniform distribution function <cit.>. We simulated the PTR accompanying the GWB with different amplitudes, and the following fitting function is determined for 𝒜_yr in the same range as above versus Δ h_3 for ζ=-2/3 and rms=100 ns:(𝒜_yr/10^-17) =aΔ h_3 + bwhere a = 68.03±11.73 and b=-321.50±65.10.IV) The width of the singularity spectrum, which quantifies the nature of multifractality, isanother benchmark for determining the amplitude of GWs superimposed on the PTRs. This measure is defined by Δ h_4(𝒜_yr,ζ) ≡ |Δα(𝒜_yr,ζ)- Δα(𝒜_yr=0)|. According to our simulations,we find:(𝒜_yr/10^-17)=aΔ h_4^b+cfor ζ=-2/3 and rms=100 ns in the range of 𝒜_yr∈[10^-17,10^-15]. Here a=106.30±7.80, b=1.62±0.42, and c= 1.52±9.74.Let us summarize our strategy based on the above criteria for searching GWs in observation. As explained in section 2, in the case of the proper value of signal-to-noise (S/N) for each observed PTR, we remove all known contributionsfrom foreground contamination. Therefore, we make regular series according to methods explained in subsection 2.3. Now we are ready to apply either AD or SVD method to extract the dominant part of the signal (the trend part) from the noise. Then, we apply the MF-DXA method to compute σ̅_×, andwe compute the spatial cross-correlation to identify the probable quadrupolar signature. In the case of finding the mentioned signature, we go through the detection of GWs. Otherwise, we can only carry out the upper-limit approach. We also apply irregular MF-DXA on the proper part of the series for all available pairs of observed PTRs to examine the temporal part of the cross-correlation function and deduce the temporal scaling exponent. In the case of the homogeneous and isotropic source of the GWB, h_× is independent from the angular separation of PTRs, while for anisotropic or different single sources of GWs, the scaling exponent of the temporal part of the cross-correlation gets various values for different pairs. Utilizingeither irregular MF-DFA or irregular MF-DMAon cleaned data leads to computing h(q). The best-fit value ofζ is then determined by using the power-spectrum exponent.Following the benchmarks, we compute Δ h_1, Δ h_2, Δ h_3, and Δ h_4 for the observed PTRs. The GWB amplitude can be conservatively readfrom the corresponding plots, as indicated in Fig. <ref> or stated by Eqs (<ref>), (<ref>), (<ref>) and (<ref>). It is worth noting that the functional form of Δ h should be determined for each value of ζ and given rms of white noise associated with observed data.Finally, we are able to compute the upper limiton 𝒜_yr using posterior analysis (see section <ref>).Fig. <ref> is a schematic representation of the pipeline.Here we emphasize some important considerations for dealing with observed PTRs. First of all, we define a relative difference between the scaling exponent computed for the observed PTRs and that computed for the PTRs without GWB to reduce the contribution of noise and trends. Finally, in our approach, the level of noise is almost no longer serious when we focus on the scaling exponent. § IMPLEMENTATION OF MULTIFRACTAL METHODS ON OBSERVED PTR DATAHere we use the MF-DFA and MF-DMA methods modified by either AD or SVD detrending procedurestoexamine the multifractal and complexitybehavior of observed pulsar timing residuals.§.§ Implementation on Observed DataAs discussed in subsection 2.3,observed PTRs datasets are in the form of irregularly sampled series, and here we use the spectralModel plug-in for the temporal smoothingalgorithmto construct equidistant regular series for further analysis <cit.>. The size of the current observed data is not large enough to use the irregular version of MF-DFA and MF-DMA introduced by Eqs. (<ref>) and (<ref>).Fig. <ref> illustrates the MF-DMA results for various observed PTRs. These results confirm that there is a crossover in fluctuation functions versus s, corresponding to s_×∼70 days. For the scaling exponent for s<s_×, we have h(q=2)∈[1.03,1.82], demonstrating that datasets have a nonstationary nature, whilefor s>s_×, we find h(q=2)∈[0.07,1.55]. In order to get rid of these crossovers and have a scaling behavior in fluctuation functions, we apply either AD or SVD separately on modified observed datasets. Then, the cleaned data will be used as input for the MF-DFA and MF-DMA algorithms. Fig. <ref> illustrates a typical observed PTR (red line) and the trend (black line) determined by AD (upper panel) and SVD (lower panel). The corresponding residual between the observed data and trend is indicated in the bottom of this figure. Fig. <ref> represents the fluctuation functions computed for a typical observed PTR by DFA and DMA applied on cleaned data provided by AD and SVD separately. The slope of the fluctuation functions for q=2 in reliable scales is h(q=2)∈ [1.56,1.87], demonstrating that all underlying series are categorized in the nonstationary class. The corresponding Hurst exponent, H=h(q=2)-1, belongs to H∈[0.56,0.87]. The value of the Hurst exponents for all observedPTRs at the 68% level of confidence is depicted in Fig. <ref>. This result confirms that the dominant part ofobserved PTRs belongs to the long-range correlated signal. The lower panel of Fig. <ref> shows the q-dependency of the generalized Hurst exponent after applying SVD on observed data and determined by MF-DMA. The results for MF-DFA are consistent with thosedetermined byMF-DMA. Since hdepends on q, we conclude that all observed PTRs are multifractal.Singularity spectra of some observed PTRs are plotted in the upper panel of Fig. <ref>. The strength of the multifractal nature of PTRs is determined by the width of the singularity spectrum, Δα=α_ max-α_ min. This value for observed data is reported in Table <ref> and is also shown in the lower panel of Fig. <ref>. The range of the mentioned singularity spectra is Δα∈ [0.89,1.79]. Other relevant exponents are reported in Table <ref>.An interesting question is, what are the sources of multifractality of observed PTRs? As explained in more detail by <cit.>,in principle, different correlation functions at small and large fluctuations can be considered as a source of multifractality. In addition,heavy-tailed probability distribution contributes to the multifractal behavior. In order to distinguish the two mentionedtypes of multifractality, we follow the method introduced in <cit.>. By shuffling the series, the scaling behavior of the ratio of fluctuation functions, ℱ_q(s)/ℱ_q^ shuf(s), is represented as:ℱ_q(s)/ℱ_q^shuf(s)∼ s^h(q)-h_shuf(q)where h_shuf(q) is the generalized Hurst exponent for shuffled data. The case of h_cor(q)≡ h(q)-h_shuf(q)=0 refers to multifractality sourced by the distribution function. In this case, we can compute h_ PDF(q)≡ h(q)-h_ sur(q). If both h_cor(q) and h_PDF(q) depend on q, both sources are playing roles in the multifractality of the data. In our samples, allPTRs have h_shuf(q)=0.50 at a 1σ confidence interval, confirming that the correlation in datasetsis almost the main source of multifractality. This property is a universal feature of all observed PTRs investigated in this paper.The multifractality responsible for observed PTRs can also be examined by our method.To this end, we have used different models for the noise component according to the SimRedNoise plug-in of TEMPO2 and applied the MF-DMA method on those series. The upper panel of Fig. <ref> indicates that the width of the singularity spectrum computed by the MF-DMA method is almost independent of the amplitude of the red-noise model. The lower panel illustrates the dependency of Δα on the exponent of the red-noise power spectrum considered as the P_red(f)=A_red(1+f^2/f_c^2)^-Q/2 model, where A_red, f_c, and Q are the amplitude of the power spectrum, corner frequency, and power-law index, respectively <cit.>. In this equation, Q=0 corresponds to white noise, and Q=2,4, and 6 are related to phase noise, frequency noise and spin-down noise. Subsequently, we can deduce that the red noise can be responsible for multifractality of observed PTRs as well as GWs. Therefore, a part of our reported multifractality is related to red noise.In Fig. <ref>, we indicate σ̅_× asfunction of Θ for 20 MSPs observed in the PPTA project (listed in Table <ref>). We have not obtained an obvious quadrupolar signature for the mentioned observed series due to the high value of rms, short length in the size of the data, unresolved foreground contamination, and systematic noise. In the next subsection, we will go through the finding upper limit for the amplitude of the probable GWB superimposed in observed PTRs.§.§ Upper bound on GWB amplitude Multifractal assessment of individual PTRs series is notadequate to make a decision on thesignificance of the stochastic GWB. Therefore, inspired by the unique signature of the GWB, i.e. the quadrupolar feature induced on the spatial correlation function of PTRs fluctuations, we apply multifractal cross-correlation analysis. This is a generalized function including spatial-temporal cross-correlation function and has some novelties compared to the standard spatial cross-correlation analysis. Our algorithm is a proper method for denoising and detrending.The irregular MF-DXA applied to the observed irregular PTRs did not yield reliable results for detecting GWB due to the limited size and low S/N of the data. Different criteria introduced in this paper will enable us to detect the footprint of possible GWs with a future generation of surveys with high-S/N observations. Now we turn to assigning an upper bound on probable GWB amplitude. Previous studies have mainly considered a model for the power spectrum of the PTR signal modulated by GWB, including the amplitudeand scaling exponent of GWB. According to priors associated with the model parameters, the Bayesian method has been adopted (<cit.> and references therein). In our approach, we proceed with our strategies for searching the GWB (subsection <ref>). The posterior probability function, 𝒫_♢(𝒜_yr|𝒟), reads as:𝒫_♢(𝒜_yr|𝒟) ∼ ℒ_♢(𝒟|𝒜_yr)𝒫_♢(𝒜_yr)= ⟨δ_D(𝒜_yr-Φ_𝒟(Δ h_♢)) ⟩Here symbol "♢" corresponds to one of four measures proposed for determining the amplitude of the stochastic GWB, and δ_D is the Dirac delta function. The Φ_𝒟(Δ h_♢) represents the functional form presented in Fig. <ref>. The integral form of Eq. (<ref>) is given by:𝒫_♢(𝒜_yr|𝒟)= ∫ d Δ h'_♢𝒫(Δ h'_♢) δ_D(Δ h'_♢-Δ h_♢)|𝒥|_Δ h'_♢=Φ_𝒟^-1(𝒜_yr)in which |𝒥| is the Jacobian computed for Δ h'_♢=Φ_𝒟^-1(𝒜_yr). Finally, the upper bound on 𝒜_yr^up-♢ can be determined by:C.L.^♢= ∫_-∞^𝒜_yr^up-♢d𝒜'_yr𝒫_♢(𝒜'_yr|𝒟)where C.L.^♢ and 𝒜_yr^up-♢are the confidence interval and upper limit associated with one of our strategies, respectively.According to the posterior function defined by Eq. (<ref>), considering {𝒟}={Δ h_♢^PTR} for a given observed pulsar called by PTR and {Υ}=𝒜_yr, we compute: χ^2_PTR(𝒜_yr)≡Δ_PTR^†.𝒞^-1_𝒜_yr.Δ_PTRwhere Δ_PTR≡[ Δ h^PTR-⟨Δ h (𝒜_yr)⟩] and 𝒞_𝒜_yr is the 4×4 covariance matrix of thefour statistical features defined by Δ h_1, Δ h_2, Δ h_3 and Δ h_4 (see Eqs. (<ref>), (<ref>), (<ref>) and (<ref>)). The ⟨Δ h(𝒜_yr)⟩ is the average of Δ h over 1000 synthetic datasets for a given 𝒜_yr, where 𝒜_yr∈[10^-16,10^-14] and with a step size of 5× 10^-16. According to the likelihood function, ℒ(Δ h^PTR|𝒜_yr)∼exp(-χ^2(𝒜_yr)/2), the 95% upper bound on 𝒜_yr^up using the observed PTRs is defined by:95%=∫_-∞^𝒜_yr^upd𝒜_yrℒ(Δ h^PTR|𝒜_yr)We report the computed upper bound for some observed pulsar timing residuals at a 95% confidence level in Table <ref>. One may note that the upper bound on𝒜_yr has not been reported for some observed PTRs. This is because, for such cases, the upper value is not in the range of 𝒜_yr∈[10^-16 , 10^-14] considered in this research.Our results are consistent with other reports <cit.>. § SUMMARY AND CONCLUSIONThe PTR is a good indicator to examine relevant physical phenomena from the interior of pulsars, as well as cosmological events. In spite of high stability in some types of pulsars, PTRs are classified as stochastic processes due to superimposed unknown trends and noises. The GWs produced by either primordial or late events affect the PTRs. Therefore, quantifying the fluctuations of PTRs can be a proper measure for GW detection. In this paper, for the first time, we utilized a multifractal approach in order to examine the statistical properties of synthetic and observed PTRs affected by trends and noises. In the presence of trends and unknown noises, only robust methods are able to recover the correct multifractal nature of underlying series. In this research, we used MF-DFA, MF-DMA, and MF-DXA modified by the preprocessors, so-called AD or SVD algorithms. The pulsar timing observations are unevenly sampled datasets.To mitigate this property, we modified some internal parts of the multifractal analysis and proposed the irregular MF-DXA method and examined its accuracy. Our results demonstrated that computed scaling exponents for anticorrelated and long-range-correlated irregular signals are consistent with the expectations. We used synthetic PTRs simulated by the TEMPO2 pulsar timing package. A template proposed by<cit.>was used to take into account the contribution of GWs. We simulated 1000 synthetic PTRs, and the MF-DFA, MF-DMA, and MF-DXA methods were implemented on the simulated series. Our results demonstrated that the ensemble average of the Hurst exponent of the simulated data is ⟨ H⟩=0.51±0.02, confirming that the pure PTRs belong to monofractal uncorrelated stationary processes. There is no crossover in fluctuation functions versus scale determined by MF-DFA and MF-DMA (Fig. <ref>). Adding mock GWB signal on pure PTRs leads to crossovers in the log-log plot of ℱ_2 as a function of s as indicated in Fig. <ref>. To examine the scaling behavior ofPTRs induced by GWs, we carried out either the SVD or AD method on the data. We found that SVD can remove the crossover on fluctuation function for any q. The time scale for crossover depends on the intensity of the GW signal. In the presence of GWs, PTRs belong to a multifractal process due to the q-dependency of the generalized Hurst exponent, h(q), (Fig. <ref>). Therefore, we were able to classify the mentioned data in the universal class of the multifractal process. The value of multifractality increased by increasing the intensity of GWs.Various components of a recorded PTR may behave as a scaling fluctuation. Therefore, applying a multifractal algorithm on individual PTRs may give spurious results in exploringGWs. We relied on quadrupolar structure associated with the impact of GWB on the spatial cross-correlation of PTRs. We carried out cross-correlation analysis by the irregular MF-DXA introduced in this paper on all available PTRs distributed in all directions. To this end, we defined a new cross-correlation function (Eq. (<ref>)) and accordingly, we computed the ensemble average of ⟨σ_×(Θ_ab)⟩_ pair for all synthetic PTRs as a function of separation angle, Θ. We obtainedan analogous behavior as a quadrupolar signature in σ̅_×.According to a model for GWB, obviously, the temporal part must be independent from the separation angle of the PTR pairs affected by isotropic GWB, while the amplitude of cross-correlation defined by the DXA method illustrates the Hellings and Downs curve (Fig. <ref>) similar to the usual spacial crosscorrelation. We proposed four criteria to quantify the footprint of GWs on pulsar timing residuals. Comparing the y-intercept of fluctuation functions with the one computed for pure PTRs is our first measure. The second measure is devoted to the generalized Hurst exponent with the one computed for pure PTRs. Comparison between h(q) and the generalized Hurst exponent computed for the Gaussian signal is the third criterion. The fourth criterion corresponds to the width of the singularity spectrum.The strategy for GWB detection in observations is as follows. After removing foreground and systematic noise by applying either SVD or AD on datasets, cleaned data that are associated with the dominant part of the signal (the trend part) will be used as input for irregular MF-DXA. Having observed relevant features for GWB on PTRs, irregular MF-DFA or irregular MF-DMA methods are applied exclusively. The type of superimposed GWs can be recognized by determining the Hurst exponent. Finally, the dimensionless amplitude of expected GWB (𝒜_yr) can be determined by inserting relevant quantities extracted by our four measures given byEqs. (<ref>), (<ref>), (<ref>) and (<ref>) for a given ζ and rms of white noise determined in observations.There is a crossover in the log-log plot of fluctuation function versus window length of observed PTRs.For s<s_× and s> s_×, the exponents h(q=2) are h(2)∈ [1.03,1.82] and h(2)∈[0.07,1.55], respectively.After applying SVD, the corresponding Hurst exponent is H∈ [0.56,0.87].The q-dependency of h(q) confirmed that all observed MSPs behave as multifractal fields. The relevant exponents for observed MSPs havebeen reported in Table <ref>. The source of multifractality is mainly the correlation in small and large scales and is a universal property of all observed pulsars examined in this paper. The contribution of red-noise model indicated the extra multifractality on observed MSPs. Consequently, the degree of multifractality reported for PPTA data sets is the upper value, and a part of this value is associated with the noise model. To infer the statistical significance of the GWB impact on the PTRs, we computed σ̅_× (Θ) for 20 MSPs observed in the PPTA project. Due to a high value of rms anda short length in the size of recorded data, we have not found a quadrupolar signature. Thereafter, we computedthe upper bound for PSRs reported in Table <ref>.Final remarks are as follows. The observed PTRs are affected by noises classifying into intrinsic and extrinsic categories <cit.>. Reliable statistical models for noise and signal were introduced. The shuffling procedure and its evaluation by multifractal detrended analysis can also be implemented in TEMPO2 and other subroutines for simulation of PTRs. It could be interesting to simulate various kinds of GWs and to consider timing noise. Evaluation of different noise models and sensitivity to frequency is beyond the scope of this paper and will be consideredelsewhere.The authors thank M. Farhang for her useful comments on the manuscript. Also, the authors appreciate R. 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"authors": [
"I. Eghdami",
"H. Panahi",
"S. M. S. Movahed"
],
"categories": [
"astro-ph.SR",
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"physics.data-an"
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"primary_category": "astro-ph.SR",
"published": "20170427143001",
"title": "Multifractal Analysis of Pulsar Timing Residuals: Assessment of Gravitational Wave Detection"
} |
printfolios=true,printccs=false,printacmref=false =1PACMPL 1 CONF1 2018 1 1noneACM-Reference-Format acmauthoryear theoremsection theoremsectioncd,automataplain theoremTheorem[section] proposition[theorem]Proposition lemma[theorem]Lemmadefinition remark[theorem]Remark automaton[node distance=1.5cm,on grid,auto,initial text=,state/.append style=inner sep=0pt,outer sep=0pt,minimum size=3.5ex]tightcenter C[1]>m#1<L[1]>m#1<R[1]>m#1<[email protected]@[email protected] College London Automata learning has been successfully applied in the verification of hardware and software. The size of the automaton model learned is a bottleneck for scalability, and hence optimizations that enable learning of compact representations are important. This paper exploits monads, both as a mathematical structure and a programming construct, to design, prove correct, and implement a wide class of such optimizations. The former perspective on monads allows us to develop a new algorithm and accompanying correctness proofs, building upon a general framework for automata learning based on category theory. The new algorithm is parametric on a monad, which provides a rich algebraic structure to capture non-determinism and other side-effects.We show that our approach allows us to uniformly capture existing algorithms, develop new ones, and add optimizations. The latter perspective allows us to effortlessly translate the theory into practice: we provide a Haskell library implementing our general framework, and we show experimental results for two specific instances: non-deterministic and weighted automata. <ccs2012> <concept> <concept_id>10003752.10003766.10003767.10003768</concept_id> <concept_desc>Theory of computation Algebraic language theory</concept_desc> <concept_significance>100</concept_significance> </concept> </ccs2012> Optimizing automata learning via monads Alexandra Silva April 25, 2017 =======================================§ INTRODUCTIONThe increasing complexity of software and hardware systems calls for new scalable methods to design, verify, and continuously improve systems. Black-box inference methods aim at building models of running systems by observing their response to certain queries. This reverse engineering process is very amenable for automation and allows for fine-tuning the precision of the model depending on the properties of interest, which is important for scalability. One of the most successful instances of black-box inference is automata learning, which has been used in various verification tasks, ranging from finding bugs in implementations of network protocols <cit.> to rejuvenating legacy software <cit.>. <cit.> has recently written a comprehensive overview of the widespread use of automata learning in verification. A limitation in automata learning is that the models of real systems can become too large to be handled by tools. This demands compositional methods and techniques that enable compact representation of behaviors. In this paper, we show how monads can be used to add optimizations to learning algorithms in order to obtain compact representations. We will use as playground for our approach the well knownalgorithm <cit.>, which learns a minimal deterministic finite automaton (DFA) accepting a regular language by interacting with a teacher, i.e., an oracle that can reply to specific queries about the target language. Monads allow us to take an abstract approach, in which category theory is used to devise an optimized learning algorithm and a generic correctness proof for a broad class of compact models.Monads also allow us to straightforwardly implement the algorithm in Haskell via the corresponding programming constructs. The inspiration for this workis quite concrete: it is a well-known fact that non-deterministic finite automata (NFAs) can be much smaller than deterministic ones for a regular language. The subtle point is that given a regular language, there is a canonical deterministic automaton accepting it—the minimal one—but there might be many “minimal” non-deterministic automata accepting the same language. This raises a challenge for learning algorithms: which non-deterministic automaton should the algorithm learn? To overcome this, <cit.> developed a version of Angluin'salgorithm, which they called , in which they use a particular class of NFAs, namely Residual Finite State Automata (RFSAs), which do admit minimal canonical representatives. Thoughindeed is a first step in incorporating a more compact representation of regular languages, there are several questions that remain to be addressed. We tackle them in this paper. DFAs and NFAs are formally connected by the subset construction. Underpinning this construction is the rich algebraic structure of languages and of the state space of the DFA obtained by determinizing an NFA. The state space of a determinized DFA—consisting of subsets of the state space of the original NFA—has a join-semilattice structure. Moreover, this structure is preserved in language acceptance: if there are subsets U and V, then the language of U ∪ V is the union of the languages of the first two. Formally, the function that assigns to each state its language is a join-semilattice map, since languages themselves are just sets of words and have a lattice structure.And languages are even richer: they have the structure of complete atomic Boolean algebras. This leads to several questions: Can we exploit this structure and have even more compact representations? What if we slightly change the setting and look at weighted languages over a semiring, which have the structure of a semimodule (or vector space, if the semiring is a field)? The latter question is strongly motivated by the widespread use of weighted languages and corresponding weighted finite automata (WFAs) in verification, from the formal verification of quantitative properties <cit.>, to probabilistic model-checking <cit.>, to the verification of on-line algorithms <cit.>. Our key insight is that the algebraic structures mentioned above are in fact algebras for a monad T. In the case of join-semilattices this is the powerset monad, and in the case of vector spaces it is the free vector space monad. These monads can be used to define a notion of T-automaton, with states having the structure of an algebra for the monad T, which generalizes non-determinism as a side-effect. From T-automata we can derive a compact, equivalent version by taking as states a set ofgenerators and transferring the algebraic structure of the original state space to the transition structure of the automaton.This general perspective enables us to generalizeto a new algorithm , which learns compact automata featuring non-determinism and other side-effects captured by a monad. Moreover,incorporates further optimizations arising from the monadic representation, which lead to more scalable algorithms.Besides the theoretical aspects, we devote large part of this paper to implementation and experimental evaluation. Monads are key for us to faithfully translate theory into practice. We provide a library that implements all aspects of our general framework, making use of Haskell monads.[The code is provided as supplementary material.] For any monad, the library allows the programmer to obtain a basic, correct-by-construction instance of the algorithm and of its optimized versions for free. This enables the programmer to experiment with different optimizations with minimal effort. Our library also allows the programmer to re-define some basic operations, if a more efficient version is available, in order to make the algorithm more amenable to real-world usage. For instance, generators can be computed efficiently in the vector space case via Gaussian elimination. One of the main challenges in applying Angluin-style algorithms to real-world systems is implementing the teacher. In fact, it is often the case that exact answers to certain queries are not available. In these cases the teacher often resorts to random testing <cit.>, with an unavoidable trade-off in terms of model accuracy (see <cit.> for a detailed discussion on this issue). Our library provides support for both exact and approximate teachers, along with a basic implementation that works for any monad. Interestingly, the exact teacher relies on bisimulation up to context <cit.>, which exploits the monad structure to efficiently determine bisimulation.§ OVERVIEW AND CONTRIBUTIONS In this section, we give an overview of our approach and highlight our main contribution. We start by explaining the originalalgorithm. We thendiscuss the challenges in adapting the algorithm to learn automata with side-effects, illustrating them through a concrete example—NFAs.§.§algorithm Thealgorithm learns the minimal DFA accepting a language ⊆ A^* over a finite alphabet A. The algorithm assumes the existence of a minimally adequate teacher, which is an oracle that can answer two types of queries:* Membership queries: given a word w ∈ A^*, does w belong to ? * Equivalence queries: given a hypothesis DFA , doesaccept ? If not, the teacher will return a counterexample, i.e., a word incorrectly accepted or rejected by . The algorithm incrementally builds an observation table.The table is made of two parts: a top part, with rows ranging over a finite set S ⊆ A^*; and a bottom part, with rows ranging over S · A (i.e., words of the form s a, with s ∈ S and a ∈ A). Columns range over a finite E ⊆ A^*. For each u ∈ S ∪ S · A and v ∈ E, the corresponding cell in the table contains 1 if and only if uv ∈. Intuitively, each row u contains enough information to fully identify the Myhill-Nerode equivalence class of u with respect to an approximation of the target language—rows with the same content are considered members of the same equivalence class. Cells are filled in by the algorithm via membership queries.As an example, and to set notation, consider the table below over A = {a,b}. It shows thatcontains the word aa and does not contain the words(the empty word), a, b, ba, aaa, and baa.[c].35c[.8pt][2pt]12ex0pt^ Ea aa 2-5[15mm[S]0 0 12-5[210mm[S · A] a 0 1 0 b 0 0 0[c].7S → 2^E(u)(v)= 1 uv ∈ S → (2^E)^A (u)(a)(v)= 1uav ∈ We use the functionsandto describe the top and bottom parts of the table, respectively. Notice that S and S · A may intersect. For the sake of conciseness, when tables are depicted, elements in the intersection are only shown in the top part.A key idea of the algorithm is to construct a hypothesis DFA from the different rows in the table. The construction is the same as that of the minimal DFA from the Myhill-Nerode equivalence, and exploits the correspondence between table rows and Myhill-Nerode equivalence classes. The state space of the hypothesis DFA is given by the set H = {(s) | s ∈ S}. Note that there may be multiple rows with the same content, but they result in a single state, as they all belong to the same Myhill-Nerode equivalence class. The initial state is (), and we use thecolumn to determine whether a state is accepting: (s) is accepting whenever (s)() = 1. The transition function is defined as (s) (s)(a). (Notice that the continuation is drawn from the bottom part of the table).For the hypothesis automaton to be well-defined,must be in S and E, and the table must satisfy two properties: * Closedness states that each transition actually leads to a state of the hypothesis. That is, the table is closed if for all t ∈ S and a ∈ A there is s ∈ S such that (s) = (t)(a). * Consistency states that there is no ambiguity in determining the transitions. That is, the table is consistent if for all s_1, s_2 ∈ S such that (s_1) = (s_2) we have (s_1) = (s_2). The algorithm updates the sets S and E to satisfy these properties, constructs a hypothesis, submits it in an equivalence query, and, when given a counterexample, refines the hypothesis. This process continues until the hypothesis is correct. The algorithm is shown in <ref>.Example Run.[subfigure]justification=centeringWe now run the algorithm with the target language = {w ∈{a}^* | |w|1}. The minimal DFA acceptingis = [initial,state,accepting] (q0) ; [state] (q1) [right of=q0] ; [state,accepting] (q2) [right of=q1] ; [->] (q0) edge node a (q1) (q1) edge node a (q2) (q2) edge [loop right] node a ();Initially, S = E = {}. We build the observation table given in <ref>. This table is not closed, because the row with label a, having 0 in the only column, does not appear in the top part of the table: the only rowhas 1. To fix this, we add the word a to the set S. Now the table ( <ref>) is closed and consistent, so we construct the hypothesis that is shown in <ref> and pose an equivalence query. The teacher replies no and informs us that the word aaa should have been accepted.handles a counterexample by adding all its prefixes to the set S. We only have to add aa and aaa in this case. The next table ( <ref>) is closed, but not consistent: the rowsand aa both have value 1, but their extensions a and aaa differ. To fix this, we prepend the continuation a to the columnon which they differ and add a · = a to E. This distinguishes () from (aa), as seen in the next table in <ref>. The table is now closed and consistent, and the new hypothesis automaton is precisely the correct one .As mentioned, the hypothesis construction approximates the theoretical construction of the minimal DFA, which is unique up to isomorphism. That is, for S = E = A^* the relation that identifies words of S having the same value inis precisely the Myhill-Nerode's right congruence. §.§ Learning non-deterministic automataAs it is well known, NFAs can be smaller than the minimal DFA for a given language. For example, the languageabove is accepted by the NFA𝒩 = [initial,state,accepting] (q0) ; [state] (q1) [right of=q0] ; [->] (q0) edge [bend left] node a (q1) (q1) edge [bend left] node a (q0) (q1) edge [loop right] node a (); which is smaller than the minimal DFA . Though in this example, which we chose for simplicity, the state reduction is not massive, it is known that in general NFAs can be exponentially smaller than the minimal DFA <cit.>. This reduction of the state space is enabled by a side-effect—non-determinism, in this case. Learning NFAs can lead to a substantial gain in space complexity, but it is challenging. The main difficulty is that NFAs do not have a canonical minimal representative: there may be several non-isomorphic state-minimal NFAs accepting the same language, which poses problems for the development of the learning algorithm.To overcome this, <cit.> proposed to use a particular class of NFAs, namely RFSAs, which do admit minimal canonical representatives. However, their ad-hoc solution for NFAs does not extend to other automata, such as weighted or alternating. In this paper we present a solution that works for any side-effect, specified as a monad.The crucial observation underlying our approach is that the language semantics of an NFA is defined in terms of its determinization, i.e., the DFA obtained by taking sets of states of the NFA as state space. In other words, this DFA is defined over an algebraic structure induced by the powerset, namely a join semilattice (JSL) whose join operation is set union. This automaton model does admit minimal representatives, which leads to the key idea for our algorithm: learning NFAs as automata over JSLs.In order to do so, we use an extended table where rows have a JSL structure, defined as follows. The join of two rows is given by an element-wise or, and the bottom element is the row containing only zeroes. More precisely, the new table consists of the two functions ^♯(S) → 2^E ^♯(S) → (2^E)^Agiven by ^♯(U) = ⋁{(s) | s ∈ U} and ^♯(U)(a) = ⋁{(s)(a) | s ∈ U}. Formally, these functions are JSL homomorphisms, and theyinduce the following general definitions:* The table isclosed if for all U ⊆ S, a ∈ A there is U' ⊆ S such that ^♯(U') = ^♯(U)(a). * The table isconsistent if for all U_1, U_2 ⊆ S s.t. ^♯(U_1) = ^♯(U_2) we have ^♯(U_1) = ^♯(U_2). We remark that our algorithm does not actually store the whole extended table, which can be quite large. It only needs to store the original table over S because all other rows in (S) are freely generated and can be computed as needed, with no additional membership queries. The only lines in <ref> that need to be adjusted are lines <ref> and <ref>, where closedness and consistency are replaced with the new notions given above. Moreover,is now built from the extended table.Optimizations.In this paper we also present two optimizations to our algorithm. For the first one, note that the state space of the hypothesis constructed by the algorithm can be very large since it encodes the entire algebraic structure. We show that we can extract a minimal set of generators from the table and compute a succinct hypothesis in the form of an automaton with side-effects, without any algebraic structure. For JSLs, this consists in only taking rows that are not the join of other rows, i.e., the join-irreducibles. By applying this optimization to this specific case, we essentially recover the learning algorithm of <cit.>. The second optimization is a generalization of the optimized counterexample handling method of <cit.>, originally intended forand DFAs. It consists in processing counterexamples by adding a single suffix of the counterexample to E, instead of adding all prefixes of the counterexample to S. This can avoid the algorithm posing a large number of membership queries.Example Revisited. We now run the new algorithm on the language = {w ∈{a}^* | |w|1} considered earlier.Starting from S = E = {}, the observation table ( <ref>) is immediately closed and consistent. (It is closed because we have ^♯({a}) = ^♯(∅).) This gives the JSL hypothesis shown in <ref>, which leads to an NFA hypothesis having a single state that is initial, accepting, and has no transitions ( <ref>). The hypothesis is obviously incorrect, and the teacher may supply us with counterexample aa. Adding prefixes a and aa to S leads to the table in <ref>. The table is again closed, but not consistent: ^♯({a}) = ^♯(∅), but ^♯({a})(a) = ^♯({aa}) ^♯(∅) = ^♯(∅)(a). Thus, we add a to E. The resulting table ( <ref>) is closed and consistent. We note that row aa is the union of other rows: ^♯({aa}) = ^♯({, a}) (i.e., it is not a join-irreducible), and therefore can be ignored when building the succinct hypothesis. This hypothesis has two states,and a, and indeed it is the correct one 𝒩. §.§ Contributions and road map of the paper After some preliminary notions in Section <ref>, our main contributions are presented as follows:*In Section <ref>, we develop a general algorithm , which generalizes the NFA one presented in Section <ref> to an arbitrary monadT capturing side-effects, and we provide a general correctness proof for our algorithm. *In Section <ref>, we describe the first optimization and prove its correctness. * In Section <ref> we describe the second optimization. We also show how it can be combined with the one of Section <ref>, and how it can lead to a further small optimization, where the consistency check on the table is dropped. *In Section <ref> we show howcan be applied to several automata models, highlighting further case-specific optimizations when available. *In Section <ref> we describe our library and explain in detail how it can be instantiated to NFAs and WFAs. The implementation of monads for these two cases is non-trivial, due to specific Haskell requirements. We also give efficient versions of both instances. To the best of our knowledge, we are the first ones to implement an Angluin-style learning algorithm for WFAs, and to provide optimizations for it. *Finally, in Section <ref> we describe experimental results for the non-deterministic and weighted cases, comparing all the optimizations enabled by our library. In particular, for NFAs we show that the Rivest and Schapire optimization, not available to <cit.>, leads to an improvement in the number of membership queries, as happens in the DFA case. § PRELIMINARIES In this section we define a notion of T-automaton, a generalization of non-deterministic finite automata parametric in a monad T. We assume familiarity with basic notions of category theory: functors (in the categoryof sets and functions) and natural transformations.Side-effects and different notions of non-determinism can be conveniently captured as a monad. A monad T = (T,η,μ) is a triple consisting of an endofunctor T onand two natural transformations: a unitη𝙸𝚍⇒ T and a multiplicationμ T^2 ⇒ T,which satisfy the compatibility lawsμ∘η_T = _T = μ∘ Tη andμ∘μ_T = μ∘ Tμ. [Monads] An example of a monad is the triple (, {-}, ⋃), wheredenotes the powerset functor associating a collection of subsets to a set, {-} is the singleton operation, and ⋃ is just union of sets. Another example is the triple ((-), e, m), where (X) is the free semimodule (over a semiring ) over X, namely {φ|φ X → having finite support}. The support of a function φ X → is the set of x ∈ X such that φ (x) ≠ 0. Then e X →(X) is the characteristic function for each x∈ X, and m((X)) →(X) is defined for φ∈((X)) and x∈ X as m(φ)(x) = ∑_ψ∈(X)φ(ψ)×ψ(x). Given a monad T, a T-algebra is a pair (X,h) consisting of a carrier set X and a function h TX → Xsuch that h ∘μ_X = h ∘ Th and h ∘η_X = _X. A T-homomorphism between two T-algebras (X,h) and (Y,k) is a function f X → Y such that f ∘ h = k ∘ Tf. The abstract notion of T-algebra instantiates to expected notions, as illustrated in the following example. [Algebras for a monad] The -algebras are the (complete) join-semilattices, and their homomorphisms are join-preserving functions. Ifis a field, -algebras are vector spaces, and their homomorphisms are linear maps. We will often refer to a T-algebra (X, h) as X if h is understood or if its specific definition is irrelevant. Given a set X, (TX, μ_X) is a T-algebra called the free T-algebra on X. One can build algebras pointwise for some operations. For instance, if Y is a set and (X, x) a T-algebra, then we have a T-algebra (X^Y, f), where fT(X^Y) → X^Y is given by f(W)(y) = (x ∘ T(_y))(W) and _yX^Y → X by _y(g) = g(y). If U and V are T-algebras and fU → V is a T-algebra homomorphism, then the image (f) of f is a T-algebra, with the T-algebra structure inherited from V.The following proposition connects algebra homomorphisms from the free T-algebra on a set U to an algebra V with functions U→ V. We will make use of this later in the section. Given a set U and a T-algebra (V, v), there is a bijective correspondence between T-algebra homomorphisms TU → V and functions U → V: for a T-algebra homomorphism fTU → V, define f^† = f ∘η U → V; for a function gU → V, define g^♯ = v ∘ TgTU → V. Then g^♯ is a T-algebra homomorphism called the free T-extension of g, and we have f^†♯ = f and g^♯† = g.We now have all the ingredients to define our notion of automaton with side-effects and their language semantics. We fix a monad (T, η, μ) with T preserving finite sets, as well as a T-algebra O that models outputs of automata. A T-automaton is a quadruple (Q, δ Q → Q^A,Q → O, ∈ Q), where the state spaceQ is a T-algebra, the transition mapδ and output map are T-algebra homomorphisms, andis the initial state. DFAs are 𝙸𝚍-automata when O = 2 = {0, 1} is used to distinguish accepting from rejecting states. For the more general case of O being any set, DFAs generalize into Moore automata. Recall that -algebras are JSLs, and their homomorphisms are join-preserving functions. In a -automaton, Q is equipped with a join operation, and Q^A is a join-semilattice with pointwise join: (f ∨ g)(a) = f(a) ∨ g(a) for a ∈ A. Since the automaton maps preserve joins, we have, in particular, δ(q_1 ∨ q_2)(a) = δ(q_1)(a) ∨δ(q_2)(a). One can represent an NFA over a set of states S as a -automaton by taking Q = ((S),⋃) and O = 2, the Boolean join-semilattice with the or operation as its join. Let ⊆ S be the set of initial states and (Q) → 2 and δ(S) →(S)^A the respective extensions (Proposition <ref>) of the NFA's output and transition functions. The resulting -automaton is precisely the determinized version of the NFA. More generally, an automaton with side-effects given by a monad T always represents a T-automaton with a free state space: by applying Proposition <ref>, we have the following. A T-automaton of the form ((TX, μ_X), δ, , ), for any set X, is completely defined by the set X with the element ∈ TX and functions δ^† X → (TX)^A ^† X → O.We call such a T-automaton a succinct automaton, which we sometimes identify with the representation (X, δ^†, ^†, ).A (generalized) language is a function A^* → O. For every T-automaton we have an observability and a reachability map, telling respectively which state is reached by reading a given word and which language each state recognizes. The reachability map of a T-automatonwith state space Q is a function r_ A^* → Q inductively defined as follows: r_() = and r_(ua) = δ(r_(u))(a).The observability map ofis a function o_ Q → O^A^* inductively defined as follows: o_(q)() = (q) and o_(q)(av) = o_(δ(q)(a))(v).The language accepted byis the function _ A^* → O given by _ = o_() = _∘ r_. For an NFArepresented as a -automaton, as seen in Example <ref>, o_(q) is the language of q in the traditional sense. Notice that q, in general, is a set of states: o_(q) takes the union of languages of singleton states. The set _ is the language accepted by the initial states, i.e., the language of the whole NFA. The reachability map r_(u) returns the set of states reached via all possible paths reading u. Given a language A^* → O, there exists a (unique) minimal T-automaton_ accepting . Its existence follows from general facts see <cit.>.Let t_ A^* → O^A^* be the function giving the residual languages of , namely t_(u) = λ v.(uv).The minimal T-automaton _ acceptinghas state space M = (t_^♯), initial state = t_(), and T-algebra homomorphisms M → O and δ M → M^A given by (t_^♯(U)) = (U) and δ(t_^♯(U))(a)(v) = t_^♯(U)(av). In the following, we will also make use of the minimal Moore automaton accepting . Although this always exists—it is defined by instantiating <ref> with T =—it need not be finite. The following property says that finiteness of Moore automata and of T-automata accepting the same language are intimately related. propositionfinitet The minimal Moore automaton acceptingis finite if and only if the minimal T-automaton acceptingis finite. § A GENERAL ALGORITHM In this section we introduce our extension ofto learn automata with side-effects. The algorithm is parametric in the notion of side-effect, represented as the monad T, and is therefore called .We fix a language A^* → O that is to be learned, and we assume that there is a finite T-automaton accepting . This assumption generalizes the requirement ofthatis regular (i.e., accepted by a specific class of T-automata, see <ref>). An observation table consists of a pair of functions S → O^ES → (O^E)^Agiven by (s)(e) = (se) and (s)(a)(e) = (sae), where S, E ⊆ A^* are finite sets with ∈ S ∩ E. For O=2, we recover exactly theobservation table. The key idea foris defining closedness and consistency over the free T-extensions of those functions. The table is closed if for all U ∈ T(S) and a ∈ A there exists a U' ∈ T(S) such that ^♯(U') = ^♯(U)(a). The table is consistent if for all U_1, U_2 ∈ T(S) such that ^♯(U_1) = ^♯(U_2) we have ^♯(U_1) = ^♯(U_2). For closedness, we do not need to check all elements of T(S) × A against elements of T(S), but only those of S × A, thanks to the following result.If for all s ∈ S and a ∈ A there is U ∈ T(S) such that ^♯(U) = (s)(a), then the table is closed. Let m (^♯) ↪ O^E be the embedding of the image of ^♯ into its codomain. According to <cit.>, the definition of closedness given in Definition <ref> amounts to requiring the existence of a T-algebra homomorphism 𝖼𝗅𝗈𝗌𝖾 making the following diagram commute: [ampersand replacement=&] T(S) [dashed]d[swap]𝖼𝗅𝗈𝗌𝖾dr^♯ (^♯)^A r[swap]m^A& (O^E)^A It is easy to see that the hypothesis of this lemma corresponds to requiring the existence of a function 𝖼𝗅𝗈𝗌𝖾' making the diagram below on the left incommute. [ampersand replacement=&] S [dashed]d[swap]𝖼𝗅𝗈𝗌𝖾'dr (^♯)^A r[swap]m^A& (O^E)^A [ampersand replacement=&] T(S) d[swap]T(𝖼𝗅𝗈𝗌𝖾')drT() T((^♯)^A) r[swap]T(m^A)d& T((O^E)^A) d (^♯)^A r[swap]m^A& (O^E)^A This diagram can be made into a diagram of T-algebra homomorphisms as on the right, where the compositions of the left and right legs give respectively 𝖼𝗅𝗈𝗌𝖾'^♯ and ^♯. This diagram commutes because the top triangle commutes by functoriality of T, and the bottom square commutes by m^A being a T-algebra homomorphism. Therefore we have that (<ref>) commutes for 𝖼𝗅𝗈𝗌𝖾 = 𝖼𝗅𝗈𝗌𝖾'^♯.For NFAs represented as -automata, the properties are as presented in Section <ref>. Recall that for T = and O = 2, the Boolean join-semilattice, ^♯ and ^♯ describe a table where rows are labeled by subsets of S. Then we have, for instance, ^♯({s_1,s_2})(e) = (s_1)(e) ∨(s_2)(e), i.e., ^♯({s_1,s_2})(e) = 1 if and only if (s_1e) = 1 or (s_2e) = 1. Closedness amounts to check whether each row in the bottom part of the table is the join of a set of rows in the top part. Consistency amounts to check whether, for all sets of rows U_1,U_2 ⊆ S in the top part of the table whose joins are equal, the joins of rows U_1 ·{a} and U_2 ·{a} in the bottom part are also equal, for all a ∈ A. If closedness and consistency hold, we can define a hypothesis T-automaton . Its state space is H = (^♯), = (), and output and transition maps are given by: H → O(^♯(U)) = ^♯(U)()δH → H^Aδ(^♯(U)) = ^♯(U). The correctness of this definition follows from the abstract treatment of <cit.>, instantiated to the category of T-algebras and their homomorphisms.We can now give our algorithm .In the same way as for the example in Section <ref>, we only have to adjust lines <ref> and <ref> in <ref>. The resulting algorithm is shown in <ref>.Correctness.Correctness foramounts to proving that, for any target language , the algorithm terminates returning the minimal T-automaton _ accepting . As in the originalalgorithm, we only need to prove that the algorithm terminates, that is, that only finitely many hypotheses are produced. Correctness follows from termination, since line <ref> causes the algorithm to terminate only if the hypothesis automaton coincides with _.In order to show termination, we argue that the state space H of the hypothesis increases while the algorithm loops, and that H cannot be larger than M, the state space of _. In fact, when a closedness defect is resolved (line <ref>), a row that was not previously found in the image of ^♯ T(S) → O^E is added, so the set H grows larger. When a consistency defect is resolved (line 9), two previously equal rows become distinguished, which also increases the size of H.As for counterexamples, adding their prefixes to S (line <ref>) creates a consistency defect, which will be fixed during the next iteration, causing H to increase. This is due to the following result, which says that the counterexample z has a prefix that violates consistency. If z ∈ A^* is such that _(z) (z) and (z) ⊆ S, then there are a prefix ua of z, with u ∈ A^* and a ∈ A, and U ∈ T(S) such that (u) = ^♯(U) and (u)(a) ^♯(U)(a). Note that (z)() = (z) (definition of )_(z) (assumption) = _(r_(z)) (Definition of _) = r_(z)() (definition of _), so (z)r_(z). Let p ∈ A^* be the smallest prefix of z satisfying (p)r_(p). We have () = _ = r_(), so p and therefore p = ua for certain u ∈ A^* and a ∈ A. Let S' ⊂ S be the set from whichwas constructed—recall that we added (z) to S after constructing . Choose any U ∈ T(S') such that ^♯(U) = r_(u), which is possible because H is the image of ^♯ restricted to the domain T(S'). By the minimality property of p we have (u) = r_(u) = ^♯(U). Furthermore, (u)(a) = (ua) (definitions ofand ) r_(ua) (ua = p and (p)r_(p)) = δ_(r_(u))(a) (definition of r_) = δ_(^♯(U))(a) (r_(u) = ^♯(U)) = ^♯(U)(a) (definition of δ_).Now, note that, by increasing S or E, the hypothesis state space H never decreases in size. Moreover, for S = A^* and E = A^*, ^♯ = t_^♯, as defined in Definition <ref>. Therefore, since H and M are defined as the images of ^♯ and t_^♯, respectively, the size of H is bounded by that of M. Since H increases while the algorithm loops, the algorithm must terminate and thus is correct. We note that the RFSA learning algorithm of Bollig et al. does not terminate using this counterexample processing method <cit.>. This is due to their notion of consistency being weaker than ours: we have shown that progress is guaranteed because a consistency defect, in our sense, is created using this method.Query complexity. The complexity of automata learning algorithms is usually measured in terms of the number of both membership and equivalence queries asked, as it is common to assume that computations within the algorithm are insignificant compared to evaluating the system under analysis in real-world applications. The complexity of answering the queries themselves is not considered, as it depends on the implementation of the teacher, which the algorithm abstracts from.Notice that, as the table is a T-algebra homomorphism, asking membership queries for rows labeled by words in S is enough to determine all other rows, for which queries need not be asked. We measure the query complexities in terms of the number of states n of the minimal Moore automaton, the number of states t of the minimal T-automaton, the size k of the alphabet, and the length m of the longest counterexample. Note that t cannot be smaller than n, but it can be much bigger. For example, when T =, t may be in 𝒪(2^n).[ This can be seen from the language {a^p}, for some p ∈ and a singleton alphabet {a}. Its residual languages are ∅ and {a^i} for all 0 ≤ i ≤ p, which means the minimal DFA accepting the language has p + 2 states. However, the residual languages w.r.t. sets of words are all the subsets of {, a, aa, …, a^p}—hence, the minimal T-automaton has 2^p + 1 states. ]The maximum number of closedness defects fixed by the algorithm is n, as a closedness defect for the setting with algebraic structure is also a closedness defect for the setting without that structure. The maximum number of consistency defects fixed by the algorithm is t, as fixing a consistency defect distinguishes two rows that were previously identified. Since counterexamples lead to consistency defects, this also means that the algorithm will not pose more than t equivalence queries. A word is added to S when fixing a closedness defect, and 𝒪(m) words are added to S when processing a counterexample. The number of rows that we need to fill using queries is therefore in 𝒪(tmk). The number of columns added to the table is given by the number of times a consistency defect is fixed and thus in 𝒪(t). Altogether, the number of membership queries is in 𝒪(t^2mk). § SUCCINCT HYPOTHESES We now describe the first of two optimizations, which is enabled by the use of monads. Our algorithm produces hypotheses that can be quite large, as their state space is the image of ^♯, which has the whole set T(S) as its domain. For instance, when T =, T(S) is exponentially larger than S. We show how we can compute succinct hypotheses, whose state space is given by a subset of S. We start by defining sets of generators for the table. A set S' ⊆ S is a set of generators for the table whenever for all s ∈ S there is U ∈ T(S') such that (s) = ^♯(U).[ Here and hereafter we assume that T(S') ⊆ T(S), and more generally that T preserves inclusion maps. To eliminate this assumption, one could take the inclusion map fS' ↪ S and write ^♯(T(f)(U)) instead of ^♯(U). ] Intuitively, U is the decomposition of s into a “combination” of generators. When T =, S' generates the table whenever each row can be obtained as the join of a set of rows labeled by S'. Explicitly: for all s ∈ S there is {s_1,…,s_n}⊆ S' such that (s) = ^♯({s_1,…,s_n}) = (s_1) ∨…∨(s_n).Recall that , with state space H, is the hypothesis automaton for the table. The existence of generators S' allows us to compute a T-automaton with state space T(S') equivalent to . We call this the succinct hypothesis, although T(S') may be larger than H. Proposition <ref> tells us that the succinct hypothesis can be represented as an automaton with side-effects in T that has S' as its state space. This results in a lower space complexity when storing the hypothesis. We now show how the succinct hypothesis is computed. Observe that, if generators S' exist, ^♯ factors through the restriction of itself to T(S'). Denote this latter function . Since we have T(S') ⊆ T(S), the image ofcoincides with (^♯) = H, andtherefore the surjection restrictingto its image has the form eT(S') → H. Any right inverse iH → T(S') of the function e (that is, e ∘ i = _H, but whereas e is a T-algebra homomorphism, i need not be one) yields a succinct hypothesis as follows. The succinct hypothesis is the following T-automaton : its state space is T(S'), its initial state is = i(()), and we define ^†S' → O^†(s) = (s)()δ^†S' → T(S')^Aδ^†(s)(a) = i((s)(a)). This definition is inspired by that of a scoop, due to <cit.>.propositionsucclang Any succinct hypothesis ofaccepts the same language as .The proof can be found in the appendix.We now give a simple procedure to compute a minimal set of generators, that is, a set S' such that no proper subset is a set of generators. This generalizes a procedure defined by <cit.> for non-deterministic, universal, and alternating automata.propositiongenalg The following algorithm returns a minimal set of generators for the table:S'S there are s ∈ S' and U ∈ T(S' ∖{s}) s.t. ^♯(U) = (s)S'S' ∖{s}S'The proof can be found in the appendix. Determining whether U as in the algorithm given in Proposition <ref> exists, one can always naively enumerate all possibilities, using that T preserves finite sets. This is what we call the basic algorithm. For specific algebraic structures, one may find more efficient methods, as we show in the following example. Consider again the powerset monad T =. We now exemplify two ways of computing succinct hypotheses, which are inspired by the definitions of canonical RFSAs <cit.>. The basic idea is to start from a deterministic automaton and to remove states that are equivalent to a set of other states. The algorithm given in Proposition <ref> computes a minimal S' that only contains labels of rows that are not the join of other rows. (In case two rows are equal, only one of their labels is kept.) In other words, as mentioned in Section <ref>, S' contains labels of join-irreducible rows. To concretize the algorithm efficiently, we use a method introduced by <cit.>, which essentially exploits the natural order on the JSL of table rows. In contrast to the basic exponential algorithm, this results in a polynomial one.[When we refer to computational complexities, as opposed to query complexities, they are in terms of the sizes of S, E, and A.] Bollig et al. determine whether a row is a join of other rows by comparing the row just to the join of rows below it. Like them, we make use of this also to compute right inverses of e, for which we will formalize the order. The function e (S') → H tells us which sets of rows are equivalent to a single state in H. We show two right inverses H →(S') for it. The first one, i_1(h) = {s ∈ S' |(s) ≤ h}, stems from the construction of the canonical RFSA of a language <cit.>. Here we use the order a ≤ ba ∨ b = b induced by the JSL structure. The resulting construction of a succinct hypothesis was first used by <cit.>. This succinct hypothesis has a “maximal” transition function, meaning that no more transitions can be added without changing the language of the automaton. The second inverse is i_2(h) = {s ∈ S' |(s) ≤ hand for alls' ∈ S's.t. (s) ≤(s') ≤ h we have (s) = (s')}, resulting in a more economical transition function, where some redundancies are removed. This corresponds to the simplified canonical RFSA <cit.>. Consider again the powerset monad T =, and recall the table in <ref>. When S' = S, the right inverse given by i_1 yields the succinct hypothesis shown below. .8[initial,state,accepting] (q0) ; [state] (q1) [right of=q0] ; [state,accepting] (q2) [right of=q1] ; [->] (q0) edge [bend left] node a (q1) (q1) edge [bend left] node a (q2) (q1) edge [bend left,swap] node a (q0) (q1) edge [loop above] node a () (q2) edge [bend left,swap] node a (q1) (q2.south) edge [bend left,swap] node[] a (q0.south) (q2) edge [loop right] node a (); Note that i_1((aa)) = {, a, aa}. When instead taking i_2, the succinct hypothesis is just the DFA (<ref>) because i_2((aa)) = {aa}. Rather than constructing a succinct hypothesis directly, our algorithm first reduces the set S'. In this case, we note that (aa) = ^♯({, a}), so we can remove aa from S'. Now i_1 and i_2 coincide and produce the NFA (<ref>). Minimizing the set S' in this setting essentially comes down to determining what <cit.> call the prime rows of the table. The algorithm in Proposition <ref> implicitly assumes an order in which elements of S are checked. Although the algorithm is correct for any such order, different orders may give results that differ in size.§ OPTIMIZED COUNTEREXAMPLE HANDLING The second optimization we give generalizes the counterexample processing method due to <cit.>, which improves the worst case complexity of the number of membership queries needed in . <cit.> proposed to add all suffixes of the counterexample to the set E instead of adding all prefixes to the set S. This eliminates the need for consistency checks in the deterministic setting. The method by Rivest and Schapire finds a single suffix of the counterexample and adds it to E. This suffix is chosen in such a way that it either distinguishes two existing rows or creates a closedness defect, both of which imply that the hypothesis automaton will grow.The main idea is finding the distinguishing suffix via the hypothesis automaton . Given a word u ∈ A^*, let q_u be the state inreached by reading u, i.e., q_u = r_(u). For each q ∈ H, we pick any U_q ∈ T(S) that yields q according to the table, i.e., such that ^♯(U_q) = q. Then for a counterexample z we have that the residual language w.r.t. U_q_z does not “agree” with the residual language w.r.t. z.The above intuition can be formalized as follows. Let A^* → O^A^* be given by (u) = t_^♯(U_q_u) for all u ∈ A^*, the residual language computation.We have the following technical lemma, saying that a counterexample z distinguishes the residual languages t_(z) and (z). If z ∈ A^* is such that _(z) (z), then t_(z)() (z)(). We have t_(z)() = (z) (definition of t_)_(z) (assumption) = (_∘ r_)(z) (definition of _) = r_(z)() (definition of _) = q_z() (definition of q_z) = ^♯(U_q_z)() (definition of U_q_z) = t_^♯(U_q_z)() (definitions ofand t_) = (z)() (definition of ). We assume that U_q_ = η(). For a counterexample z, we then have ()(z) = t_()(z) (z)(). While reading z, the hypothesis automaton passes a sequence of states q_u_0, q_u_1,q_u_2,…,q_u_n, where u_0 = ϵ, u_n = z, and u_i+1 = u_ia for some a ∈ A is a prefix of z. If z were correctly classified by , all residuals (u_i) would classify the remaining suffix v of z, i.e., such that z = u_iv, in the same way. However, the previous lemma tells us that, for a counterexample z, this is not case, meaning that for some suffix v we have (ua)(v) ≠(u)(av). In short, this inequality is discovered along a transition in the path to z. If z ∈ A^* is such that _(z) (z), then there are u, v ∈ A^* and a ∈ A such that uav = z and (ua)(v) (u)(av).To find such a decomposition efficiently, Rivest and Schapire use a binary search algorithm. We conclude with the following result that turns the above property into the elimination of a closedness witness. That is, given a counterexample z and the resulting decomposition uav from the above corollary, we show that, while currently ^♯(U_q_ua) = ^♯(U_q_u)(a), after adding v to E we have ^♯(U_q_ua)(v) ^♯(U_q_u)(a)(v). (To see that the latter follows from the proposition below, note that for all U ∈ T(S) and e ∈ E, ^♯(U)(e) = t_^♯(U)(e) and for each a' ∈ A, ^♯(U)(a')(e) = t_^♯(U)(a'e), by the definition of those maps.) The inequality means that either we have a closedness defect, or there still exists some U ∈ T(S) such that ^♯(U) = ^♯(U_q_u)(a). In this case, the rows ^♯(U) and ^♯(U_q_ua) have become distinguished by adding v, which means that the size of H has been increased. We know that a closedness defect leads to an increase in the size of H, so in any case we make progress. If z ∈ A^* is such that _(z) (z), then there are u, v ∈ A^* and a ∈ A such that ^♯(U_q_ua) = ^♯(U_q_u)(a) and t_^♯(U_q_ua)(v)t_^♯(U_q_u)(av). By Corollary <ref> we have u, v ∈ A^* and a ∈ A such that (ua)(v) (u)(av). This directly yields the inequality by the definition of . Furthermore, (U_q_ua) = q_ua(definition of U_q_ua) = r_(ua) (definition of q_ua) = δ_(r_(u))(a) (definition of r_) = δ_(q_u)(a) (definition of q_u) = δ_(^♯(U_q_u))(a) (definition of U_q_u) = ^♯(U_q_u)(a) (definition of δ_). We now show how to combine this optimized counterexample processing method with the succinct hypothesis optimization from Section <ref>. Recall that the succinct hypothesisis based on a right inverse iH → T(S') of eT(S') → H. Choosing such an i is equivalent to choosing U_q for each q ∈ H. We then redefineusing the reachability map of the succinct hypothesis. Specifically, (u) = t_^♯(r_(u)) for all u ∈ A^*.Unfortunately, there is one complication. We assumed earlier that U_q_ = η(), or more specifically ()(z) = (z). This now may be impossible because we do not even necessarily have ∈ S'. We show next that if this equality does not hold, then there are two rows that we can distinguish by adding z to E. Thus, after testing whether ()(z) = (z), we either add z to E (if the test fails) or proceed with the original method. If z ∈ A^* is such that ()(z) (z), then ^♯(_) = () and t_^♯(_)(z)t_()(z). We have ^♯(_) = ^♯(i(())) = () by the definitions of _ and i, and t_^♯(i(()))(z) = t_^♯(_)(z) (definition of _) = t_^♯(r_())(z) (definition of r_) = ()(z) (definition of )(z) (assumption) = t_()(z) (definition of t_). To see that the original method still works, we prove the analogue of Lemma <ref> for the new definition of . If z ∈ A^* is such that _(z) (z) and ()(z) = (z), then ()(z) (z)(). We have ()(z) = (z) (assumption)_(z) (counterexample) = (_∘ r_^†)(z) (definition of _) = (^♯∘ r_^†)(z)() (definition of _) = t_^♯(r_^†(z))() (definition of ^♯) = (z)() (definition of ). If z ∈ A^* is such that _(z) (z) and ()(z) = (z), then there are u, v ∈ A^* and a ∈ A such that uav = z and (ua)(v) (u)(av). Now we are ready to prove the analogue of Proposition <ref>. If z ∈ A^* is such that _(z) (z) and ()(z) = (z), then there are u, v ∈ A^* and a ∈ A such that ^♯(r_^†(ua)) = ^♯(r_^†(u))(a) and t_^♯(r_^†(ua))(v)t_^♯(r_^†(u))(av). Let u, a, and v be as in Corollary <ref>. Thus, t_^♯(r_^†(ua))(v) = (ua)(v) (u)(av) = t_^♯(r_^†(u))(av). Furthermore, since for all s ∈ S and b ∈ A we have ((^♯)^A ∘δ_^†)(s)(b) = ^♯(δ_^†(s)(b)) = (^♯∘ i)((s)(b)) (definition of δ_^†) = (s)(b) (definition of i), it follows that (^♯)^A ∘δ_ = ^♯. Therefore, ^♯(r_^†(ua)) = ^♯(δ_(r_^†(u))(a)) (definition of r_^†) = ((^♯)^A ∘δ_)(r_^†(u))(a) = ^♯(r_^†(u))(a). Recall the succinct hypothesisfrom <ref> for the table in <ref>. Note that S' = S cannot be further reduced. The hypothesis is based on the right inverse iH →(S) of e (S) → H given by i(()) = {} and i(^♯(∅)) = ∅. This is the only possible right inverse because e is bijective. For the prefixes of the counterexample aa we have r_() = {} and r_(a) = r_(aa) = ∅. Note that t_^♯({})(aa) = 1 while t_(∅)(a) = t_(∅)() = 0. Thus, ()(aa) (a)(a). Adding a to E would indeed create a closedness defect.Query complexity. Again, we measure the membership and equivalence query complexities in terms of the number of states n of the minimal Moore automaton, the number of states t of the minimal T-automaton, the size k of the alphabet, and the length m of the longest counterexample.A counterexample now gives an additional column instead of a set of rows, and we have seen that this leads to either a closedness defect or to two rows being distinguished. Thus, the number of equivalence queries is still at most t, and the number of columns is still in 𝒪(t). However, the number of rows that we need to fill using membership queries is now in 𝒪(nk). This means that a total of 𝒪(tnk) membership queries is needed to fill the table.Apart from filling the table, we also need queries to analyze counterexamples. The binary search algorithm mentioned after Corollary <ref> requires for each counterexample 𝒪(logm) computations of (x)(y) for varying words x and y. Let r be the maximum number of queries required for a single such computation. Note that for u, v ∈ A^*, and letting α TO → O be the algebra structure on O, we have(u)(v) = α(T(_v ∘ t_)(U_q_u))for the original definition ofand(u)(v) = α(T(_v ∘ t_)(r_^†(u)))in the succinct hypothesis case. Since the restricted map T(_v ∘ t_)TS → TO is completely determined by _v ∘ t_ S → O, r is at most |S|, which is bounded by n in this optimized algorithm. For some examples (see for instance the writer automata in Section <ref>), we even have r = 1. The overall membership query complexity is 𝒪(tnk + trlogm).Dropping Consistency. We described the counterexample processing method based around Proposition <ref> in terms of the succinct hypothesisrather than the actual hypothesisby showing thatcan be defined using . Since the definition of the succinct hypothesis does not rely on the property of consistency to be well-defined, this means we could drop the consistency check from the algorithm altogether. We can still measure progress in terms of the size of the set H, but it will not be the state space of an actual hypothesis during intermediate stages. This observation also explains why <cit.> are able to use a weaker notion of consistency in their algorithm.Interestingly, they exploit the canonicity of their choice of succinct hypotheses to arrive at a polynomial membership query complexity that does not involve the factor t. § EXAMPLES In this section we list several examples that can be seen as T-automata and hence learned via an instance of . We remark that, since our algorithm operates on finite structures (recall that T preserves finite sets), for each automaton type one can obtain a basic, correct-by-construction instance offor free, by just plugging the concrete definition of the monad into the abstract algorithm. However, we note that this is not howis intended to be used in a real-world context. Instead, it should be seen as an abstract specification of the operations each concrete implementation needs to perform, or, in other words, as a template for real implementations. For each instance below, we discuss whether certain operations admit a more efficient implementation than the basic one, based on the specific algebraic structure induced by the monad. We also mention related algorithms from the literature.Due to our general treatment, the optimizations of Sections <ref> and <ref> apply to all of these instances. Non-deterministic automata. As discussed before, non-deterministic automata are -automata with a free state space, provided that O = 2 is equipped with the “or” operation as its -algebra structure. We also mentioned that, as <cit.> showed, there is a polynomial time algorithm to check whether a given row is the join of other rows. This gives an efficient method for handling closedness straight away. Moreover, as shown in <ref>, it allows for an efficient construction of the succinct hypothesis. Unfortunately, checking for consistency defects seems to require a number of computations exponential in the number of rows. We recall that <cit.> use an ad-hoc version of consistency which cannot be easily captured in our framework. However, as explained at the end of Section <ref>, we can in fact drop consistency altogether.Universal automata. Just like non-deterministic automata, universal automata can be seen as -automata with a free state space. The difference, however, is that the -algebra structure on O = 2 is dual: it is given by the “and” rather than the “or” operation. Universal automata accept a word when all paths reading that word are accepting. One can dualize the optimized specific algorithms for the case of non-deterministic automata. This is precisely what <cit.> have done.Partial automata. Consider the maybe monad, given by (X) = 1 + X, with natural transformations having components η_XX → 1 + X and μ_X1 + 1 + X → 1 + X defined in the standard way. Partial automata with states X can be represented as -automata with state space (X) = 1 + X, where there is an additional sink state, and output algebra O = (1) = 1 + 1. Here the left value is for rejecting states, including the sink one. The transition map δ 1 + X → (1 + X)^A represents an undefined transition as one going to the sink state. The algorithm [] is mostly like , except that implicitly the table has an additional row with zeroes in every column. Since the monad only adds a single element to each set, there is no need to optimize the basic algorithm for this specific case.Weighted automata. Recall from Section <ref> the free semimodule monad, sending a set X to the free semimodule over a finite semiring . Weighted automata over a set of states X can be represented as -automata whose state space is the semimodule (X), the output function (X) → assigns a weight to each state, and the transition map δ(X) →(X)^A sends each state and each input symbol to a linear combination of states. The obvious semimodule structure onextends to a pointwise structure on the potential rows of the table. The basic algorithm loops over all linear combinations of rows to check closedness and over all pairs of combinations of rows to check consistency. This is an extremely expensive operation. Ifis a field, a row can be decomposed into a linear combination of other rows in polynomial time using standard techniques from linear algebra. As a result, there are efficient procedures for checking closedness and constructing succinct hypotheses. It was shown by <cit.> that consistency in this setting is equivalent to closedness of the transpose of the table. This trick is due to <cit.>, who first studied learning of weighted automata. Alternating automata. We use the characterization of alternating automata due to <cit.>. Recall that, given a partially ordered set (P,≤), an upset is a subset U of P such that, if x ∈ U and x ≤ y, then y ∈ U. Given Q ⊆ P, we write Q for the upward closure of Q, that is the smallest upset of P containing Q. We consider the monadthat maps a set X to the set of all upsets of (X).Its unit is given by η_X(x) = {{ x }} and its multiplication byμ_X(U) = {V ⊆ X |∃_W ∈ U ∀_Y ∈ W ∃_Z ∈ Y Z ⊆ V}.The sets of sets in (X) can be seen as DNF formulae over elements of X, where the outer powerset is disjunctive and the inner one is conjunctive. Accordingly, we define an algebra structure β(2) → 2 on the output set 2 by letting β(U) = 1 if { 1 }∈ U, 0 otherwise. Alternating automata with states X can be represented as -automata with state space (X), output map (X) → 2, and transition map δ(X) →(X)^A, sending each state to a DNF formula over X. The only difference with the usual definition of alternating automata is that (X) is not the full set (X), which would not give a monad in the desired way. However, for each formula in (X) there is an equivalent one in (X).An adaptation offor alternating automata was introduced by <cit.> and further investigated by <cit.>. The former found that given a row r ∈ 2^E and a set of rows X ⊆ 2^E, r is equal to a DNF combination of rows from X (where logical operators are applied component-wise) if and only if it is equal to the combination defined by Y = {{x ∈ X | x(e) = 1}| e ∈ E ∧ r(e) = 1}.In our setting, we can reuse this idea to efficiently find closedness defects and to construct the hypothesis. Notice that, even though the monadformally requires the use of DNF formulae representing upsets, in the actual implementation we can use smaller formulae, e.g., Y above instead of its upward closure. In fact, it is easy to check that DNF combinations of rows are invariant under upward closure.As with non-deterministic and universal automata, we do not know of an efficient way to ensure consistency. As in the existing algorithms mentioned above, we could drop it altogether.Writer automata. The examples considered so far involve existing classes of automata. To further demonstrate the generality of our approach, we introduce a new (as far as we know) type of automaton, which we call writer automaton. The writer monad(X) = × X for a finite monoidhas a unit η_XX →× X given by adding the unit e of the monoid, η_X(x) = (e, x), and a multiplication μ_X ×× X →× X given by performing the monoid multiplication, μ_X(m_1, m_2, x) = (m_1m_2, x). In Haskell, the writer monad is used for such tasks as collecting successive log messages, where the monoid is given by the set of sets or lists of possible messages and the multiplication adds a message.The algebras for this monad are sets Q equipped with an -action. One may take the output object to be the setwith the monoid multiplication as its action. -automata with a free state space can be represented as deterministic automata that have an element ofassociated with each transition. The semantics of these is that the encountered -elements multiply along paths and finally multiply with the output of the last state to produce the actual output.The basic learning algorithm is already of polynomial time complexity. In fact, to determine whether a given row is a combination of rows in the table, i.e., whether it is given by a monoid value applied to one of the rows in the table, one simply tries all of these values. This allows us to check for closedness, to minimize the generators, and to construct the succinct hypothesis, in polynominal time. Consistency involves comparing all ways of applying monoid values to rows and, for each comparison, at most |A| further comparisons between one-letter extensions. The total number of comparisons is clearly polynomial in ||, |S| and |A|.§ IMPLEMENTATION We have implemented the generalalgorithm in Haskell, taking full advantage of the monads provided by its standard library. Apart from the high-level implementation, our library provides* a basic implementation for weighted automata over a finite semiring, with a polynomial time variation for the case where the semiring is a field[ Despite the assumption in the present paper that the monad preserves finite set, our implementation can learn weighted automata over infinite fields and thus implements the general algorithm introduced by <cit.>, which was studied in a categorical context by <cit.> and <cit.>. ]; * an implementation for non-deterministic automata that has polynomial time implementations for ensuring closedness and constructing the hypothesis, but not for ensuring consistency; * a variation on the previous algorithm that uses the notion of consistency defined by <cit.>; * instantiations of the basic algorithm to the monad being (-) + E, for E a finite set of exceptions, and , both of which result in polynomial time algorithms; In this section we describe the main structure and ingredients of our library. After recalling monads in Haskell in Section <ref>, we start with the formalization of automata in Section <ref>. We then introduce teachers in Section <ref> before exploring the actual learning algorithm in Section <ref>. We give details for the non-deterministic and weighted case, whose monads deserve a closer analysis.§.§ Monads We note that a monad in Haskell is specified as a Kleisli triple(T, η, (-)^♯), where T assigns to every set X a set TX, η consists of a component η_XX → TX for each set X, and (-)^♯ provides for each function fX → TY an extension f^♯ TX → TY. These need to satisfyf^♯∘η = fη^♯ = (g^♯∘ f)^♯ = g^♯∘ f^♯.Kleisli triples are in a one-to-one correspondence with monads. On both sides of this correspondence we have the same T and η, which for a Kleisli triple are turned into a functor with a natural transformation by setting Tf = (η∘ f)^♯. Furthermore, (-)^♯ and μ are obtained from each other by f^♯ = μ∘ Tf and μ = ^♯. Indeed, under this correspondence the (-)^♯ operation is a specific instance of the extension operation defined for a monad, with the T-algebra codomain restricted to free T-algebras. In Haskell, the η of the Kleisli triple is written , and, given fX → TY and x ∈ TX, f^♯(x) is writtenand referred to as the bind operation. Furthermore, for any fX → Y, Tf is given by .Some basicmonads cannot directly be written down in Haskell because their definition can only be given on types equipped with an equality check, or, for reasons of efficiency, a total order. For example, thetype provided bycomes with afunction that has the following signature:One will have to use unions in one way or another in defining the bind of the powerset monad. However, since this bind needs to be of typeand does not assume aninstance on , the powerset monad cannot be defined in this way.One solution is to delay the monadic computations in a wrapper type whose constructors are used to define a monad instance: the free monad. Specifically, we endow the freer monad of <cit.> with a constraint parameter:Such a constrained free monad was first defined by George Giorgidze, but only for the specific case whereisandis .[<https://hackage.haskell.org/package/set-monad-0.2.0.0>] On the constrained free monad we can define a completeinstance:This is the same code as used by <cit.>, but we note that on the last line, sinceis the first argument ofin , we know that the appropriate constrained needed to invokeon the right-hand side, with againas its first argument, is satisfied.Finally, if there is a monad that is defined only on types satisfying a certain constraint, then we can convert from our free monad type with that constraint back to the actual “monad”:Note that operations such as equality checks foruseto delegate the operation to whatever is defined for . This means that in code that abstracts from the monad we seem to be working withas a monad.As an example, the “monad” becomesWe may then useas the monad.To implement the free semimodule monad in Haskell, we use thetype from . Note that the monad will be defined in the first argument for that type, so we need to create an auxiliary type to swap the arguments.Defining the monad again requiresconstraints.The functionscales a map by an element from the semiring;adds two maps together. Both operations are pointwise. The monad we can use is . §.§ Automata We model an automaton as a simple deterministic automaton.For such automata, we can easily implement reachability and language functions, as well as bisimulation. Bisimulation is used to realize exact equivalence queries for the teachers that hold an automaton accepting the language to be learned. To optimize for the monad in the same way the learning algorithm is optimized, we use bisimulation up to context <cit.>.Hererepresents the monad that we optimize for. Up to context means that, when considering a pairof next states and the current relation , the pairdoes not need to be added to the relation if it can be obtained as a combination of the elements of , using the free algebra structures ofand . The first argument ofis a function that should determine this. Because of this abstraction, we do not actually need to constrainto be a monad here. For themonad, one can simply useas the first argument. The second argument is the alphabet.Succinct automata optimized by a monadenjoy a more concrete representation involving maps.This is the type of the automata that theimplementation learns. The concrete representation allows the automaton to be displayed and exported. Of course, one can determinize a succinct automaton using -algebras forand .The typeis defined to be . We allow an arbitrary algebra onrather than assuming the componentof the distributive law used in earlier sections because this allows us to run the delayed monadic computations discussed earlier, which would otherwise pile up and cause serious performance issues. §.§ Teaching A teacher in our implementation is an object that comprises membership and equivalence functions. It also records the alphabet.objects are parameterized by a monadthat serves a different purpose than optimizing the learning algorithm: it is the monad of side-effects allowed by the implementation of queries. Whereas themonad suffices for a predefined automaton, one may have to use themonad to interact with an actual black-box system. By allowing an arbitrary monad rather than assuming themonad, we are able to build features such as query counters and a cache on top of any teacher through the use of monad transformers. A monad transformer provides for any monad a new monad into which the original one can be embedded. For example, themonad adds a state with values into an existing monad . This is the transformer that enables the addition of query counters and a cache to a teacher: The most basic teacher holds an automaton that it uses to determine membership and equivalence, the latter of which is implemented through bisimulation.It implements afor any monadbecause it does not have any side-effects.We also provide a teacher that implements equivalence queries through random testing. Its first argument is the number of tests per equivalence query, while the second argument samples test words:is a random number generator. Once more we use themonad transformer, in this case to add a random number generator state to the monadthat the membership query function, which is the last argument, may use. This query function is used both for membership queries and for generated test queries. Note that this particular teacher does not give any guarantees on the validity of positive responses to equivalence queries. We do also provide the random sampling teacher suggested by <cit.>, which guarantees that on a positive answer the hypothesis is probably approximately correct, a notion introduced by <cit.>.Here the first argument is the accuracy ϵ, while the second one is the confidence ∂. Both should be values between 0 and 1. If dA^* → [0, 1] is the distribution represented by the third argument (converting between Haskell types and sets for convenience) and l_1, l_2A^* → O are the languages of the hypothesis and the target, the guarantee is that, with probability at least 1 - ∂, ∑_u ∈ A^*, l_1(u)l_2(u) d(u) ≤ϵ. Compared to , anhas been added to the state because the number of tests depends on the number of equivalence queries that have already been asked. §.§ Learning We define atype that allows us to switch between variations onand to optimize certain specific procedures.The functiontakes an observation table, a list of labels , and a row , and determines whethercan be obtained as a combination of the rows with labels in . If this is the case, it returns the combination, which has type . This function is used to check closedness, to minimize the labels used as states for the hypothesis, and to construct the hypothesis. Ifis set to , it indicates that consistency should be solved by solving closedness for what we call the transpose of the table (swapping S and E and reversing their words while considering the reverse of the target language as the target language); otherwise, it contains a function that given an observation table produces a new column to fix one of its consistency defects, unless the table is already consistent. Solving closedness for the transpose of the table always ensures consistency, but in general it may add more columns than necessary. Lastly,is a type that enumerates our adaptations of the three counterexample handling methods: the original one by <cit.>, the one by <cit.>, and the one by <cit.>.To enable basic implementations ofandthat work for any monad T (preserving finite sets), we need to be able to loop over the values of TS. In order to facilitate this, there is a classwhose only member function turns a list of values of any type into a list of values with typeapplied to that type. It is intended to be the concrete application of a functor to a set (represented as a list). We provide the functionsand , both conditioned with aconstraint, which directly enable a basic version of the learning algorithm.To optimize the algorithm in a specific setting, a programmer only has to adjust these two functions. We provide such optimized functions for the cases of non-deterministic and weighted automata (over a field). Regarding the former case, we provideand , which are essentially the right inverses corresponding to the canonical and simplified canonical RFSA, respectively, as explained in <ref>. Our optimized weighted algorithm uses Gaussian elimination in a function calledand solves consistency by solving closedness for the transpose of the table, a method readily available regardless of the monad.Enabling our adaptation of the counterexample handling method due to Rivest and Schapire requires an additional condition. Recall that this method requires us to pose membership queries for combinations of words, which can be done by extending the membership query function (the language) of typeto one of typeusing the algebra structure defined on . However, our membership query function actually has type , and there is no reason to assume any interaction betweenand .As a workaround, we will assume an instance offor the monad , whereis a class defined as follows:Given anyand , we requireto be such that the computation ofonly evaluateson the elements of . Naturally, we wantto be as small as possible: it should contain exactly those elements of typethat are present in . As an example, recall that the free semimodule monad with values in a semiringcan be defined on a typeas , where we identify a missing value for an element with that element being assigned zero. Given ,is given by the keys of the mapthat are assigned a non-zero value. Using the instance for a monad , the membership query function can be extended by querying the words in the support of a given element ofsequentially, constructing a partial membership query function defined only on that support, and evaluating the extension of that function. This method works because we assume that the side-effects exhibited bydo not influence future membership queries.Finally, our generalimplementation has the following signature: § EXPERIMENTS In this section we analyze the performance, in terms of number of queries, of several variations of our algorithm by running them on randomly generated WFAs, NFAs, and plain Moore automata. Our aim is to show the effect of exploiting the right monad and of using our adapted optimized counterexample handling method. The experiments are run using the implementation discussed in Section <ref>. In all cases we use an alphabet of size 3. Random Moore automata are generated by choosing for each state an output and further for each input symbol a next state using uniform distributions. The WFAs are over the field of size 5. Here the outputs are chosen in the same way, and for each pair of states and each input symbol, we create a transition symbol from the first to the second state with a random weight chosen uniformly. We take the average of 100 iterations for each of the sizes for which we generate automata. Membership query results in tables will be rounded to whole numbers. We use bisimulation to find counterexamples in all experiments, exploiting the fact that the target automaton will be known. We cache membership queries so that the counts exclude duplicates.experiments/dfa/lstar_mq.datexperiments/dfa/lstarMP_mq.datexperiments/dfa/lstarRS_mq.datexperiments/dfa/lstar_eq.datexperiments/dfa/lstarMP_eq.datexperiments/dfa/lstarRS_eq.dat[subfigure]justification=centeringFor reference, <ref> comparesand the two counterexample handling variations by Maler and Pnueli (denoted ) and by Rivest and Schapire (denoted ),on randomly generated DFAs of size 20 through 200 with increments of 20. Compared to , both ^𝙼𝙿 and ^𝚁𝚂 remove the need for consistency checks. Interestingly, whereas ^𝚁𝚂 compared toimproves in membership queries and worsens in equivalence queries, the situation is reversed for ^𝙼𝙿.experiments/wfa/vangluin_mq.datexperiments/wfa/vmp_mq.datexperiments/wfa/vrs_mq.datexperiments/wfa/weighted_mq.datexperiments/wfa/vangluin_eq.datexperiments/wfa/vmp_eq.datexperiments/wfa/vrs_eq.datexperiments/wfa/weighted_eq.dat §.§ [] In <ref> we compare the performance ofwith that of []. (Recall thatis the free vector space monad.)Hereis the obvious generalization of the originalalgorithm to learn Moore automata—DFAs with outputs in an arbitrary set, which here is the field with five elements. Thus, as opposed to [],ignores the vector space structure on the output set. In both cases we consider the three different counterexample handling methods. The algorithms are run on randomly generated WFAs of sizes 1 through 4. As expected, each [] variation provides a massive gain over the correspondingvariation in terms of membership queries, and a more modest one in terms of equivalence queries. Comparing the results of thevariations, we see that the membership query results ofand ^𝚁𝚂 are extremely close together. Other than that, the ordering of the counterexample handling methods is the same as with the DFA experiments. The [] variations will be compared in more detail later.experiments/wfa/vangluinDFA_mq.datexperiments/wfa/angluinDFA_mq.datexperiments/wfa/mpDFA_mq.datexperiments/wfa/mpmDFA_mq.datexperiments/wfa/rsDFA_mq.datexperiments/wfa/rsmDFA_mq.datexperiments/wfa/vangluinDFA_eq.datexperiments/wfa/angluinDFA_eq.datexperiments/wfa/mpDFA_eq.datexperiments/wfa/mpmDFA_eq.datexperiments/wfa/rsDFA_eq.datexperiments/wfa/rsmDFA_eq.dat[subfigure]justification=centeringNow we runand variations of [] on randomly generated Moore automata of sizes 5 through 50 with increments of 5. We chose to compare the [] variations only tobecause of its average performance in between ^ and ^ as seen in <ref>. The results are shown in <ref>. We see that, in terms of membership queries, bothandcounterexample handling methods improve over the one by Angluin in this setting, andperforms best in terms of either query type. In these experiments,performs much better than the algorithms that attempt to take advantage of the non-existent vector space structure. Together with the results in <ref>, this is consistent with the findings of <cit.>: they found that for DFAs and non-deterministic, universal, and alternating automata, the adaptation ofthat takes advantage of the exact type of structure of the randomly generated target automata performs the best.experiments/wfa/angluin_mq.datexperiments/wfa/mp_mq.datexperiments/wfa/mpm_mq.datexperiments/wfa/rs_mq.datexperiments/wfa/rsm_mq.datexperiments/wfa/angluin_eq.datexperiments/wfa/mp_eq.datexperiments/wfa/mpm_eq.datexperiments/wfa/rs_eq.datexperiments/wfa/rsm_eq.dat[subfigure]justification=centering <ref> illustrates the performance of [] variations on randomly generated WFAs. Here we generated WFAs of sizes 5 through 30 with increments of 5. We emphasize that in Table <ref> we could not go beyond size 4, because of performance issues with . There is hardly any difference between the use of Angluin's counterexample handling method and , neither in terms of membership queries, nor in terms of equivalence queries. Interestingly, while themethod performs worse than the other methods in terms of equivalence queries, as usual, it provides no significant gain in terms of membership queries. We ran these experiments also with the variations on theandalgorithms where we drop the consistency checks. In both cases the differences were negligible.experiments/nfa/lstar_mq.datexperiments/nfa/lstarMP_mq.datexperiments/nfa/lstarRS_mq.datexperiments/nfa/nlstarMP_mq.datexperiments/nfa/nlstarMPNC_mq.datexperiments/nfa/nlstarRS_mq.datexperiments/nfa/nlstarRSNC_mq.datexperiments/nfa/lstar_eq.datexperiments/nfa/lstarMP_eq.datexperiments/nfa/lstarRS_eq.datexperiments/nfa/nlstarMP_eq.datexperiments/nfa/nlstarMPNC_eq.datexperiments/nfa/nlstarRS_eq.datexperiments/nfa/nlstarRSNC_eq.datexperiments/nfa/lstar_size.datexperiments/nfa/nlstarMP_size.dat §.§We now consider learning algorithms for NFAs. To generate random NFAs, we use the strategy introduced by <cit.> with a transition density of 1.25, meaning that for each input symbol there are on average 1.25 transitions originating from each state. According to Tabakov and Vardi, this density results in the largest equivalent minimal DFAs. Like Tabakov and Vardi, we let half of the states be accepting. We ran several variations ofandon randomly generated NFAs of sizes 4 through 16 with increments of 4. The results are shown in <ref>. Here ^ refers to the original algorithm by <cit.>, with their notion of consistency; ^ is the same algorithm, but using the counterexample handling method that we adapted from Rivest and Schapire's. The variations ^ and ^ drop the consistency checks altogether. Unfortunately, doing the full consistency check was not computationally feasible. As expected, thealgorithms yield a great improvement overin terms of membership queries, and in most cases they also improve in terms of equivalence queries. This was already observed by Bollig et al. The exception is ^, which, despite having the best membership query results, requires by far the most equivalence queries. As happened toon DFAs, switching withinfrom theto thecounterexample handling method improves the performance in terms of membership queries and worsens it in terms of equivalence queries. Dropping consistency altogether turns out to increase both query numbers. § CONCLUSION We have presented , a general adaptation ofthat uses monads to learn an automaton with algebraic structure, as well as a method for finding a succinct equivalent based on its generators. Furthermore, we adapted the optimized counterexample handling method of <cit.> to this setting and discussed instantiations to non-deterministic, universal, partial, weighted, alternating, and writer automata. We have provided a prototype implementation in Haskell, using which we obtained experimental results confirming that exploiting the algebraic structure reduces the number of queries posed. The results also reveal that the best counterexample handling method depends on the type of automata considered and the algebraic structure exploited by the algorithm. We found that there is a significant gain in membership queries compared to thealgorithm by <cit.> when using our adapted optimized counterexample handling method. Related Work. This paper builds on and extends the theoretical toolkit of <cit.>, who are developing a categorical automata learning framework (CALF) in which learning algorithms can be understood and developed in a structured way.An adaptation ofthat produces NFAs was first developed by <cit.>. Their algorithm learns a special subclass of NFAs consisting of RFSAs, which were introduced by <cit.>. <cit.> unified algorithms for NFAs, universal automata, and alternating automata, the latter of which was further improved by <cit.>. We are able to provide a more general framework, which encompasses and goes beyond those classes of automata. Moreover, we study optimized counterexample handling, which <cit.> do not consider.The algorithm for weighted automata over a (not necessarily finite) field was studied in a category theoretical context by <cit.> and elaborated on by <cit.>. The algorithm itself was introduced by <cit.>. The present paper provides the first, correct-by-construction implementation of the algorithm.The theory of succinct automata used for our hypotheses is based on the work of <cit.>, revamped to more recent category theory.Our library is currently a prototype, which is not intended to compete with a state-of-the-art tool such as LearnLib <cit.> or other automata learning libraries like libalf <cit.>. Our Haskell implementation does not provide the computational efficiency achieved by LearnLib, which furthermore includes the TTT-algorithm with its optimized data structure that replaces the observation table by a tree <cit.>. Such optimization is ad-hoc for DFAs, and an extension to other classes of automata is not trivial. First steps in this direction have been done by <cit.>, who have studied the tree data structure in a more general setting.We intend to further pursue investigation in this direction, in order to allow for optimized data structures in a future version of our library. We note that, although libalf supports NFAs, none of the existing tools and libraries offers the flexibility of our library, in terms of available optimizations and classes of models that can be learned. Future Work. Whereas our general algorithm effortlessly instantiates to monads that preserve finite sets, a major challenge lies in investigating monads that do not enjoy this property. In fact, although the algorithm for weighted automata generalizes to an infinite field <cit.>, for an infinite semiring in general we cannot guarantee termination. This is because a finitely generated semimodule may have an infinite chain of strict submodules. Intuitively, this means that while fixing closedness defects increases the size of the hypothesis state space semimodule, an infinite number of steps may be needed to resolve all closedness defects. There are however subclasses of semirings for which a generalization should be possible, e.g., Noetherian or, more generally, proper semirings, which were recently studied by <cit.>.Moreover, we expect thatcan be generalized from the category of sets to locally finitely presentable categories. As a result of the correspondence between learning and conformance testing <cit.>, it should be possible to include in our framework the W-method <cit.>, which is often used in case studies deploying <cit.>.We defer a thorough investigation of conformance testing to future work.§ OMITTED PROOFS *The left to right implication is proved by freely generating a T-automaton from the Moore one via the monad unit, and by recalling that T preserves finite sets. The resulting T-automaton acceptsand is finite, therefore any of its quotients, including the minimal T-automaton accepting , is finite. Analogously, the right to left implication follows by forgetting the algebraic structure of the T-automaton: this yields a finite Moore automaton accepting . * Assume a right inverse iH → T(S') of eT(S') → H. We first prove o_∘ e = o_, by induction on the length of words. For all U ∈ T(S'), we have o_(e(U))() = _(e(U)) (definition of o_) = _(^♯(U)) (definition of e) = ^♯(U)() (definition of _) = _(U) (definition of _) = o_(U)() (definition of o_). Now assume that for a given v ∈ A^* and all U ∈ T(S') we have o_(e(U))(v) = o_(U)(v). Then, for all U ∈ T(S') and a ∈ A, o_(e(U))(av) = o_(δ_(e(U))(a))(v) (definition of o_) = o_(δ_(^♯(U))(a))(v) (definition of e) = o_(^♯(U)(a))(v) (definition of δ_) = (o_∘ e ∘ i)(^♯(U)(a))(v) (e ∘ i = _H) = (o_∘ i)(^♯(U)(a))(v) (induction hypothesis) = o_(δ_(U)(a))(v) (definition of δ_) = o_(U)(av) (definition of o_). From this we see that o_(_) = (o_∘ i ∘)() (definition of _) = (o_∘ e ∘ i ∘)() (o_∘ e = o_) = (o_∘)() (e ∘ i = _H) = o_(_) (definition of _). * Minimality is obvious, as S' not being minimal would make the loop guard true. We prove that the returned set is a set of generators. For clarity, we denote by d_S' S → T(S') the function associated with a set of generators S'. The main idea is incrementally building d_S' while building S'. In the first line, S is a set of generators, with d_S = η_SS → T(S). For the loop, suppose S' is a set of generators. If the loop guard is false, the algorithm returns the set of generators S'. Otherwise, suppose there are there are s ∈ S' and U ∈ T(S' ∖{s}) such that ^♯(U) = (s). Then there is a function fS' → T(S' ∖{s}) f(s') = Uif s' = s η(s')if s's that satisfies (s') = ^♯(f(s')) for all s' ∈ S', from which it follows that ^♯(U') = ^♯(f^♯(U')) for all U' ∈ T(S'). Then we can set d_S' ∖{s} to f^♯∘ d_S' S → T(S' ∖{s}) because (s') = ^♯(d_S' ∖{s}(s')) for all s' ∈ S. Therefore, S' ∖{s} is a set of generators. | http://arxiv.org/abs/1704.08055v4 | {
"authors": [
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unsrt #1#2#3#4#1 #2, #3 (#4) Nuovo Cimento Nucl. Instrum. Methods Nucl. Instrum. Methods A Nucl. Phys. B Phys. Lett.B Phys. Rev. Lett. Phys. Rev. D Z. Phys. Cϵ^' ε → π^+π^-γ p K^0 K̅^̅0̅ α α̅CP-1.80em/ Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20–364, CDMX 01000, México PRECISION ELECTROWEAK MEASUREMENTS AT RUN 2 AND BEYOND JENS ERLER December 30, 2023 ====================================================== After reviewing the key features of the global electroweak fit, I will provide updated results and offer experimental and theoretical contexts.I will also make the case for greater precision and highlight future directions.§ INTRODUCTIONTo chase out the elephants in the room, I recall that with the Higgs boson discovery the Standard Model (SM) is now complete, and with very few marginal exceptions it passed all the tests. Furthermore, the LHC did not yet find any convincing evidence for physics beyond the SM. Nevertheless, if nothing else does, at least dark matter provides a solid observational hint at the presence of new physics, and it may quite plausibly linger near the electroweak (EW) scale.Perhaps we are witnessing a revival of the times where precision physics is guiding high energy physics, like in the era of LEP. It could be that the renormalizable SM is merely the leading set of terms in a non-renormalizable effective quantum field theory, where the former gives rise to the (relatively) long-range physics. Here I review and update the global electroweak (EW) fit, restricting myself to the CP-even and flavor-diagonal part of the SM. For more flavorful observables I refer to the contribution by Jure Zupan <cit.>. I will also allow certain model-independent parameters describing new physics. § PRECISE INPUTS FOR THE ELECTROWEAK FIT§.§ Bosonic sectorThe EW fit needs five input variables to define the bosonic sector of the SM, namely the three gauge couplings associated with SU(3)_C × SU(2)_L × U1)_Y and the two parameters entering the Higgs potential. It is inessential which parameters or observables we call inputs and which ones output because there is no fundamental distinction between those in a global fit. Nevertheless, one may think of the most precise ones as inputs parameters and these are listed in Table <ref>. §.§ Top quark massGreater precision in the top quark mass, m_t, still matters in EW fits. Indeed, the small change from the value used about 18 months ago <cit.>, m_t = 173.34 ± 0.81 GeV, reduces the fitted Higgs boson mass by about 3 GeV. Very recently, ATLAS <cit.>, CMS <cit.>, and the Tevatron EW Working Group <cit.>, each released combinations of their various top quark mass determinations.The results are listed in Table <ref>. For the grand average, needed in the fits below,I assumed that there is a systematic uncertainty of 0.29 GeV that is common among all three. It is the sum (in quadrature) of the error components induced by the Monte Carlo generator, parton distribution functions, and QCD, as obtained by ATLAS. For comparison, the Tevatron modeling plus theory error amounts to 0.38 GeV and the CMS modeling error is 0.41 GeV. Other uncertainties are assumed uncorrelated between collaborations. Notice, that the statistical precision [Precision is defined as the inverse of the square of the uncertainty.]of the grand average is not simply the sum of the statistical precisions of the individual combinations, as is sometimes assumed.Rather, the procedure developed in Ref. <cit.> should be applied.To the total experimental error one has to add a common theory error because the quoted values areinterpretedto either representthe top quark pole mass, m_t, or some other operational mass definition supposedly coincidingwith the pole mass roughly within the strong interaction scale Λ_ QCD (taken here as 500 MeV). Thus, the constraint used in the fits is m_t = 172.97 ± 0.28_ uncorr.± 0.29_ corr.± 0.50_ theory = 172.97 ± 0.64 ,where I have split the experimental error into uncorrelated and correlated components. The uncertainty of O(Λ_ QCD) is assumed to also account for the uncertainty in the relation between the top quark pole and MS mass definitions. By accounting for the leading renormalon contribution in this relation,it may ultimately be possible to reduce this uncertainty to about 70 MeV <cit.>.§.§ Charm and bottom quark massesI should mention the increasing importance the charm and bottom quark masses, m_c and m_b, have on the EW fit. If they are known very precisely, one can use perturbative QCD to calculate the heavy quark contributions to the renormalization group evolution of α from the Thomson limit to the Z pole <cit.>, and conversely of the weak mixing angle which has been measured precisely near the Z pole (see Sec. <ref>) to lower energies <cit.>.Similarly, m_c and m_b enter the SM prediction <cit.> of the anomalous magnetic moment of the muon, g_μ - 2. While I do not cover it here, I recall that g_μ - 2 deviates by more than 4 standard deviations if one includes τ decay spectral functions corrected for γ–ρ mixing <cit.>. The latter brings τ decays into agreement with e^+ e^- annihilation and radiative return data.Even though the charm quark is technically decoupling, its numerical effect enters at the same level into g_μ - 2 as the hadronic light-by-light contribution, and an uncertainty of 70 MeV in m_c would induce an error comparable to the anticipated uncertainties in upcoming experiments at Fermilab and J–PARC. Thus, one would like to know m_c an order of magnitude more accurately than this.Finally, the linear relationship <cit.> between Higgs couplings and masses of the particles in the single Higgs doublet SM can be studied precisely at future lepton colliders. To match the projections of the charm and bottom Yukawa coupling measurements from the corresponding Higgs branching ratios one needs knowledge of m_c and m_b to 8 MeV and 9 MeV, respectively. Interestingly, Ref. <cit.> calibrated the m_c uncertainty in the first-principle relativistic QCD sum rule approachand fortuitously found the minimally required 8 MeV accuracy (not accounting for the parametric uncertainty induced by α_s which should become negligible in the future).§ THE WEAK MIXING ANGLE§.§ High-energy measurementsThe weak mixing angle, sin^2θ_W, is one of two observables at the heart of the EW fit. As a derived quantity, the strategy is to compute it and to compare it with Z pole asymmetry measurementsat LEP, the Tevatron and the LHC, from which the effective weak mixing angle for leptons, sin^2θ_W^ eff, is obtained. An important application is to models with extra Z^' bosons,in which sin^2θ_W constrains the Z–Z^' mixing angle typically to the 10^-2 level <cit.>. The hadron collider measurements shown in Table <ref> agree well with each other, but the two most precise Z pole determinations are deviating by about 3 standard deviations as illustrated in Fig. <ref>. §.§ Low-energy measurementsOne can also compare the measurements of sin^2θ_W near the Z pole with off-pole determinations (see Fig. <ref>) to isolate possible new contact interactions. This works because any four-fermion operator would be almost hopelessly suppressed under the Z resonance, but off the pole — one could go to higher energies as well, but there are much more precise data at lower energies — there is a milder power suppression.Thus, if a significant difference between on-pole and off-pole measurements of sin^2θ_W^ eff is observedit may be due to an effective contact interaction induced by TeV scale new physics.§ BOSON MASSES§.§ W boson massThe other observable at the heart of the EW fit is the W boson mass, M_W. Its measurements at LEP 2 average to M_W = 80.376 ± 0.033 GeV <cit.>, while the Tevatron combination yields M_W = 80.387 ± 0.016 GeV <cit.>. The ATLAS result, M_W = 80.3695 ± 0.0185 GeV <cit.>, represents the first at the LHC,and while it is based on only 4.6 fb^-1 of 7 TeV data,it is already at the level of the most precise result at the Tevatron. For what follows I assume a common PDF error of 7 MeV between the Tevatron and ATLAS uncertaintiesand will work with the average,M_W = 80.379 ± 0.012 GeV,corresponding to the weak mixing angle in the on-shell scheme, sin^2θ_W^ on-shell≡ 1 - M_W^2/M_Z^2 = 0.22301 ± 0.00023 .However, the physics of M_W and sin^2θ_W is quite different, and while it is popularto convert one into the other, this is a fairly pointless exercise, especially in the context of new physics. Rather, the two observables are complementary and provide, for example,constraints on the oblique parameters (see Sec <ref>) that are linearly independent. The global fit returnsM_W = 80.362 ± 0.005 GeV,which is now driven by the directly measured M_H and somewhat lower than the world average.M_W is of interest as it is easily affected by new physics in general and Higgs sector modifications in particular,but it needs the top quark mass, m_t, as an input.Fig. <ref> compares the direct measurements of M_W and m_t from the colliders to everything else in precision EW physics, including the direct value of M_H. It is quite interesting that the measured M_W is somewhat high, because most kinds of new physics models addressing the EW hierarchy problem can easily affect the M_W prediction.This includes the Minimal Supersymmetric Standard Model, where the shift in M_W is predicted to be positivethroughout parameter space <cit.>, in agreement with what is currently favored by the data. §.§ Higgs boson massThere are three different methods to determine M_H. One employs Higgs boson branching ratios <cit.>, since especially the branching fractions into pairs of gauge boson feature a strong M_H dependence.Using furthermore ratios of branching ratios, such as B(H →γγ) relative toB(H → WW) or B(H → ZZ), cancels the dominant production uncertainties, and we find <cit.>,M_H = 126.1 ± 1.9.The global EW fit including updates presented at this meeting favors the rather low range,M_H = 94^+18_-16.This is about 1.7 σ below the direct kinematic reconstruction result <cit.>,M_H = 125.09 ± 0.24 .Thus, while M_H is now known, it still provides a very valuable cross-check of the SM.Before discussing the prospects at future LHC runs,it is entertaining to review how previous experimental projections compare with the actual achievements.Table <ref> shows projections <cit.> at the time of the Snowmass 2001 gatheringon what was then thought to be the future of high-energy physics. As one can see at the example of the Tevatron, with less than the expected integrated luminosity the goals were either achieved orsurpassed and the finalized uncertainties may well turn out to be smaller, yet. Similarly, the uncertainty of the first measurement of M_W at the LHC with only one detector and just a few fb^-1 of data is approaching the 100 fb^-1 projection. And m_t from the LHC is already more accurate than anticipated.The result in Eq. (<ref>) is dominated by M_W, which by itself corresponds to M_H = 89^+22_-19 GeV. A hypothetical measurement of M_W = 80.376 ± 0.008 GeV (the assumed central value is adjustedso as to reproduce the current best fit value for M_H and the error is motivated by Ref. <cit.>)at the LHC after the accumulation of 150 fb^-1 of data yields M_H = 94^+17_-15.For this I assumed that the total m_t error will be completely dominated by the QCD uncertainty in Eq. (<ref>).And I neglected the theoretical error in the prediction of M_W, but to compensate I did not assume any improvement in other parameterslike α_s or the electromagnetic coupling at the Z scale. Similarly, the hypothetical result sin^2θ_W^ eff = 0.23135 ± 0.00020 <cit.>would yield M_H = 94^+47_-32. Adding these improvements to the current data gives M_H = 90^+13_-12. Finally, at the high-luminosity LHC (HL–LHC) the uncertainty in M_W may optimistically be reduced to 5 MeV, and the one in sin^2θ_W^ eff to 1.4 × 10^-4, which would then result in M_H = 89 ± 10 GeV.§ CONSTRAINTS ON PHYSICS BEYOND THE SM§.§ Oblique physics beyond the SMThe oblique parameters, S, T and U, describe corrections to the W and Z boson self-energies. The SM contributions are subtracted out by definition, so that S, T and U are new physics parameters, whereS and T (see in Fig. <ref>) correspond to dimension six operators in the effective field theory, and U is of dimension eight. T breaks the custodial SO(4) symmetry of the Higgs potential.A multiplet of heavydegenerate chiral fermions contributes a fixed amount to S,Δ S = N_C/3π∑_i (t_3L^i - t_3R^i)^2, where t_3L and t_3R are the third components of weak isospin of the extra left and right-handed fermions, respectively. Thus, for example, an additionaldegenerate fermion family yieldsΔ S = 2/3π≈ 0.21 The updated EW fit with S and T allowed simultaneously gives a range of valuesS = 0.06 ± 0.08 T = 0.09 ± 0.06 Δχ^2 = - 4.0which are in marginal agreement with the SM but the decrease in χ^2 relative to the SM is not insignificant.§.§ Implications of the T parameterThe T parameter has the same effect as the ρ parameter — the ratio of interaction strengths of the neutral and charged currents — as it is proportional to ρ - 1, but T is often quoted for loop effects. The ρ parameter constrains vacuum expectation values of higher dimensional Higgs representations to ≲ 1 GeV. There is also sensitivity todegenerate scalar doublets up to 2 TeV, a result based on an effective field theory approach <cit.>.Most importantly, non-degenerate doublets of additional fermions or scalars contribute an amount <cit.>, Δρ = G_F/√(2)∑_i C_i/8 π^2Δ m_i^2 Δ m_i^2 ≥ (m_1 - m_2)^2,where C_i is the color factor. Δ m_i^2 is not exactly m_1^2 - m_2^2, where m_i are the masses of the two members of the doublet, but is a more complicated function bounded by (m_1 - m_2)^2 and thus gives rise to a positive-definite contribution.Despite the appearance of this form which seems to suggest that there is sensitivity to mass splittingseven when the m_i increase all the way to the Planck scale,there is decoupling of these heavy fermions or scalars, because in models one will always face a see-saw type suppression of Δ m_i^2 for very large m_i. I updated the one-parameter fit — just allowing ρ (or T) in addition to the SM parameters — with the result that ρ is now 1.9 σ above the SM prediction of unity, ρ = 1.00036 ± 0.00019,and thus one can constrain the sum of contributions of any additional EW doublet, ∑_i C_i/3Δ m_i^2 ≤ (46 )^2(95%).Looking ahead, the LHC after the accumulation of 150 fb^-1 of data (with the same assumptions as in Sec. <ref>) could reduce the error in ρ to imply a stronger constraint on the mass splittings, ρ = 1 ± 0.00014 ⟹∑_i C_i/3Δ m_i^2 ≤ (27 )^2.Or assuming that there is no change in the central value from today, one would actually obtain a precise measurement of Δ m_i^2, ρ = 1.00036 ± 0.00014 ⟹∑_i C_i/3Δ m_i^2 = (34^+6_-7)^2.Finally, turning to the HL–LHC one would find even stronger constraints, ρ = 1 ± 0.00012 ⟹∑_i C_i/3Δ m_i^2 ≤ (25 )^2.§.§ Compositeness scales from low energiesReturning to the contact interactions in Sec. <ref> that may be derived by comparing on-pole and off-pole measurements of sin^2θ_W, Fig. <ref> shows constraints on effective couplings corresponding to various parity-violating effective four-fermion operators (as before, the couplings are defined to vanish in the absence of new physics). These can be translated into compositeness scales that can be tested [The numerical values of such scales are convention dependent.We use those laid out in Ref. <cit.>.]. As shown in Fig. <ref> the new physics reach already surpassed 40 TeV and will increase beyond 50 TeVwhen the future experimental results from polarized electron scattering briefly mentioned in Sec. <ref>are combined with measurements of atomic parity violation (APV). § CONCLUSIONSThe SM remains in remarkable health. It is over-constrained, as M_W, sin^2θ_W, g_μ-2, and many other quantitieshave been simultaneously computed and measured.If there is strongly coupled new physics its energy scale can be tested up to O(50 TeV)through parity-violating four-fermion operators. There are some inconclusive, yet interesting deviations. M_H extracted from the EW fit is 1.7 σ below the direct value, and it is therefore mandatory to increase the precision in m_t further and to obtain mutual consistency among different experiments.In a one-parameter fit (S= U = 0) the T parameter appears 1.9 σ high, and future measurements of M_W at the LHC may increase this to a 3 σ effect. Given that M_W is particularly sensitive to physics beyond the SM and theoretically clean, one may argue that a deviation in M_W may be even more tantalizing than the current 4 σ SM discrepancy in g_μ-2. Thus, greater precision in M_W is a must, with or or without an LHC discovery. § ACKNOWLEDGMENTSI would like to thank the organizers for the kind invitation to a very enjoyable meeting in a beautiful location. This work is supported by CONACyT (México) project 252167–F. 99Zupan Jure Zupan, these proceedings.Bouchendira:2010esR. Bouchendiraet al., Phys. Rev. Lett.106, 080801 (2011). Webber:2010zfMuLan Collaboration: D. M. Webberet al., Phys. Rev. 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Henning, X. Lu and H. Murayama, JHEP 1601, 023 (2016). Veltman:1977khM. J. G. Veltman, Nucl. Phys. B123, 89 (1977).Erler:2013xhaJ. Erler and S. Su, Prog. Part. Nucl. Phys.71, 119 (2013). Erler:2014fqaJ. Erler, C. J. Horowitz, S. Mantry and P. A. Souder,Ann. Rev. Nucl. Part. Sci.64, 269 (2014). Wang:2014bbaPVDIS Collaboration: D. Wanget al.,Nature506, 67 (2014). | http://arxiv.org/abs/1704.08330v1 | {
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INAF - Osservatorio Astrofisco di Arcetri, largo E. Fermi 5, 50127 Firenze, Italy [email protected] Department of Electrical Engineering and Center of Astro Engineering, Pontificia Universidad Catolica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile Observatorio Nacional, Rua José Cristino 77, 20921-400 Rio de Janeiro, Brasil Dipartimento di Fisica e Astronomia, Università di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy INAF - Osservatorio Astronomico di Bologna, viaGobetti 93/2, 40129 Bologna, Italy Laboratoire AIM, CEA/IRFU/Service d’Astrophysique, CNRS, Université Paris Diderot, Bat. 709, 91191 Gif-sur-Yvette, France Department of Astronomy, University of Virginia, 530 McCormick Road, Charlottesville, VA 22904, USA We study the physical and dynamical properties of the ionized gas in the prototypical HII galaxy Henize 2-10 using MUSE integral field spectroscopy. The large scale dynamics is dominated by extended outflowing bubbles, probably the results of massive gas ejection from the central star forming regions. We derive a mass outflow rate Ṁ_out∼ 0.30 M_ yr^-1, corresponding to mass loading factor η∼0.4, in range with similar measurements in local LIRGs. Such a massive outflow has a total kinetic energy that is sustainable by the stellar winds and Supernova Remnants expected in the galaxy. We use classical emission line diagnostic to study the dust extinction, electron density and ionization conditions all across the galaxy, confirming the extreme nature of the highly star forming knots in the core of the galaxy, which show high density and high ionization parameter. We measure the gas phase metallicity in the galaxy taking into account the strong variation of the ionization parameter, finding that the external parts of the galaxy have abundances as low as12 + log(O/H) ∼ 8.3, while the central star forming knots are highly enriched with super solar metallicity. We find no sign of AGN ionization in the galaxy, despite the recent claim of the presence of a super massive active Black Hole in the core of He 2-10. We therefore reanalyze the X-ray data that were used to propose the presence of the AGN, but we conclude that the observed X-ray emission can be better explained with sources of a different nature, such as a Supernova Remnant. Cresci et al.The MUSE view of He 2-10:no AGN ionizationbut a sparkling starburst This work is based on observations made at the European Southern Observatory, Paranal, Chile (ESO program 095.B-0321)G. Cresci1,L. Vanzi2, E. Telles3, G. Lanzuisi4,5, M. Brusa4,5, M. Mingozzi5,1, M. Sauvage6, K. Johnson7Received ; accepted================================================================================================================================================================================================ § INTRODUCTION Nearby HII galaxies represent an unique test bench to study in detail the physical mechanisms that drive star formation and galaxy evolution in nearly pristine environments, resembling those in high-z galaxies. In fact, their low chemical abundances, high gas fractions, high specific star formation rate (sSFR) and the dominance of very young stellar populations in massive star clusters allow a closer comparison with the conditions present in primordial star forming galaxies at high redshift. Given their proximity, these processes can be studied at much higher spatial resolution than at high-z, thus representing a fundamental test in our understanding of galaxies and of galaxy evolution. For these reasons our group has undertaken a long term program to investigate the internal physical properties of these star forming galaxies through spatially resolved spectroscopy in the optical and in the near-IR, in order to assess their kinematics, physical conditions and their relation to their local star formation properties (see e.g. Vanzi et al. , ; Cresci et al. ; Lagos et al. ; Telles et al. ).With the advent of the new large field Multi Unit Spectroscopic Explorer (MUSE, Bacon et al. ) optical integral field spectrometer, we target Henize 2-10, one of the prototypical HII galaxies (Allen et al. ). It is located at just 8.23 Mpc (Tully et al. , providing a scale of 40 pc arcsec^-1), and its optical extent is less than 1 kpc, showing a complex and irregular morphology. Despite its low mass (M_*=3.7× 10^9 M_, Reines et al. , although Nguyen et al.revised this value to 10± 3×10^9 M_ using deep optical imaging of the outer regions), it hosts an intense burst of star formation (Star Formation Rate SFR=1.9 M_ yr^-1, Reines et al. ), as traced by classical indicators such as strong optical and infrared emission lines excited by young stars (Vacca & Conti , Vanzi & Rieke ), the detection of Wolf-Rayet features (Schaerer et al. ), intense mid-IR emission (Sauvage et al. , Vacca et al. ) and far-IR continuum (Johansson ).The galaxy is gas rich, with a molecular gas mass of ∼ 1.6× 10^8 M_and an atomic gas mass of 1.9× 10^8 M_ (Kobulnicky et al. , suggesting a gas fraction of about 3-10%). Apart from the central 500 pc characterized by blue colors and irregular morphology, the outer part of the galaxy appears to be redder and dominated by an older stellar population, consistent with an early-type system (Nguyen et al. ). The stellar component seems to be dispersion dominated also in the central part, further supporting the spheroidal nature of the older stellar population (Marquart et al. ). The recent burst (∼ 10^7 yr, Beck et al. ) was probably triggered by a massive gas infall due to an interaction, or to a merger with a companion dwarf, as inferred by the tidal CO and HI plumes extended up to 30" to the southeast and northeast of the core (Kobulnicky et al. , Vanzi et al. ).The galaxyis also surrounded by a complex kiloparsec-scale superbubble centered on the most intense star forming core, and evident in HST narrowbandimages (Johnson et al. ). The estimate of the energy of the bubbles is compatible with the expected mechanical energy released by Supernovae (SNe) and stellar winds in the central starburst, suggesting a stellar driven galaxy wind (Méndez et al. ). The outflowing material was also detected in absorption as blueshifted interstellar lines up to -360 km/s, that were used to compute a lower limit on the mass of the outflowing material of M_out≳10^6 M_ (Johnson et al. ).The most actively star forming regions are concentrated in the core of the galaxy, in an area of about 3" in radius (120 pc). Here optical and UV studies have detected an arc-like structure of resolved young super massive star clusters, withmasses up to 10^5 M_ and ages of ∼10 Myr (e.g. Conti & Vacca , Johnson et al. , Cresci et al. ), prompting for the first time the question of whether this is a dominant mode of star formation in galaxies. These clusters lie at the center of a cavity depleted of both cold molecular gas and warm line emitting ionized gas, whose emission is instead concentrated in two regions East and West of the arc (knot 4 and 1+2 in fig. <ref>), harboring actively star forming, dust embedded young star clusters, prominent at mid-IR wavelength (ages ≲ 5× 10^6 yr and masses ≳ 5× 10^5 M_, Vacca et al. , Cabanac et al. ). The peculiar configuration of a cavity with older clusters surrounded by gas rich regions was interpreted by Cresci et al.as an example of stellar `positive feedback', as the estimates of the shock shell velocity agree with the measured ages of the young clusters in the IR sources. All these IR clusters are associated with extended radio sources, and their properties are mostly explained as Ultra-Dense HII regions (UDHII) powering the radio emission (Kobulnicky & Johnson , Johnson & Kobulnicky ).The only notable exception is a non-thermal radio source located in the line emitting bridge between the twobrightest clumps (designated as knot 3 in Johnson & Kobulnicky , see Fig <ref>. Note that Cabanac et al.detected in L band two infrared source at this location). This source is very compact, with a physical scale of ≤ 3 pc × 1 pc (Reines & Deller ), and has been associated by Reines et al. <cit.> to a compact X-ray emission detected with Chandra observations by Kobulnicky & Martin <cit.>. Using their combination of radio and X-ray fluxes, Reines et al. <cit.> concluded that the most likely explanation for the radio source was an actively accreting super massive Black Hole (SMBH), with log(M/M_)=6.3±1.1. As the presence of very few SMBHs have been previously inferred in low mass highly star forming galaxies (see e.g. Barth et al. ), and given that He 2-10 shows no sign of a bulge or massive cluster related to the putative SMBH position, this claim has important implications on our picture of SMBH and galaxy assembly and coevolution, suggesting that black hole seeds may evolve faster than their hosts.However, recent and deeper follow-up Chandra observations presented in Reines et al. <cit.> have shown that the hard X-ray emission previously identified was dominated by an additional source that is distinct from the compact radio source and compatible with an off-center X-ray binary. If the interpretation of the compact radio source as a SMBH is maintained, their estimate of its hard X-ray radiation without contamination brings down the accretion of the candidate SMBH well below the Eddington limit (∼10^-6 L_Edd).Here we present new MUSE optical integral field observations of He 2-10, to study for the first time the global ionization, dynamics and physical properties of the warm ionized gas on large scales. The paper is organized as follows: in Sect. <ref> we describe the MUSE observations, data reduction, and the method used for the spectral fitting of the continuum and of the emission lines; in Sect. <ref> we discuss the obtained gas dynamics, the ionization and metallicity properties are discussed in Sect. <ref>, and in Sect. <ref> we discuss the evidences of the presence of a SMBH in He 2-10 from our MUSE data as well as archive Chandra X-ray ones. Our conclusions follow in Sect. <ref>.§ OBSERVATIONS AND DATA REDUCTIONHe 2-10 was observed with MUSE on May 28, 2015, under program 095.B-0321 (PI Vanzi).The galaxy was observed with four dithered pointings of 30 s each, for a total of 2 minutes on source, with the background sky sampled with 3 equal exposures in between. The data reduction was performed using the recipes from thelatest version of the MUSE pipeline (1.6.2), as well as a collection of custom IDL codes developed to improve the sky subtraction, response curve and flux calibration of the data. Further details on the data reduction can be found in Cresci et al. <cit.> and references therein.The final datacube consists of 321×328 spaxels, for a total of over 100000 spectra with a spatial sampling of 0.2"×0.2" and a spectral resolution going from 1750 at 465 nm to 3750 at 930 nm. The field of view is ∼ 1'×1', sufficient to sample the optical extent of the source (see Fig. <ref>). The average seeing during the observations, derived directly from foreground stars in the final datacube, was FWHM=0.68"±0.02". §.§ Emission line fittingThe obtained datacube was analyzed using a set of custom python scripts, developed to subtract the stellar continuum and fit the emission lines with multiple Gaussian components where needed. The details about the procedures used are described in Venturi et al. (in prep.); here we just summarize the steps followed to obtain the emission lines fluxes, velocities and velocity dispersions. The underlying stellar continuum was subtracted using a combination of MILES templates (Sánchez-Blázquez et al. ) in the wavelength range 3525-7500 Å covered by the stellar library. This wavelength interval covers most of the emission lines considered in the following, with the exception of , for which we used a polynomial fit to the local continuum given that this line is not contaminated by underlying absorptions. The continuum fit was performed using the pPXF code (Cappellari & Emsellem ) on binned spaxels using a Voronoi tessellation (Cappellari & Copin ) to achieve a minimum S/N>50 on the continuum under 5530 Å rest frame. The main gas emission lines included in the selected range (, , , , , , , , , ) were fitted simultaneously to the stellar continuum using multiple Gaussian components to better constrain e.g. the absorption underlying the Balmer lines. Fainter lines as well as regions affected by sky residuals were masked out of the fitting region. The fitted stellar continuum emission in each bin is then subtracted on spaxel to spaxel basis, rescaling the continuum model to the median of the observed continuum in each spaxel. The continuum subtracted datacube was finally used to fit the emission lines in each spaxel. The velocity and widths of the Gaussian components were bound to be the same for each emission line of the different species, while the intensities were left free to vary with the exception of the [NII]λ6548,84 and [OIII]λ4959,5007 doublets, were the intrinsic ratio between the two lines was used. We verified that this assumption, corresponding to assume that the different emission lines come from similar environments in the galaxy, applies in our data randomly inspecting the line profiles of the different emission lines in several spaxels (see also Fig. <ref> for an extreme case). Each fit was performed three times, with 1, 2 and 3 Gaussian components per each emission line, in order to reproduce peculiar line profiles where needed.A selection based on the reduced χ^2 obtained using each of the three different fits in each spaxel was used to select the spaxels where a multiple components fit was required to improve the fit. This choice allows us to use the more degenerate multiple components fits only where it is really needed to reproduce the observed spectral profiles: these conditions apply only in the central part of the galaxy and in some knots of the outflowing filaments where double peaked lines are detected, as discussed in the next Section. § GAS AND STELLAR DYNAMICSThe velocity (first moment) and velocity dispersion (second moment) of the total emission line profiles in each pixel are used to study the gas dynamics in He 2-10. Fig. <ref> shows the dynamical maps for the whole MUSE field of view: the most striking features are the high velocity and high velocity dispersion regions corresponding to the bubble structure NE (redshifted, Fig. <ref>, left panel) and SW (blueshifted, Fig. <ref>, central panel) of the center, evident in themaps (see e.g. Fig. <ref>). These structures are very extended, up to ∼18” (∼720 pc) projected from the central star forming regions, both to the NE and to the SW. The multiple bubbles form a complex structure (see Fig. <ref>), that can be interpreted as due to multiple gas ejections from the central highly star forming region.While the bulk of the gas has velocity dispersions of ∼50-60 km/s, corrected for instrument broadening, the bubbles show dispersions as high as ≳ 180 km/s. The line profile is in fact broadened towards blue (SW) or red (NE) velocities, with the first moment of the lines clearly showing high velocity shifts, up to ± 130 km/s. The line profile in the bubbles region is particularly complex, showing double peaks with velocity differences as high as ∼ 310 km/s (see Fig. <ref>, right panel), consistent with the blueshift inferred from UV interstellar features detected by Johnson et al. <cit.>. In the following, we will assume that this velocity shift is representative of the average outflow velocity v_out, i.e. the de-projected velocity of the outflow, and regard the lower velocities observed as due to projection effects. Given that the escape velocity from He 2-10 is v_esc≈160 km/s (Johnson et al. ), it is clear that most of the gas detected in the bubbles has enough kinetic energy to escape the potential of the galaxy. Given the measured radius and velocity of the outflow, the dynamical time of the outflow is t_d ∼ R_out/v_out = 2.3 Myr. In order to compute the total gas mass in the ionized outflow, we extracted a stellar continuum subtracted spectrum integrated in an aperture of 120 spaxels in radius (24” or 960 pc), covering the full extent of the outflowing bubbles. We fit theemission with a set of two Gaussian profiles, a broad (FWHM_b=311 km/s; F_b(Hα)=3.47±0.15×10^-12 erg s^-1 cm^-2) and a narrow component (FWHM_n=148 km/s; F_n(Hα)=1.93±0.07×10^-12 erg s^-1 cm^-2). Using the average derived extinction value on the outflow region from the Balmer decrement (see Sect. <ref>, A_V∼0.5), we derived an extinction corrected luminosity for the broad component of L_b(Hα)=1.77 × 10^40 erg s^-1.We note that this is a lower limit, as especially part of the redshifted component may be more absorbed by the intervening dust in the interposed galaxy. We assume that the broad component in the integrated spectrum is fully due to the outflowing bubble detected as kinematical features in the analysis above. As comparison, a stricter lower limit on the flux from the outflowing bubbles is given by a spatial analysis where we integrate theemission only in the external regions where the high velocity gas is detected: in this case we find that theflux is ∼7×10^-13 erg s^-1 cm^-2, but missing all the emission from high velocity gas in the central region of the galaxy. Assuming the simplified outflow model by Genzel et al. <cit.>, we find that the mass in the ionized outflow is given by:M_out=3.2×10^5 (L_b(Hα)/10^40 erg/s) (100 cm^-3/n_e) M_ = 2.3×10^5 M_Although this value is only taking into account the warm ionized gas, it is in broad agreement with the lower limit derived from UV absorption spectroscopy by Johnson et al. (, M_out≈10^5-10^6) for the cold component. Assuming a biconical outflow distribution with velocity v_out=310 km/s out to a radius R_out=720 pc for the ionized wind, uniformly filled with outflowing clouds, the mass outflow rate is given by (see Cresci et al. ):Ṁ_out≈ρ_out_V ·Ω R^2_out· v_out = 3 v_out M_out/R_out = 0.30 M_ yr^-1where ρ_out_V is the volume-average density of the gas and Ω the opening angle of the (bi-)cone. It is interesting to compare this outflow value with the SFR in the galaxy. The ratio between the two, i.e. the mass loading factor η, for the outflow in He 2-10 is therefore:η=Ṁ_out / SFR = 0.39using the total SFR=0.76 M_ yr^-1 fromemission in our MUSE data (see Sect. <ref>). This value for the mass loading factor is in the range derived by Arribas et al. <cit.> for local LIRGs (η=0.3) and ULIRGs (η=0.5), confirming the extreme nature of the starburst in this galaxy. The total kinetic energy in the outflow is E_out(kin)∼2.2×10^53 erg, an order of magnitude lower than the energy of 3.5×10^54 erg that the ∼3750 Supernova Remnants (SNR), estimated by Méndez et al. <cit.> from radio observations, can inject in the interstellar medium. An additional, comparable energy contribution is expected from stellar winds by massive stars (Méndez et al. ). The difference is probably due to radiative losses during the expansion of the bubble, as well as to the fact that our outflowing mass estimate is probably a lower limit. In any case, the starburst present in the central region of the galaxy is able to provide all the energy required to sustain the observed outflow. In addition to the velocity structures due to the expanding bubbles, there is a velocity gradient in the SE (blueshift) - NW (redshift) direction, in the regions with lower velocity dispersion, consistent both in amplitude and orientation with the velocity field of the HI disk reported by Kobulnicky et al. <cit.>. This is therefore the first detection of a large scale rotating warm ionized gas body in He 2-10.The star kinematics from stellar absorption features is instead different from what we derived for the ionized gas. Although the S/N on the continuum is much lower than in the line emission, the stellar velocity and velocity dispersion mapped in the Voronoi bins described in Sect. <ref> are mostly flat, with σ∼45 km/s and no detectable velocity gradient or obvious signatures of past merging events, with velocity differences ≲10 km/s. This is in agreement with the smaller field of view but higher S/N and spectral resolution IFU observations of Marquart et al. <cit.>, as well as the near-IR data of Nguyen et al.<cit.>, confirming that the stars and the gas in this galaxy are at least partly dynamically decoupled in this galaxy.§ GAS PROPERTIESThe excitation, physical conditions, dust and metal content of the interstellar gas in He 2-10 can be explored using selected ratios between the measured emission lines. Thanks to our IFU observations, we are able to spatially map the line emission (see e.g. Fig. <ref>) and line ratio diagnostics across the MUSE field of view.§.§ Dust extinction and electron densityFirst of all we use the Balmer decrement / to derive the dust extinction map, assuming a Calzetti et al. <cit.> attenuation law and a fixed temperature of 10^4 K: the resulting extinction map is shown in Fig. <ref> (left panel), for the spaxels where theline was detected with S/N > 3. The dust attenuation appear to be highest in the two star forming clumps detached ∼18” to the SW of the main galaxy and in the Eastern region where the line and continuum emission is less prominent. These locations correspond to the position of the CO gas (Kobulnicky et al. ). Moreover, the extinction is high on one of the central star forming regions (A_V=2.3, knot 1+2 in the radio notation by Kobulnicky & Johnson ). It is interesting to compare this extinction map with the one obtained by near-IR IFU data for the central region of the galaxy by Cresci et al. <cit.> from the Br12/Brγ line ratio, where the extinction towards the two brightest star forming region at the center of the galaxy was A_V∼7-8. The difference is probably due to the fact that IR observations are capable to probe deeper in the highly embedded star forming clusters. The electron density was estimated using the / ratio (e.g. Osterbrock & Ferland ). We compute the line ratio in each spaxel where the [SII] lines are detected with S/N>3, and convert it to an electron density using the IRAF task , assuming a temperature of 10^4 K as before. The resulting electron density map is shown in Fig. <ref> (right panel). In this case, the highest density is obtained in the Western central star forming region (knot 4) where densities of n_e=1500 cm^-3 are reached. The density distribution is instead flat in the rest of the galaxy, with densities ∼ 100 cm^-3.This evidence confirms that the central regions of He 2-10 host dense, dust embedded, young and highly star forming star clusters, a common feature in starburst galaxies (see e.g. Vanzi & Sauvage ) and possibly in galaxies in general (e.g. Förster Schreiber et al. ).§.§ Gas excitationWe investigate the dominant ionization source for the line emitting gas in each MUSE spaxel using the so called BPT diagrams (Baldwin et al. <cit.>). Along with the classical diagnostic using the / versus / line ratios (N-BPT in the following), we also explore the alternative versions using(S-BPT) orinstead of(O-BPT, see e.g. Kewley et al. ; Lamareille ). In all these diagrams, galaxies or spatially resolved regions of galaxies dominated by star formation, AGN (Seyfert-type), Low Ionization Emission line Regions (LIERs, Belfiore et al. ), or shocks populate different regions. The three BPT diagrams for the spaxels in He 2-10 were the S/N>3 for all the lines involved are shown in Fig. <ref> (left panels). The dominant source of ionization is marked with different colors in each plot, blue for star formation, magenta for intermediate regions in the N-BPT diagram, green for AGN like ionizing spectra, red for LIER/shocks. The position of the different source of ionization in the map of He 2-10 is reported as well in Fig. <ref> (right panels), where each spaxel is plotted with the color corresponding to its dominating ionization source. Clearly, all the line emitting gas in the galaxy is dominated by ionization from young stars, as different sources are limited to few noisier spaxels at the edges of the galaxy.Given that all the line emitting gas is ionized by young stars, we use an integrated stellar continuum subtracted spectrum, extracted in an aperture of 120 spaxels in radius (24” or 960 pc) to compute the total SFR in the galaxy. We derive a total L(Hα)=1.48 × 10^41 erg s^-1, after correcting for the dust extinction using the Balmer decrement (A_Hα=1.32). This value for theluminosity converts into a SFR=0.76 M_ yr^-1, using the calibration by Kennicutt & Evans <cit.>. §.§ Metallicity and ionizationAs according to diagnostic diagrams discussed in the previous section the gas ionization is dominated by young stars across all the galaxy, it is possible to use selected line ratios of the most intense lines to derive the chemical enrichment of the interstellar gas (the so-called “strong line methods”, see e.g. Curti et al.and references therein). The integrated metal abundance of He 2-10 was already derived by Kobulnicky et al. <cit.> using this method, yielding a super solar abundance of 12+log(O/H)=8.93. Such a high valueis not unexpected given the relative higher mass of He 2-10 and the well known relation between mass and metallicity: as an example, assuming the mass metallicity relation of Tremonti et al. <cit.> and a stellar mass M_*=3.7× 10^9 M_, we expect 12+log(O/H)∼8.82, that is increased to 8.95 using the higher mass M_*=1× 10^10 M_ by Nguyen et al. <cit.>. If we also consider the SFR as third parameter in the relation,Mannucci et al. <cit.> would predict 12+log(O/H)∼8.70, somehow lower than measured. However, Esteban et al. <cit.> recently obtained a new measure of the oxygen metallicity integrated on the central 8”×3” of 12+log(O/H)=8.55±0.02, using faint pure recombination lines. Such difference between the different methods to measure metallicity and different calibrations is a well known effect (see e.g. the discussion in Kewley & Ellison ), and mostly due to a mismatch between direct methods, based on electron temperature T_e or recombination lines, and strong line calibrations based on photoionization models, that are offset by ∼0.3-0.5 dex at high metallicity. This mismatch is now being solved by the advent of new “strong line” diagnostics, fully empirical calibrated using direct methods up to high metallicities (see e.g. Brown et al. , Curti et al. ). The new metallicity value by Esteban et al. <cit.> is again comparable with what is expected by a computation of the Mass Metallicity Relation obtained with such new diagnostics, as Andrews & Martini <cit.> would predict 12+log(O/H)∼8.65 using only the stellar mass and 12+log(O/H)∼8.50 taking into account the SFR as well.Here we attempt to spatially map the metallicity in He 2-10 using this new calibrations and our IFU data. In Fig. <ref> (first panel) we show the spatial variation of one of these diagnostic ratios, O3N2=/·/, and the corresponding metallicity map, using the calibrations by Curti et al. <cit.> (second panel). It can be seen how the global average metallicity in the disk is compatible with the value derived by Esteban et al. <cit.> with recombination lines, while the Eastern region of the galaxy and the central highly star forming regions show lower metallicity values (12+log(O/H)=8.2-8.3). Highly star forming regions with lower gas metallicity than the surrounding galaxy have been interpreted as signatures of pristine, low metallicity gas accretion, that dilutes the metal content of the ISM and boosts star formation, both locally (e.g. Sánchez Almeida et al. ) and at high-z (Cresci et al. ). However, in the case of He 2-10 the abundance gradient amplitudes is of the same order of magnitude of the intrinsic scatter in the calibrations (e.g. 0.2 dex for O3N2, probably due to variation in the ionization in different sources). Moreover, it varies depending on the diagnostic ratio used: as an example, the gradient using / is Δ Z∼0.15 dex, using O32 is Δ Z∼0.25 dex, while using a simultaneous fit with a combination of the above diagnostics and of / is Δ Z∼0.12 dex, although the different indicators are supposed to be cross-calibrated. Moreover, we use the line ratio [SIII]/[SII] to derive a map of the ionization parameter U, using the calibrations of Kewley & Dopita <cit.> (Fig. <ref>, third panel). The ionization parameter is the ratio between the number density of photons at the Lyman edge and the number density of Hydrogen, U=Q_H^0/(4π R_s^2 c n_H), where Q_H^0 is the flux of ionizing photons produced by the exciting stars above the Lyman limit, n_H the number density of Hydrogen atoms, c the speed of light and R_s the Strömgren radius of the nebula, and it is a measure of the intensity of the radiation field. It can be seen from Fig. <ref> (third panel) that this parameter is one order of magnitude higher in the two nuclear star forming region, further suggesting that the line ratio variation is due to ionization effects in extreme environments such as these embedded, highlystar forming regions and not to a metallicity variation. To further explore this, we map the MUSE spaxels on the new diagnostic diagram by Dopita et al. <cit.>, which makes use of , , ,andlines to constrain both metallicity and U using photoionization models. The result is shown in Fig. <ref> (left panel): the different grids shows the variation of metallicity and logU for different values of ISM pressure. The spaxels corresponding to the central star forming regions define two clear sequences towards high metallicity and high U, plotted in magenta and blue. The location of these spaxels are plotted in the upper right panel superimposed to themap of He 2-10. The spaxels with metallicity 12+log(O/H)<8.55 are marked in green, both in the left and in the right panel. This confirms that the Eastern part of the galaxy has lower metallicity than the rest by ∼0.2 dex, possibily in agreement with the merger interpretation of the origin of He 2-10, while the different line ratios in the central star forming regions are due to higher ionization parameter (see also Fig. <ref>, fourth panel). In the Dopita et al. <cit.> diagram the two star forming regions actually show very high supersolar metallicities, probably due to efficient metal enrichment in those extreme environments. As shown in Fig. <ref>, lower right panel, these regions form a definite structure in the BPT diagram as well. This result suggests caution in the interpretation of a single line ratio as a variation in metallicity of the ISM, and confirms the importance of a large wavelength range to exploit multiple physical diagnostics.§ NO COHERENT EVIDENCE FOR A SMBH IN HE 2-10: A SUPERNOVA REMNANT ORIGIN FOR THE COMPACT RADIO EMISSION?As discussed in the introduction, Reines et al. <cit.> identified as an accreting SMBH the compact radio source detected by Johnson & Kobulnicky (, their knot 3 located between the two strongest star forming regions), based on the combination of radio and X-ray data. In particular, they used the ratio between the radio to X-ray luminosity, R_X=ν L_ν (5 GHz)/L_X(2-10 keV) to discriminate between different scenarios: their data provided log R_X∼-3.6 for the radio source 3, that is in the range expected for low luminosity AGNs (log R_X≃-2.8÷-3.8, Ho et al. ). A stellar-mass black hole X-ray binary system was excluded, as this kind of sources are too weak in the radio (log R_X<-5.3) to explain the observed ratio. Conversely, Supernova Remnants (SNR) were excluded as they are relatively weak X-ray sources, with log R_X≳-2.7. We therefore extracted a spectrum integrated over an aperture of 2 pixels radius (0.4") around the location of the compact radio source 3 and putative black hole to check for any sign of AGN ionization in the MUSE data. The spectrum is shown in Fig. <ref>. The position of the line ratios extracted from this spectrum on the BPT diagrams is shown in Fig. <ref> as a yellow cross, showing that the ionizing radiation is completely dominated by young stars. Moreover, despite the very high S/N obtained, no high ionization lines typical of AGNs such as [FeVII]λ5721 and [FeVII]λ6087 are detected (e.g. Vanden Berk et al. ). We derive an upper limit of 7·10^-3 for the [FeVII]λ6087/ ratio, while it is typically ∼0.1 in Seyfert galaxies (see e.g. Netzer ). We therefore conclude that even if a SMBH is present in He 2-10, it is not contributing to the gas ionization.Deeper (200 ks) Chandra observations recently obtained by Reines et al. <cit.> have shown, on the basis of an accurate spectral, spatial and temporal analysis, that the hard X-ray radiation previously attributed to the compact radio source 3 in the shallower archive data (20 ks) is instead due to a varying source located in one of the star forming region (knot 4), compatible with a massive X-ray binary. They also reported the discovery of a faint source coincident with the radio source 3, with a tentative detection of a ∼9 hours periodicity which can in principle be ascribed to instabilities in the accretion disc flow, although the light curve can be adequately reproduced by a simple constant value. Its X-ray spectrum is very steep, and the ∼180 X-ray counts are fitted with an unobscured power law with Γ=2.9. However, as noted also by Reines et al. <cit.>, the low counting statistics and the high local background due to the extended emission does not allow an unique characterisation of the X-ray emission, given that also a thermal plasma model with kT∼1.1 keV can also well reproduce the data.The luminosity associated with the radio source 3 over the full (0.3-10 keV) X-ray range, assuming the Γ=2.9 fit, is∼10^38 erg s^-1. This translates into an hard (2-10 keV) X-ray luminosity of L_2-10 keV∼1.4×10^37 erg s^-1.We reanalyzed the spectrum of the radio source 3 in the longest Chandra observation available (160 ks), extracted from a 0.5 radius circle around the radio centroid. We retrieve a consistent hard X-ray luminosity when using the same power-law model adopted by Reines et al. <cit.>,but with residuals at E>5 keV at ∼ 3σ. We therefore conservatively added an additional power law component to the fit, in the hypothesis that we see emission from an (obscured) AGN. This returns an hard X-ray luminosity of 1.0×10^38 erg s^-1. However, a contamination from the variable high mass X-ray Binary north of the nuclear source is still present, as all the detected photons with E>5 keV are located in a 0.5” radius from the contaminating source. For these reasons, the L_X derived from this two components fit should be considered as a very conservative upper limit to the maximum 2-10 keV luminosity allowed for the radio source.The upper limit on the hard X-ray luminosity adding a second power law to the fit corresponds to a radio to X-ray ratio R_X>-2.4, assuming the revised ν L_ν (5 GHz)=4.1×10^35 erg s^-1, an order of magnitude higher than what is measured in local low luminosity AGNs, and even a factor of 2.5 higher than transitional objects between HII galaxies and LINERs (Ho et al. ). A much higher value of R_X=-1.6, more than an order of magnitude higher than any kind of local low luminosity AGN, is instead obtained from the single power-law fit X-ray luminosity.Both these new values for R_X are actually now compatible with the range observed in SNRs (R_X≳-2.7, Reines et al. , Vink ). This scenario is further supported by the detection of high [FeII]/Brγ at the location of the compact radio source by Cresci et al. <cit.>, consistent with thermal excitation in SNR shocks. In this respect we note that a high [FeII] emission also corresponds to the location of a second compact radio source in He 2-10 detected by Reines & Deller <cit.> and therein classified as a SNR. Moreover, the radio spectral index α∼-0.5 (Reines & Deller ) is the typical spectral index measured in SNR (e.g. Reynolds et al. ), and its radio luminosity is high but still compatible with a young SNR, as well as its compact size (see e.g. Fenech et al. ).On the other hand, if we assume that both the radio and hard X-ray luminosity is due to a SMBH, the fundamental plane of BH activity (Merloni et al. ) would predict a BH mass of log(M_BH/M_)=7.65±0.62 using the single component X-ray spectrum fit. This BH mass would be incompatible with the stellar dynamical measurements obtained by Nguyen et al. <cit.> with near-IR IFU observations at Adaptive Optics assisted spatial resolution (0.15", comparable with the sphere of influence of a 10^6 M_ BH of 4 pc, given the local stellar σ=45 km s^-1). They placed a firm upper limit of log(M_BH/M_)<7 at 3 σ level based on their data, finding no dispersion peak or enhanced rotation around the compact radio source, where actually the dispersion is lower than the surrounding area.Such a massive BH is also excluded by measurements of the dynamical map in the central 70 pc by CO observations of Kobulnicky et al. <cit.> (3-48·10^6 M_). We note that if we assume the maximum 2-10 keV luminosity allowed from the fit with an obscured AGN component (L_X<1×10^38 erg s^-1), the Merloni et al. (2003) relation gives a lower limit to the SMBH powering the AGN, log(M_BH/M_)>7, that is still ruled out by the dynamical measurements. Summarizing, the gas excitation in the MUSE data do not show any evidences of an active BH in He 2-10. This is further supported by the analysis of [FeII]/Brγ ratio suggesting thermal excitation in SNR shocks. Moreover, the new Chandra observations that revised the hard X-ray flux from the putative location of the BH provide a flux ∼ 200 times smaller than the estimate reported in Reines et al. <cit.> based on the first 20 ks of Chandra data. The corresponding luminosity is now compatible with a SNR origin for the non thermal emission, and so it is the Radio to X-ray loudness ratio (>-2.4), much higher than that observed in local low luminosity AGNs. In case the X-ray and radio emission are still attributed to an accreting BH, the resultinglog(M_BH/M_)=7.65 would be too large to fit the dynamical information available, as well as the known correlations between BH mass and bulge mass (e.g. Kormendy & Ho ), given the dynamical mass of ∼7·10^9 M_ (Kobulnicky et al. ). Although a smaller BH would be still allowed by the large scatter in the relations, the resulting Eddington ratio for this putative SMBHwith log(M_BH/M_)∼7 would be L_bol/L_Edd≲1×10^-6, at least two order of magnitudes lower than the value adopted for actively accreting BHs (see e.g. Merloni & Heinz ). Therefore, an actively accreting SMBH origin for the radio source 3 is excluded, and a young SNR seems easier to fit the gathered evidences. The only feature that cannot be simply ascribed to a SNR is the periodicity observed in the X-rays which, however, is reported as tentative and possibly due to random fluctuations (Reines et al. ). Further high resolution multi-wavelength observations are needed to definitely assess the nature of this interesting system.§ CONCLUSIONSWe have presented MUSE integral field observations of He 2-10, a prototypical nearby HII galaxy. The MUSE data allowed a detailed study of the dynamics and physical conditions of the warm ionized gas in the galaxy. Our main results can be summarized as follows:- The gas dynamic is characterized by a complex and extended (up to 720 pc projected from the central star forming region) high velocity expanding bubbles system, with both blue and redshifted gas velocities v_max> 500 km/s, higher than the galaxy escape velocity. We derive a mass outflow rate Ṁ_out∼ 0.30 M_ yr^-1, corresponding to mass loading factor η∼0.4, in range with similar measurements in local LIRGs. Such a massive outflow has a total kinetic energy that is sustainable by the stellar winds and Supernova Remnants expected in the galaxy, with no additional energy source required.- The lower velocity dispersion regions (σ∼55 km/s), where the outflowing gas is not the dominant component, show a velocity gradient of ± 35 km/s consistent with the HI disk detected in He 2-10, thus tracing a rotating gaseous disk.The stellar kinematics is instead remarkably flat, suggesting a decouple between the bulk of the stellar population in the galaxy and the warm ionized gas.- The central star forming regions where the most embedded star cluster reside show the highest values of electron density (n_e=1500 cm^-3) and ionization parameter (logU=-2.5), confirming the extreme conditions in these environments. The dust extinction as derived by the Balmer decrement is high in the Western star forming knot (A_V=2.3) and in a region to the SE of the nucleus, corresponding to the location of an accreting CO cloud (A_V∼2.5). Higher values of the dust extinction are obtained in the near-IR, probably because the optical diagnostic are just looking at the external shells of the embedded star forming regions. - Given the large variation of the ionization parameter across the galaxy, the use of a single line ratio to derive the metallicity of the galaxy is not recommended, as we show that the line ratio variation is due to a combination of metallicity and ionization variations. We use a diagram proposed by Dopita et al <cit.> to account simultaneously to the variation of these two parameters, finding a large metallicity gradient across the galaxy, with the central star forming regions having super solar metallicity, while the external part of the galaxy chemical abundances as low as 12+log(O/H)∼8.3.- The ionization all across the galaxy is dominated by young stars, as described by the so called BPT diagrams, with no sign of AGN ionization. This seems in contrast to the claim of the presence of an accreting SMBH in He 2-10 (Reines et al. , ), based on the combination of radio and X-ray data. However, we show that the X-ray radio-loudness parameters R_X obtained with a revised estimate of the X-ray flux are much higher than expected from local low luminosity AGN but consistent with a SNR origin, as suggested also by the high [FeII]/Brγ ratio observed at the location of the compact radio source. Therefore, the constraints given by the hard X-ray flux are not enough to confirm the presence of a SMBH in He 2-10, as the data can be explained with different kind of sources such as a young SNR.This work confirms the unique capabilities of large field integral field spectroscopy to explore the details and physical properties of the interstellar medium and star formation in nearby starburst galaxies. Upcoming similar observations of a larger sample of nearby HII and dwarf star forming galaxies will allow to shed light on the different mechanisms that regulates and trigger the star formation in these extreme environments.§ ACKNOWLEDGMENTS MUSE data were obtained from observations made with the ESO Telescopes at the Paranal Observatory. We are grateful to the ESO staff for their work and support. We also show our gratitude to E. Amato, R. Bandiera and N. Bucciantini for useful suggestions and discussion on Supernova Remnants, to A. Comastri on X-ray spectral fitting, and to A. Marconi and G. Venturi for sharing the python scripts used for the fitting. LV acknowledges support by the project CONICYT Anillo ACT-1417. MB acknowledges support from the FP7 Career Integration Grant “eEASy” (CIG 321913). GC, MB and GL acknowledge financial support from INAF under the contracts PRIN-INAF-2014 (“Windy Black Holes combing Galaxy evolution"). GC is also grateful to F. Chiesa for reminding the importance of never giving up. [(1976)]allen76 Allen, D. A., Wright, A. E., & Goss, W. M. 1976, , 177, 91[2013]andrews13 Andrews, B. H., & Martini, P. 2013, , 765, 140[2014]arribas14 Arribas, S., Colina, L., Bellocchi, E., Maiolino, R., & Villar-Martín, M. 2014, , 568, A14 [2010]bacon10 Bacon, R., Accardo, M., Adjali, L., et al. 2010, , 7735, 773508 [1981]baldwin81 Baldwin, J. A., Phillips, M. M., & Terlevich, R. 1981, , 93, 5[2004]barth04 Barth, A. J., Ho, L. C., Rutledge, R. E., & Sargent, W. L. W. 2004, , 607, 90[1997]beck97 Beck, S. C., Kelly, D. M., & Lacy, J. 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"authors": [
"Giovanni Cresci",
"Leonardo Vanzi",
"Eduardo Telles",
"Giorgio Lanzuisi",
"Marcella Brusa",
"Matilde Mingozzi",
"Marc Sauvage",
"Kelsey Johnson"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170426222740",
"title": "The MUSE view of He 2-10: no AGN ionization but a sparkling starburst"
} |
[][email protected] [][email protected] ^1Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland ^2ICFO - Institut de Ciences Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, SpainEigenstate-preserving multi-qubit parity measurements lie at the heart of stabilizer quantum error correction, which is a promising approach to mitigate the problem of decoherence in quantum computers. In this work we explore a high-fidelity, eigenstate-preserving parity readout for superconducting qubits dispersively coupled to a microwave resonator, where the parity bit is encoded in the amplitude of a coherent state of the resonator. Detecting photons emitted by the resonator via a current biased Josephson junction yields information about the parity bit. We analyse theoretically the measurement back-action in the limit of a strongly coupled fast detector and show that in general such a parity measurement, while approximately Quantum Non-Demolition (QND) is not eigenstate-preserving. To remediate this shortcoming we propose a simple dynamical decoupling technique during photon detection, which greatly reduces decoherence within a given parity subspace. Furthermore, by applying a sequence of fast displacement operations interleaved with the dynamical decoupling pulses, the natural bias of this binary detector can be efficiently suppressed. Finally, we introduce the concept of a heralded parity measurement, where a detector click guarantees successful multi-qubit parity detection even for finite detection efficiency.Towards a heralded eigenstate preserving measurement of multi-qubit parity in circuit QED Simon E. Nigg^1 December 30, 2023 ===========================================================================================§ INTRODUCTIONQuantum computers are open quantum systems: The quantum information carriers – qubits – inevitably couple to the outside world and this coupling leads to decoherence. Quantum error correction (QEC), which aims at correcting the errors induced by decoherence, is thus necessary for quantum computation. Stabilizer codes <cit.> are among the most promising quantum error correction codes. A common feature of all stabilizer codes is that errors happening on the physical qubits can be detected by repeatedly measuring a set of mutually commuting multi-qubit operators called stabilizer operators. Every detectable error needs to anti-commute with at least one stabilizer operator. Typically, stabilizer operators are chosen as elements of the Pauli group, represented by tensor products of single-qubit operators in the set {, X, Y, Z}. Here X, Y and Z are the three spin-1/2 Pauli matrices andis the identity operator. If in the system under consideration all qubits can be addressed and controlled individually, then measuring arbitrary multi-qubit Pauli operators is equivalent, up to single-qubit rotations, to measuring arbitrary tensor products of operators in the reduced set {, Z}. The latter task, which we call parity measurement, is what we focus on in this work.To be useful for the purpose of quantum error correction, parity measurements need to be eigenstate-preserving, e.g. measuring the parity of the two-qubit state (1/√(2))(|ee⟩+|gg⟩) must not destroy the superposition. Note that this is a stronger requirement than asking the measurement to be QND, which only requires that repeated measurements yield always the same result <cit.>. Developing multi-qubit parity measurements in superconducting circuits is a very active area of research and has been discussed in a number of previous works <cit.>. <cit.> succesfully measured the parity of an arbitrary subset of three superconducting transmon qubits in an approximately eigenstate-preserving fashion based on the theoretical proposal of <cit.>. In this approach in a first stage, the parity bit is first mapped onto the phase of a coherent state of a microwave field dispersively coupled to the qubits. In a second stage, the parity bit is mapped onto an ancilla qubit and in a final third stage, the parity bit is read out by homodyne measurement of the ancilla qubit.In the present work we discuss an alternative approach to parity readout. This work is motivated by the desire to improve upon two current limitations of the scheme presented in <cit.>, namely the reduction of parity detection fidelity due to photon leakage and finite ancilla qubit lifetime. Our proposal can also be seen as an extension of <cit.>, where it was proposed to correlate the parity of multiple qubits with the amplitude of a coherent state (either the vacuum state or a coherent state with finite amplitude) and then detect the emission of a photon with a microwave photon detector based on a current biased Josephson junction (CBJJ) <cit.>. A click of the detector corresponds to the switching of the CBJJ to the resistive state. Such an event indicates a certain parity of the multi-qubit state, while the absence of a photon detection indicates the other parity with some probability that depends on the measurement time and the detector efficiency. As presented in <cit.>, this elegant scheme however suffers from two important deficiencies. First, while QND, it leads in general to intra parity-subspace decoherence because a randomly emitted photon carries with it more information than just the parity-bit of the multi-qubit state. For the purpose of QEC it is crucial to limit such information leakage to avoid intra parity-subspace decoherence. Second, because one of the two parities is correlated with a bright state while the other is correlated with the vacuum in the cavity, the parity detection is inherently asymmetric: While a click of the photon detector guarantees the correct parity detection, a no-click event is ambiguous and the wrong parity can be inferred if the measurement time is too short or if the detection efficiency is below unity.The parity readout proposed in the present work addresses both of these shortcomings: First, by combining photon detection with dynamical decoupling, the measurement induced decoherence is reduced. Second, by periodically swapping the encoding between the two parities and the dark and bright states of the cavity, the bias of the detector is reduced. With these two modifications, the detection of a photon genuinely heralds a successful parity detection with minimal back-action induced intra parity-subspace decoherence. Hence, multi-qubit parity measurements via direct photon detection become a viable alternative for QEC in an architecture where fixed frequency qubits are dispersively coupled to a common bosonic mode. The manuscript is organized as follows. In Section <ref>, we start by reviewing different methods to encode the parity of multiple qubits into the state of microwave photons. In Section <ref>, we briefly review previous work on how to read out the encoded parity information. In Section <ref> we introduce our model for parity readout and make a first qualitative discussion of the measurement back-action in Section <ref>. In Section <ref> we simplify our model. In Section <ref>, the bulk of this work, we analyze the back-action in the photon detection based parity detection scheme and find that while QND, the parity measurement is in general not eigenstate preserving, i.e. it induces intra parity-subspace decoherence. In Section <ref> we propose the use of a simple dynamical decoupling scheme to evade the back-action and quantify the ideal fidelities in this modified scheme numerically. In Section <ref> we propose a way to reduce the detector bias and show that with these modifications, heralded and unbiased parity measurement with minimal back-action induced intra parity-subspace decoherence is feasible. We conclude this section with an estimate of achievable fidelities for realistic parameter values. Finally, we conclude with some remarks in Section <ref>. § REVIEW OF PARITY ENCODINGThe parity operator of N qubits is defined asP_N=∏_n=1^Nσ_n^z.The eigenstates of σ_n^z are the computational basis states. For a superconducting qubit, they typically correspond to the two lowest energy eigenstates |g⟩ and |e⟩ so that σ^z=|e⟩⟨e|-|g⟩⟨g|. Sine ( P_N)^2=, the eigenvalue is either +1 or -1 and can be interpreted as the parity of the number of qubits in state |g⟩.We say that an N-qubit state is an even (odd) parity state if it is an eigenstates of P_N with eigenvalue +1 (-1).There exist several methods to encode the parity of a multi-qubit state into the state of an electromagnetic field. In <cit.> this was achieved by utilizing the dispersive interaction of superconducting transmon qubits with the quantized field of a microwave resonator <cit.>. This interaction is characterized by the Hamiltonian termH_ disp = ∑_i=1^N_i^σ_i^z a^†a,where ^_i is the dispersive frequency shift, while a and a^† are the annihilation and creation operators of the microwave field. The central idea of the approach of <cit.>, is to apply pairs of coherent π-pulses, which effectively corresponds to the application of σ_x≡|e⟩⟨g|+|g⟩⟨e|, to each individual qubit properly spaced in time such as to control its contribution to the total phase shift of the cavity, which is initially prepared in a coherent state with amplitude α. Specifically, if the time delay t_j between two π-pulses on qubit j is chosen as t_j=T/2-π/2 _j^, then under the action of (<ref>) at time T the parity of all qubits becomes entangled with the phase of the cavity state as|ϕ(T)⟩=|α_N⟩+ P_N/2|ψ_N⟩+|-α_N⟩- P_N/2|ψ_N⟩.Here α_N=(-i)^Nα and |ψ_N⟩ denotes the initial multi-qubit state. (± P_N)/2 are the projectors onto the even (+) and odd (-) parity subspaces. Selectivity to an arbitrary subset 𝒮 of qubits can be achieved by instead choosing t_j=T/2 for j∉𝒮 <cit.>.In <cit.> an alternative method for parity encoding was proposed, which also makes use of the dispersive interaction (<ref>). Instead of applying control pulses to the qubits, one applies frequency multiplexed drives to the cavity initially in the vacuum to selectively displace the cavity state out of the vacuum conditioned on a specific parity of the multi-qubit state. This is possible when the frequency shifts of all the even parity states differ from all the frequency shifts of the odd parity states but may require slow pulses to ensure proper frequency selectivity. The encoding thus generated can be written as|φ(T)⟩=|0⟩+ P_N/2|ψ_N⟩+|β⟩- P_N/2|ψ_N⟩.The amplitude β is controlled by the envolop of the applied drive pulses. Note that the final encodings (<ref>) and (<ref>) are equivalent up to a displacement operation D(-β/2)=exp[-(β/2) a^†+(β^*/2) a] with β=-2α_N. § REVIEW OF PARITY READOUTTo complete the parity measurement, the parity information encoded in the cavity, as per Eqs. (<ref>) or (<ref>), must be read out. We next briefly review two ways to achieve this. In Refs. <cit.> the cavity state is swapped onto that of an ancilla qubit. The ancilla is initialized in its ground state and is dispersively coupled to the cavity field encoding the parity of the remaining qubits according to (<ref>). The swapping of the parity onto the ancilla is achieved in two steps. In the first step, a conditional π-pulse is applied to the ancilla qubit conditioned on the vacuum state of the cavity <cit.>. This step results in the tripartite entangled state|φ(T)⟩=|0⟩|e⟩_A+ P_N/2|ψ_N⟩+|β⟩|g⟩_A- P_N/2|ψ_N⟩,where |g⟩_A and |e⟩_A denote the ground and excited states of the ancilla. In the second step, the cavity is disentangled either via a conditional displacement of amplitude -β conditioned on the ground state of the ancilla qubit <cit.>, or by inverting the unitary encoding operations <cit.>. This results in the state|φ(T)⟩=|0⟩(|e⟩_A + P_N/2|ψ_N⟩+|g⟩_A- P_N/2|ψ_N⟩),where the parity is encoded in the state of the ancilla. The latter can subsequently be read out via standard homodyne detection <cit.>.An advantage of this readout via an ancilla qubit is that after the entanglement swapping, the cavity is back in the vacuum state and no further information about the multi-qubit state can leak out from the cavity. However, the decoherence of the ancilla does limit the fidelity of the parity mapping and readout as observed in <cit.>.<cit.> proposed an alternative readout based on direct photon detection via a CBJJ capacitively coupled to the cavity. The basic idea of this readout, the physical mechanism of which is explained in details in Section <ref>, is as follows: In the state of Eq. (<ref>), if a photon is detected, then the multi-qubit parity is inferred to be even. If a photon is not detected, then the parity is inferred to be odd with some probability that depends on the measurement time and the detector efficiency. In <cit.> it was shown that this approach leads approximately to a quantum non-demolition parity readout under the condition that the dispersive shifts of all qubits are equal. However, for the purpose of stabilizer quantum error correction, QNDness of parity measurements while necessary is not a sufficient condition. Indeed the kind of parity measurements required must preserve the coherence within each parity subspace. This property has recently been coined eigenstate preserving QND (EP-QND) <cit.>. One of the main goals of the present work is to analyze in detail the back-action of the parity measurement based on photon detection <cit.>. In Section <ref>, we show that it is in general not EP-QND because the emitted photons contain more information than the parity bit alone. To a lesser extent, this also affects the parity readout used in <cit.>, because the parity encoding and the swapping of the parity information onto the ancilla take a finite amount of time during which photons may escape the cavity. In the following we focus on the readout stage of the parity measurement, once the parity bit has been encoded in a photonic state such as in Eq. (<ref>). § SYSTEM AND MODELWe consider a specific circuit quantum electrodynamics (cQED) architecture, where a set of superconducting qubits, such as e.g. transmon qubits, are dispersively coupled to the quantized field of a microwave resonator. A concrete realization of this type of architecture with 3D-transmons and cylindrical microwave resonators is provided in <cit.> but realizations with coplanar waveguide resonators are also common <cit.>. In the following we consider an abstract model that applies to both implementations. The considered system is shown in Fig. <ref> and consists of a high-Q microwave resonator dispersively coupled to N transmon qubits and furthermore capacitively coupled to a CBJJ, which serves as a microwave photon detector <cit.>. The Hamiltonian we use to model this system isH= ( ω_c+ ∑_i=1^ N _i σ_i^z ) a^†a+ g_J ( a|2⟩⟨1| + a^†|1⟩⟨2|)+ ω_12|2⟩⟨ 2| - ω_20|0⟩⟨0|.Here σ_i^z=|e⟩⟨e|_i-|g⟩⟨g|_i denotes the Pauli Z operator for qubit i. The inevitable dissipation in the CBJJ associated with the photo-detection process is accounted for by the Lindblad master equation,ρ̇ = -i [H, ρ] + κ_J 𝒟[ |0⟩⟨2|]ρ,where 𝒟[c ]ρ =c ρ c^† - 1/2(c^† c ρ + ρ c^† c ).Here we have reduced the CBJJ to an effective three-level system <cit.>. The states | 1 ⟩ and | 2 ⟩ represent the two states localized inside a well of the tilted washboard potential of the CBJJ (see Fig. <ref>). Via the dc current bias, the transition frequency between | 1 ⟩ and | 2 ⟩ is tuned in resonance with the bare cavity frequency ω_c. Furthermore, by suitably designing the junction capacitance, it is possible to make the upper level | 2 ⟩ couple strongly to the continuum, which is modeled here as an additional state | 0 ⟩. A photon leaving the cavity towards the CBJJ coherently populates level | 2 ⟩, which incoherently decays at a rate κ_J into the continuum state | 0 ⟩. The tunnel coupling of the lower level | 1 ⟩ to the continuum state | 0 ⟩ is exponentially smaller than the coupling between | 2 ⟩ and | 0 ⟩ and will be neglected in the following. Note however that this coupling will lead to dark counts and thus negatively impact the parity readout fidelity. For a discussion of this effect see e.g. <cit.>.§ QUALITATIVE DISCUSSION OF THE MEASUREMENT BACK-ACTIONIn this section we briefly discuss the dynamics of of the full system depicted in Fig. <ref>, obtained by numerically solving the Lindblad master equation Eq. (<ref>). To illustrate the effect of the measurement we show in Fig. <ref> the time evolution of a two-qubit state coupled to a cavity with amplitude α and the CBJJ. The system is initially in the state |ψ⟩ = (|gg⟩+ |ee⟩)/ √(2)⊗|α⟩⊗|1⟩_ CBJJ. For simplicity we let the dispersive shifts from Eq. (<ref>) be equal i.e. _1 = _2 =.As a measure for phase coherence within a given parity subspace we use the expectation values ⟨σ_1^iσ_2^i|$⟩ fori=x,y. The hermitian part of the Lindblad master equation (<ref>) leads to a periodical change of⟨σ_1^xσ_2^x|$⟩ and ⟨σ_1^yσ_2^y|$⟩ which is a consequence of the entanglement of the qubits with the cavity due to the dispersive interaction. We will refer to the periodical reappearance of maxima in these expectation values as the revival of coherence <cit.>. The decrease in the amplitude of these revivals is a direct measure of the intra parity-subspace decoherence and is a consequence of the non-hermitian part of Eq. (<ref>), which describes the effect of the measurement when ignoring the measurement record. In Section <ref> we will show that at the level of the individual quantum trajectories, the loss of a photon out of the cavity leads to a random phase kick on the qubit parity-subspaces. Because in the master equation one averages over all such random events, this results in the observed suppression of the revival amplitudes. Furthermore we obtain from the numerics in Fig. <ref> that the cavity decay stops after the loss of one photon. This is because the CBJJ is trapped in the continuum state on a much longer time scale than that of the actual photon decay.The revival time scale can be estimated in the coherent limit, i.e. by considering a reduced system, where the two qubits are coupled to a cavity, without CBJJ and leakage. The unitary time evolution through the HamiltonianH=ω_c a^†a +(σ_1^z + σ_2^z)a^†aleads to the time dependent entanglement of the qubit and the cavity in the form|ψ(t) ⟩= 1 / √(2) (|gg⟩ ⊗|αe^+ 2i t ⟩+ |ee⟩ ⊗| αe^-2 it⟩ ). The revival occurs if the cavity and the qubit are disentangled, i.e. are separable again, hence the time of the revival ist_rev = π/ (2 ). § ADIABATIC ELIMINATION OF THE CBJJ The parity readout discussed in this work takes place in the limit where the effective Rabi coupling between the cavity and the CBJJ is small compared with the decay rate of the metastable state of the CBJJ.In this regime, the population of the metastable state of the CBJJ remains small. This allows us to adiabatically eliminate the CBJJ in the spirit of a Weisskopf-Wigner approximation. In this way we obtain an analytically tractable and physically transparent model of the detector where a photon detection event corresponds simply to a photon loss event out of the cavity. The rate at which such an event takes place is calculated as follows.Consider a system ofNqubits in the computational basis state|j_1,j_2, ⋯, j_N⟩wherej_i ∈{e,g }are fixed but arbitrary. The associated total state dependent dispersive shift isΔ=∑_i=1^Nσ_i_i, whereσ_i=+1ifj_i=eandσ_i=-1ifj_i=g. Since the dispersive coupling commutes withσ_i^z, the projected Hamiltonian becomesH= ( ω_c + Δ) a^†a + g_J ( a|2⟩⟨1| + a^†|1⟩⟨2|)+ ω_C| 2 ⟩⟨ 2 | - ω_20| 0 ⟩⟨ 0 |.Here we already tuned the CBJJ on resonance with the bare cavity frequencyω_12 = ω_c. To solve the master equation for this Hamiltonian we notice that the interaction only couples a closed set of states:|n+1,1 ⟩,|n, 2 ⟩and|n, 0 ⟩, where|n ⟩is a Fock state with the photon numbernand|1 ⟩,|2 ⟩and|0 ⟩represent the states of the CBJJ. If we truncate the Hamiltonian to this reduced set of basis states and defineρ := [ ρ_11 ρ_12 ρ_10; ρ_21 ρ_22 ρ_20; ρ_01 ρ_02 ρ_00;],where the subscripts0,1,2are again representing the states of the CBJJ, the master equation (<ref>) yields a set of coupled linear differential equations,ρ̇_11 =-i g_J √(n+1)(ρ_21-ρ_12) ρ̇_22= -i g_J √(n+1)(ρ_12-ρ_21) - κ_J ρ_22ρ̇_00 = κ_J ρ_22ρ̇_12 = -i g_J √(n+1)(ρ_22-ρ_11) -κ_J/2ρ_12 -i Δρ_12ρ̇_21 = -i g_J √(n+1)(ρ_11-ρ_22)-κ_J/2ρ_21 +i Δρ_21.In a similar manner as in <cit.> these equations can be solved by Laplace transformation (See Appendix <ref>) and yieldρ_00^(n)=1-exp{-4tg_J^2(n+1)/κ_J[ 1-( Δ/κ_J)^2 ] }.Here we have added a superscript(n)to emphasize the dependence on the photon numbern. The solution for a coherent state|α⟩in the cavity is obtained by averaging (<ref>) over the Poissonian photon number distribution <cit.>. In the large amplitude limit|α|^2≫1, we can neglect the relative photon number fluctuations and perform the replacementn+1→n̅=|α|^2. We then obtainρ_00≃1-exp( -κ_eff^CBJJt ), withκ_ eff^ CBJJ = 4 Ω_n̅^2/κ_J[ 1 - 𝒪(Δ^2/κ_J^2)].This approximation holds in the limitΩ_n̅≡g_J √(n̅+1)≪κ_JandΔ≪Ω_n̅wheren̅=|α|^2andΩ_n̅is the effective Rabi frequency. The first inequality defines the regime of an overdamped CBJJ, that directly decays from its excited state|2⟩to the continuum state|0⟩, without Rabi flopping with the cavity states|n+1⟩and|n⟩. The second relation embodies that the energy is transferred from the cavity to the CBJJ fast on the time scale characterizing the multi-qubit dynamics. Note that previous work by <cit.> focused on an intermediate regime whereκ_J≃g_J.A caveat of the adiabatic elimination is that we have lost the saturation effect due to the long relaxation time of the CBJJ (see Section <ref>). This can however be accounted for a posteriori by matching the effective cavity decay⟨a^†a|=⟩|α|^2e^-κ_eff^cavtwith the saturation behavior of the CBJJ via⟨a^†a ⟩= |α|^2 - ρ_00(t). Expanding the population decay of the cavity on the left hand side andρ_00(t)on the right hand side of this equation for short times, we find the effective decay rateκ_ eff^ cav = 4 g_J^2/κ_J(1 - 𝒪( Δ^2/κ_J^2) ).This form is reminiscent of the resonant vacuum Purcell decay rate. The second term in the parenthesis represents the effect of the qubit state dependent detuning on the decay rate. It is of order∼𝒪((Δ/κ_J)^2)and is therefore negligible as long as the relationκ_J ≫Ω_n̅ ≫Δholds. In Appendix <ref>, we discuss the consequence of this higher-order term on the measurement back-action. Here we focus on the leading order measurement back-action, which is independent of the multi-qubit state and characterized by the effective detection rateκ_eff^cav=4g_J^2/κ_J.§ CHARACTERIZATION OF INTRA PARITY-SUBSPACE DECOHERENCE In the effective model derived in Section <ref>, whereNqubits are coupled to a cavity with the effective photon detection rateκ_eff^cavthe Hamiltonian reduces to H=ω_c a^†a + ∑_i^N _i σ_i^za^†aand the dynamics of the dissipative system can be described by the Lindblad master equation,ρ̇ = -i [H, ρ] + κ^ cav_ eff D [ a] ρ. However, a master equation is the average over infinitely many measurements and ignores the outcome of individual measurements. A clearer picture of the measurement back-action is obtained from a quantum trajectory analysis which keeps track of the measurement outcome. Because this measurement is based on photo-detection, a trajectory consists of a (pure) state conditioned on the presence or absence of a photon detection event random in time. The corresponding unraveling of the master equation (<ref>) is obtained in a standard fashion <cit.> by introducing the measurement operatorsM_0= 1- [ iH + κ_eff/2 a^†a ]dtandM_1= √(κ_eff dt ) a. If no photon is detected in a given time step the conditional state evolves according to:|ψ (t + dt)⟩ = 1/√(⟨ψ| M^†_0 M_0 |ψ⟩) M_0|ψ (t)⟩.If a photon jump occurs the state evolves according to the jump dynamics|ψ (t + dt)⟩ =1/√(⟨ψ| M^†_1 M_1 |ψ⟩) M_1|ψ (t)⟩.For parity detection, the initial state is of the form|ψ_0⟩=|α⟩P_o|ψ⟩_N+|0⟩P_e|ψ⟩_N(see Eq. (<ref>)), where|α⟩is a coherent state andP_e=(+P_N)/2(P_o=(- P_N)/2) is the projector onto the even (odd) parity subspace. For compactness we introduce the following notation for anNqubit basis state|σ_1,σ_2,…,σ_N⟩=|(-1)^n_1,(-1)^n_2,…,(-1)^n_N⟩≡|n⟩, wherenis the integer with binary representationn_1n_2…n_N. With this notation, the multi-qubit state is|ψ⟩_N=∑_n=0^2^N-1c_n|n⟩with∑_n|c_n|^2=1and the parity defined in Eq. (<ref>) corresponds to the Hamming weight of the binary representation of the numbern. The state right before (-) and right after (+) a detection event taking place at timet_Jcan be written explicitly as|ψ⟩_-= 1/𝒩_-∑_n=0^2^N-1c_n P_ o|n⟩| e^-(i ω_c+ i Δ_n + κ_ eff/2)t_J α⟩ + P_ e|ψ⟩_N|0⟩, |ψ⟩_+=1/𝒩_+∑_n c_n e^ -i (ω_c +Δ_n) t_JP_ o|n⟩⊗| e^-(iω_c+i Δ_n+κ_ eff/2)t_J α⟩ .HereΔ_n=∑_i_i(-1)^n_idenotes the total dispersive shift of theN-qubit basis state|n⟩,𝒩_+= √(_N^⟨ψ|P_o|ψ|_⟩N)and𝒩_-= √( e^- |α|^2 (1- e^- κ_eff t) _N^⟨ψ|P_o|ψ|_⟩N + _N^⟨ψ|P_e|ψ|_⟩N). The back-action is now clear. Following a photon loss event, the state undergoes a phase kickwhich depends on the associated multi-qubit state, i.e. each component of the multi-qubit state acquires adifferent phase. In addition the amplitude of the cavity state is exponentially suppressed at the rateκ_eff^cav. Crucially, because the phase kicks are random, the dephasing they induce between the multi-qubit componentswithin a given parity subspace results, after averaging, in intra parity-subspace decoherence. The simple physical picture is that an emitted photon carries more information about the multi-qubit state than only the parity bit which is encoded in the presence or absence of a photon. This additional information, which is encoded in the phase of the emitted photon, isin principle accessible and hence its presence must reduce quantum coherence in the same way as for example which-path information suppresses the ability of a quantum particle to interfere with itself in a double-slit experiment.To confirm this simple interpretation of the dominant source of intra parity-subspace decoherence, we compare, in Fig. <ref>, the analytic predictions with a numerically exact Monte Carlo quantum trajectory simulation of the full system including the CBJJ dynamics. The system is initialized in the pure state1 / √(2)(|gg⟩+|ee⟩) ⊗|α⟩ ⊗|1⟩. At the random jump timet_Ja jump occurs (vertical dashed line) at which the qubit receives a kick.The 2-qubit state is initially polarized inX-direction (⟨ σ_1^x ⊗σ_s^x|=⟩ 1).At the revival, where we can neglect the cavity dynamics the expectation values for the 2-qubit state after the jump are⟨ σ_1^x ⊗σ_2^x|=⟩ cos(2 t_J)and⟨σ_1^y ⊗σ_2^y|=⟩ sin(2 t_J). We will refer to this values as X- and Y-Kick. This agrees with the numerical solution obtained in Fig. <ref> at the revival times (marked with dots). We emphasize that in contrast to the master equation result of Fig. <ref>, in the trajectory picture of Fig. <ref>, the revival height is not damped since the state remains pure along the trajectory.Having understood the dominant source of back-action in photo-detection based parity measurement, we next turn to the question of how to suppress it. One option would be to use the acquired phase information in a coherent feedback loop to combat decoherence of the multi-qubit state as shown in <cit.>. This should also work in the case where homodyne detection is used instead of photo-detection via the CBJJ <cit.>. The phase information gathered in this way could then be used in a coherent feedback loop to combat decoherence of the multi-qubit state. However, homodyne detection in the weak measurement limit, would suffer even more from the entangling dynamics due to the dispersive interaction that is always on. Previous work <cit.> addressed a similar problem by utilizing squeezing to “hide unwanted information” in the enhanced noise of an anti-squeezed quadrature. Alternatively, the unwanted entanglement dynamics in the readout phase of the measurement could be suppressed by using a high-Q tunable resonator, which after the encoding phase is strongly detuned from the qubits. While progress has recently been achieved with the fabrication of tunable high-Q microwave resonators <cit.>, further improvements are necessary to make this approach viable. Here we discuss a simpler and more direct alternative that works for fixed frequency resonators and uses dynamical decoupling to minimize the back-action of the CBJJ detector.§ BACK-ACTION EVASION VIA DYNAMICAL DECOUPLING On the one hand the dispersive interaction of the qubits with the cavity is crucial for the entanglement of the parity state with the cavity state during the encoding stage of the measurement. On the other hand it is not desirable during the readout stage, because it causes the qubit state dependent detuningΔ_nand therefore the random phase kicks, which induce decoherence.Typically, in a high-coherence architecture the dispersive coupling is not tunable and cannot simply be turned off after the encoding stage. If high-fidelity single-qubit rotations are available, as is the case in state-of-the-art superconducting circuits architectures, we can however effectively cancel the effect of the dispersive interaction on the system dynamics by periodically flipping all the qubits on a time scale shorter than the time scale of the entanglement dynamics∼π/|Δ_n|.This can be achieved by repeatedly applying the pattern U X U U X U|ψ⟩on the state, whereU = exp( -i H τ)is the unitary time evolution operator,X = ⊗_i=1^N σ_i^xis theN-qubit flip operator and2 τis the time between two consecutive flips (except the first flip of a measurement, which is applied afterτ).After each flip, the direction of phase rotation of the cavity state, caused by the dispersive term of the Hamiltonian, is reversed. Figure <ref> illustrates the phase dynamics of a multi-qubit state e.g. the state1 / √(2) (|gg⟩ + |ee⟩) ⊗|α⟩in the rotating frame of the bare cavity frequencyω_c<cit.>. During the timeτthe cavity state entangles with the substates|gg⟩(|ee⟩) and gains a phaseϕ(- ϕ) according to the time evolution throughU. It evolves therefore from position A to B (A to C). At this point we flip the qubits by applying the operatorX, so that during the next unitary time evolutionUthe cavity state rotates back to its initial position A. Since the qubits are still flipped the cavity will continue to rotate in the same direction during the next time stepτand the cavity state gains a phase of- ϕ(ϕ) and evolves from A to C (A to B). At this position we apply againXand let it once more evolve according toU. This pattern then will be repeated until a photon jump occurs.This technique of dynamical decoupling <cit.> can be applied on any piecewise constant HamiltonianH(t), which is in our case the Hamiltonian of Eq. (<ref>) repeatedly interrupted by an instantaneous spin flip. If we use the anti-commutation relation{ σ_i^x , σ_i^z } = 0 we find that the sequenceUXUUXUsimplifies toexp( -4 iω_c a^†a τ), therefore the system will evolve according to the average HamiltonianH̅ = ω_c a^†a. However this result is only exact, if we can neglect dissipation. If a photon jump occurs, the assumption of piecewise constant Hamiltonian does not hold anymore and errors will be introduced. Furthermore, for simplicity the qubit flips are here assumed to be instantaneous but more complex sequences of pulses can be designed to account for finite flip durations <cit.>.In Figure <ref> we show that the measurement fidelity can be high if the phase between the initial cavity state|α_init⟩and the cavity state at the photon jump|α_meas⟩is small (Δτ≪2 π). We compare the fidelity of the initial qubit state with the state after the photon jump for differentΔat a fixed flip time intervalτ(dotted line) and for the case where we do not apply dynamical decoupling for differentΔat a fixed measurement timet_M(solid line). Each data point is averaged over8000trajectories. The black dashed line represents a single trajectory at differentΔat a fixed time interval of qubit flipsτand illustrates the random character of photodetection forΔτ≈2 π. In this limit of fast cavity rotations the dynamical decoupling breaks down, if the cavity is far rotated from its initial direction when the random jump happens. These numerical results provide an upper bound for the achievable fidelities of about98%. In Table <ref> we estimate achievable fidelities compatible with state-of-the-art superconducting circuit architectures and the corresponding qubit flip timesτ: The less phase the cavity gains during a flip, the higher is the fidelity. For a total dispersive shift ofΔ=5 MHzfidelities above90%are reached for switching times on the order of10 ns.§ DETECTOR BIAS SUPPRESSION AND HERALDED PARITY DETECTION The multi-qubit parity measurement via direct photon detection has a bias towards one of the parities. Due to finite measurement timest_Mthe parity associated with the vacuum cannot be inferred with the same confidence as the parity associated with the bright cavity state. If we do not detect a photon, there is always a non-zero probability that the cavity is bright and the measurement time was too short to detect a photon decay. If we also include detector efficienciesη< 1the measurement bias towards the parity associated with the bright cavity gets even stronger. In order to suppress this bias we apply a sequence of displacement operations to swap the encoding (even↔bright, odd↔dark) with (even↔dark, odd↔bright) hence "symmetrizing" the roles of the two parities. Preferably this displacement should be applied only if a qubit flipping sequenceU X U U X Uis finished, therefore at integer multiples of4 τ. In this case we know that the cavity amplitude in the rotating frame of the bare cavity frequencyω_cis simply|α(t)|^2 = |α|^2 exp( - κ_eff^cav t ). This procedure will lead to the possibility of a heralded parity detection: If a photon is detected we know that the qubits are in the parity state that is associated with the bright cavity state according to the cavity encoding at the time of detection. If we do not detect a photon during the measurement timet_Mwe have to ignore the result, reset and repeat the measurement. Figure <ref> shows numerical results averaged over 20000 successful measurement runs for different displacement periodst / t_M. The probability to not measure a photon (Missed Detections) if the cavity initially is in the vacuum state (solid line) decreases for faster cavity displacements. Fort / t_M = 1, if we do not displace the cavity at all, the probability to not detect a photon is100 %because the cavity is dark. Also the probability to miss a photon if the cavity is initially in a bright cavity state increases (dashed line). This stems from the fact that an initial bright cavity does not stay bright for the entire measurement durationt_Mbut rather switches between the vacuum and|α(t)⟩effectively decreasing the time where one can measure a photon tot_M / 2. Therefore, for increasing displacement frequencies the measurement bias is suppressed at the cost of an increasing number of failed measurements where no photon was detected. The occurrence of the latter events on the other hand can be reduced by a longer measurement timet_M. § CONCLUSION In conclusion we have characterized an eigenstate preserving multi-qubit parity measurement scheme based on direct microwave photo-detection via a current biased Josephson junction. By dynamically decoupling the dispersive term of the Hamiltonian during readout qubit decoherence is suppressed. Furthermore, by periodically swapping the encoding of the parity onto bright and dark states the measurement bias can be reduced. The detection of a photon then heralds a successful parity measurement. We estimated numerically that high fidelities can be obtained with switching rates on the order of100 MHzfor dispersive couplings of the order of5 MHz. Finally, we note that although we focused here on a simple microwave photon detector, the CBJJ, the presented parity measurement also works with more sophisticated detectors such as <cit.>. Acknowledgments:This work was financially supported by the Swiss National Science Foundation and the Spanish MINECO (FOQUS FIS2013- 46768, QIBEQI FIS2016-80773-P and Severo Ochoa Grant No. SEV-2015-0522), Fundación Cellex, and Generalitat de Catalunya (Grant No. SGR 874 and 875, and CERCA Programme).P.H also acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 665884. We thank Christoph Bruder for useful discussions and the anonymous referee for suggesting to us the idea of using a tunable high-Q resonator to mitigate the impact of measurement back-action. The numerical computations were performed in a parallel computing environment at sciCORE (<http://scicore.unibas.ch/>) scientific computing core facility at University of Basel using the Python library QuTip (<http://qutip.org/>). § LAPLACE TRANSFORMATION AND DERIVATION OF Ρ_00^(N) To derive the effective decay rate caused by the cavity-CBJJ interaction we describe the system with the Hamiltonian Eq. (<ref>) and solve the master equation (<ref>). We can rewrite the Hamiltonian in the reduced set of basis sates | n+1 , 1 ⟩ , | n,2 ⟩ , | n,0 ⟩ asH_n = [ (n+1) (ω_c + Δ) √(n+1)g_J 0; √(n+1)g_J (n+1) ω_c + n Δ 0; 0 0 - ω_0; ]The master equation (<ref>) then yields the differential equations (<ref> - <ref>). To simplify the calculation we split ρ_12 into its imaginary and real partsρ̇_12^R =1/2 (ρ_12 + ρ_21) = -κ_J/2ρ_12^R + Δρ_12^Iρ̇_12^I =1/2 i (ρ_12- ρ_21) = - g_J √(n+1)(ρ_22-ρ_11)-κ_J/2ρ_12^I - Δρ_12^R.We next apply a Laplace transformation, with ρ_11 (0) = 1 and findρ_12^R = Δ/s + κ/2ρ_12^I ρ_22 = 2 Ω_n/s + κρ_12^I = 2 Ω_n/s + κρ_12^Iρ_11 = 1/s - 2 Ω_n/s ρ_12^Iρ_12^I = Ω_n/s( Δ^2/s + κ/2 +2 Ω_n^2 (1/s+1/s+κ))Here we used the shorthand notation Ω_n=g_J √(n+1). We are interested in ρ_00 which can be obtained by integrating ρ_22 (See Eq. (<ref>)). To find ρ_22 we substitute ρ_12^I into Eq. (<ref>). The inverse Laplace transform is obtained from the integral:ρ_kl(t) = 1/2 π i∫_γ - i ∞^γ + i ∞ ds ρ_kl(s) e^st,which can easily be solved by summing over the residua of the integrand. The poles of ρ_12^I are: s_0 = 1/2( + √(+ √((16 Ω_n^2+κ ^2+4 Δ ^2)^2-64 Ω_n^2 κ ^2 )-16 Ω_n^2+κ ^2-4 Δ ^2)/√(2)-κ)s_1 = 1/2( - √(- √((16 Ω_n^2+κ ^2+4 Δ ^2)^2-64 Ω_n^2 κ ^2 )-16 Ω_n^2+κ ^2-4 Δ ^2)/√(2)-κ)s_2 = 1/2( + √(- √((16 Ω_n^2+κ ^2+4 Δ ^2)^2-64 Ω_n^2 κ ^2 )-16 Ω_n^2+κ ^2-4 Δ ^2)/√(2)-κ)s_3 = 1/2( - √(+ √((16 Ω_n^2+κ ^2+4 Δ ^2)^2-64 Ω_n^2 κ ^2 )-16 Ω_n^2+κ ^2-4 Δ ^2)/√(2)-κ).The Laplace transfom of ρ_22 then takes the formρ_22(s)=2 Ω_n(s+ κ/2 ) (s+κ)/(s-s_0)(s-s_1)(s-s_2)(s-s_2),and the inverse transform is ρ_22 (t) = ∑_s_i Res(ρ_22(s) e^st, s_i). Finally the occupation of the continuum state is obtained from ρ_00(t)=κ_J ∫_0^t ρ_22(τ) d τ. By inspection of the residua we see that Res(ρ_22, s_3) is the dominant contribution to ρ_22 which simplifies the expression for ρ_00 toρ_00 = 1-exp[ t/4( √(-32 Ω_n ^2+ 2 √(256 Ω_n ^4+32 Ω_n ^2 (4 Δ ^2-κ_J ^2)+(κ_J ^2+4 Δ ^2)^2)+2 κ_J ^2- 8 Δ ^2)-2 κ_J ) ] ≈ 1- exp(- 4 Ω_n^2/κ t ( 1 - 4 Δ^2/κ_J^2) ),where we made use of the limits Δ≪Ω_n ≪κ_J. § HIGHER-ORDER DECOHERENCE In Section <ref> we derived the detuning dependence of the effective decay rate, with a fixed detuning Δ. Because the detuning Δ = ∑_i_iσ_i depends on the multi-qubit state this means that the effective decay rate can be different for different multi-qubit state components evenwithin a given parity subspace. As a consequence, in addition to random phase kicks that correspond to amplitude preserving random rotations of the multi-qubit state around the logical Z axis, these higher-order terms will lead to random rotations out of the logical XY plane. Because the detuning dependence is quadratic and for two qubits Δ_ even=-Δ_ odd we must consider at least three qubits to observe this higher-order effect. In Fig. <ref> we numerically solve for a quantum trajectory of the full system with the Hamiltonian from Eq. (<ref>) for the odd three-qubit state |ψ⟩ = 1 / √(2)(|egg⟩ + |eee⟩) coupled to a cavity with α = 3 and the CBJJ initially in the state |1⟩. We define the logical Z-operator σ^z̃= |egg⟩⟨egg|- |eee⟩⟨eee|. Its expectation displays a clear jump when a photon loss event occurs. Upon averaging over many trajectories this results in an additional contribution to the measurement induced decoherence rate. It remains an open question how to extend the dynamical decoupling scheme to compensate also for such higher-order effects.apsrev | http://arxiv.org/abs/1704.08734v3 | {
"authors": [
"Patrick Huembeli",
"Simon E. Nigg"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170427202329",
"title": "Towards a heralded eigenstate preserving measurement of multi-qubit parity in circuit QED"
} |
Anosov representations and convex cocompact actions]Projective Anosov representations, convex cocompact actions, and rigidity Department of Mathematics, University of Chicago, Chicago, IL 60637. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706 [email protected] [2010]In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any projective Anosov representation of that group acts convex cocompactly on some properly convex domain in real projective space. We also show that if a projective Anosov representation preserves a properly convex domain, then it acts convex cocompactly on some (possibly different) properly convex domain. We then give three applications. First, we show that Anosov representations into general semisimple Lie groups can be defined in terms of the existence of a convex cocompact action on a properly convex domain in some real projective space (which depends on the semisimple Lie group and parabolic subgroup). Next, we prove a rigidity result involving the Hilbert entropy of a projective Anosov representation. Finally, we prove a rigidity result which shows that the image of the boundary map associated to a projective Anosov representation is rarely a C^2 submanifold of projective space. This finalrigidity result also applies to Hitchin representations.[ Andrew Zimmer December 30, 2023 =====================§ INTRODUCTION If G is a connected simple Lie group with trivial center and K ≤ G is a maximal compact subgroup, then X=G/K has a unique (up to scaling) Riemannian symmetric metric g such that G = _0(X,g). The metric g is non-positively curved and X is simply connected, hence every two points in X are joined by a unique geodesic segment. A subset ⊂ X is called convex if for every x,y ∈ the geodesic joining them is also in .Finally, a discrete group Γ≤ G is said to be convex cocompact if there exists a non-empty closed convex set ⊂ X such that γ() = for all γ∈Γ and the quotient Γ\ is compact.In the case in which G has real rank one, there are an abundance of examples of convex cocompact subgroups and one has the following characterization: Suppose G is a real rank one simple Lie group with trivial center, (X,g) is the symmetric space associated to G, and Γ≤ G is a discrete subgroup. Then the following are equivalent:* Γ≤ G is a convex cocompact subgroup,* Γ is finitely generated and for some (hence any) x ∈ X the map γ∈Γ→γ· x induces a quasi-isometric embedding of Γ into X,* Γ is word hyperbolic and there exists an injective, continuous, Γ-equivariant map ξ: ∂Γ→ X(∞). For a proof of this theorem see Theorem 5.15 in <cit.> which relies on results in <cit.>. When G has higher rank, the situation is much more rigid: Suppose G isa simple Lie group with real rank at least two and Γ≤ G is a Zariski dense discrete subgroup. If Γ is convex cocompact, then Γ is a cocompact lattice in G.Although the most natural definition of convex cocompact subgroups leads to no interesting examples in higher rank, it turns out that the third characterization in Theorem <ref> can be used to define a rich class of representations called Anosov representations. This class of representations was originally introduced by Labourie <cit.> and then extended by Guichard-Wienhard <cit.>.Since then several other characterizations have been given by Kapovich-Leeb-Porti <cit.>, Kapovich-Leeb <cit.>, Guéritaud-Guichard-Kassel-Wienhard <cit.>, and Bochi-Potrie-Sambarino <cit.>.We refer the reader to <cit.> for a precise definition of Anosov representations, but informally: if Γ is word hyperbolic, G is a semisimple Lie group, and P is a parabolic subgroup, thena representation ρ: Γ→ G is called P-Anosov if there exists an injective, continuous, ρ-equivariant map ξ: ∂Γ→ G/P satisfying certain dynamical properties. In the case in which G has real rank one, every two parabolic subgroups are conjugate and the quotient G/P can naturally be identified with X(∞). Recently, Danciger, Guéritaud, and Kasselestablished a close connection between Anosov representations into (p,q) and convex cocompact actions. However, the convex cocompact action is not on the associated symmetric space X=(p,q)/ P((p) ×(q)), but on a properly convex domain in the projective model of the pseudo-Riemannian hyperbolic space ^p,q-1. In this context convex cocompactness can be defined as follows:<cit.>Let^p,q-1 = { [x]∈(^p+q) : x,x_p,q < 0}where ·, ·_p,q is the standard bilinear form of signature (p,q). Then an irreducible discrete subgroup Λ≤(p,q) is called ^p,q-1-convex cocompact if there exists a non-empty properly convex subsetof (^p+q) such that*is a closed subset of ^p,q-1, * Λ acts properly discontinuously and cocompactly on ,*has non-empty interior, and* \ contains no projective line segments.Danciger, Guéritaud, and Kassel then proved the following theorem.For p,q ∈^* with p+q ≥ 3, let Λ be an irreducible discrete subgroup of (p,q) and let P^p,q_1 ≤(p,q) be the stabilizer of an isotropic line in (^p+q, ·, ·_p,q).* If Λ is ^p,q-1-convex cocompact, then it is word hyperbolic and the inclusion representation Λ↪(p,q) is P^p,q_1-Anosov.* Conversely, if Λ is word hyperbolic, ∂Λ is connected, and Λ↪(p,q) is P^p,q_1-Anosov, then Λ is either ^p,q-1-convex cocompact or ^q,p-1-convex cocompact (after identifying (p,q) with (q,p)).The special case when q=2 and Λ is the fundamental group of a closed hyperbolic p-manifold follows from work of Mess <cit.> for p = 2 andwork of Barbot-Mérigot <cit.> for p ≥3.In this paper we further explore connections between Anosov representations and convex cocompact actions on domains in real projective space.In the general case, we make the following definition. Suppose V is a finite dimensional real vector space, Ω⊂(V) is a properly convex domain, and Λ≤(Ω) is a discrete subgroup. Then Λ is a convex cocompact subgroup of (Ω) if there exists a non-empty closed convex subset ⊂Ω such that g() = for all g ∈Λ and the quotient Λ\ is compact. In the context of Anosov representations a more refined notion of convex cocompactness seems to be necessary: there exist properly convex domains Ω⊂(V) with convex cocompact subgroups Λ≤(Ω) which are not word hyperbolic. To avoid such examples, we make the following stronger definition. Suppose V is a finite dimensional real vector space, Ω⊂(V) is a properly convex domain, and Λ≤(Ω) is a discrete subgroup. Then Λ is a regular convex cocompact subgroup of (Ω) if there exists a non-empty closed convex subset ⊂Ω such that g() = for all g ∈Λ, the quotient Λ\ is compact, and every point in ∩∂Ω is a C^1 extreme point of Ω.* When Λ is an irreducible subgroup of (V), we show that every point in ∩∂Ω is a C^1 point of Ω if and only if every point in ∩∂Ω is a extreme point of Ω (see Theorem <ref> below).* It turns out that ^p,q-1-convex cocompact subgroups always satisfy this stronger condition. In particular, by Proposition 1.14 in <cit.>: If Γ≤(p,q) is irreducible, discrete, and ^p,q-1-convex cocompact, then there exists a properly convex domain Ω⊂(^p+q) such that Λ is a regular convex cocompact subgroup of (Ω). Finally we are ready to state our first main result.(see Section <ref>) Suppose G is a semisimple Lie group with finite center and P ≤ G is a parabolic subgroup. Then there exists a finite dimensional real vector space V and an irreducible representation ϕ:G →(V) with the following property: if Γ is a word hyperbolic group and ρ:Γ→ G is a Zariski dense representation with finite kernel, then the following are equivalent: * ρ is P-Anosov,* there exists a properly convex domain Ω⊂(V) such that (ϕ∘ρ)(Γ) is a regular convex cocompact subgroup of (Ω). Properly convex domains and their projective automorphism groups have been extensively studied, especially in the case in which there exists a discrete group Γ≤(Ω) such that Γ\Ω is compact. Such domains are called convex divisible domains and have a number of remarkable properties, see the survey papers by Benoist <cit.>, Marquis <cit.>, and Quint <cit.>.Theorem <ref> provides a way to use the rich theory of convex divisible domains to study general Anosov representations. For instance, the proofs of Theorem <ref> and Theorem <ref> below are inspired byrigidity results for convex divisible domains.§.§ Projective Anosov representations The first step in the proof of Theorem <ref> is to use a result of Guichard and Wienhard to reduce to the case of projective Anosov representations. A projective Anosov representation is simply an P-Anosov representation in the special case when G=_d() and P ≤_d() is the stabilizer of a line. This special class of Anosov representations can be defined as follows. Suppose that Γ is a word hyperbolic group, ∂Γ is the Gromov boundary of Γ, and ρ: Γ→_d() is a representation.Two maps ξ: ∂Γ→(^d) and η: ∂Γ→(^d*) are called: * ρ-equivariant if ξ∘γ= ρ(γ) ∘ξ and η∘γ = ρ(γ)∘η for allγ∈Γ,* dynamics-preserving if for every γ∈Γ of infinite order with attracting fixed point x_γ^+ ∈∂Γ the points ξ(x^+_γ) ∈(^d) and η(x^+_γ) ∈(^d*) are attracting fixed points of the action of ρ(γ) on (^d) and (^d*), and* transverse if for every distinct pair x, y ∈∂Γ we have ξ(x) + η(y) = ^d. Given an element g ∈_d() let λ_1(g) ≥…≥λ_d(g)denote the absolute values of the eigenvalues (counted with multiplicity) of some (any) lift g̃∈_d() of g with g̃=± 1.Suppose that Γ is word hyperbolic, S is a finite symmetric generating set, and d_S is the associated word metric on Γ. Then for γ∈Γ, let ℓ_S(γ) denote the minimal translation distance of γ acting on the Cayley graph of (Γ,S), that isℓ_S(γ) = inf_x ∈Γ d_S(γ x, x).A representation ρ: Γ→_d() is then called a projective Anosov representation if there exist continuous, ρ-equivariant, dynamics preserving, and transverse maps ξ: ∂Γ→(^d), η: ∂Γ→(^d*) and constants C,c>0 such thatlogλ_1(ρ(γ))/λ_2(ρ(γ))≥ C ℓ_S(γ) -cfor all γ∈Γ. This is not the initial definition of Anosov representations given by Labourie <cit.> or Guichard-Wienhard <cit.>, but a nontrivial characterization proved in <cit.>. We use this characterization as our definition because it is more elementary to state than the original definition, but it is not necessary for any of the proofs in the paper. We should also note that this is far from the simplest definition of Anosov representations, for instance if one replaces the estimate in Equation (<ref>) with a similar estimate on singular values, then it follows from work of Kapovich, Leeb, and Porti <cit.> that one does not need to assume that the maps η,ξ exist or even that Γ is a word hyperbolic group (only finite generation is required). But since many of the results that follow involve these boundary maps and eigenvalues, it seems like this definition is the most natural in the context of this paper.Guichard and Wienhard proved the following connection between general Anosov representations and projective Anosov representations.Suppose G is a semisimple Lie group with finite center and P ≤ G is a parabolic subgroup. Then there exist a finite dimensional real vector space V_0 and an irreducible representation ϕ_0:G →(V_0) with the following property: if Γ is a word hyperbolic group and ρ:Γ→ G is a representation, then the following are equivalent: * ρ is P-Anosov,* ϕ_0 ∘ρ is projective Anosov. Proofs of this theorem can also be found in <cit.> and <cit.>. Using Theorem <ref>, the proof of Theorem <ref> essentially reduces to the case of projective Anosov representations. In this case, we consider the following two questions.* Given a properly convex domain Ω⊂(V) and a convex cocompact subgroup Λ≤(Ω), what geometric conditions on Ω imply that the inclusion representation Λ↪(V) is a projective Anosov representation?* Given a projective Anosov representation ρ: Γ→(V) what conditions on ρ or Γ imply that ρ(Γ) acts convex cocompactly on a properly convex domain in (V)? Projective Anosov representations are closely related to the representations studied by Danciger, Guéritaud, and Kassel <cit.> in the (p,q) case. In particular, if ρ: Γ→(p,q) is a representation of a word hyperbolic group, then (by definition) the following are equivalent: * ρ isP^p,q_1-Anosov where P^p,q_1 ≤(p,q) be the stabilizer of an isotropic line in (^p+q, ·, ·_p,q), * ρ is projective Anosov when viewed as a representation into _p+q(). Thus Theorem <ref> provides answers to the above questions for projective Anosov representations whose images preserve a non-degenerate bilinear form.§.§ When a convex cocompact action leads to a projective Anosov representation When Ω⊂(^d) and Λ≤(Ω) is a discrete group which acts cocompactly on Ω, Benoist has provided geometric conditions on Ω so that the inclusion representation Λ↪_d() is projective Anosov. Suppose Ω⊂(^d) is a properly convex domain and Λ≤(Ω) is a discrete group which acts cocompactly on Ω. Then the following are equivalent: * Λ is word hyperbolic, * ∂Ω is a C^1 hypersurface, * Ω is strictly convex.Moreover, when these conditions are satisfied the inclusion representation Λ↪_d() is projective Anosov.There exist examples of properly convex domains Ω⊂(^d) with discrete subgroups Λ≤(Ω) where Λ acts co-compactly on Ω and Λ is not word hyperbolic, see <cit.> and <cit.>.The case of convex cocompact actions is more complicated as the next example shows. Let C = { (x_1, x_2, y) ∈^3 : y> √(x_1^2 + x_2^2)}.Then C is a properly convex cone and the group _0(1,2) preserves C. Let Λ_0 ≤_0(1,2) be a cocompact lattice. Next consider the properly convex domainΩ = { [(v_1, v_2)] ∈(^6) : v_1 ∈ C, v_2 ∈ C}and the discrete group Λ = {[ φ 0; 0 φ ]∈_6() : φ∈Λ_0 }.Let _0 = { [(v,v)] ∈(^d) : v ∈ C} and forr > 0 let _r = { p ∈Ω :d_Ω(p,_0) ≤ r }⊂Ω.Then each _r is convex (see <cit.> or <cit.>) and the quotient Λ\_r is compact. This example has the following properties: * Λ is word hyperbolic (since Λ_0 is word hyperbolic),* the inclusion representation Λ↪_6() is not projective Anosov, * Λ≤(Ω) is a convex cocompact subgroup,* when r>0there exist line segments in ∂Ω∩_r, and* every point in ∂Ω∩_r is not a C^1 point of ∂Ω. Despite examples like these, we will prove the following analogue of Benoist's theorem for convex cocompact subgroups (see Section <ref>) Suppose Ω⊂(^d) is a properly convex domain and Λ≤(Ω) is a discrete irreducible subgroup of _d(). If Λ preserves and acts cocompactly on a closed convex subset ⊂Ω, then the following are equivalent: * every point in ∩∂Ω is a C^1 point of ∂Ω, * every point in ∩∂Ω is an extreme point of Ω.Moreover, when these conditions are satisfied Λ is word hyperbolic and the inclusion representation Λ↪_d() is projective Anosov. * Theorem <ref> can be seen as a generalization of Theorem <ref> part (1) to the case when the representation is not assumed to preserve a non-degenerate bilinear form (see Remarks <ref> and <ref>). * This result was established independently by Danciger, Guéritaud, and Kassel, see Theorems 1.4 and 1.15 in <cit.> and Subsection <ref> below.§.§ When a projective Anosov representation acts convex cocompactly In general a projective Anosov representation will not preserve a properly convex domain: Consider a cocompact lattice Λ≤_2() and consider the representation ρ: _2() →_3() given by ρ(g) = [ g; 1 ].Then the representation ρ|_Λ: Λ→_3() is projective Anosov and the image of the boundary map is :={ [ x_1 : x_2 : 0 ] ∈(^3) : x_1, x_2 ∈, (x_1, x_2) ≠ 0}.From this, it is easy to see that ρ(Λ) cannot preserve a properly convex domain Ω because then we would have ⊂∂Ω.The above example is simple to construct, but is not an irreducible representation. To obtain an example of an irreducible projective Anosov representation which does not preserve a properly convex domain, one can consider Hitchin representations of surface groups in _2d(), see Proposition 1.7 in <cit.>.With some mild conditions on Γ we can prove that every projective Anosov representation of Γ acts convex cocompactly on a properly convex domain. (see Section <ref>) Suppose Γ is a non-elementary word hyperbolic group which is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface. If ρ: Γ→_d() is an irreducible projective Anosov representation, then there exists a properly convex domain Ω⊂(^d) such that ρ(Γ) is a regular convex cocompact subgroup of (Ω).Work of Stallings implies that Γ is not commensurable to a non-trivial free product if and only if ∂Γ is connected <cit.>. So Theorem <ref> can be seen as an analogue of Theorem <ref> part (2) in the case when the representation is not assumed to preserve a non-degenerate bilinear form.We can also prove that once the image acts on some properly convex domain, then it acts convex cocompactly on some (possibly different) properly convex domain:(see Section <ref>) Suppose Γ is a word hyperbolic group. If ρ: Γ→_d() is an irreducible projective Anosov representation and ρ(Γ) preserves a properly convex domain in (^d), then there exists a properly convex domain Ω⊂(^d) such that ρ(Γ) is a regular convex cocompact subgroup of (Ω).This result was established independently by Danciger, Guéritaud, and Kassel, see Theorems 1.4 and 1.15 in <cit.> and Subsection <ref> below.Using Theorem <ref>, we can construct a convex cocompact action for any projective Anosov representation by post composing with another representation.Let _d() be the vector space of symmetric d-by-d real matrices and consider the representation S: _d() →( _d())given by S(g) X = gXtgThen := { [X] ∈( _d() ): X > 0}is a properly convex domain in (_d()) and S(_d()) ≤().Combining Theorem <ref> with the above examples establishes the following corollary.(see Section <ref>)Suppose Γ is a word hyperbolic group and ρ: Γ→_d() is an irreducible projective Anosov representation. Let V = _{ξ(x)tξ(x): x ∈∂Γ}⊂_d(). Then there exists a properly convex domain Ω⊂(V) such that (S∘ρ)(Γ) is a regular convex cocompact subgroup of (Ω).In the context of Theorem <ref>, it is also worth mentioning a theorem of Benoist which gives a necessary and sufficient condition for a subgroup of _d() to preserve a properly convex cone. Before stating Benoist theorem we need some terminology. An element g ∈_d() is called proximal if it has a unique eigenvalue of maximal absolute value and a proximal element g ∈_d() is called positively proximal if its unique eigenvalue of maximal absolute value is positive. Then a subgroup G ≤_d() is called positively proximal if G contains a proximal element and every proximal element in G is positively proximal. With this language, Benoist proved the following theorem.If G ≤_d() is an irreducible subgroup, then the following are equivalent:* G is positively proximal* G preserves a properly convex cone ⊂^d. As an application, we will apply Theorem <ref> and Benoist's theorem to Hitchin representations in certain dimensions. Suppose that Γ≤_2() is a torsion-free cocompact lattice and ι: Γ↪_2() is the inclusion representation. For d > 2, let τ_d : _2() →_d() be the unique (up to conjugation) irreducible representation. Then the connected component of τ_d ∘ι in (Γ, _d()), denoted _d(Γ), is called the Hitchin component of Γ in _d().Labourie <cit.> proved that every representation in _d(Γ) is projective Anosov (it is actually B-Anosov where B ≤_d() is a minimal parabolic subgroup).Suppose that Γ≤_2() is a torsion-free cocompact lattice and ρ: Γ→_d() is in the Hitchin component. If d is odd, then there exists a properly convex domain Ω⊂(^d) such that ρ(Γ) is a regular convex cocompact subgroup of (Ω).This result was also established independently by Danciger, Guéritaud, and Kassel, see Proposition 1.7 in <cit.> and Subsection <ref> below. In the case where d is even, Danciger, Guéritaud, and Kassel showed that ρ(Γ) cannot even preserve a properly convex domain in (^d).Since the proof is short we include it here. If we identify ^d with the vector space of homogenous polynomials P: ^d → of degree d-1, then the representation τ_d : _2() →_d() is given by τ_d(g) · P = P ∘ g^-1 Since d is odd, _d() = _d() and if g ∈_2() has eigenvalues with absolute values λ, λ^-1 then τ_d(g) has eigenvalues λ^d-1, λ^d-3, …, λ^-(d-1).Hence each eigenvalue of τ_d(g) is positive. Now fix some ρ∈_d(Γ). Since _d(Γ) is connected, we see that ρ(Γ) is positively proximal. So by Benoist's theorem ρ(Γ) preserves a properly convex cone ⊂^d. So by Theorem <ref>,there exists a properly convex domain Ω⊂(^d) so that ρ(Γ) is a regular convex cocompact subgroup of (Ω).§.§ Other applications: §.§.§ Entropy rigidity: Suppose that Γ is a group and let [Γ] be the conjugacy classes of Γ. Given a representation ρ: Γ→_d() define the Hilbert entropy to be H_ρ= lim sup_r →∞1/rlog#{ [γ] ∈ [Γ] : 1/2log(λ_1(ρ(γ))/λ_d(ρ(γ)))≤ r}. We will prove the following upper bound on entropy.(see Section <ref>) Suppose Γ is a word hyperbolic group and ρ:Γ→_d() is an irreducible projective Anosov representation. If ρ(Γ) preserves a properly convex domain in (^d), then H_ρ≤ d-2with equality if and only if ρ(Γ) is conjugate to a cocompact lattice in (1,d-1). Theorem <ref> shows that Theorem <ref> applies to many Anosov representations.Theorem <ref> is a generalization of a theorem of Crampon. Suppose Ω⊂(^d) is a properly convex domain and Λ≤(Ω) is a discrete word hyperbolic group which acts cocompactly on Ω. If ι: Λ↪_d() is the inclusion representation, then H_ι≤ d-2with equality if and only if Λ is conjugate to a cocompact latticein (1,d-1). In the context of Theorem <ref>, Theorem <ref> implies that ι is a projective Anosov representation and so Theorem <ref> is a true generalization of Theorem <ref>. Recently, Theorem <ref> was also generalized in a different direction in <cit.>. Theorem <ref> also improves, in some cases, bounds due to Sambarino. Suppose Γ is a convex cocompact group of a (-1) space X and let ρ:Γ→_d() be an irreducible projective Anosov representation with d ≥ 3. Then α H_ρ≤δ_Γ(X)where the boundary map ξ: ∂Γ→(^d) is α-Hölder and δ_Γ(X) is the Poincaré exponent of Γ acting on X.In Theorem <ref>, ξ is Hölder with respect to a visual metric of X restricted to the limit set of Γ and a distance on (^d) induced by a Riemannian metric. Sambarino also proves a rigidity result in the case when α H_ρ = δ_Γ(X) and X is real hyperbolic k-space, for details see Corollary 3.1 in <cit.>.If Γ satisfies the hypothesis of Theorem <ref> andα < δ_Γ(X)/d-2,then Theorem <ref> can be used to provide a better upper bound on entropy§.§.§ Regularity rigidityIn this subsection we describe some rigidity results related to the regularity of the limit curve of a projective Anosov representation. We should note that if the boundary of a word hyperbolic group is a topological manifold, then it actually must be a sphere (see for instance <cit.>).For certain types of projective Anosov representations, the image of the boundary map is actually a C^1 submanifold. Suppose Ω⊂(^d) is a properly convex domain and Λ≤(Ω) is a discrete group which acts cocompactly on Ω. If Λ is word hyperbolic, then Theorem <ref> implies that the inclusion representation Λ↪_d() is projective Anosov. The image of the associated boundary map is ∂Ω which is a C^1 submanifold of (^d) by Theorem <ref>.Suppose that Γ≤_2() is a torsion-free cocompact lattice and ρ : Γ→_d() is in the Hitchin component. If ξ: ∂Γ→(^d) is the boundary map associated to ρ, then ξ(∂Γ) is a C^1 submanifold of (^d). This follows from the fact that ξ is a hyperconvex Frenet curve, see <cit.>. In both of theses cases it is known that the image of the boundary map cannot be too regular unless the representation is very special. Suppose Ω⊂(^d) is a properly convex domain and Λ≤(Ω) is a discrete group which acts cocompactly on Ω. If ∂Ω is a C^1,α hypersurface for every α∈ (0,1), then Ω is projectively isomorphic to the ball and hence Λ is conjugate to a cocompact lattice in (1,d-1). Suppose that Γ≤_2() is a torsion-free cocompact latticeand ρ : Γ→_d() is in the Hitchin component. If ξ: ∂Γ→(^d) is the associated boundary map and ξ(∂Γ) is a C^∞ submanifold of (^d), then there exists a representation ρ_0 : Γ→_2() such that ρ is conjugate to τ_d ∘ρ_0.Using Theorem <ref>, we will prove the following.(see Section <ref>) Suppose d > 2, Γ is a word hyperbolic group, and ρ:Γ→_d() is an irreducible projective Anosov representation with boundary map ξ: ∂Γ→(^d). If* M = ξ(∂Γ) is a C^2 k-dimensional submanifold of (^d) and* the representation ∧^k+1ρ : Γ→(∧^k+1^d) isirreducible,thenλ_1(ρ(γ))/λ_2(ρ(γ)) =λ_k+1(ρ(γ))/λ_k+2(ρ(γ))for all γ∈Γ. * Notice that the regularity assumption concerns the set ξ(∂Γ) and not the mapξ: ∂Γ→(^d). * As before, λ_1(g)≥…≥λ_d(g) denote the absolute values of the eigenvalues (counted with multiplicity) of some (any) lift g̃∈_d() of g with g̃=± 1. * Theorem <ref> is only needed in the case when k >1. When ρ: Γ→_d() has Zariski dense image, then ρ and ∧^k+1ρ areirreducible. Moreover in this case the main result in <cit.> implies that there exists some γ∈Γ such that λ_1(ρ(γ))/λ_2(ρ(γ))≠λ_k+1(ρ(γ))/λ_k+2(ρ(γ)).So we have the following corollary of Theorem <ref>.Suppose d > 2, Γ is a word hyperbolic group, and ρ:Γ→_d() is a Zariski dense projective Anosov representation with boundary map ξ: ∂Γ→(^d). Then ξ(∂Γ) is not a C^2 submanifold of (^d).The proof of Theorem <ref> can also be used to prove the following rigidity result for Hitchin representations. (see Section <ref>) Suppose that Γ≤_2() is a torsion-free cocompact latticeandρ : Γ→_d() is in the Hitchin component. If ξ: ∂Γ→(^d) is the associated boundary map and ξ(∂Γ) is a C^2 submanifold of (^d), then λ_1(ρ(γ))/λ_2(ρ(γ)) = λ_2(ρ(γ))/λ_3(ρ(γ))for all γ∈Γ. This corollary greatly restricts the Zariski closure of ρ(Γ) when ρ is Hitchin and ξ(∂Γ) is a C^2 submanifold (see <cit.> again). In particular, the corollary implies that in this case: * ρ(Γ) cannot be Zariski dense, * if d = 2n>2, then the Zariski closure of ρ(Γ) cannot be conjugate to (2n,), * if d = 2n+1 > 3 then the Zariski closure of ρ(Γ) cannot be conjugate to (n,n+1), and* if d=7, then the Zariski closure of ρ(Γ) cannot be conjugate to the standard realization of G_2 in _7().See Section <ref> in the appendix for details.Guichard has announced that these are the only possibilities for the Zariski closure of ρ(Γ) when ρ is Hitchin but not Fuchsian (that is conjugate to a representation of the form τ_d ∘ρ_0), see for instance <cit.>.§.§ Convex cocompactness in the work of Danciger, Guéritaud, and KasselAfter I finished writing this paper, Danciger, Guéritaud, and Kassel informed me of their preprint <cit.> which has some overlapping results with this paper. They consider a class of subgroups of _d() which they call strongly convex cocompact which (using the terminology of this paper) are discrete subgroups Γ≤_d() which act convex cocompactly on a properly convex domain which is strictly convex and has C^1 boundary. This notion appears to be first studied in work of Crampon and Marquis <cit.>. Danciger, Guéritaud, and Kassel also show (stated with different terminology) that if Λ≤(Ω) is a regular convex cocompact subgroup (as in Definition <ref>), then it is actually a strongly convex cocompact subgroup of _d(), that is there exists a possibly different properly convex domain Ω^' where Λ≤(Ω^') is a convex cocompact subgroup and Ω^' is a strictly convex domain with C^1 boundary (see Theorem 1.15 in <cit.>). Danciger, Guéritaud, and Kassel also study a notion of convex cocompact actions on general properly convex domains (see Definition 1.11 in <cit.>) that is different than the one we consider in Definition <ref> above.The main overlap in the two papers is in Theorems <ref>, <ref>, and Corollary <ref> above and Theorems 1.4, 1.15 and Proposition 1.7 in <cit.>.§.§ AcknowledgementsI would like to thank Thomas Barthelmé and Ludovic Marquis for many helpful conversations. In particular, we jointly observed the fact that an argument due to G. Liu could be used to prove Proposition <ref> during the course of writing our joint paper Entropy rigidity of Hilbert and Riemannian metrics <cit.>.I would also like to thank the referees for their careful reading of this paper and their many helpful comments and corrections. This material is based upon work supported by the National Science Foundation under grants DMS-1400919 and DMS-1760233.§ PRELIMINARIES In this section we recall some facts that we will use in the arguments that follow. §.§ Some notations * If M ⊂(^d) is a C^1 k-dimensional submanifold of (^d) and m ∈ M we will let T_m M ⊂(^d) be the k-dimensional projective subspace of (^d) which is tangent to M at m. * If V ⊂^d is a linear subspace, we will let (V) ⊂(^d) denote its projectivization. In most other cases, we will use [o] to denote the projective equivalence class of an object o, for instance:* if v ∈^d∖{0}, then [v] denotes the image of v in (^d), * if ϕ∈_d(), then [ϕ] denotes the image of ϕ in _d(), and * if T ∈(^d) ∖{0}, then [T] denotes the image of T in ((^d)).* A line segment in (^d) is a connected subset of a projective line. Given two points x,y ∈(^d) there is no canonical line segment with endpoints x and y, but we will use the following convention: if Ω is a properly convex domain and x,y ∈Ω, then (when the context is clear) we will let [x,y] denote the closed line segment joining x to y which is contained in Ω. In this case, we will also let (x,y)=[x,y]∖{x,y}, [x,y)=[x,y]∖{y}, and (x,y]=[x,y]∖{x}.§.§ Gromov hyperbolicity Suppose (X,d) is a metric space. If I ⊂ is an interval, a curve σ: I → X is a geodesic if d(σ(t_1),σ(t_2)) = t_1-t_2for all t_1, t_2 ∈ I.A geodesic triangle in a metric space is a choice of three points in X and geodesic segmentsconnecting these points. A geodesic triangle is said to be δ-thin if any point on any of the sides of the triangle is within distance δ of the other two sides.A proper geodesic metric space (X,d) is called δ-hyperbolic if every geodesic triangle is δ-thin. If (X,d) is δ-hyperbolic for some δ≥0 then (X,d) is called Gromov hyperbolic. We will use the following (probably well known) characterization of Gromov hyperbolicity. Suppose (X,d) is a proper geodesic metric space, δ > 0, and there exists a map (x,y) ∈ X × X →σ_x,y∈ C([0,d(x,y)],X)where σ_x,y is a geodesic segment joining x to y. If for every x,y,z ∈ X distinct, the geodesic triangle formed by σ_x,y, σ_y,z, σ_z,x is δ-thin, then (X,d) is Gromov hyperbolic.We begin the proof with a definition and a lemma. Define the Gromov product of x,y ∈ X with respect to o∈ X to be (x | y)_o := 1/2( d(x,o)+d(o,y) - d(x,y) ).Suppose (X,d) is a metric space, x,y,o ∈ X, and σ:[0,T] → X is a geodesic with σ(0)=x and σ(T)=y. Then (x|y)_o ≤ d(o,σ) : = inf{ d(o,σ(t)) : t ∈ [0,T] }. For t ∈ [0,T],d(x,y) = d(x, σ(t)) + d(σ(t), y)and so the triangle inequality implies that: 2(x|y)_o = d(x,o)+d(o,y)-d(x,y)≤ 2d(o, σ(t)).We start by proving the following claim: Claim: If x,y,o ∈ X and t ≤ (x|y)_o - δ, thend(σ_ox(t), σ_oy(t)) ≤ 2 δ. It is enough to consider the case when t < (x|y)_o -δ. In this case d(σ_ox(t), σ_xy) ≥ d(o, σ_xy)-d(σ_ox(t), o) ≥ (x|y)_o - t > δ.So by the thin triangle condition, there exists s such that d(σ_ox(t), σ_oy(s)) ≤δ. Then δ≥ d(σ_ox(t), σ_oy(s)) ≥d(σ_ox(t), o) - d(o,σ_oy(s)) = t-s.So d(σ_ox(t), σ_oy(t)) ≤ d(σ_ox(t), σ_oy(s))+d(σ_oy(s), σ_oy(t)) ≤ 2 δand the claim is established. By Proposition 1.22 in Chapter III.H in <cit.>, (X,d) is Gromov hyperbolic if and only if there exists some δ_0 >0 such that(x|y)_o ≥min{ (x|z)_o, (y|z)_o} - δ_0 for all o,x,y,z ∈ X. Fix o,x,y,z ∈ X. We claim that(x|y)_o ≥min{ (x|z)_o, (y|z)_o} - 3δ.Let m =min{ (x|z)_o, (y|z)_o}. Since (x|y)_o ≥ 0, the inequality is trivial when m ≤δ. So we can assume m > δ. Then the triangle inequality implies thatmin{ d(x,o), d(y,o), d(z,o)}≥ m > δ.Then let x^' = σ_ox(m-δ), y^' = σ_oy(m-δ), and z^' = σ_oz(m-δ). Then by the claimd(x^', y^') ≤ d(x^', z^')+d(z^', y^') ≤ 4 δ.Then 2(x|y)_o = d(x,o)+d(o,y)-d(x,y) = d(x,x^')+d(x^',o)+d(o,y^')+d(y^',y)-d(x,y)≥ d(x^',o)+d(o,y^') - d(x^', y^')≥ m-δ + m-δ - 4δ = 2m - 6δ.So (x|y)_o ≥min{ (x|z)_o, (y|z)_o} - 3δ.By combining several deep theorems from geometric group theory we can deduce the following. Suppose Γ is a non-elementary word hyperbolic group which does not split over a finite group and is not commensurable to the fundamental group of a closed hyperbolic surface. Then* ∂Γ is connected, * ∂Γ∖{x} is connected for every x ∈∂Γ, and* there exist u,w ∈∂Γ distinct such that ∂Γ∖{u,v} is connected.The argument below comes from the proof of Theorem 3.1 in <cit.>.By work of Stallings, ∂Γ is disconnected if and only if Γ splits over a finite group <cit.>. So ∂Γ must be connected. Then a theorem of Swarup <cit.> implies that ∂Γ∖{x} is connected for every x ∈∂Γ.Now suppose for a contradiction that ∂Γ∖{u,v} is disconnected for every u,v ∈∂Γ distinct. Then ∂Γ is homeomorphic to the circle by <cit.>. But then by work of Gabai <cit.> and Tukia <cit.>, Γ is commensurable to the fundamental group of a closed hyperbolic surface.§.§ Properly convex domainsIn this subsection we review some basic definitions involving convexity in real projective space. * A set Ω⊂(^d) is called a domain if Ω is open and connected* A set Ω⊂(^d) is called convex ifL ∩Ω is connected and L ∩Ω≠ L for every projective line L ⊂(^d). * A convex domain Ω⊂(^d) is called a properly convex domain if L ∩Ω≠ L for every projective line L ⊂(^d).When Ω⊂(^d) is a properly convex domain, there exists an affine chart 𝔸⊂(^d) which contains Ω as a bounded convex domain (see for instance <cit.>).Given a properly convex domain Ω⊂(^d), a hyperplane H ⊂(^d) is a supporting hyperplane of Ω at x ∈∂Ω if x ∈ H and H ∩Ω = ∅.One of the most important properties of properly convex domains is that every boundary point is contained in at least one supporting hyperplane (which follows from the supporting hyperplane characterization of convexity in Euclidean space).Suppose that Ω⊂(^d) is a properly convex domain. Then * a point x ∈∂Ω is a C^1 point of Ω if x is contained in a unique supporting hyperplane of Ω. In this case, we let T_x ∂Ω denote this unique supporting hyperplane. * a point x ∈∂Ω is an extreme point of Ω if there does not exist a line segment (p,q) in ∂Ω with x ∈ (p,q). It is straightforward to show that x ∈∂Ω is a C^1 point of Ω (in the sense above) if and only if ∂Ω is locally the graph of a function which is differentiable at x. Moreover, in this case if x_n ∈∂Ω is a sequence converging to x and H_n is a supporting hyperplane at x_n, then lim_n →∞ H_n=T_x ∂Ω. Given a properly convex domain Ω⊂(^d) the dual set is defined to be:Ω^* = { f ∈(^d*): f(x) ≠ 0for allx ∈Ω}.The set Ω^* is a properly convex domain in (^d*) and the two sets have the following relation. If f ∈∂Ω^*, then ( f) is a supporting hyperplane of Ω. §.§ The Hilbert metric For distinct points x,y ∈(^d) let xy be the projective line containing them. Suppose Ω⊂(^d) is a properly convex domain. If x,y ∈Ω are distinct let a,b be the two points in xy∩∂Ω ordered a, x, y, b along xy. Then define the Hilbert distance between x and y to bed_Ω(x,y) = 1/2log [a, x,y, b] where[a,x,y,b] = x-by-a/x-ay-b is the cross ratio. Using the invariance of the cross ratio under projective maps and the convexity of Ω it is possible to establish the following (see for instance <cit.>).Suppose Ω⊂(^d) is a properly convex domain. Then d_Ω is a complete (Ω)-invariant metric on Ω which generates the standard topology on Ω. Moreover, if p,q ∈Ω, then there exists a geodesic joining p and q whose image is the line segment [p,q]. We will use the following observation (which follows immediately from the definition of d_Ω). [see Lemma 3 in <cit.>] Suppose Ω⊂(^d) is a properly convex domain and p_n, q_n ∈Ω are sequences. If p_n → p ∈Ω and d_Ω(p_n, q_n) → 0, then q_n → p.We will also use the following estimate.Suppose Ω⊂(^d) is a properly convex domain and [a,b], [c,d] ⊂Ω are line segments. If p ∈ [a,b], thend_Ω(p, [c,d]) ≤ d_Ω(a,c) + d_Ω(b,d).We will also consider the Gromov product induced by the Hilbert metric: given a properly convex domain Ω⊂(^d) define the Gromov product of p,q ∈Ω based at o ∈Ω to be (p|q)_o^Ω = 1/2( d_Ω(p,o)+d_Ω(o,q)-d_Ω(p,q) ). Karlsson and Noskov established the following estimates.<cit.> Suppose Ω⊂(^d) is a properly convex domain, o∈Ω, p_n ∈Ω is a sequence with p_n → p ∈∂Ω, and q_m ∈Ω is a sequence with q_m → q ∈∂Ω. * If p= q, then lim_n,m →∞ (p_n|q_m)_o^Ω = ∞. * If lim sup_n,m →∞ (p_n|q_m)_o^Ω = ∞, then [p,q] ⊂∂Ω. Since the proof is short we include it.Both parts are consequences of the fact that every line segment in Ω can be parametrized to be a geodesic. First suppose that p_n, q_m ∈Ω both converge to some p ∈∂Ω. Let σ_n : _≥ 0→Ω and σ_m:_≥ 0→Ω be the geodesic rays in (Ω, d_Ω) whose images are line segments with σ_n(0)=σ_m(0)=o and σ_n(T_n) = p_n, σ_m(S_m) = q_m for some T_n, S_m ∈_≥0. Then if t ≤min{T_n, S_m} we have2(p_n|q_m)_o^Ω = 2t + d_Ω(p_n,σ_n(t))+d_Ω(σ_m(t), q_m) - d_Ω(p_n,q_m) ≥ 2t-d_Ω(σ_n(t), σ_m(t)).Since lim sup_n,m →∞ d_Ω(σ_n(t), σ_m(t)) = 0for any fixed t > 0, we then see that lim_n,m →∞ (p_n|q_m)_o^Ω = ∞. Next suppose that p_n → p ∈∂Ω, q_m → q ∈∂Ω, and lim sup_n,m →∞ (p_n|q_m)_o^Ω = ∞.By passing to subsequences we can suppose that lim sup_n →∞ (p_n|q_n)_o^Ω = ∞.Next let σ_n : [0,R_n] →Ω be a geodesic joining p_n to q_n whose image is a line segment.Then for any t ∈ [0,R_n] we have2(p_n|q_n)_o^Ω =d_Ω(p_n,o)+d_Ω(o,q_n)-d_Ω(p_n,σ_n(t)) - d_Ω(σ_n(t),q_n) ≤ 2d_Ω(o, σ_n(t))and so(p_n|q_n)_o^Ω≤inf_t ∈ [0,R_n] d_Ω(o,σ_n(t)).Since the image of σ_n is the line segment [p_n,q_n] we then must have [p,q] ⊂∂Ω. §.§ A fact about Anosov representationsIn this subsection we describe the behavior of sequences of elements in a projective Anosov representation.When a matrix is proximal, its iterates have the following behavior.Suppose g ∈_d() is proximal. Viewing _d() as a subset of ((^d)), the limitT = lim_n →∞ g^nexists in ((^d)). Moreover, the image of T is the eigenline of g corresponding to the eigenvalue with maximal modulus. By changing coordinates we can assume that g = [ λ 0; 0 A ]where [1:0:…:0] is the eigenline of g corresponding to the eigenvalue with maximal modulus and A is a (d-1)-by-(d-1) matrix. Theng^n = [10;0 1/λ^nA^n ]and the observation immediately follows from Gelfand's formula (see Theorem <ref>).Notice that if g ∈_d() is proximal, then the representation m ∈→ g^m is projective Anosov. A well known analogue of the above observation holds forgeneral projective Anosov representations. Suppose that Γ is a word hyperbolic group. Let ρ: Γ→_d() be an irreducible projective Anosov representation with boundary maps ξ: ∂Γ→(^d) and η : ∂Γ→(^d*). Assume γ_n ∈Γ is a sequence such that γ_n → x^+ ∈∂Γ and γ_n^-1→ x^- ∈∂Γ. Then viewing _d() as a subset of ((^d)), T=lim_n →∞ρ(γ_n)where (T) = ξ(x^+) and T = η(x^-). In particular, ξ(x^+) = lim_n →∞ρ(γ_n)vfor all v ∈(^d) ∖(η(x^-)) and the convergence is uniform on compact subsets of (^d) ∖(η(x^-)).Since the proof is short we include it.We first consider the case in which #∂Γ = 2. Then since ρ is irreducible and ρ preserves ξ(∂Γ) we see that d=2. Then the lemma follows easily from the dynamics of 2-by-2 matrices acting on (^2).So suppose that #∂Γ >2. Then #∂Γ = ∞ and ∂Γ is a perfect space. Since ( (^d)) is compact it is enough to show that every convergent subsequence of ρ(γ_n) converges to T. So suppose that ρ(γ_n) → S in ( (^d)). We first claim that (S) = ξ(x^+). Since ρ: Γ→_d() is irreducible, there exists x_1, …, x_d∈∂Γ such thatξ(x_1), …, ξ(x_d) spans ^d. Since ∂Γ is a perfect space, we can perturb the x_i (if necessary) and assume thatx^- ∉{ x_1, …, x_d}.Then γ_n x_i → x^+ and since ξ is ρ-equivariant, we then see that ρ(γ_n)ξ(x_i) →ξ(x^+). Since ξ(x_1), …, ξ(x_d) spans ^d this implies that (S) = ξ(x^+). Next view ^tρ(γ_n) as an element of ((^d*)). Then ^tρ(γ_n) converges to ^tS in ((^d*)). Since ^tρ(γ_n) η(x) = η( γ_n^-1 x),repeating the argument above shows that (^tS) = η(x^-).But this implies that S = η(x^-).This lemma has the following corollary. Suppose that Γ is a word hyperbolic group. Let ρ: Γ→_d() be an irreducible projective Anosov representation with boundary maps ξ: ∂Γ→(^d) and η : ∂Γ→(^d*). If ρ(Γ) preserves a properly convex domain Ω⊂(^d), then ξ(∂Γ) ⊂∂Ω and η(∂Γ) ⊂∂Ω^*. Fix some x ∈∂Γ. Then there exists γ_n ∈Γ such that γ_n → x. Now suppose that γ_n^-1→ x^-. Since Ω is open, there exists some v ∈Ω∖(η(x^-)). Then ξ(x) = lim_n →∞ρ(γ_n) v ∈Ω.On the other hand, ρ has finite kernel (by definition) and discrete image by Theorem 5.3 in <cit.>. Further, since (Ω) preserves the Hilbert metric on Ω, (Ω) acts properly on Ω. So we must have ξ(x) ∈∂Ω. Since x ∈∂Γ was an arbitrary point we then have ξ(∂Γ) ⊂∂Ω. Repeating the same argument on Ω^* shows that η(∂Γ) ⊂∂Ω^*. § CONSTRUCTING A CONVEX COCOMPACT ACTION In this section we establish Theorems <ref> and <ref> from the introduction. The argument has two parts: first we show that we can lift the boundary maps ξ, η to maps into ^d, ^d* and then we will show that whenever we can lift ξ,η we obtain a regular convex cocompact action.§.§ Lifting the maps Before stating the theorem we need some notation: fix a norm · on ^d, this induces a norm on ^d* by f = max{f(v) : v=1}.Then let S^d-1⊂^d and S^(d-1)*⊂^d* be the unit spheres relative to these norms. In the statement and proof of the next theorem we will use the standard action of _d() on S^d-1 and S^(d-1)* given by g · v = gv/gv andg · f =f ∘g^-1/f ∘g^-1.Finally let_d^±() = { g ∈_d() :g = ± 1 }.Suppose Γ is a word hyperbolic group. Let ρ: Γ→_d() be an irreducible projective Anosov representation with boundary maps ξ: ∂Γ→(^d) and η:∂Γ→(^d*). If one of the following conditions hold: * there exists a properly convex domain Ω_0 ⊂(^d) such that ρ(Γ) ≤(Ω_0) or* Γ is a non-elementary word hyperbolic group which is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then there exist lifts ρ: Γ→^±_d(), ξ: ∂Γ→ S^d-1, η: ∂Γ→ S^(d-1)* of ρ, ξ, η respectively such that ξ and η are continuous, ρ-equivariant, and η(y) ( ξ(x) ) > 0for all x,y ∈∂Γ distinct. We will consider each case separately.Case 1: Suppose that there exists a properly convex domain Ω_0 ⊂(^d) such that ρ(Γ) ≤(Ω_0).Let π: ^d∖{0}→(^d) be the natural projection. Since Ω_0 is properly convex, π^-1(Ω_0) has two connected components _1 and _2. Moreover, _1 and _2 are properly convex cones and _1 = -_2. By Corollary <ref>, we see that ξ(∂Γ) ⊂∂Ω_0 and η(∂Γ) ⊂∂Ω_0^*. Now for x ∈∂Γ let ξ(x) ∈ S^d-1 be the unique representative of ξ(x) such that ξ(x) ∈_1 and let η(x) ∈ S^(d-1)* be the unique representative of η(x) such that η(x)(v) > 0for all v ∈_1. Then by construction, η(x) (ξ(y) ) ≥ 0with equality if and only if x=y. Moreover, uniqueness implies that ξ and η are continuous. Now for γ∈Γ let ρ(γ) ∈_d^±() be the unique lift that preserves _1. Then ρ : Γ→_d^±() is a homomorphism and ξ and η are ρ-equivariant. Case 2: Suppose that Γ is a non-elementary word hyperbolic group which is not commensurable to a non-trivial free product or a fundamental group of a closed hyperbolic surface. We will reduce to Case 1 by constructing a properly convex domain Ω_0 ⊂(^d) such that ρ(Γ) ≤(Ω_0).Let Λ = ρ(Γ). Then by Selberg's lemma Λ has a torsion-free finite index subgroup Λ_0. Moreover, Λ_0 is commensurable to Γ and ∂Λ_0 is homeomorphic to ∂Γ. Since Λ_0 is torsion-free, the condition on Γ implies that Λ_0 does not split over a finite group and is not commensurable to the fundamental group of a closed hyperbolic surface. Hence by Theorem <ref>, we see that* ∂Γ is connected, * ∂Γ∖{x} is connected for every x ∈∂Γ, and* there exist u,w ∈∂Γ distinct such that ∂Γ∖{u,w} is connected. The space (^d) ∖ (( η(u)) ∪(η(w) )) has two connected components which we denote by A^+ and A^-. Since ξ(∂Γ∖{u,w}) is connected, by relabelling we can assume that ξ(∂Γ∖{u,w}) ⊂ A^+. Then ξ(∂Γ) ⊂A^+. Next define C : = ∩_γ∈Γρ(γ)A^+. By construction C is closed, ξ(∂Γ) ⊂ C, and ρ(γ) C = C for every γ∈Γ. Let C_0 denote the connected component of C which contains ξ(∂Γ) and let Ω_0 denote the interior of C_0. We claim that Ω_0 is a properly convex domain and ρ(Γ) ≤(Ω_0). By construction, ρ(γ) Ω_0 = Ω_0 for every γ∈Γ and so it is enough to show that Ω_0 is a properly convex domain. To accomplish this we recall the following terminology: a subset E ⊊(^d) is called linearly convex if for every x ∈(^d) ∖ E there exists a hyperplane H ⊂(^d) such that H ∩Ω = ∅. We also recall the following basic properties of these sets:*every convex set is linearly convex, *every connected component of a linearly convex set is convex, *the intersection of a collection of linearly convex sets is linearly convex, and *if E ⊂(^d) is linearly convex and g ∈_d(), then gE is linearly convex. Proofs of Properties <ref> and <ref> can be found in <cit.>. Properties <ref> and <ref> are direct consequences of the definition. Since A^+ is projectively equivalent to { [1:x_1:…:x_d-1] : x_1 > 0}, we see that A^+ is linearly convex. Thus by Properties <ref>, <ref>, and <ref>, C_0 is convex. Since ρ is irreducible, {ξ(x) : x ∈∂Γ} spans ^d. Since ξ(∂Γ) ⊂ C_0, this implies that C_0 has non-empty interior. So Ω_0 is a non-empty convex domain. Since Ω_0 ⊂ A^+, we see that Ω_0 ∩(η(u)) = ∅. Since Γ· u ⊂∂Γ is dense, η is continuous, and ρ(γ) Ω_0 = Ω_0 for every γ∈Γ, we then haveΩ_0 ∩( η(x)) = ∅for all x ∈∂Γ. Since ρ is irreducible, η(∂Γ) spans ^d* and so Ω_0 must be properly convex.It is easy to construct examples of “half spaces” E_1, E_2 ⊂(^d) such that E_1 ∩ E_2 is disconnected (and hence not convex). For instance, let E_1 be the connected component of (^d) ∖ ( {x_1 = 0}∪{x_2=0}) which contains [1:1:0:…:0]. And let E_2 be the connected component of(^d)∖( { x_1-x_2=0}∪{ 2x_1 - x_2 =0})which contains [1:3:0:…:0]. Then E_1 ∩ E_2 = { [1:x_2 : … :x_d] : x_2 ∈ (0,1) ∪ (2,∞) }.Examples like these are why we consider linearly convex sets in the proof of Theorem <ref>.§.§ Showing the action is convex cocompact Suppose Γ is a word hyperbolic group. Let ρ: Γ→_d() be an irreducible projective Anosov representation with boundary maps ξ: ∂Γ→(^d) and η:∂Γ→(^d*). If there exist lifts ρ: Γ→^±_d(), ξ: ∂Γ→ S^d-1, η: ∂Γ→ S^(d-1)* of ρ, ξ, η respectively such that ξ and η are continuous, ρ-equivariant, and η(y) ( ξ(x) ) > 0for all x,y ∈∂Γ distinct, then there exists a properly convex domain Ω⊂(^d) such that ρ(Γ) is a regular convex cocompact subgroup of (Ω).For the rest of this subsection let Γ, ρ, ξ, η, ρ, ξ, and η satisfy the hypothesis of Theorem <ref>. DefineΩ := { [v] ∈(^d) : η(x)(v) > 0for allx ∈∂Γ}.With the notation above, Ω is a properly convex domain, ρ(Γ) ≤(Ω), and if N > 1; λ_1,…, λ_N > 0;andx_1, …, x_N ∈∂Γ are distinct, then[ ∑_i=1^N λ_i ξ(x_i) ] ∈Ω. If N > 1; λ_1,…, λ_N > 0;andx_1, …, x_N ∈∂Γ are distinct, thenη(y) ( ∑_i=1^N λ_i ξ(x_i) )>0for all y ∈∂Γ. So [ ∑_i=1^N λ_i ξ(x_i) ] ∈Ω.In particular, Ω is non-empty.We now show that Ω is open. Suppose p_0 ∈Ω. Then there exists v_0 ∈^d such that p_0=[v_0] and η(x)(v_0) > 0 for all x ∈∂Γ. Since ∂Γ is compact and η:∂Γ→ S^(d-1)* is continuous, we have0< r:=inf_x ∈∂Γη(x)(v_0).So { [v] ∈(^d) : v-v_0 < r}⊂Ω.HenceΩ is open. By construction Ω is a convex domain andΩ∩(η(x)) = ∅for all x ∈∂Γ. Since ρ is irreducible, η(∂Γ) spans ^d* and so Ω must be properly convex. Finally, since ρ(γ)[v] = [ρ(γ)(v)]when v ∈^d and γ∈Γ, we see that ρ(Γ) ≤(Ω). With the notation above, ξ(∂Γ) ⊂∂Ω and η(∂Γ) ⊂∂Ω^*.This follows immediately from Corollary <ref>, but here is a direct proof: by the definition of Ω we see that η(∂Γ) ⊂Ω^*. Moreover, if x,y ∈∂Γ are distinct, then ξ(x) = lim_λ→∞[ λξ(x) + ξ(y) ] ∈Ω.So ξ(∂Γ) ⊂Ω. Then, since η(x)(ξ(x)) = 0for allx ∈∂Γ we see that ξ(∂Γ) ⊂∂Ω and η(∂Γ) ⊂∂Ω^*.Next letbe the closed convex hull of ξ(∂Γ) in Ω.With the notation above, ρ(Γ) acts cocompactly on . Proposition <ref> follows from either a recent result of Kapovich and Leeb <cit.> or a recent result of Kapovich, Leeb, and Porti <cit.>. In particular, the action of ρ(Γ) on ξ(∂Γ) is a uniform convergence action and so ρ(Γ) acts cocompactly onby Theorem 1.9 in <cit.>. Alternatively, one can useto construct an invariant set in the space of flags of the form (line, hyperplane) and then apply Theorem 1.5 in <cit.> to see that ρ(Γ) acts cocompactly on . We will provide a proof of Proposition <ref> that only uses elementary properties of convex sets. This direct argument requires a few preliminary lemmas.Given a set A ⊂Ω and a point p ∈Ω defined_Ω(p,A) := inf_a ∈ A d_Ω(p,a).Then given two sets A,B ⊂Ω define the Hausdorff distance in d_Ω between A and B to be:d_Ω^(A,B) := max{sup_a ∈ A d_Ω(a,B), sup_b ∈ B d_Ω(b,A) }. Next fix a finite, symmetric generating set S of Γ and let d_S be the induced word metric on Γ.With the notation above, suppose that p_0 ∈Ω. Then there exists some R >0 with the following property: if g_1, …, g_N ∈Γ is a geodesic in (Γ, d_S), then d_Ω^( {ρ(g_1)p_0, …, ρ(g_N)p_0}, [ρ(g_1) p_0, ρ(g_N) p_0] ) ≤ R. We first claim that there exists some R_1 > 0 with the following property: if g_1, …, g_N ∈Γ is a geodesic in (Γ, d_S), then max_1 ≤ i ≤ N d_Ω( ρ(g_i) p_0, [ρ(g_1) p_0, ρ(g_N) p_0] ) ≤ R_1.Suppose not, then after possibly translating by elements in Γ we can assume: for any n > 0 there exists a geodesic g_-M_n^(n), g_-M_n+1^(n), …, g_N_n^(n)in (Γ,d_S) such that g_0^(n)=𝕀 andd_Ω( ρ(g_0^(n))p_0, [ρ(g_-M_n^(n)) p_0, ρ(g_N_n^(n)) p_0] )=d_Ω( p_0, [ρ(g_-M_n^(n)) p_0, ρ(g_N_n^(n)) p_0] ) > n.Notice that we must have M_n, N_n →∞. Now by passing to a subsequence, we can suppose that the limitsg_i = lim_n →∞ g_i^(n)exists for all i. Then …, g_-2, g_-1, g_0=𝕀, g_1, g_2, …is a geodesic in (Γ,d_S). So there exist x^+,x^- ∈∂Γ distinct such that lim_i →±∞ g_i = x^±.By the standard geodesic ray definition of the topology on Γ∪∂Γ, we havelim_n →∞ g_N_n^(n) = x^+andlim_n →∞ g_-M_n^(n) = x^-. Now(η(x^-)) ∩Ω = ∅ and so Lemma <ref> implies thatlim_n →∞ρ(g_N_n^(n)) p_0 = ξ(x^+).The same reasoning implies thatlim_n →∞ρ(g_-M_n^(n)) p_0 = ξ(x^-).Since x^+,x^- ∈∂Γ are distinct, Lemma <ref> implies that (ξ(x^-), ξ(x^+)) ⊂Ω and so d_Ω( p_0, (ξ(x^-), ξ(x^+)) ) < ∞.Then since ∞ = lim_n →∞ d_Ω( p_0, [ρ(g_-M_n^(n)) p_0, ρ(g_N_n^(n)) p_0] ) =d_Ω( p_0, (ξ(x^-), ξ(x^+)) ) < ∞we have a contradiction. Hence, there exists some R_1 > 0 such that: if g_1, …, g_N ∈Γ is a geodesic in (Γ, d_S), then max_1 ≤ i ≤ N d_Ω( ρ(g_i) p_0, [ρ(g_1) p_0, ρ(g_N) p_0] ) ≤ R_1. Now let C = max{ d_Ω(p_0, ρ(g) p_0) : g ∈ S}.We claim that: if g_1, …, g_N ∈Γ is a geodesic in (Γ, d_S) and if p ∈ [ρ(g_1) p_0, ρ(g_N) p_0], then d_Ω( p, {ρ(g_1)p_0, …, ρ(g_N)p_0}) ≤ 2R_1 + C/2.For each 1 ≤ i ≤ N, let p_i be a closest point to ρ(g_i)p_0 in [ρ(g_1) p_0, ρ(g_N) p_0]. Then d_Ω(p_i, p_i+1) ≤ d_Ω(p_i, ρ(g_i)p_0) + d_Ω(ρ(g_i)p_0, ρ(g_i+1)p_0) + d_Ω(ρ(g_i+1)p_0, p_i+1) ≤ R_1 + C+R_1 = 2R_1+C.Since p_1 = ρ(g_1) p_0 and p_N = ρ(g_N) p_0 we see that: for any p ∈ [ρ(g_1) p_0, ρ(g_N) p_0] min_1 ≤ i ≤ N d_Ω(p, p_i) ≤1/2 ( 2R_1+C) = R_1 +C/2and so d_Ω( p, {ρ(g_1)p_0, …, ρ(g_N)p_0}) ≤ 2R_1 + C/2.So R = 2R_1 + C/2 satisfies the conclusion of the lemma.With the notation above, suppose that p_0 ∈Ω. For any N ≥ 2 there exists C_N > 0 such that: ifp = [ ∑_i=1^N λ_i ξ(x_i)]where λ_1, …, λ_N > 0 and x_1, …, x_N ∈∂Γ are distinct, then d_Ω(p, ρ(Γ) · p_0) ≤ C_N. We induct on N. For the remainder of the proof let R be the constant from Lemma <ref>.For the N=2 case suppose that x_1,x_2 ∈∂Γ are distinct. Then there exist sequences g_n, h_n ∈Γ such that g_n → x_1 and h_n → x_2. By Lemma <ref> ρ(g_n)p_0 →ξ(x_1)and ρ(h_n)p_0 →ξ(x_2).So if p =[ λ_1ξ(x_1)+λ_2ξ(x_2)]for some λ_1,λ_2 >0, then there exists a sequence p_n ∈ [ρ(g_n)p_0, ρ(h_n)p_0] such that p_n → p.Lemma <ref> implies that d_Ω(p_n, ρ(Γ) · p_0) ≤ Rand so d_Ω(p, ρ(Γ) · p_0) ≤ R. Next suppose that N > 2 and consider p = [ ∑_i=1^N λ_i ξ(x_i)]where λ_1, …, λ_N > 0 and x_1, …, x_N ∈∂Γ are distinct. We claim that d_Ω(p, ρ(Γ) · p_0) ≤ 2C_N/2+R.Letp_1 = [ ∑_1 ≤ i < N/2λ_i ξ(x_i) + 1/2λ_N/2ξ( x_N/2)]andp_2 = [1/2λ_N/2ξ( x_N/2)+∑_N/2 < i ≤ N λ_i ξ(x_i)].Then, by induction there exist elements g_1, g_2 ∈Γ such that d_Ω(p_i, ρ(g_i) · p_0) ≤ C_N/2.Now p ∈ [p_1, p_2] and so by Lemma <ref> there exists q ∈ [ρ(g_1) · p_0, ρ(g_2) · p_0] such that d_Ω(p,q) ≤ 2C_N/2.Then Lemma <ref> implies thatd_Ω(q, ρ(Γ) · p_0) ≤ Rand henced_Ω(p, ρ(Γ) · p_0) ≤ 2C_N/2+R. By Carathéodory's convex hull theorem any p ∈ can be written as p = [ ∑_i=1^Nλ_i ξ(x_i)]where 2 ≤ N ≤ d+1; λ_1, …, λ_N >0; and x_1, …, x_N ∈∂Γ are distinct. Thus by the previous lemma there exists some M > 0 such that = ∪_g ∈Γρ(g) (B_Ω(p_0;M) ∩)where B_Ω(p_0;M) is the closed metric ball of radius M in (Ω, d_Ω).With the notation above, if f ∈Ω^*, then there exists 1 ≤ N ≤ d+1; λ_1, …, λ_N > 0; and x_1, …, x_N ∈∂Γ distinct so that f = [ ∑_i=1^N λ_i η(x_i) ]. By the definition of Ω, the set Ω^* is the image of ConvexHull{η(x) : x ∈∂Γ}⊂^d*in (^d*). Then since η: ∂Γ→^d* is continuous, Carathéodory's convex hull theorem implies that f can be written as f =[ ∑_i=1^N λ_i η(x_i) ]for some 1 ≤ N ≤ d+1; λ_1, …, λ_N > 0; and x_1, …, x_N ∈∂Γ.With the notation above, ξ(∂Γ) = ∩∂Ω,every point in ∩∂Ω is a C^1 extreme point of Ω, andT_ξ(x)∂Ω =( η(x))for all x ∈∂Γ.Lemma <ref> and the definition ofimply that ξ(∂Γ) = ∩∂Ω. So suppose that x ∈∂Γ. We first show that ξ(x) is a C^1 point of Ω. Suppose that H is a supporting hyperplane of Ω at ξ(x). Then H=( f) for some f ∈Ω^*. By Lemma <ref>f = [ ∑_i=1^N λ_i η(x_i) ]for some 1 ≤ N ≤ d+1; λ_1, …, λ_N > 0; and x_1, …, x_N ∈∂Γ distinct. Since f(ξ(x))=0, we then have 0=∑_i=1^N λ_i η(x_i)(ξ(x)).By hypothesisη(y)(ξ(z)) > 0when y,z ∈∂Γ are distinct and so we must have N=1 and x_1 = x. Thus f = η(x) and H = (η(x)). Since H was an arbitrary supporting hyperplane of Ω at ξ(x) we see that ξ(x) is a C^1 point of ∂Ω and T_ξ(x)∂Ω = (η(x)). We next show that ξ(x) is an extreme point of Ω. This follows immediately from Lemma <ref> below, but we will provide a direct argument. Suppose for a contradiction that ξ(x) is not an extreme point, then there exists p^', q^'∈∂Ω such that ξ(x) ∈ (p^', q^') ⊂∂Ω.Fix a point c_0 ∈ and consider a sequence of points q_n along the line [c_0, ξ(x)) which converge to ξ(x). Since ρ(Γ) acts cocompactly on , there exist some M_1 >0 and elements γ_n ∈Γ such that d_Ω(ρ(γ_n) c_0, q_n) ≤ M_1.Next fix some p ∈ (p^', q^') ⊂∂Ω with p ≠ξ(x). Then, by the definition of the Hilbert metric, we can find a sequence of points p_n along the line [c_0,p) such that M_2:=sup_n ≥ 0 d_Ω(p_n, q_n) < +∞.Next let k_n = ρ(γ_n)^-1 q_n and ℓ_n = ρ(γ_n)^-1 p_n. Then k_n, ℓ_n ∈B_Ω(c_0; M_1+M_2)where B_Ω(c_0; M_1+M_2) is closed metric ball of radius M_1+M_2 in (Ω, d_Ω). Since the Hilbert metric is proper, we can pass to a subsequence such that k_n → k ∈Ω and ℓ_n →ℓ∈Ω. Then lim_n →∞ d_Ω(ρ(γ_n) k, ρ(γ_n) k_n) =lim_n →∞ d_Ω(k, k_n)= 0which implies from the definition of the Hilbert metric, see Observation <ref>, that lim_n →∞ρ(γ_n) k =lim_n →∞ρ(γ_n) k_n =ξ(x).The same reasoning shows thatlim_n →∞ρ(γ_n) ℓ =lim_n →∞ρ(γ_n) ℓ_n =p. Next view _d() as a subset of ((^d)) and pass to a subsequence so that ρ(γ_n) converges to some T in ((^d)). By Lemma <ref>, T has image ξ(x^+) and kernel η(x^-) for some x^+, x^- ∈∂Γ. Since (η(x^-))∩Ω = ∅ we see that ξ(x^+)=T(k) = lim_n →∞ρ(γ_n)k= ξ(x).However, by the same reasoning we haveξ(x^+) = T(ℓ) = lim_n →∞ρ(γ_n)ℓ = p.Hence ξ(x) = p which is a contradiction. Thus ξ(x) is an extreme point of Ω. §.§ Proof of Corollary <ref> For the rest of this subsection suppose that Γ is a word hyperbolic group and ρ: Γ→_d() is an irreducible projective Anosov representation. Let ξ: ∂Γ→(^d) and η : ∂Γ→(^d*) denote the boundary maps associated to ρ. Then define V = _{ξ(x)⊗ξ(x) : x ∈∂Γ}⊂_d()where we make the identification v ⊗ v = v tv∈_d() when v ∈^d. Let ρ_S : Γ→(V) be the representation ρ_S(γ) X = ρ(γ)Xtρ(γ).Using Theorem <ref> it is enough to show that ρ_S is an irreducible projective Anosov representation and there exists a properly convex domain Ω_0 ⊂(V) such that ρ_S(Γ) ≤(Ω_0).There exists a properly convex domain Ω_0 ⊂(V) such that ρ_S(Γ) ≤(Ω_0). As in Example <ref>, let := { [X] ∈( _d() ): X > 0}.Thenis a properly convex domain in (_d()). Since ρ is irreducible, there exists x_1, …, x_d ∈∂Γ such that ξ(x_1),…, ξ(x_d) span ^d. If v_1, …, v_d ∈^d are representatives of ξ(x_1),…, ξ(x_d) respectively, then [∑_i=1^d v_i ⊗ v_i ]∈∩ V.So Ω_0: = ∩(V) is a non-empty properly convex domain in (V) and by construction ρ_S(Γ) ≤(Ω_0).Given γ∈Γ with infinite order, let x^+_γ∈∂Γ be the attracting fixed point of γ. And given a vector space W andg ∈(W) proximal let ℓ_g^+ ∈(W) be the eigenline of g corresponding to the eigenvalue of maximal modulus.If γ∈Γ has infinite order, then g = ρ_S(γ) is proximal and ℓ_g^+ = ξ(x_γ^+)⊗ξ(x_γ^+). If λ_1 > λ_2 ≥…≥λ_d are the absolute values of the eigenvalues of ρ(γ) normalized to have product one, then there exists C > 0 such that some subset ofCλ_i λ_jfor1 ≤ i ≤ j ≤ dare the the absolute values of the eigenvalues of g=ρ_S(γ) normalized to have product one. By construction ξ(x^+_γ) ⊗ξ(x^+_γ) ∈ V and is the eigenline corresponding to C λ_1^2, so g is proximal andℓ_g^+ = ξ(x_γ^+)⊗ξ(x_γ^+). ρ_S is irreducible.Let G be the Zariski closure of ρ(Γ) in _d() and consider the representation τ : G →(V)given by τ(g) X =gXtg. Since ρ is an irreducible representation, G acts irreducibly on ^d. So G acts minimally on the set {ℓ^+_g : g ∈ Gis proximal}⊂(^d),see for instance <cit.>. So τ(G) acts minimally on the set X={ℓ^+_g ⊗ℓ_g^+: g ∈ Gis proximal}⊂(_d()).Since X ∩(V) ≠∅, τ(G) acts minimally on X, and τ(G) · V = V, we see that X ⊂(V). Further, X spans V by the definition of V.Since G is semisimple (see for instance <cit.>), we can decompose V= ⊕_i=1^m W_i where each W_i ≤ V is τ(G)-invariant and the induced representation G →(W_i) is irreducible (see for instance <cit.>). Fix some γ∈Γ with infinite order and let h =ρ(γ). Then τ(h) ≤(V) is proximal by Lemma <ref>. Viewing (V) as a subset of ((V)), Observation <ref> implies that T = lim_n →∞ϕ(h)^nin ((V)) and the image of T is ℓ^+_h⊗ℓ_h^+. By relabeling the W_i, we can suppose that there exists some element w ∈ W_1 ∖ T. Then ℓ^+_h⊗ℓ_h^+ = T([w]) = lim_k →∞ϕ(h)^n_k [w] ⊂ W_1.Then since τ(G) acts minimally on the set X={ℓ^+_g⊗ℓ^+_g : g ∈ Gis proximal}and X spans V, we see that W_1 = V. Hence τ: G →(V) is an irreducible representation. Since ρ(Γ) is Zariski dense in G and ρ_S = τ∘ρ, we then see that ρ_S is also irreducible. ρ_S is projective Anosov.We define boundary maps ξ_S : ∂Γ→(V) and η_S : ∂Γ→(V^*) as follows. First, letξ_S(x) = ξ(x)⊗ξ(x).Next, let f ∈^d* be a lift of η(x) and pick w ∈^d such that f(v) = tw v. Then define η_S(x) byη_S(x)(X ) =tw X w.By construction the maps ξ_S, η_S are ρ_S-equivariant and continuous. Since the maps ξ, η are transverse andη_S(x)(ξ_S(y) ) = η(x) ( ξ(y) )^2,the maps ξ_S, η_S are also transverse. Thus ρ_S is projective Anosov by Proposition 4.10 in <cit.>.§ BASIC PROPERTIES OF CONVEX COCOMPACT ACTIONS In this section we establish some basic properties of convex cocompact actions on properly convex domains.§.§ Quasi-isometries The fundamental lemma of geometric group theory (see <cit.>)immediately implies the following.Suppose Ω⊂(^d) is a properly convex domain and Λ≤(Ω) is a discrete convex cocompact group. Then Λ is finitely generated and for any p_0 ∈Ω the map φ∈Λ→φ p_0induces an quasi-isometric embedding Λ→ (Ω, d_Ω).§.§ Rescaling Given a finite dimensional real vector space V, let K(V) denote the set of all compact subsets in (V) equipped with the Hausdorff topology (with respect to a distance on (V) induced by a Riemannian metric). Next let (V) denote the set of properly convex open sets in (V). Then the map Ω∈(V) →Ω∈ K(V)is injective and so (V) has a natural topology coming from K(V). Finally, we let _0(V) = { (Ω, x): Ω∈(V), x ∈Ω}equipped with the product topology.In the 1960's Benzécri proved the following theorem.The group (V) acts properly and cocompactly on _0(V). Moreover, if Ω⊂(V) is a properly convex domain and (Ω) acts cocompactly on Ω, then the orbit (V) ·Ω is closed in (V).In this section we will use a result of Benoist to prove an analogue of Benzécri's theorem for convex cocompact actions.Suppose Ω⊂(^d) is a properly convex domain, G ≤(Ω) is a subgroup, and there exists a closed convex subset ⊂Ω such that g= for all g ∈ G and G \ is compact. Assume V ⊂(^d) is a subspace that intersects , c_n ∈∩ V, and h_n ∈(V) satisfy * h_n(Ω∩ V) →Ω_V where Ω_V is a properly convex domain in (V),* h_n(∩ V) →_V where _V is a properly convex closed set in (V),* h_n(c_n) → p_∞∈Ω_V.Then there exists some φ∈_d() so that φ(Ω) ∩ V= Ω_Vand φ() ∩ V ⊃_V.Before starting the proof of the theorem we make two observation about the Hausdorff topology.Suppose Ω_n →Ω in (^d) and K ⊂Ω is a compact set. Then K ⊂Ω_n for n sufficiently large. We can pick an affine chart 𝔸⊂(^d) such that Ω is relatively compact in 𝔸. Then for n sufficiently large, Ω_n is also relatively compact in 𝔸. Then we can identify 𝔸 with ^d-1 and view Ω_n, Ω as convex subsets of ^d-1 (at least for n sufficiently large). Then Ω_n→Ω is the Hausdorff distance induced by the Euclidean distance on ^d-1. Now suppose, for a contradiction, that there exist n_j →∞ and k_j ∈ K such that k_j ∉Ω_n_j. By passing to a subsequence we can assume that k_j → k. Now since Ω is open, there exists some ϵ > 0 such that { x ∈^d-1 : k-x≤ϵ}⊂Ω.Since each Ω_n_j is convex, we can find an real hyperplane H_j such that k_j ∈ H_j and Ω_n_j∩ H_j = ∅. Then for j sufficiently large, there exists some x_j ∈^d∖Ω_n_j such that d_ Euc(x_j, H_j) ≥ϵ/2 and k-x_j≤ϵ. But then x_j ∈Ω and sod_ Euc^ Haus(Ω_n_j, Ω) ≥ d_ Euc(Ω_n_j, x_j) ≥ϵ/2which is a contradiction.Suppose Ω_n →Ω in (^d).If V ⊂(^d) is a subspace and V ∩Ω≠∅, then Ω_n ∩ V →Ω∩ V in (V).Since K(V) is compact, it is enough to show that every convergent subsequence of Ω_n ∩ V converges to Ω∩ V. So suppose that Ω_n ∩ V→ C in K(V). Then by the definition of the Hausdorff topology we have C ⊂Ω∩ V. Since Ω is convex, we have Ω∩ V = Ω∩ V. So we can pick a sequence K_m ⊂Ω∩ V of compact sets such that ∪ K_m = Ω∩ V.Fix m. Then K_m ⊂Ω_n for n sufficiently large by Observation <ref>. So K_m ⊂ C. Since m was arbitrary Ω∩ V =∪ K_m⊂ C.Hence C = Ω∩ V. By Lemma 2.8 in <cit.> there exists g_n ∈_d() and a properly convex domain Ω^'⊂(^d) such that* g_n|_V = h_n, * Ω_n := g_nΩ→Ω^', and* Ω^'∩ V = Ω_V. Now fix a point p_0 ∈. Then there exist R ≥ 0 and a sequence γ_n ∈ G such that d_Ω(c_n, γ_n p_0) ≤ R.Next consider the element φ_n = g_n γ_n. Note thatd_Ω_n(φ_np_0, g_n c_n) = d_Ω(γ_np_0, c_n) ≤ R.Then since lim_n →∞ g_nc_n = lim_n →∞ h_n c_n = p_∞∈Ω^'and d_Ω_n converges locally uniformly to d_Ω^' we can pass to a subsequence so that φ_np_0 → q_∞∈Ω^'. Then φ_n(Ω, p_0) → (Ω^', q_∞) and since _d() acts properly on _0(^d), we can pass to a subsequence such that φ_n →φ∈_d(). Then by the Observation <ref>φ(Ω) ∩ V = lim_n →∞φ_n(Ω) ∩ V = lim_n →∞ g_n(Ω) ∩ V = Ω^'∩ V = Ω_V. By passing to a subsequence we can suppose that the sequence g_n() ∩ V converges in K(V). Then, by the definition of the Hausdorff topology, φ() ∩ V ⊃lim_n →∞φ_n() ∩ V= lim_n →∞ g_n() ∩ V ⊃lim_n →∞ h_n(∩ V) ∩ V =lim_n →∞ h_n(∩ V) = _V.§ REGULAR CONVEX COCOMPACTNESS IMPLIES PROJECTIVE ANOSOVNESS In this section we prove Theorem <ref> from the introduction. The proof uses many ideas from Benoist's work on the Hilbert metric <cit.>. Suppose Ω⊂(^d) is a properly convex domain and Λ≤(Ω) is a discrete convex cocompact subgroup. Letbe a closed convex subset of Ω such that g = for all g ∈Λ and Λ\ is compact. If Λ is an irreducible subgroup of _d(), then the following are equivalent:* every point in ∩∂Ω is a C^1 point of ∂Ω, * every point in ∩∂Ω is an extreme point of ∂Ω Moreover, when these conditions are satisfied Λ is word hyperbolic and the inclusion representation Λ↪_d() is projective Anosov.In the special case when Ω =, Theorem <ref> was established by Benoist <cit.>, see Theorem <ref> in the introduction. For the rest of the section fix a properly convex domain Ω⊂(^d), a discrete convex cocompact subgroup Λ≤(Ω), and a closed convex subset ⊂Ω which satisfy the hypothesis of Theorem <ref>.Notice thathas non-empty interior since Λ is irreducible and preserves the subspace _{ c: c ∈}. With the notation above, if each q ∈∂Ω∩ is a C^1 point of ∂Ω, then each q ∈∂Ω∩ is an extreme point of Ω. Suppose for a contradiction that there exists a point q ∈∂Ω∩ which is not an extreme point of Ω. Then after making a change of coordinates we can assume the following:* q=[1:0:…:0] ∈∂Ω∩, * [1:0:1:0:…:0] ∈, * Ω⊂{ [1:x_1:x_2:… : x_d-1] ∈(^d): x_2 >0}, and* { [1:t:0:…:0] ∈(^d) : t ∈ [-1,1]}⊂∂Ω. Now let V = { [x_1 : x_2 : x_3: 0 : … : 0]∈(^d) : x_1, x_2, x_3 ∈},c_n = [ 1: 0:1/n : 0 : … : 0] ∈∩ V, and h_n ∈(V) be given byh_n[x_1:x_2:x_3:0:…:0] = [x_1:x_2:nx_3:0:…:0].Then h_nc_n → [1:0:1:0:…:0], h_n(Ω∩ V) →Ω_V := { [1:s:t:0:…:0] ∈(^d) : [1:s:0:…:0] ∈∂Ω andt > 0},andh_n(∩ V) →_V := { [1:s:t:0:…:0] ∈(^d) : [1:s:0:…:0] ∈∂Ω∩ andt > 0}.Clearly Ω_V is properly convex and so by Theorem <ref>, there exists some φ∈_d() such that φ(Ω) ∩ V = Ω_V and _V ⊂φ() ∩ V. But then[0:0:1:0:…:0] is a not a C^1 point of ∂Ω_V and hence φ^-1[0:0:1:0:…:0] ∈∩∂Ω is not a C^1 point of ∂Ω. So we have a contradiction. With the notation above, if each q ∈∂Ω∩ is an extreme point of Ω, then each q ∈∂Ω∩ is a C^1 point of ∂Ω. Suppose for a contradiction that there exists a point q ∈∂Ω∩ which is not a C^1 point of ∂Ω. Then there exist two different hyperplanes H_1, H_2 such that q ∈ H_1 ∩ H_2 and H_1 ∩Ω = H_2 ∩Ω = ∅. Sincehas non-empty interior, there exists a two dimensional subspace V ⊂(^d) so that V intersects the interior of , and V ∩ H_1 ≠ V ∩ H_2.By making a change of coordinates, we can assume that* q = [1:0:…:0], * V = { [x_1 : x_2 : x_3: 0 : … : 0]∈(^d) : x_1, x_2, x_3 ∈},* Ω∩ V ⊂{ [1:x_1:x_2:0:…:0] ∈(^d) : x_2 > 0},* [1:0:1:…:0] is contained in the interior of , and* there exists α_1 < 0 < α_2 such that H_i ∩ V = { [1:t:α_i t:0:…:0] ∈(^d): t ∈}∪{[0:1:α_i]}.Now since [1:0:1:…:0] is contained in the interior of , there exists ϵ>0 and β_1 < 0 < β_2 such that { [1:t:β_2 t:0:…:0] ∈(^d) : 0 < t < ϵ}⊂and { [1:t:β_1 t:0:…:0] ∈(^d) : -ϵ < t < 0 }⊂. Next consider the points c_n = [1:0:1/n:0:…:0] and let h_n ∈(V) be given byh_n[x_1:x_2:x_3:0:…:0] = [x_1: n x_2:n x_3:0:…:0].Then h_nc_n → [1:0:1:0:…:0], h_n(Ω∩ V) converges to the tangent cone 𝒯𝒞_q(Ω∩ V) of Ω∩ V at q, and h_n(∩ V) converges to the tangent cone 𝒯𝒞_q(∩ V) of ∩ V at q. By construction 𝒯𝒞_q(Ω∩ V) is a properly convex domain in V. So by Theorem <ref>, there exists some φ∈_d+1() such that φ(Ω) ∩ V = 𝒯𝒞_g(Ω∩ V) and φ() ∩ V ⊃𝒯𝒞_g(∩ V). But then φ^-1{[0:1:s :0:…:0] ∈(^d) : β_1 ≤ s ≤β_2 }⊂∩∂Ωwhich contradicts the fact that every point in ∩∂Ω is an extreme point. For the remainder of the section we assume, in addition, that* every point in ∩∂Ω is a C^1 point of ∂Ω and* every point in ∩∂Ω is an extreme point of ∂Ω. With the notation above, Λ is word hyperbolic.Fix a finite symmetric generating set S of Λ. By Proposition <ref>, (Λ, d_S) is quasi-isometric to (, d_Ω) and so it is enough to show that (, d_Ω) is Gromov hyperbolic. Now for each x,y ∈ let σ_x,y be the geodesic joining x to y which parametrizes the line segment joining them. By Proposition <ref> it is enough to show that there exists an δ > 0 such that every geodesic triangle in (, d_Ω) of the formσ_x,y, σ_y,z, σ_z,x is δ-thin. Suppose not. Then for every n > 0 there exists points x_n, y_n, z_n, u_n ∈ such that u_n ∈σ_x_n, y_n and d_Ω(u_n, σ_y_n,z_n∪σ_z_n, x_n) > n.By replacing the the points x_n, y_n, z_n,u_n by g_n x_n, g_n y_n, g_n z_n, g_n u_n for some g_n ∈Λ we canassume that the sequence u_n is relatively compact in . Then by passing to a subsequence we can suppose that u_n → u ∈. By passing to another subsequence we can assume that x_n, y_n, z_n → x,y,z ∈. Since d_Ω(u_n, {x_n,y_n,z_n}) > nwe must have x,y,z ∈∩∂Ω. The image of σ_x_n,y_n converges to a line segment containing x,y,u. Since u ∈ and x,y ∈∂Ω we must have x ≠ y. Then either z ≠ x or z ≠ y. By relabeling we may assume that z ≠ x. Then the image of σ_x_n,z_n converges to the line segment [x,z]. Since every point in ∩∂Ω is an extreme point of ∂Ω and x ≠ z, we must have (x,z) ⊂Ω. So ∞ = lim_n →∞ d_Ω(u_n, σ_z_n,x_n) = d_Ω(u, (z,x)) < ∞.So we have a contradiction and hence Λ is word hyperbolic. With the notation above, there exists a Λ-equivariant homeomorphism ξ : ∂Λ→∩∂Ω.Since every point in ∩∂Ω is an extreme point, this follows from Lemma <ref>.With the notation above, the inclusion representation Λ↪_d() is projective Anosov.Let ξ : ∂Λ→∩∂Ω be the Λ-equivariant homeomorphism from the previous lemma. Since every point in ∩∂Ω is a C^1 point, the map η: ∂Λ→(^d*) with(η(x)) = T_ξ(x)∂Ωis well defined, continuous, and Λ-equivariant. We claim that ξ(x) + η(y) = ^dfor x,y ∈∂Γ distinct. If not, then [ξ(x), ξ(y)] ⊂∩(η(y)) =∩ T_ξ(y)∂Ω⊂∩∂Ω.But since each q ∈∂Ω∩ is an extreme point of ∂Ω we see that this is impossible. Then Proposition 4.10 in <cit.> implies that the inclusion representation is projective Anosov.§ PROOF OF THEOREM <REF> We now prove Theorem <ref> from the introduction:Suppose G is a semisimple Lie group with finite center and P ≤ G is a parabolic subgroup. Then there exists a finite dimensional real vector space V and an irreducible representation ϕ:G →(V) with the following property: if Γ is a word hyperbolic group and ρ:Γ→ G is a Zariski dense representation with finite kernel, then the following are equivalent: * ρ is P-Anosov,* there exists a properly convex domain Ω⊂(V) such that (ϕ∘ρ)(Γ) is a regular convex cocompact subgroup of (Ω). For the rest of the section fix G a semisimple Lie group with finite center and P ≤ G a parabolic subgroup.By Theorem <ref>, there exist a finite dimensional real vector space V_0 and an irreducible representation ϕ_0:G →(V_0) with the following property: if Γ is a word hyperbolic group and ρ:Γ→ G is a representation, then the following are equivalent: * ρ is P-Anosov,* ϕ_0 ∘ρ is projective Anosov. We will construct a new representation of G by taking the tensor product of ϕ_0 with itself. In general, this will not produce an irreducible representation and so we will construct a subspace of V_0 ⊗ V_0 where ϕ_0 ⊗ϕ_0 acts irreducibly.For a proximal element g ∈(V_0) let ℓ^+_g ∈(V_0) be the eigenline of g corresponding to the eigenvalue of largest absolute value.Then consider the vector space V =_{ℓ^+_g⊗ℓ^+_g: g ∈ϕ_0(G)is proximal}and the representation ϕ: G →(V) given by ϕ(g)(v ⊗ v) = (ϕ_0(g)v) ⊗ (ϕ_0(g)v).Notice that we can assume that V ≠ (0), for otherwise there is nothing to prove. With the notation above, if g ∈ G and ϕ_0(g) is proximal, then ϕ(g) is proximal and ℓ_ϕ_0(g)^+ ⊗ℓ_ϕ_0(g)^+ is the eigenline of ϕ(g) corresponding to the eigenvalue of largest absolute value. The argument is similar to the proof of Lemma <ref>. With the notation above, ϕ: G →(V) is an irreducible representation.The argument is similar to the proof of Lemma <ref>.We now complete the proof of the theorem. With the notation above, if Γ is a word hyperbolic group and ρ:Γ→ G is a Zariski dense representation with finite kernel, then the following are equivalent: * ρ is P-Anosov,* there exists a properly convex domain Ω⊂(V) such that (ϕ∘ρ)(Γ) is a regular convex cocompact subgroup of (Ω).If ρ is P-Anosov, then ϕ_0 ∘ρ is projective Anosov representation by our choice of ϕ_0. Let ξ_0: ∂Γ→(V_0) and η_0: ∂Γ→(V_0^*) be the associated boundary maps. Since ϕ_0: G →(V_0) is irreducible and ρ(Γ) ≤ G is Zariski dense, we see that ϕ_0 ∘ρ : Γ→(V_0) is irreducible. So by Corollary <ref>, if V^' = {ξ_0(x) ⊗ξ_0(x) : x ∈∂Γ},then there exists a properly convex domain Ω⊂(V^') so that(ϕ∘ρ)(Γ) is a regular convex cocompact subgroup of (Ω).Since ϕ: G →(V) is irreducible and ρ(Γ) ≤ G is Zariski dense, we see that ϕ∘ρ : Γ→(V) is irreducible. Then since V^'⊂ V, we must have V^' = V. Next suppose that there exists some properly convex domain Ω⊂(V) such that (ϕ∘ρ)(Γ) ≤(Ω) is a regular convex cocompact subgroup.Since ϕ: G →(V) is irreducible and ρ(Γ) ≤ G is Zariski dense, we see that ϕ∘ρ : Γ→(V) is irreducible. Hence Theorem <ref>implies that ϕ∘ρ is a projective Anosov representation. Let ξ: ∂Γ→(V) and η: ∂Γ→(V^*) be the associated boundary maps. We claim that there exist maps ξ_0: ∂Γ→(V_0) and η_0: ∂Γ→(V_0^*) such that ξ(x) = ξ_0(x) ⊗ξ_0(x)and η(x) = η_0(x) ⊗η_0(x)for all x ∈∂Γ. Since ρ(Γ) is Zariski dense in G and ϕ_0(G) contains proximal elements, there exists some φ∈Γ such that (ϕ_0 ∘ρ)(φ) is proximal, see for instance <cit.>. Let x^+ ∈∂Γ be the attracting fixed point of φ in ∂Γ. Then ξ(x^+) is the eigenline of (ϕ∘ρ)(φ) whose eigenvalue has maximal absolute value. Since(ϕ_0 ∘ρ)(φ) is proximal, Lemma <ref> says thatξ(x^+) = ℓ^+ ⊗ℓ^+where ℓ^+ ∈(V) is the eigenline of (ϕ_0 ∘ρ)(φ) whose eigenvalue has maximal absolute value. Now* ξ: ∂Γ→(V) is continuous and (ϕ∘ρ)-equivariant, * the set A={ [v ⊗ v] : v ∈ V_0 ∖{0}}⊂(V)is closed and ϕ(G)-invariant, and * the set Γ· x^+ is dense in ∂Γ.Since ξ(x^+) ∈ A, the three properties above imply that ξ(∂Γ) ⊂ A. Hence there exists a map ξ_0: ∂Γ→(V_0) such that ξ(x) = ξ_0(x) ⊗ξ_0(x)for all x ∈∂Γ. Since [v] ∈(V_0) → [v ⊗ v] ∈ Ais a diffeomorphism, the map ξ_0 is continuous. Finally, by construction, the map ξ_0 is (ϕ_0 ∘ρ)-equivariant. Applying this same argument to η yields a continuous (ϕ_0 ∘ρ)-equivariant map η_0: ∂Γ→(V_0^*) such that η(x) = η_0(x) ⊗η_0(x)for all x ∈∂Γ. If x,y ∈∂Γ, thenη(y) (ξ(x)) = η_0(y) ⊗η_0(y)( ξ_0(x) ⊗ξ_0(x) ) = η_0(y)(ξ_0(x))^2.Since ξ and η are transverse, this implies that ξ_0 and η_0 are transverse. Finally, since the representation ϕ_0: G →(V_0) is irreducible and ρ(Γ) ≤ G is Zariski dense, we see that ϕ_0 ∘ρ : Γ→(V_0) is irreducible. Hence by Proposition 4.10 in <cit.> we see that ϕ_0 ∘ρ: Γ→ G is a projective Anosov representation. Thus by our choice of ϕ_0 we see that ρ: Γ→ G is P-Anosov. § ENTROPY RIGIDITY The proof of Theorem <ref> has three steps: first we use results of Coornaert-Knieper, Coornaert, and Cooper-Long-Tillmann to transfer to the Hilbert metric setting, then we use a result of Tholozan to transfer to the Riemmanian metric setting, and finally we use an argument of Liu to prove rigidity. This general approach is based on the arguments in <cit.>.It will also be more notationally convenient in this section to work with (^d+1) instead of (^d).§.§ Some notation Suppose (X,d) is a proper metric space and x_0 ∈ X is some point. If G ≤(X,d) is a discrete subgroup, then define the Poincaré exponent of G to be δ_G(X,d) := lim sup_r →∞1/rlog#{ g ∈ G : d(x_0, g x_0) ≤ r }.Notice that δ_G(X,d) does not depend on x_0. If X has a measure μ one can also define the volume growth entropy relative to μ ash_vol(X,d,μ) := lim sup_r →∞1/rlogμ( { x ∈ X : d(x, x_0) ≤ r }).If the measure μ is (X,d)-invariant, finite on bounded sets, and positive on open sets, then δ_G(X,d) ≤ h_vol(X,d,μ)by the proof of Proposition 2 in <cit.>. In the case in which (X,g) is a Riemannian manifold, we will let h_vol(X,g):=h_vol(X,d, )where d is the distance induced by g andis the Riemannian volume associated to g. §.§ Transferring to the Hilbert metric settingAs in the introduction, we define the Hilbert entropy of a representation ρ: Γ→_d() to be H_ρ= lim sup_r →∞1/rlog#{ [γ] ∈ [Γ] : 1/2log(λ_1(ρ(γ))/λ_d(ρ(γ)))≤ r}where [Γ] is the set of conjugacy classes of Γ. By combining results of Coornaert-Knieper, Coornaert, and Cooper-Long-Tillmann, we will establish the following proposition. Suppose Γ is a word hyperbolic group, ρ: Γ→_d+1() is an irreducible projective Anosov representation, andΩ⊂(^d+1) is a properly convex domain such that ρ(Γ) ≤(Ω) is a regular convex cocompact subgroup. ThenH_ρ = δ_ρ(Γ)(Ω, d_Ω).Moreover, for any p_0 ∈Ω there exists C ≥ 1 such that1/C e^H_ρ r≤#{γ∈Γ : d_Ω(p_0, ρ(γ) p_0) ≤ r }≤ C e^H_ρ r. Let ⊂Ω be a closed convex subset such that g = for all g ∈ρ(Γ), ρ(Γ) \ is compact, and every point in ∩∂Ω is a C^1 extreme point of Ω.Using Selberg's lemma, we can find a finite index subgroup Γ_0 ≤Γ such that ρ(Γ_0) is torsion free.Then H_ρ = H_ρ|_Γ_0, δ_ρ(Γ_0)(Ω, d_Ω)=δ_ρ(Γ)(Ω, d_Ω), and ρ(Γ_0) \ is compact.For γ∈Γ_0 define τ(γ) = inf_c ∈ d_Ω(ρ(γ)c,c).Since (, d_Ω) is a proper geodesic metric space, ρ(Γ_0) acts cocompactly on , Γ_0 is word hyperbolic, and ρ is finite, Theorem 1.1 in <cit.> says thatδ_ρ(Γ_0)(Ω, d_Ω) =lim_r →∞1/rlog#{ [γ] ∈ [Γ_0] : τ(γ) ≤ r }.Next we claim that τ(γ) = 1/2log(λ_1(ρ(γ))/λ_d+1(ρ(γ)))for every γ∈Γ_0. Fix some γ∈Γ_0. Then Proposition 2.1 in <cit.> says that inf_x ∈Ω d_Ω(ρ(γ)x,x) = 1/2log(λ_1(ρ(γ))/λ_d+1(ρ(γ))).and so τ(γ) ≥1/2log(λ_1(ρ(γ))/λ_d+1(ρ(γ))).Since γ has infinite order we see that ρ_0(γ) is biproximal, that is ρ_0(γ) and ρ_0(γ)^-1 are proximal. So if ℓ^+ and ℓ^- are the attracting and repelling eigenlines of ρ_0(γ) respectively, then Corollary <ref> implies that ℓ^+, ℓ^- ∈∩∂Ω. Since every point in ∩∂Ω is an extreme point of Ω, we then see that (ℓ^+, ℓ^-) ⊂. But if p ∈ (ℓ^+, ℓ^-), thend_Ω(ρ_0(γ) p,p) = 1/2log(λ_1(ρ(γ))/λ_d+1(ρ(γ)))by the definition of the Hilbert distance. Hence τ(γ) = 1/2log(λ_1(ρ(γ))/λ_d+1(ρ(γ))). Then by Equations (<ref>) and (<ref>)δ_ρ(Γ_0)(Ω, d_Ω) =lim_r →∞1/rlog#{ [γ] ∈ [Γ_0] : τ(γ) ≤ r }=lim_r →∞1/rlog#{ [γ] ∈ [Γ_0] : 1/2log(λ_1(ρ(γ))/λ_d+1(ρ(γ))) ≤ r } =H_ρ|_Γ_0and soH_ρ = H_ρ|_Γ_0=δ_ρ(Γ_0)(Ω, d_Ω)=δ_ρ(Γ)(Ω, d_Ω).Finally by Théorème 7.2 in <cit.>, for any p_0 ∈Ω there exists C ≥ 1 such that1/C e^H_ρ r≤#{γ∈Γ : d_Ω(p_0, ρ(γ) p_0) ≤ r }≤ C e^H_ρ r.§.§ Transferring to the Riemannian settingAssociated to every properly convex domain Ω⊂(^d+1) is a Riemannian distance B_Ω on Ω called the Blaschke distance (see, for instance, <cit.>). This Riemannian distance is (Ω)-invariant and by a result of Calabi <cit.> has Ricci curvature bounded below by -(d-1). Since the Ricci curvature is bounded below by -(d-1), the Bishop-Gromov volume comparison theorem implies that h_vol(Ω, B_Ω) ≤ d-1. Benzécri'stheorem (see Theorem <ref>) provides a simple proof that the Hilbert distance and the Blaschke distance are bi-Lipschitz (see for instance <cit.>) and Tholozan recently proved the following refined relationship between the two distances:<cit.> If Ω⊂(^d+1) is a properly convex domain, then B_Ω < d_Ω +1.In particular,if Γ≤(Ω) is a discrete group, then δ_Γ(Ω, d_Ω) ≤δ_Γ(Ω, B_Ω) ≤ h_vol(Ω, B_Ω) ≤ d-1. §.§ Rigidity in the Riemannian settingThe Bishop-Gromov volume comparison theorem implies that amongst the class of Riemannian d-manifolds with ≥ -(d-1) the volume growth entropy is maximized when (X,g) is isometric to real hyperbolic d-space. There are many other examples which maximize volume growth entropy, but if X has “enough” symmetry then it is reasonable to suspect that h_vol(X,g)=d-1 if and only if X is isometric to real hyperbolic d-space. This was proved by Ledrappier and Wang when X covers a compact manifold:Let (X,g) be a complete simply connected Riemannian d-manifold with ≥ -(d-1). Suppose that X is the Riemannian universal cover of a compact manifold. Then h_vol(X,g)=d-1 if and only if X is isometric to real hyperbolic d-space.Later Liu <cit.> provided an alternative proof of Ledrappier and Wang's result and Liu's argument can be adaptedto prove the following. Let (X,g) be a complete simply connected Riemannian d-manifold with ≥ -(d-1) and bounded sectional curvature. Suppose Γ≤(X,g) is a discrete subgroup and there exist C,r_0>0 and x_0 ∈ X such thatC e^(d-1)r≤#{γ∈Γ : d_X(x_0, γ x_0) ≤ r } for every r > r_0. Then X is isometric to real hyperbolic d-space.We will prove this result in Section <ref> of the appendix. §.§ Proof of Theorem <ref> Suppose Γ is a finitely generated word hyperbolic group, ρ:Γ→_d+1() is an irreducible projective Anosov representation, and ρ(Γ) preserves a properly convex domain in (^d+1). Using Theorem <ref>, there exists a properly convex domainΩ⊂(^d+1) such that ρ(Γ) ≤(Ω) is a regular convex cocompact subgroup. Combining Proposition <ref> and Theorem <ref> we see that H_ρ = δ_ρ(Γ)(Ω, d_Ω) ≤ d-1. Now suppose that H_ρ = d-1. By Theorem <ref> and Proposition <ref> there exists some C_0 > 0 such that C_0 e^(d-1)r≤#{γ∈Γ : B_Ω(p_0, ρ(γ) p_0) ≤ r }for all r ≥ 0.Moreover, Benzécri's theorem implies that B_Ω has bounded sectional curvature (see for instance <cit.>). So by Proposition <ref>, (Ω, B_Ω) is isometric to the real hyperbolic d-space. Hence (Ω, d_Ω) is projectively equivalent to the Klein-Beltrami model of hyperbolic space (see <cit.>). In particular, by conjugating ρ(Γ) we can assume Ω = { [1:x_1 : … : x_d] ∈(^d) : ∑_i=1^d x_i^2 < 1}and (Ω)=(1,d). Then ρ(Γ) is a convex cocompact subgroup of (1,d) in the classical sense. Sinceδ_ρ(Γ)(Ω, d_Ω) = d-1,Theorem D in <cit.> implies that ∂Ω∩ = ∂Ω.Then sinceis convex we see that =Ω. Then since ρ(Γ) \= ρ(Γ) \Ω is compact, we see that ρ(Γ)≤(1,d) is a co-compact lattice.§ REGULARITY RIGIDITY In this section we will prove Theorems <ref> and <ref> from the introduction. The proof of both theorems are based on the following observation. Suppose g ∈_d() is proximal and ℓ_g^+ ∈(^d) is the eigenline of g corresponding to the eigenvalue of largest absolute value. Let d_ is a distance on (^d) induced by a Riemannian metric. If v ≠ℓ^+_g and g^n v →ℓ^+_g, thenlogλ_2(g)/λ_1(g)≥lim sup_n →∞1/nlog d_(g^n v, ℓ^+_g ).Moreover, there exists a proper subspace V ⊂(^d) such that: if v ∈(^d) ∖ V and g^n v →ℓ^+_g, thenlogλ_2(g)/λ_1(g) = lim_n →∞1/nlog d_(g^n v, ℓ^+_g ).We give a proof of the observation in Appendix <ref>. §.§ Proof of Theorem <ref>We begin by recalling the theorem.Suppose d>2, Γ is a word hyperbolic group, and ρ:Γ→_d() is an irreducible projective Anosov representation with boundary map ξ: ∂Γ→(^d). If* M = ξ(∂Γ) is a C^2 k-dimensional submanifold of (^d) and* the representation ∧^k+1ρ : Γ→(∧^k+1^d) is irreducible,thenλ_1(ρ(γ))/λ_2(ρ(γ)) = λ_k+1(ρ(γ))/λ_k+2(ρ(γ))for all γ∈Γ.For the rest of the subsection, fix a word hyperbolic group Γ and a projective Anosov representation ρ: Γ→_d() which satisfy the hypothesis ofTheorem <ref>.Define a map Φ: M →(∧^k+1^d) by Φ(m) = [v_1 ∧ v_2 ∧…∧ v_k+1]where T_mM = (_{ v_1, …, v_k+1}). Since M is a C^2 submanifold, Φ is a C^1 map.With the notation above, Φ: M →(∧^k+1^d) is a C^1 immersion.We break the proof into two cases: when k = 1 and when k >1. Case 1: Assume k=1.We first consider the case when d(Φ)_m = 0 for every m ∈ M. Then there exists a two dimensional subspace V ⊂^d such that T_mM = (V) for all m. Then we must have M ⊂(V). Since ρ is irreducible, the elements in M span ^d and so d ≤ 2. Thus we have a contradiction. So d(Φ)_m ≠ 0 on an open set in M. But sinceΦ∘ρ(γ) =( ∧^2ρ(γ)) ∘Φfor every γ∈Γ and Γ acts minimally on M, we see that d(Φ)_m ≠ 0 for every m. Case 2: Assume k > 1. Then by Theorem <ref> there exists a properly convex domain Ω⊂(^d) such that ρ(Γ) ≤(Ω) is a regular convex cocompact subgroup. Suppose ⊂Ω is a closed convex subset such that g = for all g ∈ρ(Γ), ρ(Γ) \ is compact, and every point in ∂Ω∩ is a C^1 extreme point of Ω. We first claim that Φ is injective. By Lemma <ref> we have ξ(∂Γ) ⊂∂Ω∩and η(∂Γ) ⊂∂Ω^*.Then since ξ(x) is a C^1 point of ∂Ω we haveT_ξ(x)∂Ω = (η(x)).Further since M ⊂∂Ωwe see thatT_ξ(x) M ⊂ T_ξ(x)∂Ω = (η(x))for every x ∈∂Γ. Now suppose that T_ξ(x)M =T_ξ(y)M for some x,y ∈∂Γ. Then ξ(x) ∈ T_ξ(x)M= T_ξ(y)M⊂(η(y)).So x=y and hence Φ is injective. Since Φ is injective and C^1, d(Φ) must have full rank at some point. By continuity, d(Φ) has full rank on an open set. But sinceΦ∘ρ(γ) =( ∧^k+1ρ(γ)) ∘Φfor every γ∈Γ and Γ acts minimally on M, we see that d(Φ) has full rank everywhere. Hence, since M is compact and Φ is injective, Φ is a C^1 embedding.Next fix distances d_1 on (^d) and d_2 on (∧^k+1^d) which are induced by Riemannian metrics. Since Φ is a C^1 immersion, there exists C ≥ 1 such that 1/C d_1(m_1, m_2) ≤ d_2(Φ(m_1), Φ(m_2)) ≤ C d_1(m_1, m_2)for all m_1, m_2 ∈ M sufficiently close. Now fix some γ∈Γ with infinite order and let g ∈_d() be a lift of ρ(γ) with g=± 1. Suppose that λ_1 ≥λ_2 ≥…≥λ_dare the absolute values of the eigenvalues of g. Then the absolute values of the eigenvalues of ∧^k+1 g have the form λ_i_1λ_i_2⋯λ_i_k+1for 1 ≤ i_1 < i_2 < … < i_k+1≤ d. In particular, λ_1 λ_2 ⋯λ_k+1is the absolute value of the largest eigenvalue of ∧^k+1 g and λ_1 λ_2 ⋯λ_k λ_k+2is the absolute value of the second largest eigenvalue of ∧^k+1 g. Next let x^+, x^- be the attracting and repelling fixed points of γ in ∂Γ.With the notation above,∧^k+1ρ(γ) is proximal with attracting fixed point Φ(ξ(x^+)).We first show that Φ(ξ(x^+)) is an eigenline of ∧^k+1 g whose eigenvalue has absolute value λ_1 ⋯λ_k+1.Fix a norm on (∧^k+1^d). Then we can find a sequence n_m →∞ such that 1/(∧^k+1 g)^n_m(∧^k+1g)^n_m converges to some T ∈(∧^k+1^d). ThenT(v) = lim_m →∞ (∧^k+1ρ(γ))^n_m vfor every v ∈(∧^k ^d) ∖( T). By Observation <ref>, every element in the image of T is a sum of generalized complex eigenvectors of ∧^k+1 g whose eigenvalue has maximal absolute value (that is, λ_1 ⋯λ_k+1). We will show that the image of T is Φ(ξ(x^+)) and hence Φ(ξ(x^+)) is an eigenline of ∧^k+1 g whose eigenvalue has absolute value λ_1 ⋯λ_k+1.Now since ∧^k+1ρ : Γ→(∧^k+1^d) is irreducible, there exists x_1, …, x_N ∈∂Γ such that Φ(ξ(x_1)), …, Φ(ξ(x_N))span ∧^k+1^d. By perturbing the x_i (if necessary) we can also assume thatx^- ∉{x_1, …, x_N}.Next by relabelling the x_i we can also assume that there exists 1 ≤ m ≤ N such thatΦ(ξ(x_1)) + … + Φ(ξ(x_m)) +T = ∧^k+1^dand ( Φ(ξ(x_1)) + … + Φ(ξ(x_m)) ) ∩ T = (0).Then by Equation (<ref>) T ( Φ(ξ(x_i)) ) = lim_m →∞ (∧^k+1ρ(γ))^n_mΦ(ξ(x_i)) =lim_m →∞Φ( ξ( γ^n_m x)) = Φ(ξ(x^+))for 1 ≤ i ≤ m. So the image of T is Φ(ξ(x^+)) and so Observation <ref> implies that Φ(ξ(x^+)) is an eigenline of ∧^k+1 g whose eigenvalue has absolute value λ_1 ⋯λ_k+1.We next argue that ∧^k+1ρ(γ) is proximal. Suppose not, then by Observation <ref> there exists a proper subspace V ⊂(∧^k+1^d) such that: if v ∈(∧^k+1^d) ∖ V, then 0= lim_n →∞1/nlog d_2((∧^k+1ρ(γ))^n v, Φ(ξ(x^+))).Since ∧^k+1ρ : Γ→(∧^k+1^d) is irreducible there exists x ∈∂Γ such that Φ(ξ(x)) ∉ V. Then by perturbing x (if necessary) we can also assume that x ≠ x^-. Then ρ(γ)^n ξ(x) = ξ(γ^nx) →ξ(x^+)and(∧^k+1ρ(γ))^nΦ(ξ(x)) =Φ(ξ(γ^nx))→Φ(ξ(x^+)).So by Observation <ref> applied to ρ(γ)0 > logλ_2/λ_1 ≥lim sup_n →∞1/nlog d_1(ρ(γ)^n ξ(x), ξ(x^+)) = lim sup_n →∞1/nlog d_1(ξ(γ^nx), ξ(x^+)) =lim sup_n →∞1/nlog d_2( Φ(ξ(γ^nx)),Φ( ξ(x^+)))= lim sup_n →∞1/nlog d_2( ( ∧^k+1ρ(γ) )^nΦ(ξ(x)),Φ( ξ(x^+))) =0.Notice that we used Equation (<ref>) in the second equality. So we have a contradiction and hence ∧^k+1ρ(γ) is proximal. By Observation <ref>, there exists a proper subspace V_1 ⊂(^d) such thatlogλ_2/λ_1 = lim_n →∞1/nlog d_1(ρ(γ)^n v, ξ(x^+))for all v ∈(^d) ∖ V_1 with ρ(γ)^n v →ξ(x^+). By the same observation, there exists a proper subspace V_2 ⊂(∧^k+1^d+1) such that logλ_k+2/λ_k+1=logλ_1 λ_2 ⋯λ_k λ_k+2/λ_1 λ_2 ⋯λ_k+1 = lim_n →∞1/nlog d_2((∧^k+1ρ(γ))^n w, Φ(ξ(x^+_γ)))for all w ∈(∧^k+1^d) ∖ V_2 with (∧^k+1ρ(γ))^n w →Φ(ξ(x^+_γ)). Since ρ is irreducible, {ξ(x) : x∈∂Γ} spans ^d. So we can pick some x ∈∂Γ such that ξ(x) ∉ V_1. By perturbing x (if necessary) we can also assume that x ≠ x^-. Then γ^n x → x^+ and so logλ_2/λ_1 =lim_n →∞1/nlog d_1(ρ(γ)^n ξ(x), ξ(x^+)) = lim_n →∞1/nlog d_1( ξ(γ^n x), ξ(x^+))=lim_n →∞1/nlog d_2( Φ(ξ(γ^n x)),Φ( ξ(x^+))) = lim_n →∞1/nlog d_2( ( ∧^k+1ρ(γ) )^nΦ(ξ(x)),Φ( ξ(x^+))).Notice that we used Equation (<ref>) in the third equality. Then applying Observation <ref> to ∧^k+1ρ(γ) we havelogλ_2/λ_1≤logλ_k+1/λ_k. We prove the opposite inequality in exactly the same way. Since ∧^k+1ρ is irreducible, {Φ(ξ(x)) : x∈∂Γ} spans ^d. So we can pick some x ∈∂Γ such that Φ(ξ(x)) ∉ V_2. By perturbing x (if necessary) we can assume that x ≠ x^-. Then γ^n x → x^+ and sologλ_k+1/λ_k = lim_n →∞1/nlog d_2( ( ∧^k+1ρ(γ) )^nΦ(ξ(x)),Φ( ξ(x^+))) =lim_n →∞1/nlog d_2( Φ(ξ(γ^n x)),Φ( ξ(x^+)))= lim_n →∞1/nlog d_1( ξ(γ^n x), ξ(x^+)) =lim_n →∞1/nlog d_1(ρ(γ)^n ξ(x), ξ(x^+)) ≤logλ_2/λ_1.Hence λ_2/λ_1= λ_k+1/λ_kand since γ∈Γ was an arbitrary element with infinite order this proves the theorem. §.§ Proof of Theorem <ref>We begin by recalling the theorem.Suppose that Γ≤_2() is a torsion-free cocompact latticeandρ : Γ→_d() is in the Hitchin component. If ξ: ∂Γ→(^d) is the associated boundary map and ξ(∂Γ) is a C^2 submanifold of (^d), then λ_1(ρ(γ))/λ_2(ρ(γ)) = λ_2(ρ(γ))/λ_3(ρ(γ))for all γ∈Γ.For the rest of the section suppose that Γ≤_2() is a torsion-free cocompact latticeandρ : Γ→_d() is in the Hitchin component.Let (^d) denote the full flag manifold of ^d. Then by Theorem 4.1 and Proposition 3.2 in <cit.> there exists a continuous, ρ-equivariant map F=(ξ^(1),…, ξ^(d)): ∂Γ→(^d) such that: *ξ=ξ^(1). *If x,y,z ∈∂Γ are distinct, k_1,k_2,k_3 ≥ 0, and k_1+k_2+k_3 =d, thenξ^(k_1)(x) + ξ^(k_2)(y) + ξ^(k_3)(z) = ^d is a direct sum.*If x,y,z ∈∂Γ are distinct and 0≤ k < d-2, then ξ^(k+1)(y) + ξ^(d-k-2)(x) + (ξ^(k+1)(z) ∩ξ^(d-k)(x) )= ^d is a direct sum. *If γ∈Γ∖{1}, then the absolute values of the eigenvalues of ρ(γ) satisfy λ_1(ρ(γ)) > … > λ_d(ρ(γ)).*If γ∈Γ∖{1} and x^+_γ∈∂Γ is the attracting fixed point of γ, then ξ^(k)(x^+_γ) is the span of the eigenspaces of ρ(γ) corresponding to the eigenvaluesλ_1(ρ(γ)), …, λ_k(ρ(γ)). Throughout the following argument we will identify a k-dimensional subspace W = { w_1, …, w_k} of ^d with the point [w_1 ∧…∧ w_k] ∈(∧^k ^d). Next fix distances d_1 on (^d) and d_2 on (∧^2^d) which are induced by Riemannian metrics. With the notation above, if γ∈Γ∖{1} and x ∈∂Γ∖{x^+_γ, x^-_γ}, then logλ_2(ρ(γ))/λ_1(ρ(γ)) = lim_n →∞1/nlog d_1(ξ(γ^nx), ξ(x^+_γ))and logλ_3(ρ(γ))/λ_2(ρ(γ)) =lim_n →∞1/nlog d_2(ξ^(2)(γ^nx), ξ^(2)(x^+_γ)) Fix γ∈Γ∖{1} and let λ_i = λ_i(ρ(γ)). Then let v_1, …, v_d ∈^d be eigenvectors of ρ(γ) corresponding to λ_1, …, λ_d. Then by Property (<ref>)ξ^(k)(x_γ^+) = { v_1,…, v_k}and ξ^(k)(x_γ^-) = { v_d-k+1,…, v_d}.Further, if w ∉{ v_1,v_3,…, v_d} then logλ_2/λ_1 = lim_n →∞1/nlog d_1(ρ(γ)^n[w], ξ(x^+_γ)).Notice, if x ∈∂Γ∖{x^+_γ, x^-_γ} then Property (<ref>) implies that ξ(x) ∉(ξ(x^+_γ) ⊕ξ^(d-2)(x^-_γ)) = ({ v_1,v_3,…, v_d})and so logλ_2/λ_1 = lim_n →∞1/nlog d_1(ξ(γ^nx), ξ(x^+_γ)). For the second equality, notice that v_i ∧ v_j areeigenvectors of ∧^2 ρ(γ). So λ_1 λ_2 is the absolute value of the largest eigenvalue of ∧^2 ρ(γ) andλ_1 λ_3 is the absolute value of the second largest eigenvalue of ∧^2 ρ(γ). So if w ∉{ v_i ∧ v_j: {i,j}≠{ 1,3}}, then logλ_3 /λ_2 =logλ_1λ_3 /λ_1λ_2 = lim_n →∞1/nlog d_2( (∧^2ρ(γ))^n[w], ξ^(2)(x^+_γ)). Now we claim that ξ^(2)(x)∉({ v_i ∧ v_j: {i,j}≠{ 1,3}}) when x ∈∂Γ∖{x^+_γ, x^-_γ}. Suppose that ξ^(2)(x) = [ w_1 ∧ w_2] where w_1 = ∑_i=1^d α_i v_iandw_2 = ∑_i=1^d β_i v_i.Then ξ^(2)(x) = [ ∑_1 ≤ i < j ≤ d (α_i β_j - α_j β_i) v_i ∧ v_j ].Now Property (<ref>) implies that ξ^(2)(x) +ξ^(d-3)(x^-_γ) + (ξ^(2)(x^+_γ) ∩ξ^(d-1)(x^-_γ) ) = ^dis direct. Since ξ^(d-3)(x^-_γ) + (ξ^(2)(x^+_γ) ∩ξ^(d-1)(x^-_γ) )= {v_2, v_4,…, v_d}we see that α_1 β_3 - α_3 β_1 ≠ 0. Thus ξ^(2)(x)∉({ v_i ∧ v_j: {i,j}≠{ 1,3}}). Sologλ_3 /λ_2 = lim_n →∞1/nlog d_2( ξ^(2)(γ^n x), ξ^(2)(x^+_γ)).Now assume that M = ξ(∂Γ) is a C^2 submanifold in (^d). Then define a map Φ: M →(∧^2^d) byΦ(m) = [v_1 ∧ v_2]where T_mM = (_{ v_1,v_2}). Since M is a C^2 submanifold, Φ is a C^1 map.With the notation above, Φ(ξ(x)) = ξ^(2)(x) for all x ∈∂Γ.Since { x^+_γ : γ∈Γ∖{1}} is dense in ∂Γ, it is enough to show that Φ(ξ(x_γ^+)) = ξ^(2)(x_γ^+) for γ∈Γ∖{1}. By property (5) above, ξ^(k)(x_γ^+) is the span of the eigenspaces of ρ(γ) corresponding to the eigenvaluesλ_1(ρ(γ)),…,λ_k(ρ(γ)) while ξ^(k)(x_γ^-) is the span of the eigenspaces of ρ(γ) corresponding to the eigenvaluesλ_d-k+1(ρ(γ)), …, λ_d(ρ(γ)).Now fix y ∈∂Γ∖{x^+_γ, x^-_γ}. By Properties (<ref>) and (<ref>), ξ(y) ∉(ξ(x_γ^+) ⊕ξ^(d-2)(x_γ^-))and so ξ(γ^n y) = ρ(γ)^n ξ(y) approaches ξ(x^+_γ) along an orbit tangential to ξ^(2)(x^+_γ). Which implies that Φ(ξ(x^+_γ)) = ξ^(2)(x^+_γ).With the notation above, Φ: M →(∧^2^d) is a C^1 embedding. By the previous lemma and Property (<ref>), Φ is injective. Since Φ is also C^1, d(Φ)_m ≠ 0 for some m ∈ M. So d(Φ)_m ≠ 0 on an open set. But sinceΦ∘ρ(γ) =( ∧^2ρ(γ)) ∘Φfor every γ∈Γ and Γ acts minimally on M, we see that d(Φ)_m ≠ 0 for all m ∈ M. Hence, since M is compact and Φ is injective, Φ is a C^1 embedding.Since Φ is a C^1 embedding, there exists C ≥ 1 such that 1/C d_1(m_1, m_2) ≤ d_2(Φ(m_1), Φ(m_2)) ≤ C d_1(m_1, m_2)for all m_1, m_2 ∈ M. Then by Lemma <ref> we have λ_3(ρ(γ))/λ_2(ρ(γ)) = λ_2(ρ(γ))/λ_1(ρ(γ))for all γ∈Γ. § AN ARGUMENT OF LIU In this section we explain how an argument of Liu <cit.> can be adapted to prove the following.Let (X,g) be a complete simply connected Riemannian d-manifold with ≥ -(d-1) and bounded sectional curvature. Suppose Γ≤(X,g) is a discrete subgroup and there exist C,r_0>0 and x_0 ∈ X such thatC e^(d-1)r≤#{γ∈Γ : d_X(x_0, γ x_0) ≤ r } for every r > r_0. Then X is isometric to real hyperbolic d-space.Essentially the only change in Liu's argument is replacing the words “by a standard covering technique” with the proof of Lemma <ref> below. Suppose for the rest of the section that (X,g) is a Riemannian manifold and Γ≤(X,g) is a discrete subgroup which satisfy the hypothesis of the theorem. Let d_X:X → X → be the distance, Vol denote the volume form, ∇ denote the gradient, and let Δ denote the Laplace-Beltrami operator on (X,g). Also, for x ∈ X and r > 0 defineB_r(x) = { y ∈ X : d_X(x,y) < r}. We begin by recalling a result of Ledrappier and Wang. <cit.> If there exists a C^∞ function u: X → such that ∇ u≡ 1 and Δ u ≡ d-1, then X is isometric to real hyperbolic space.Define ϕ = e^(d-1)u. Then ϕ is positive and by the chain ruleΔ(ϕ) = e^(d-1)u( (d-1)^2∇ u^2 - (d-1) Δ u) = 0.Further, ∇logϕ =(d-1)∇ u≡ d-1. So by Theorem 6 in <cit.>, X is isometric to real hyperbolic space. Next fixa point x_0 ∈ X and some very large R >0. Let d_0 : X → be the function d_0(x) = d_X(x,x_0). Next let _0⊂ X denote the cut locus of x_0. Then d_0 is smooth on X ∖ (_0∪{x_0}) and (_0) = 0.There exists r_n →∞ such that: if A_n = { x ∈ X : r_n-50R ≤ d_X(x_0, x) ≤ r_n + 50R },then lim_n →∞1/(A_n)∫_A_n ∖_0Δ d_0(x)dV = d-1. This is essentially claim 1 and claim 2 from <cit.>. First, the Laplacian comparison theorem (see Theorem <cit.>) immediately implies that lim sup_n →∞1/(A_n)∫_A_n ∖_0Δ d_0(x) dV ≤ d-1and so we just have to prove lim inf_n →∞1/(A_n)∫_A_n ∖_0Δ d_0(x) dV ≥ d-1. Let S_x_0 X denote the unit tangent sphere at x_0. For v ∈ S_x_0 X let τ(v) = min{ t > 0 : exp_x_0(tv) ∈_0}.Next for r >0 define C(r) = { v ∈ S_x_0 X : r < τ(v) }.Let J(r,v) be the non-negative function defined on ∪_r > 0{r }× C(r) such that: if φ∈ L^1(X, dV), then ∫_X φ(x) dV = ∫_0^∞∫_C(r)φ(exp_x_0(rv) ) J(r,v) dμ(v)drwhere dμ is the Lebesgue meaure on S_x_0X. For r > 0 let S_r = ∫_C(r) J(r,v) dμ(v).Then by Fubini's theorem∫_0^R S_r dr=Vol(B_g(x_0, R))for every R > 0. We claim that there exists r_n →∞ such that lim inf_n →∞S_r_n+50R/S_r_n-50R≥ e^100(d-1)R.Suppose such a sequence does not exist, then there exists ϵ > 0 and R_0 > 0 such that S_r+50R/S_r-50R≤ e^100(d-1)R(1-ϵ)for every r > R_0. But then an iteration argument implies that S_r≤ C(1-ϵ)^r/100R e^(d-1)rfor some C > 0 which is independent of r. But then Equation (<ref>) implies that h_vol(X,g) < (d-1). So we have a contradiction and hence there exists r_n →∞ such that lim inf_n →∞S_r_n+50R/S_r_n-50R≥ e^100(d-1)R.Next for v ∈ S_x_0X and r ∈(0,τ(v)), define H(r,v) = (Δ d_0)( exp_p(rv)). We have the following well known relationship between J and H, see for instance <cit.>,H(r,v) J(r,v) = ∂/∂ r J(r,v).Next definea_n(v): = min{τ(v), r_n-50R} andb_n(v) := min{ r_n+50R, τ(v) }.Then by Equation (<ref>)∫_A_n ∖_0 Δ d_0(x) dV= ∫_S_x_0 X∫_a_n(v)^b_n(v) H(r,v)J(r,v) dr dμ(v) = ∫_S_x_0 X∫_a_n(v)^b_n(v)∂ J/∂ r(r,v) dr dμ(v)= ∫_S_x_0 X J(b_n(v), v) - J(a_n(v), v) d μ(v)= S_r_n+50R-S_r_n-50R + ∫_{r_n-50R < τ(v) < r_n+50R} J(b_n(v), v) d μ(v)≥ S_r_n+50R - S_r_n - 50R.By using the volume comparison theorem for annuli, see <cit.>, we havelim sup_n →∞(A_n)/S_r_n-50R_n≤1/d-1( e^100(d-1)R_n-1 )and solim inf_n →∞ 1/(A_n)∫_A_n ∖_0Δ d_0(x) dV≥lim inf_n →∞S_r_n+50R - S_r_n - 50R/(A_n)≥(lim inf_n →∞S_r_n+50R/S_r_n-50R-1 )lim inf_n →∞S_r_n-50R/(A_n)≥ d-1.Next let M_n ⊂Γ be a maximal set such that * if γ∈ M_n, then γ B_R( x_0) ⊂ A_n, * if γ_1, γ_2 ∈ M_n are distinct, then γ_1 B_R(x_0) ∩γ_2B_R(x_0) = ∅. Then let E_n = M_n · B_R(x_0) ⊂ A_n. For R>0 sufficiently large, lim inf_n →∞(A_n)/e^(d-1)r_n >0andlim inf_n →∞(E_n)/(A_n) >0. We prove the second inequality first. Fix some δ∈ (0,R) such that: if γ_1, γ_2 ∈Γ andγ_1 B_δ(x_0) ∩γ_2 B_δ(x_0) ≠∅,then γ_1 x_0= γ_2x_0. Then let s_0 = #{γ∈Γ : γ x_0 = x_0} andN_n = {γ∈Γ : r_n -49R < d(γ x_0,x_0) ≤ r_n+49R }Then ( N_n · B_δ(x_0)) = (B_δ(x_0))/s_0# N_n.Moreover, since M_n was chosen maximally, we haveN_n · B_δ(x_0) ⊂ M_n · B_2R+δ(x_0).Then since (M_n · B_2R+δ(x_0)) / (M_n · B_R(x_0))≤ (B_2R+δ(x_0)) /(B_R(x_0))we havelim inf_n →∞(E_n)/# N_n >0.So it is enough to show that lim inf_n →∞# N_n/(A_n) >0. Now #N_n= #{γ∈Γ : d_X(x_0, γ x_0) ≤ r_n+49R } - #{γ∈Γ : d_X(x_0, γ x_0) ≤ r_n-49R }≥ C e^(d-1)(r_n+49R)-s_0/ B_δ(x_0) B_r_n-49R(x_0).By the Bishop volume comparison theorem, see <cit.>,there exists V_0 > 0 so that B_r_n-49R(x_0) ≤ V_0 e^(d-1)(r_n-49R)for all n > 0. So#N_n ≥ e^(d-1)r_n( Ce^49R - s_0V_0/ B_δ(x_0)e^-49R).Now C, s_0,V_0, δ do not depend on R>0 and R is some very large number so we may assume that( Ce^49R - s_0V_0/ B_δ(x_0)e^-49R) ≥ 1.Then#N_n ≥ e^(d-1)r_n.Finally, by the volume comparison theorem for annuli (see <cit.>) we havelim inf_n →∞ e^(d-1)r_n/(A_n) >0and solim inf_n →∞# N_n/(A_n) >0.This proves the second inequality. To prove the first inequality, notice that (A_n) ≥ B_δ(x_0)/s_0# N_n and then use Equation <ref>. There exists a sequence ϵ_n > 0 with lim_n →∞ϵ_n = 0 such that: if φ: A_n → [0,1] is a C^∞ function compactly supported in A_n, then∫_X d_0 Δφ dV - ∫_X ∖_0φΔ d_0 dV ≤ϵ_n (A_n). When d_0 is smooth on X ∖{x_0} and φ is compactly supported in X ∖{x_0}, then∫_X d_0 Δφ dV = ∫_X φΔ d_0 dVby integration by parts. So Lemma <ref> says that we can still do integration by parts in the case when d_0 is not smooth, but at the cost of some additive error which depends on the support of φ.Let τ(v), J(r,v), a_n(v), and b_n(v) be as in the proof of Lemma <ref>. Let I =∫_X d_0 Δφ dV. Since integration by parts holds for Lipschitz functions, we haveI = - ∫_X ∇ d_0 ·∇φ dV = -∫_S_x_0 X∫_a_n(v)^b_n(v)∂φ/∂ r(r,v)J(r,v) dr dμ(v)where φ(r,v) = φ(exp_x_0(rv)). Integrating by parts again and using Equation (<ref>)I = ∫_S_x_0 X∫_a_n(v)^b_n(v)φ(r,v) ∂ J/∂ r(r,v) dr dμ(v) -. ∫_S_x_0 Xφ(r,v) J(r,v)|_a_n(v)^b_n(v)dμ(v) = ∫_X ∖ C_0φΔ d_0 d V -. ∫_S_x_0 Xφ(r,v) J(r,v)|_a_n(v)^b_n(v)dμ(v). Next we estimate the absolute value of the second term in last equation. If τ(v) > r_n+50R, thenφ(a_n(v), v) = φ(b_n(v), v)=0since φ is compactly supported in A_n. Further, if τ(v) < r_n - 50R, then a_n(v) = b_n(v). Hence, if. φ(r,v) J(r,v)|_a_n(v)^b_n(v)≠ 0,then we must have τ(v) ∈ [r_n-50R, r_n+50R]. By the volume comparison theorem, there exists J_0> 0 such that J(r,v) ≤ J_0e^(d-1)r.Then since φ≤ 1, we have. ∫_S_x_0 Xφ(r,v) J(r,v)|_a_n(v)^b_n(v)dμ(v) ≤ J_0e^(d-1)rμ({ v : τ(v) ∈[r_n-50R, r_n+50R] }).Since μ is a finite measure, we havelim_n →∞μ({ v : τ(v) ∈[r_n-50R, r_n+50R] }) = 0.Then by the first part of Lemma <ref>,ϵ_n := J_0e^(d-1)r/(A_n)μ({ v : τ(v) ∈[r_n-50R, r_n+50R] })satisfies the conclusion of the lemma.Let {χ_i : i ∈} be a partition of unity for B_R(x_0). Then define ϕ_n = ∑_i=1^n χ_i. Then each ϕ_n is smooth, maps into [0,1], and has compact support in B_R(x_0). Moreover, ϕ_1 ≤ϕ_2 ≤… and if K ⊂ B_R(x_0) is a compact set, then K ⊂ϕ_n^-1(1) for n sufficiently large. Next let ϕ_n = ∑_γ∈ M_nϕ_n ∘γ^-1. Then ϕ_n is compactly supported in E_n andlim_n →∞1/(E_n)∫_X 1_E_n-ϕ_n dV = 0.There exists a sequence γ_n ∈ M_n such that lim_n →∞∫_Xd_0(γ_n x) Δϕ_n(x) dV = (d-1)(B_R(x_0)) Let c_n = 1/(B_R(x_0))max_γ∈ M_n∫_Xd_0(γ x) Δϕ_n(x) dV.By the Laplacian comparison theoem (see Theorem <cit.>),lim sup_x →∞Δ d_0(x) ≤ d-1in the sense of distributions, solim sup_n →∞ c_n ≤ d-1. And we just have to prove thatlim inf_n →∞ c_n ≥ d-1. Using Lemma <ref> and the Laplacian comparison theorem we haved-1 = lim_n →∞1/(A_n)∫_A_n ∖_0Δ d_0 dV = lim_n →∞1/(A_n)( ∫_(A_n ∖ E_n) ∖_0Δ d_0 dV + ∫_E_n ∖_0Δ d_0 dV) ≤lim inf_n →∞1/(A_n)( (d-1) (A_n ∖ E_n)+ ∫_E_n ∖_0Δ d_0(x) dV)=lim inf_n →∞1/(A_n)( (d-1) (A_n ∖ E_n)+ ∫_X ∖_0ϕ_n Δ d_0 dV).In the last equality above we used Equation (<ref>) and the fact that Δ d_0 is uniformly bounded. Then by Lemma <ref> and the definition of c_nd-1≤lim inf_n →∞1/(A_n)( (d-1) (A_n ∖ E_n) + ∫_X ∖_0d_0 Δϕ_ndV) ≤ (d-1)+lim inf_n →∞(c_n-d+1)(E_n)/(A_n).So by Lemma <ref>, we must have lim inf_n →∞ c_n ≥ d-1. Next consider the functions f_n : B_R(x_0) → given by f_n(x) = (d_0 ∘γ_n)(x) - (d_0 ∘γ_n)(x_0) = d(x_0,γ_n x) - d(x_0, γ_n x_0)Then each f_n is 1-Lipschitz and f_n(x_0)=0, so we can pass to a subsequence such that f_n converges locally uniformally to a function f:B_R(x_0) →. f is C^∞, Δ f ≡ d-1, and ∇ f≡ 1.Using elliptic regularity, to show the first two assertions it is enough to verify that Δ f ≡ d-1 in the sense of distributions on B_R(x_0). Let φ be a positive C^∞ function compactly supported in B_R(x_0). We can assume that φ≤ 1. Then∫_X f(x) Δφ(x) dV= lim_n →∞∫_B_R(x_0) d_0(γ_n x)Δφ(x) dV.So by the Laplacian comparison theorem (see Theorem <cit.>)∫_X f(x) Δφ(x) dV ≤ (d-1) ∫_X φ(x) dV. By Lemma <ref>lim_n →∞ ∫_X d_0(γ_n x) Δ (ϕ_n-φ)(x) dV = (d-1)(B_R(x_0)) - ∫_X f(x) Δφ(x) dV.Since φ≤ 1 and is compactly supported in B_R(x_0), the function ϕ_n - φ is non-negative for large n and so by the Laplacian comparison theorem lim_n →∞ ∫_X d_0(γ_n x) Δ (ϕ_n-φ)(x) dV ≤ (d-1)lim_n →∞∫_X (ϕ_n-φ) dV= (d-1)(B_R(x_0)) - (d-1)∫_X φ(x) dVThus∫_X f(x) Δφ(x) dV ≥ (d-1)∫_X φ(x) dV.Hence Δ f ≡ d-1 on B_R(x_0).Finally, by construction f is the restriction of some Busemann function to B_R(x_0) and so ∇ f≡ 1 on B_R(x_0) by Lemma 1 part (1) in <cit.>.Now we fix a sequence R_n →∞ and repeat the above argument to obtain functions h_n : B_R_n(x_0) → which satisfy ∇ h_n≡ 1 and Δ h_n ≡ d-1 on B_R_n(x_0). Since each h_n is 1-Lipschitz and h_n(x_0)=0, we can pass to a subsequence so that h_n → h where h : X → satisfies ∇ h≡ 1 and Δ h ≡ d-1. Then Xis isometric to real hyperbolic space by Lemma <ref>.§ EIGENVALUES OF CERTAIN SUBGROUPSSuppose d ≥ 3, Λ≤_d() is a discrete subgroup, and G ≤_d() is the Zariski closure of Λ. If* G = _d(), * d = 2n>2 and G is conjugate to (2n,), * d = 2n+1 > 3 and G is conjugate to (n,n+1), or* d=7 and G is conjugate to the standard realization of G_2 in _7(),then there exists some γ∈Λ such that λ_1(γ)/λ_2(γ)≠λ_2(γ)/λ_3(γ). By conjugating, we can assume that either G=_d(), d = 2n>2 and G=(2n,), d = 2n+1 > 3 and G=(n,n+1), or d=7 and G coincides with the standard realization of G_2 in _7().By the main theorem in <cit.> it is enough to find some element g ∈ G such that λ_1(g)/λ_2(g)≠λ_2(g)/λ_3(g).This is clearly possible when G = _d() and d ≥ 3.Consider the case when d = 2n>2 and G = (2n,). Then for any σ_1, …, σ_n ∈, G contains the matrix [e^σ_1 ; ⋱;e^σ_n ;e^-σ_1;⋱ ;e^-σ_n ].So picking σ_1 > σ_2> …> σ_n > 0 with σ_1-σ_2 ≠σ_2-σ_3 does the job. Consider the case when d=2n+1 > 3 and G = (n,n+1).Then for any σ_1, …, σ_n ∈, G contains a matrix g which is conjugate to the block diagonal matrix [ cosh(σ_1) sinh(σ_1); sinh(σ_1) cosh(σ_1); ⋱; cosh(σ_n) sinh(σ_n); sinh(σ_n) cosh(σ_n); 1 ].Notice that this matrix has eigenvalues e^σ_1, e^-σ_1, …, e^σ_n, e^-σ_n, 1. So picking σ_1 > σ_2> …> σ_n > 0 with σ_1-σ_2 ≠σ_2-σ_3 does the job when n ≥ 3 and picking σ_1 > σ_2 > 0 with σ_1 - σ_2 ≠σ_2 does the job when n=2. Finally consider the case when d=7 and G coincides with the standard realization of G_2 in _7(). The standard realization of G_2 in _7() can be described as follows. First let = { a_1 + a_2 i + a_3 j + a_4 k : a_1, …, a_4 ∈}be the quaternions. Then define the split Cayley algebra ℭ^' = ⊕ e with multiplication (a+be)(c+de) = (ac+db)+(bc+da)e.This is an 8-dimensional algebra overwith conjugation (a+be) = a - be.Next let _2 be the -linear transformations of ℭ^' which satisfyα(xy) = α(x)α(y).Then for α∈_2 and x ∈ℭ^' it is straightforward to verify that α(x) = α(x) (see for instance <cit.>). So _2 preserves the subspace _{ i, j, k, e, ie, je, ke}of purely imaginary elements. Since α(1) = 1 for every α∈_2,if we identify i, j, k, e, ie, je, ke withe_1, …, e_7 the standard basis of ^7 we obtain an embedding _2 ↪_7(). Now if t,s ∈ a tedious calculation shows that [ cosh(t) 0 0 0 sinh(t) 0 0; 0 cosh(s) 0 0 0 sinh(s) 0; 0 0 cosh(s+t) 0 0 0 sinh(s+t); 0 0 0 1 0 0 0; sinh(t) 0 0 0 cosh(t) 0 0; 0 sinh(s) 0 0 0 cosh(s) 0; 0 0 sinh(s+t) 0 0 0 cosh(s+t) ]is contained in the image of this embedding. This matrix has eigenvalues e^t, e^-t, e^s, e^-s, e^s+t, e^-(s+t), 1. So picking t > s > 0 with s ≠ t -s does the job.§ FACTS ABOUT LINEAR TRANSFORMATIONS In this section we describe some basic properties of the action of _d() on (^d). These facts are used in Section <ref> and are all simple consequences of Gelfand's formula. In this section we let v denote the Euclidean norm of a vector v ∈^d. For a non-zero d-by-d real matrix A let λ_d(A) ≤…≤λ_1(A)to be the absolute values of the eigenvalues of A (counting multiplicity) and let σ_d(A) ≤…≤σ_1(A)denote the singular values of A.Suppose that A is a non-zero d-by-d real matrix. Then λ_1(A) = lim_n →∞σ_1(A^n)^1/n.Moreover, there exists a proper subspace V ⊂^d such that logλ_1(A) = lim_n →∞1/nlogA^nvfor all v ∈^d ∖ V.Since the “moreover” part is usually not included in statements of Gelfand's formula we sketch the proof.Notice that the first part of Gelfand's formula implies thatlim sup_n →∞1/nlogA^nv ≤lim sup_n →∞1/nlog(σ_1(A^n)v ) = logλ_1(A)for nonzero v ∈^d. So we just have to show that there exists a proper subspace V ⊂^d such that lim inf_n →∞1/nlogA^nv ≥logλ_1(A).for all v ∈^d ∖ V. Using the Jordan decomposition we can write A as a product of three commuting matrices A = E S U where E is elliptic, S is real diagonalizable, and U is unipotent. Let χ_1, …, χ_k be the eigenvalues of S (not counting multiplicity) and let ^d = ⊕_i=1^k V_i denote the corresponding eigenspace decomposition. Then letV = ⊕{ V_i : χ_i≠λ_1(S)}.Also, define a new norm ·_* on ^d by w_* = √(∑_i=1^k v_i^2)where w = ∑_i=1^k w_i and w_i ∈ V_i. Since E is elliptic, there exists C > 1 such that:1/Cw≤E^nw≤ C wfor all n ∈ and w ∈^d. Further, since U^-1 is unipotent, Gelfand's formula implies thatlim_n →∞1/nlogσ_1(U^-n) =0Then if v ∈^d ∖ V we have lim inf_n →∞ 1/nlogA^nv= lim inf_n →∞1/nlogE^nS^nU^nv= lim inf_n →∞1/nlogU^nS^nv ≥lim inf_n →∞1/nlog( 1/σ_1(U^-n) S^nv )= lim inf_n →∞1/nlogS^n v.Then, by the equivalence of finite dimensional norms, lim inf_n →∞1/nlogS^n v = lim inf_n →∞1/nlogS^n v_* = lim inf_n →∞1/nlog( λ_1(A)^n v)= logλ_1(A).For the rest of the section, let d_ be a distance on (^d) induced by a Riemannian metric. We will use the following estimate. Suppose 𝔸⊂(^d) is an affine chart and ι : ^d-1→𝔸 is an affine automorphism. Then for any compact set K ⊂^d-1 there exists C>1 such that 1/Cv-w≤ d_(ι(v), ι(w)) ≤ C v-wfor all v,w ∈ K. This follows from a compactness argument.Suppose g ∈_d() is proximal and ℓ_g^+ ∈(^d) is the eigenline of g corresponding to the eigenvalue of largest absolute value.If v ≠ℓ^+_g and g^n v →ℓ^+_g, thenlogλ_2(g)/λ_1(g)≥lim sup_n →∞1/nlog d_(g^n v, ℓ^+_g ).Moreover, there exists a proper subspace V ⊂(^d) such that: if v ∈(^d) ∖ V and g^n v →ℓ^+_g, thenlogλ_2(g)/λ_1(g) = lim_n →∞1/nlog d_(g^n v, ℓ^+_g ). By changing coordinates we can assume that g = [ λ 0; 0 A ], ℓ_g^+ = [1:0:…:0], λ = λ_1(g), and λ_1(A) = λ_2(g). Through out the proof we will use the notation [v_1 : v_2] ∈(^d) where v_1 ∈ and v_2∈^d-1. With this notation g^n · [v_1 : v_2] = [ λ^n v_1 : A^nv_2 ] = [ v_1 : A^n/λ^n v_2 ].By Gelfand's formula A^n/λ^n→ 0 and so g^n · v →ℓ_g^+ if and only if v_1 ≠ 0. Next we fix a small neighborhood U of ℓ_g^+ such that U⊂{ [v_1 : v_2] : v_1 ≠ 0}.By Observation <ref> there exists C > 1 such that if v=[v_1 : v_2] and w = [w_1 : w_2] are in U, then 1/C v_2/v_1 - w_2/w_1≤ d_(v,w) ≤ Cv_2/v_1 - w_2/w_1.So if v=[v_1 : v_2] ∈(^d) and g^n v →ℓ_g^+, then by Equations (<ref>) and (<ref>) we havelim sup_n →∞ 1/nlog d_(g^n v, ℓ^+_g ) =lim sup_n →∞1/nlog( 1/λ^nA^n v_2)≤lim sup_n →∞1/nlog( 1/λ^nσ_1(A^n) )= logλ_2(g)/λ_1(g). Using the “moreover” part of Gelfand's formula, there exists a proper subspace V_0 ⊂^d-1 such that logλ_1(A) = lim_n →∞1/nlogA^nvfor all v ∈^d-1∖ V_0. Then let V = { [v_1 : v_2 ] ∈(^d) : v_2 ∈ V_0}.Then if v=[v_1:v_2] ∈(^d)∖ V andg^n v →ℓ_g^+, Equations (<ref>) and (<ref>) imply that lim_n →∞ 1/nlog d_(g^n v, ℓ^+_g ) =lim_n →∞1/nlog(1/λ^nA^n v_2)= logλ_2(g)/λ_1(g). Suppose that A ∈_d() and there exists n_k →∞ such that T = lim_k →∞1/A^n_k A^n_kin (^d). If v ∈ Im(T), then there exists generalized eigenvectors v_1, …, v_m ∈^d of A such that v= v_1 + … + v_mand the eigenvalues corresponding to v_1, …, v_m all have absolute value λ_1(A). By changing coordinates we can assume that A = [ A_1 0; 0 A_2 ]where A_1 ∈_k(), A_2 ∈_d-k(), every eigenvalue of A_1 has absolute value λ_1(A), and every eigenvalue of A_2 has absolute value strictly less than λ_1(A). Then every v ∈{ e_1,…, e_k} can be written as a linear combination of generalized eigenvectors in ^d whose corresponding eigenvalues have absolute value λ_1(A). Further by Gelfand's formula0=lim_k →∞1/A^n_k A_2^n_k.and soT = [ T_1 0; 0 0 ]for some k-by-k matrix T_1. Suppose that g ∈_d(), λ_1(g) = λ_2(g), and v_0 ∈^d is an eigenvector of g whose eigenvalue has absolute value λ_1(g). Then there exists a proper subspace V ⊂(^d) such that:0=lim_n →∞ 1/nlog d_(g^n v, [v_0] )for every v ∈(^d) ∖ V.Suppose that gv_0 = λ v_0. Let e_1,…, e_m be the standard basis of ^d. By making a change of coordinates we can assume that v_0 = e_1 andg = [ J 0; 0 A ]where J is a m-by-m upper triangular matrix with λ, …, λ down the diagonal. By Observation <ref>, we can fix a small neighborhood U of [e_1] and C > 1 such that: if w=[w_1:…:w_d] ∈ U, then1/C (w_2/w_1,…, w_d/w_1) ≤ d_([e_1],w) ≤ C(w_2/w_1,…, w_d/w_1) .Then fix δ > 0 such that: if w ∉ U, then d_(w,[e_1]) ≥δ. We consider two cases:Case 1: m> 1. Since J is upper triangular with λ, …, λ on the diagonal, ge_i ∈λ v_i + { e_1,…, e_i-1} fori = 1,…, m.LetV = [{e_1,…, e_m-1, e_m+1, …, e_d }]. Suppose that v=t(v_1,…,v_d)∈^d and [v] ∉ V. Then v_m ≠ 0. Let t(v_1^(n), …, v_d^(n)) := g^n v.Then v_1^(n)≤g^n v≤σ_1(g^n)vand v_m^(n) = λ^n v_m. Since d_ has finite diameter we see that0 ≥lim sup_n →∞ 1/nlog d_(g^n [v], [e_1] ).If g^n [v] ∉ U, then1/n log d_(g^n [v], [e_1] ) ≥1/nlogδ.And if g^n [v] ∈ U, then by Equation (<ref>) 1/n log d_(g^n v, [e_1] ) ≥-1/nlog(C) +1/nlog v_m^(n)/v_1^(n)≥-1/nlog(C) +1/nlogλ -1/nlogσ_1(g^n)-1/nlogv.Hence Gelfand's formula implies that 0 ≤lim inf_n →∞ 1/nlog d_(g^n [v], [e_1] ).So 0= lim_n →∞ 1/nlog d_(g^n [v], [e_1] ). Case 2: m = 1. Theng = [ λ 0; 0 A ]where A ∈_d-1(). Since λ_1(g) = λ_2(g), we see that λ_1(A) = λ_1(g). By the “moreover” part of Gelfand's formula there exists some proper subspace V_0 ⊂^d-1 such thatlogλ_1(A) = lim_n →∞1/nlogA^nvfor all v ∈^d-1∖ V_0. We will use the notation [v_1 : v_2] ∈(^d) where v_1 ∈ and v_2∈^d-1. With this notation g^n · [v_1 : v_2] = [ λ^n v_1 : A^nv_2 ] = [ v_1 : A^n/λ^n v_2 ].Then defineV = { [v_1 : v_2 ] ∈(^d) : v_2 ∈ V_0}. Fix some v ∈(^d) ∖ V. Since d_ has finite diameter we see that0 ≥lim sup_n →∞ 1/nlog d_(g^n [v], [e_1] ).If g^n [v] ∉ U, then1/n log d_(g^n [v], [e_1] ) ≥1/nlogδ.And if g^n [v] ∈ U, then by Equation (<ref>)1/nlog d_(g^n [v], [e_1] )≥ - 1/nlog C +1/nlog1/λ^n A^nv_2= - 1/nlog C +- logλ + 1/nlog A^nv_2.So by Equation (<ref>)lim inf_n →∞ 1/nlog d_(g^n [v], [e_1] ) ≥ 0. So 0= lim_n →∞ 1/nlog d_(g^n [v], [e_1] ).plain | http://arxiv.org/abs/1704.08582v3 | {
"authors": [
"Andrew Zimmer"
],
"categories": [
"math.DG",
"math.GT"
],
"primary_category": "math.DG",
"published": "20170427140654",
"title": "Projective Anosov representations, convex cocompact actions, and rigidity"
} |
^aLeung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 10617, Taiwan^bDepartment of Physics, National Taiwan University, Taipei 10617, Taiwan^cGraduate Institute of Astrophysics, National Taiwan University, Taipei 10617, Taiwan^dKavli Institute for Particle Astrophysics and Cosmology, SLAC National Accelerator Laboratory, Stanford University, Stanford, California 94305, USAWe investigate the entanglement entropy and the information flow of two-dimensional moving mirrors. Here we point out that various mirror trajectories can help to mimic different candidate resolutions to the information loss paradox following the semi-classical quantum field theory: (i) a suddenly stopping mirror corresponds to the assertion that all information is attached to the last burst, (ii) a slowly stopping mirror corresponds to the assertion that thermal Hawking radiation carries information, and (iii) a long propagating mirror corresponds to the remnant scenario. Based on such analogy, we find that the last burst of a black hole cannot contain enough information, while slowly emitting radiation can restore unitarity. For all cases, there is an apparent inconsistency between the picture based on quantum entanglements and that based on the semi-classical quantum field theory. Based on the quantum entanglement theory, a stopping mirror will generate a firewall-like violent emission which is in conflict with notions based on the semi-classical quantum field theory. Entropy evolution of moving mirrors and the information loss problemPisin Chen^a,b,c,d[[email protected]] and Dong-han Yeom^a[[email protected]]December 30, 2023 ================================================================================================§ INTRODUCTION The black hole information loss problem <cit.> is one of the most difficult yet essential paradoxes to be resolved toward the final theory of quantum gravity. The essence of the problem is this: how can we reconcile general relativity with unitary quantum mechanics. There is no commonly accepted answer yet, but here we provide a list of candidate resolutions to this problem <cit.>. 1. Information may not be conserved and the unitarity can be violated after evaporation <cit.>. However, this is contradictory with current investigations on holography and AdS/CFT <cit.>. Regarding this, there is still a debate, e.g., see <cit.>.2. Information can be preserved by Hawking radiation <cit.>. However, this may cause inconsistency <cit.>. Then, how can we overcome these paradoxes?3. Information is not with the Hawking radiation but with other objects <cit.>, e.g., via the following possibilities: 3-1. A last burst of the evaporation carries information <cit.> or the resolution/regularization of the singularity explains the whereabouts of the information <cit.>.3-2. A long-lifetime remnant carries all information <cit.>.3-3. A bubble universe <cit.> or large interior inside the black hole <cit.> carries all information.3-4. Some special quantum correlations can carry information <cit.>.Usual counter-arguments against 3-1 and 3-2 are related to the entropy bound <cit.>. In general, the last burst, the last stage of the black hole evaporation, or the Planck scale objects have too small an amount of the Bekenstein-Hawking entropy. Usual counter-arguments against 3-3 and 3-4 are about their generality <cit.>; these scenarios may not be applicable for the most general examples.4. There may be other options, for examples, effective loss of information <cit.>, etc. It is still unclear which is the correct answer. However, at least we can study their self-consistency by using some theoretical methods. One good toy model for this purpose is the moving mirror <cit.> (for historical remarks, see <cit.>). The moving mirror is a surface that has the reflective boundary condition. Due to this boundary condition, as the mirror accelerates, it can create thermal particles that mimic Hawking radiation from a black hole <cit.>. In addition, even though energy and entropy will be transmitted from the mirror, the causal structure is trivial and hence there is no way to lose unitarity. Therefore, we can see how the unitarity can be preserved by using this toy model and check which is the most probable idea among candidates of the resolution on the information loss problem <cit.>.One more fascinating point of moving mirrors is that they can be realized by tabletop experiments. Recently, Chen and Mourou <cit.> suggested that accelerating plasma mirrors can be generated by plasma-laser interactions. If one impinges a strong laser pulse into a plasma medium, a plasma mirror with a controllable reflective index can be generated. By adjusting the densities of plasma layers, one can even control the speed or acceleration of the mirror. Then one can find a parameter space where the thermal radiation is detectable, through which various quantum field theoretical expectations can be validated or falsified. Of course, there remain experimental challenges, but regarding this potentially possible experiments, it is definitely worthwhile to provide theoretical guidances. In this paper, especially we will focus on the information loss problem. The most important step toward solving the information paradox is to understand the entanglement entropy. In the simplest case, the two-dimensional moving mirror, people already have investigated the entanglement entropy <cit.>. Usually, the entanglement entropy depends on the cutoff scales. However, for two-dimensional systems with the conformal symmetry, one may apply the regularization and renormalization method to obtain a well-defined entanglement entropy, which was first shown by Holzhey, Larsen and Wilczek <cit.> and was later simplified by Bianchi and Smerlak <cit.>. Thanks to this formula, we can see quantitative details on the restoration of unitarity by the radiation from the mirror. By adjusting different mirror trajectories, one can mimic different black hole evaporation scenarios associated with different resolutions <cit.>, through which the self-consistency of these proposals can be examined.This paper is organized as follows. In SEC. <ref>, we review the physics of two-dimensional moving mirrors and the entanglement entropy formula. In SEC. <ref>, we show that the mirror can mimic different resolutions by adjusting the trajectory and it can be shown that some candidates are not suitable while some others are viable to explain the information loss problem. In SEC. <ref>, we discuss more about the consistency and figure out the tension between the quantum entanglement and the semi-classical quantum field theory. Finally, in SEC. <ref>, we summarize and discuss what one can learn and expect from the moving mirror experiments. § TWO DIMENSIONAL MOVING MIRRORS Let us consider a two-dimensional moving mirror in a conformal field theory with the metricds^2 = - α^2(u,v) dudv,where u is the retarded time (left-moving) and v is the advanced time (right-moving). There is a mirror at v = p(u), where we impose the reflective boundary condition on the mirror following the trajectory.First we separate this space-time into two subsystems (left of FIG. <ref>), where one corresponds to A = [u_0, u], while the other, B, is its complementary. In two dimensions, every right-moving modes will be bounced by the mirror as one traces backward in time. Therefore, one can interpret that A corresponds to the Hawking-like thermal radiation that is already radiated from the mirror, while B corresponds to the vacuum fluctuating modes complementary to the thermal radiation in A that comove with the mirror, which are still not radiated yet.Under this setting, it has been known that the entanglement entropy between A and B is formally defined from the density matrix ρ_A for A, whereρ_A = Tr_Bρand ρ = | Ψ⟩⟨Ψ | with | Ψ⟩ the quantum state of the total system. Using this, the entanglement entropy is defined byS(A|B) = - Tr_Aρ_Alogρ_A.For the two-dimensional conformal field theory, following Holzhey, Larsen and Wilczek <cit.>, this can be evaluated byS(A|B)= c/12log(u-u_0)^2/ϵ^2= c/12log(p(u)-p(u_0))^2/p'(u) p'(u_0) δ u δ u_0,where c is the central charge that, for simplicity, can be taken as unity and ϵ is the UV-cutoff. Here, δ u and δ u_0 both are introduced for the covariant regularization at u and u_0 <cit.>, while both should approach zero and hence still there remains a divergence.There is a cutoff dependence in this formula, but this can be avoided by subtracting a counter term. In Holzhey, Larsen and Wilczek <cit.>, this term is subtracted by the corresponding term in the stationary mirror limit, i.e., p(u) = u and p(u_0) = u_0, and they obtained the renormalized entropyS(A|B) = c/12log(p(u)-p(u_0))^2/(u-u_0)^2 p'(u) p'(u_0).Bianchi and Smerlak <cit.> further simplify this expression by choosing u_0→ - ∞ and they obtainedS(A|B) = - c/12log p'(u).The relation between the entanglement entropy S and the position of the mirror as a function of t, say x(t), can be presented by <cit.>S(t) = -1/6tanh^-1ẋ(t),where p'(u) = (1 + ẋ(t))/(1 - ẋ(t)). By using this relation, one can further estimate the out-going energy flux according to the Davies-Fulling-Unruh formula <cit.> as a functional of the entanglement entropy <cit.>:F(t) = 1/2π e^-12 S( 6 Ṡ^2tanh 6S + S̈) cosh^2 6S. There have been some attempts to generalize this result. First, one can generalize the entanglement entropy-flux relation for higher dimensional black hole systems <cit.>. Eq. (<ref>) works consistently except near the Page time and the end point of evaporation. Second, one can apply it to dynamical causal structures of black hole physics <cit.>. In this paper, we will not generalize this formula to various black holes (e.g., <cit.>) but focus on the moving mirror itself.§ ENTROPY EVOLUTION BY MOVING MIRRORS In this section, now we consider the entropy evolution associated with moving mirrors in two-dimensions. In addition to this, we will connect various trajectories of moving mirrors with candidate resolutions to the information loss problem.Before we discuss details, in order to describe a closed system with finite number of states <cit.>, we define and assume as follows: – Boltzmann entropy of the mirror side: S̅(M).– Boltzmann entropy of the radiation: S̅(R).– The total Boltzmann entropy of the radiation at u = ∞ is S̅_m, where we assume that S̅_m = log N is a constant for a closed system with a given constant N. Now we apply the entanglement entropy formula, where previous A corresponds to R and B corresponds to M in our new notations. The entanglement entropy and information will have the following properties: – The entanglement entropy is S(R|M), where due to unitarity S(R|M)=S(M|R).– We can define the information measure of M by I_M = S̅(M) - S(M|R).– Radiation will generate the mutual information between mirror and future infinity, where it can be measured by I_m = S(R|M) + S(M|R) - S(M ∪ R). Here, the entanglement entropy of the total system is S(M ∪ R) = 0. Hence, for the unitary system, I_m = 2 S(R|M).– We can define information of radiation R at future infinity by I_R = S̅(R) - S(R|M).Note that the total information is always conserved: I = I_M + I_m + I_R = log N = const (FIG. <ref>) <cit.>. In terms of the entanglement entropy, after the mirror stops to move (i.e., ẋ = 0), it becomes zero again and hence we can regard that information is transmitted from the mirror M to the future infinity by radiation R.In two-dimensional cases, the black hole entropy is proportional to the mass, but the exact proportionality relation depends on the model parameters of the black hole system <cit.>. Therefore, the exact estimation of the thermal entropy via radiation (in terms of energy) is less clear. Keeping this in mind, we illustrate possible mirror histories as well as corresponding information transmission. §.§ Partially reflective mirror If the role of the mirror (the reflective boundary condition) fails by any reason, then we may not be able to apply this entanglement entropy formula. This possibility was discussed in <cit.> when the mirror is not totally reflective. This may mimic the bubble universe scenario (3-3) <cit.> in the sense that the mirror takes a part of information (e.g., the left side of the future infinity, right of FIG. <ref>) that cannot be accessed by the other side (e.g., the right side of the future infinity). §.§ Totally reflective mirror We use the following model:dS(t)/dt =A sin^2πt/t_P 0 ≤ t < t_P,=- A t_P/t_f-t_Psin^2πt-t_P/t_f-t_P t_P≤ t < t_f,so that S”(t) is continuous at t_P (may not be differentiable but this does not matter for later discussions). We choose A = 1 and t_P = 10, remaining t_f as a free parameter. Of course, at once we fix a shape of S(t), then we can reconstruct the corresponding trajectory x(t) or p(u).Using this simplified model, we can name several different mirror trajectories, so-called (left of FIG. <ref>) – Suddenly stopping mirror: t_f = 15,– Slowly stopping mirror: t_f = 20,– Long propagating mirror: t_f = 50. §.§.§ Suddenly stopping mirror For the suddenly stopping mirror case, it should release a very strong energy burst. As the decelerating time becomes shorter and shorter, the last burst energy increases more and more. Hence, it is not unreasonable to assume that the last burst can have enough capacity to restore all information within a short time.However, it seems that the last burst of the moving mirror would emit too much energy (right of FIG. <ref>). In the moving mirror case, this is not a problem, since the mirror is accelerating and this implies that there is an external source to provide the energy. The situation is different, however, in the case of the black hole. Due to the evaporation process, the last burst energy cannot exceed more than the Planck mass.Therefore, this mirror trajectory cannot realize the idea of 3-1 for black holes. If the final burst or emission happens during a very short time with a bounded energy <cit.>, then it will not be helpful to explain the information loss problem.§.§.§ Slowly stopping mirror For the slowly stopping mirror case, one can imagine that after the Page time <cit.>, the mirror begins to decrease its velocity. Then the radiation around the stopping phase transmits information, and hence the thermal or Hawking-like radiation will carry information. Typically, the radiated energy before the Page time and after the Page time should be of the similar order and hence this is consistent with the usual picture of black hole complementarity.Then, the interesting question is whether there is any inconsistency, such as black hole complementarity <cit.>. In the moving mirror case, there is no formal interior of the black hole and hence it may not be easy to do the same duplication experiment <cit.>, but it will be still possible to apply the Almheiri-Marolf-Polchinski-Sully (AMPS) thought experiment <cit.>. We leave this topic to the next section.§.§.§ Long propagating mirror For the long propagating mirror case, after the Page time, it takes a very long time to finish the deceleration <cit.>. This mimics a very long lifetime remnant. One interesting observation is to see the total amount energy as a function of t_f (FIG. <ref>). As the lifetime increases, we can reduce the radiated energy after the Page time. This means that if the late time radiation is slow enough, even though the radiated energy after the Page time is very small, it can contain enough information. So, this mimics the possibility that a negligible energy (e.g., vacuum) carries all correlations <cit.> or a remnant preserves information <cit.>.There are some valuable comments. In the literature, there exist several models for two-dimensional dynamical black holes, especially by introducing a dilaton field. One famous example is the Callan-Giddings-Harvey-Strominger (CGHS) model <cit.>. In this model, we can introduce the exact form (up to the one-loop order) of the renormalized energy-momentum tensor <cit.> and can include semi-classical effects. By solving this system using numerical simulations <cit.>, we can see the detailed causal structures. One interesting numerical observation was that as the black hole finishes its evaporation, the total mass of the black hole approaches a universal constant <cit.>. Perhaps, one can interpret that this limit implies the breakdown of the semi-classical approach. However, several authors suggested interpreting that this final universal constant mass implies a kind of Planck scale remnants <cit.>. Still, it is not possible to conclude, but at least our mirror calculation supports that such a remnant picture can functionally work in terms of the entanglement entropy.There are some cautious comments for the generalization of this argument. First, we ignored non-perturbative effects. For the mirror case, it is less clear what is the role of non-perturbative effects. For black hole cases, the existence of remnants may cause the infinite production problem <cit.> (although for two-dimensional cases, there is a support in <cit.>). Second, we considered only for two-dimensional mirrors. Perhaps, for black hole cases, as the dimension increases, such a universal end-point mass may not exist.In summary, we cannot conclude that the remnant picture is completely viable. However, we are certain that, at least for two-dimensions, the remnant picture is viable and it may even be the real answer to the information loss problem.§ CONSISTENCY CHECK: IS THERE A FIREWALL FROM A MIRROR? Up to this section, we have investigated the entanglement entropy and the energy flux released from the mirror. The former is related to the unitary quantum mechanics whereas the latter is related to the semi-classical quantum field theory. In previous sections, we implicitly assumed that these two aspects are consistent. But are they really consistent?In this section, we first summarize the original version of the AMPS though experiment. We then apply similar thought experiments in the context of the mirror dynamics. We will conclude that such a thought experiment is indeed possible and this seems to reveal the tension between the quantum entanglement and the semi-classical quantum field theory even with a moving mirror. §.§ AMPS thought experiment: the original version Black hole complementarity assumes <cit.>: – A1. Unitarity. There exists an observer who can recover all information/correlation of collapsed matter into the event horizon.– A2. For an asymptotic observer, the local quantum field theory, i.e., semi-classical descriptions on Hawking radiation is a good description.– A3. For a free-falling observer, general relativity is a good description. Hence, the in-falling observer can probe inside the black hole until the observer touches the singularity.In addition, although usually people do not explicitly present, we need further two assumptions <cit.>: – A4. The area of the black hole is proportional to the coarse-grained entropy of the black hole. Hence, information begins to come out from the black hole by Hawking radiation around the Page time <cit.> (still the black hole is semi-classical).– A5. There is an ideal in-falling observer who satisfies A1-A4 and counts information/states.Now black hole complementarity argues that these assumptions are consistent and no observer can see the violation of the previous assumptions.Let us assume that a black hole is unitary and (after the Page time) Hawking radiation should contain information. If we denote that the earlier part (before the Page time) of radiation is ℰ and the later part (after the Page time) of radiation is ℒ (left of FIG. <ref>), then the entanglement entropy between ℰ and ℒ should strictly decrease as the black hole evaporates. On the other hand, the internal degrees of freedom is also independent according to general relativity, unless there is a special event inside the horizon. Therefore, the in-falling counterpart of later Hawking radiation ℱ_ℒ will form another independent system.Then AMPS suggested to check the strong subadditivity <cit.>, where it should be satisfied in general:S_ℰℒ + S_ℒℱ_ℒ≥ S_ℒ + S_ℰℒℱ_ℒ,where S_𝒳 is the von Neumann entropy of the system 𝒳. For two systems 𝒳 and 𝒴, S_𝒳𝒴≡ S_𝒳∪𝒴.Keeping this in mind, by relying on A5, let us assume that there exists a free-falling observer 𝒪 who satisfies A1 to A4. Then according to 𝒪, the following logic should be true. – L1. We know that after the Page time (by A1, A2, and A4), as the black hole evaporates, the entropy of the outside the black hole (ℰ∪ℒ) should gradually decrease from the maximum value S_ℰ to zero.Hence, 𝒪 obtains S_ℰℒ < S_ℰ.– L2. If there is no special event for 𝒪 (by A3), then a creation of a Hawking particle pair can be regarded as a localized unitary quantum evolution. Hence, ℒ∪ℱ_ℒ should be in a pure state and S_ℒℱ_ℒ = 0 for 𝒪. From the subadditivity relation|S_ℰ - S_ℒℱ_ℒ| ≤ S_ℰℒℱ_ℒ≤ S_ℰ + S_ℒℱ_ℒ,we obtain that S_ℰℒℱ_ℒ = S_ℰ.– L3. By plugging L1 and L2 to the strong subadditivity relation, we obtain the relationS_ℰ > S_ℰ + S_ℒ,where this is impossible.Therefore, the assumptions of black hole complementarity are inconsistent, if there is an observer who counts ℰ and ℒ outside the black hole, falls into the black hole after the Page time, and check the strong subadditivity inside the horizon. §.§ AMPS thought experiment: the mirror version For the moving mirror case, A1, A2, A3, and A5 would be satisfied. It is less clear whether A4 is true or not, but thanks to the entanglement entropy formula, we can define the Page time. Then the question is this: can the moving mirror overcome inconsistency? Regarding this, the crucial point is as follows: before the Page time, the mirror should accelerate, while after the Page time, the mirror should decelerate.Before the Page time, the mirror accelerates and there appears a horizon. Hence, a separation between the in-going mode and the out-going mode are possible <cit.> (right of FIG. <ref>). Let us distinguish that the out-going radiation before the Page time is ℰ, while its in-going counterpart is ℱ_ℰ <cit.>. On the other hand, during the decelerating phase (after the Page time), there is no well-defined horizon and hence the separation between ℒ and its counterpart ℱ_ℒ are not well-defined.Then, we can find three independent systems ℰ, ℱ_ℰ, and ℒ. Now we can apply the same analysis, so to speak L1', L2', and L3' as follows. – L1'. We know that after the Page time (by A1, A2, and A4), the entropy of the outside the mirror (ℰ∪ℒ) should gradually decrease from the maximum value S_ℰ to zero. Hence, S_ℒℰ < S_ℰ.– L2'. If there is no special event for 𝒪 (by A3), then a creation of a Hawking particle pair can be regarded as a localized unitary quantum evolution. Hence, ℰ∪ℱ_ℰ should be in a pure state and S_ℰℱ_ℰ = 0. From the subadditivity relation|S_ℒ - S_ℰℱ_ℰ| ≤ S_ℒℰℱ_ℰ≤ S_ℒ + S_ℰℱ_ℰ,we obtain that S_ℒℰℱ_ℰ = S_ℒ.– L3'. By plugging L1' and L2' to the strong subadditivity relation S_ℒℰ + S_ℰℱ_ℰ≥ S_ℰ + S_ℒℰℱ_ℰ, we obtain the relationS_ℰ > S_ℰ + S_ℒ,where this is impossible.Therefore, we can repeat the similar thought experiment of AMPS in the mirror case.The physical meaning is clear. In the black hole case, before and after the Page time, due to the unitarity, there is a pair of maximally entangled particles, where one is in the later part of Hawking radiation (let us call it α) and the other is in the earlier part of Hawking radiation (let us call it β). At the same time, according to the semi-classical quantum field theory, there is an entanglement between α and its counterpart α'. However, this is impossible, since α is impossible to be entangled with both β and α' at the same time.For the mirror case, due to the trajectory of the mirror, there is no α'. However, there is an accessible counterpart of β that we can call it β'. Therefore, now β causes an inconsistency since β is entangled with both β' and α. This is a simple explanation of the inconsistency of the moving mirror. §.§ Then, what happens? If this inconsistency is real, then what is wrong? Since there is a tension between the unitary quantum mechanics (i.e., the entanglement entropy formula) and the semi-classical quantum field theory (i.e., the radiation formula according to the mirror trajectory), there are basically two possibilities. The first possibility is that our entanglement entropy formula is wrong and even the unitarity itself would not be preserved. The other possibility is that the radiation formula breaks down after the Page time.The latter possibility may be a more conservative assumption. The Davies-Fulling-Unruh formula <cit.> is true in the context of the semi-classical analysis. However, quantum mechanically, there should be a partner mode of the out-going flux <cit.>, where the quantum interactions between the partner mode and the mirror itself are less clear in the semi-classical analysis. The interaction between the mirror and the partner mode will be turned on after the mirror begins to decelerate (i.e., after the Page time). If the interaction between the mirror and the in-going partner mode cannot be covered by the semi-classical analysis, then it is not so surprising that the interaction between the in-going partner mode and the mirror generates a violent effect.At the theoretical level, this corresponds to a quantum gravitational effect, the so-called firewall phenomena <cit.> that would be seen by an outside observer <cit.>. In the experimental connection, it is impossible for tabletop experiments to reach the quantum gravitational scale, but such violent effects may indeed burn or break the moving mirror. Therefore, in the tabletop experiment, if our interpretations are correct, then we conservatively expect that there will be a firewall-like effect that burns the moving mirror. § DISCUSSION: WHAT CAN WE LEARN FROM MIRROR EXPERIMENTS? In this paper, we investigated the entropy and the information transmission process in two-dimensional moving mirrors. This can in principle be fabricated by future laser-plasma experiments <cit.>. The task of this paper is to show a connection between the mirror experiments and the information loss problem of black holes.By designing the mirror trajectory, we can mimic possible candidates of resolutions to the information loss problem. We can summarize as follows. 1. A suddenly stopping mirror mimics the case that the final burst carries all information. However, this mirror analogy shows that in order to recover all information, the final burst should have a large amount of energy. Hence, we conclude that the final burst is not enough to carry all information in the realistic black hole evaporation.2. A slowly stopping mirror mimics the case that thermal Hawking radiation carries information.3. A long propagating mirror mimics a Planck scale remnant. Even though the remnant has a small amount of energy, as long as the lifetime is long enough, it can restore information. Therefore, this partly demonstrates the possibility of the remnant scenario. On the other hand, we also checked the apparent inconsistency between the scenarios based on the unitary quantum mechanics (the entanglement entropy formula) and the semi-classical quantum field theory (the Davies-Fulling-Unruh formula). This opens a possibility that the interaction between the in-going partner mode and the decelerating mirror can cause a violent effect that can be interpreted as the firewall-like phenomena. This can be definitely demonstrated by Chen-Mourou plasma-laser experiments <cit.>. This should happen for all possible decelerating trajectories, and hence, the analysis of the previous paragraph would remain tentative.For all cases, there is a period of rapid energy decrease around the Page time, though it is unclear whether we can detect the negative energy flux or not <cit.>. In any case, such a rapid energy drop can be a clear expectation from theoretical calculations that needs to be confirmed by future experiments.One additional question for the future investigation is as follows: how can we measure the entanglement entropy in realistic experiments? This should be in principle possible, but it is fair to say that it will be a challenge. If we can find a way to measure the entanglement entropy, then we can confirm or falsify the renormalized entanglement entropy formula experimentally. Of course, even if the formula is not very correct, we reasonably guess that the qualitative properties of the entanglement entropy should not be so different from the formula, and hence experimental results should not be very different from the present paper. We leave these topics as future tasks that need to be confirmed by upcoming experiments.§ ACKNOWLEDGMENTThe authors would like to thank Bill Unruh for critical comments about this work. 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"authors": [
"Pisin Chen",
"Dong-han Yeom"
],
"categories": [
"hep-th",
"gr-qc"
],
"primary_category": "hep-th",
"published": "20170427150135",
"title": "Entropy evolution of moving mirrors and the information loss problem"
} |
1ICRA and Dipartimento di Fisica, Sapienza Universitàdi Roma and ICRA, Piazzale Aldo Moro 5, 00185 Roma, Italy 2Université de Nice Sophia-Antipolis, Grand Château Parc Valrose, Nice, CEDEX 2, France3ICRANet, P.zza della Repubblica 10, 65122 Pescara, Italy 4ICRANet-Rio, Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, 22290–180 Rio de Janeiro, Brazil 5ASI Science Data Center, Via del Politecnico s.n.c., 00133 Rome, Italy 6Dept. of Physical Sciences, Earth and Environment, University of Siena, Via Roma 56, 53100 Siena, Italy Theoretical and observational evidences have been recently gained for a two-fold classification of short bursts: 1) short gamma-ray flashes (S-GRFs), with isotropic energy E_iso<10^52 erg and no BH formation, and 2) the authentic short gamma-ray bursts (S-GRBs), with isotropic energy E_iso>10^52 erg evidencing a BH formation in the binary neutron star merging process. The signature for the BH formation consists in the on-set of the high energy (0.1–100 GeV) emission, coeval to the prompt emission, in all S-GRBs. No GeV emission is expected nor observed in the S-GRFs. In this paper we present two additional S-GRBs, GRB 081024B and GRB 140402A, following the already identified S-GRBs, i.e., GRB 090227B, GRB 090510 and GRB 140619B. We also return on the absence of the GeV emission of the S-GRB 090227B, at an angle of 71^o from the Fermi-LAT boresight. All the correctly identified S-GRBs correlate to the high energy emission, implying no significant presence of beaming in the GeV emission. The existence of a common power-law behavior in the GeV luminosities, following the BH formation, when measured in the source rest-frame, points to a commonality in the mass and spin of the newly-formed BH in all S-GRBs. GRB 081024B and GRB 140402A: two additional short GRBs from binary neutron star mergers Y. Aimuratov1,2, R. Ruffini1,2,3,4, M. Muccino1,3, C. L. Bianco1,3, A. V. Penacchioni3,5,6, G. B. Pisani1,3, D. Primorac1,3, J. A. Rueda1,3,4, Y. Wang1,3 December 30, 2023 =============================================================================================================================================================§ INTRODUCTIONGamma-ray bursts (GRBs) have been historically divided into a two-fold classification based on the observed T_90 duration of their prompt emission: short GRBs with T_90≲2 s and long GRBs with T_90≳2 s <cit.>.The progenitor systems of short bursts are traditionally identified with binary neutron star (NS)and NS-black hole (BH) mergers <cit.>. This assumption has received observational supports by their localization, made possible by the X-ray emission of the afterglow, with large off-sets from their hosts galaxies, both late and early type galaxies with no star formation evidence <cit.>.A vast activity of numerical work on relativistic magnetohydrodynamical (MHD) simulation using the largest facilities in the world (equipped by supercomputers with peak performances of 6.8 PFLOPS[The acronym PFLOPS means Peta (10^15) Floating Point Operations per second.], see , 13.3 PFLOPS, see , and 10.51 PFLOPS, see ) have been developed with the declared goal of finding a jetted emission which they considered, without convincing observational support, to be a necessary step to develop short GRB models in merging binary NS-NS or binary BH-NS systems <cit.>. It is interesting that they themselves recognized the shortcoming of their approach: “...there is microphysics that we do not model here, such as the effects of a realistic hot, nuclear EOS [equation of state] and neutrino transport” <cit.>. They also expected such models would be further confirmed by the observation associated with gravitational waves (GWs) of aLIGO <cit.>.There is no observational signature for the role of MHD activities in GRBs, nor, as we show in this paper, for jetted emission in the X- and γ-rays, as well as in the ultrarelativistic GeV emission of short bursts (see Sec. <ref>). On the contrary, also in the case of short GRBs we have strong evidence for the necessary occurrence of hypercritical accretion process as already shown in long GRBs with the fundamental role of neutrino emission <cit.> and the value of the NS critical mass M_ crit^ NS (, see also ). We also established firm upper limits on the observation of GWs from short GRBs by aLIGO <cit.>.Our approach is markedly different from the traditional ones. Since <cit.> we started: a) daily systematic and independent analyses of the GRB data in the X-, γ-rays and GeV emission from Beppo-SAX <cit.>, Swift <cit.>, Fermi <cit.>, Konus-WIND <cit.>, and AGILE <cit.>. We extended our data analysis to the optical and radio data. b) We have developed theoretical and astrophysical models based on quantum and classical relativistic field theories. c) At every step we have verified that the theoretical considerations be consistent with the observational data.In this article we mainly address the study of NS–NS mergers and only at the end we refer to BH–NS binaries.In <cit.> a further division of the short bursts into two different sub-classes has been proposed, and specific observable criteria characterizing this division have been there given: * The first sub-class of short bursts is characterized by isotropic energies E_ iso≲ 10^52 erg and rest-frame spectral peak energies E_p,i≲2 MeV <cit.>. In this case the outcome of the NS–NS merger is a massive NS (MNS) with additional orbiting material <cit.>. An alternativescenario leads to a new binary system composed by a MNS and a less massive NS or a white dwarf (WD). For specific mass-ratios a stable mass-transfer process may occur from the less massive to the MNS <cit.>. Consequently, the donor NS moves outward by loosing mass and may also reach the beta-decay instability becoming a low-mass WD. In view of their moderate hardness and their low energetics, we have indicated such short bursts as short gamma-ray flashes<cit.>. There, the local rate of S-GRFs has been estimated to be ρ_0=3.6^+1.4_-1.0 Gpc^-3 yr^-1. * The second sub-class corresponds to the authentic short GRBs (S-GRBs) with E_ iso≳ 10^52 erg and E_p,i≳2 MeV <cit.>. In this system the NS–NS merger leads to the formation of a Kerr BH with additional orbiting material, in order to conserve energy and angular momentum <cit.>. A further characterizing feature of S-GRBs absent in S-GRFs is the presence of the 0.1–100 GeV emission, coeval to their prompt emission and evidencing the activity of the newly-born BH. In <cit.> the local rate of this S-GRB has been estimated to be ρ_0=(1.9^+2.8_-1.1)×10^-3 Gpc^-3 yr^-1. The impossibility of detecting the observed short GRB 140619B from LIGO was evidenced <cit.>. We return again in this article on the issue of non-detectability of GWs for S-GRBs. The above relative rate of these two sub-classes of short bursts has been discussed and presented in <cit.>. There, it has been shown that the S-GRFs are the most frequent events among the short bursts. This conclusion is in good agreement with the NS–NS binariesobserved within our Galaxy: only a subset of them has a total mass larger than M_ crit^ NS and can form a BH in their merging process <cit.>. There, in Fig. 3, it has been assumed M_ crit^ NS=2.67M_⊙ for a non-rotating NS, imposing global charge neutrality and using the NL3 nuclear model <cit.>. Similar conclusions have been also independently reached by <cit.> and <cit.>.We have identified three authentic S-GRBs: GRB 090227B <cit.>, GRB 090510 <cit.>, and GRB 140619B <cit.>. All of them populate the high energy part of the E_p,i–E_iso relation for short bursts <cit.> and have E_iso>10^52 erg. We have analyzed the above three S-GRBs within the fireshell model <cit.>. The transparency emission of the e^+e^- plasma (the P-GRB emission), the on-set of the prompt emission, the correlation between the spike emission of the prompt and CBM inhomogeneities have led to the most successful test and applicability of the fireshell model.A further and independent distinguishing feature between S-GRFs and S-GRBs has been found thank to the Fermi data: when these three S-GRBs fall within the Fermi-LAT field of view (FoV), a GeV emission occurs, starting soon after the P-GRB emission, related to the emission from a newly-born BH.In this paper, we present two additional S-GRBs: GRB 081024B and GRB 140402A.The S-GRB 081024 is historically important since that source gave the first clear detection of a GeV temporal extended emission from a short burst<cit.>. From the application of the fireshell model to this S-GRB we theoretically derived its redshift z=3.12±1.82 and, therefore, E_iso=(2.6±1.0)×10^52 erg, E_p,i=(9.6±4.9) MeV, and E_ LAT=(2.79±0.98)×10^52 erg. For the S-GRB 140402A, we theoretically derived a redshift z=5.52±0.93 which provides E_iso=(4.7±1.1)×10^52 erg and E_p,i=(6.1±1.6) MeV. A long-lived GeV emission within 800 s has been reported <cit.>. The total energy of the brightest GeV emission is E_ LAT=(4.5±2.2)×10^52 erg.We also updated the analysis of the GeV emission of the S-GRB 090227B.The apparent absence of the GeV emission has been already discussed in <cit.>, recalling that this source was outside the nominal LAT FoV, and only photons in the LAT low energy (LLE) channel and a single transient-class event with energy above 100 MeV were associated with this GRB <cit.>. A further updated analysis would indicate that, in view of the missing observations, in no way the absence of the GeV emission before ∼40 s in the source rest-frame can be inferred. From the analyses of the two additional S-GRB 081024B and S-GRB 140402A and the further check for the GeV emission associated to the S-GRB 090227B, we conclude that all S-GRBs correlate to the high energy emission implying no significance presence of beaming in the GeV emission.In Sec. <ref> we briefly recall the fireshell model and its implications for S-GRBs. In Secs. <ref> and <ref> we report the data analyses of the S-GRBs 081024B and 140402A, respectively, and show their theoretical interpretation within the fireshell model: from the theoretical inference of their cosmological redshift, their transparency emission parameters, to the details of the circumburst media where they occurred. In Sec. <ref> we summarize the properties of the GeV emission of all S-GRBs and show the characteristic common power-law behavior of ther rest-frame 0.1–100 GeV luminosity light curves. We discuss also the minimum Lorentz factor of the GeV emission Γ^ min_ GeV obtained by requiring that the outflow must be optically thin to GeV photons (namely to the pair creation process), as well as its possible energy source, i.e., the matter accretion onto the new formed BH. In Sec. <ref> we indicate that there is no evidence in favor or against a common behavior of the X-ray afterglows of the S-GRBs in view of the limited observations. In Sec. <ref>we shortly address the issue of the possible emission of short bursts from BH-NS binaries leading to the ultrashort GRBs <cit.> In Sec. <ref> we infer our conclusions.§ THE FIRESHELL MODELIn the fireshell model <cit.>, the GRB acceleration process consists in the dynamics of an optically thick e^+e^- plasma of total energy E_e^+e^-^tot – the fireshell.Its expansion and self-acceleration is due to the gradual e^+e^- annihilation, which has been described in <cit.>.The effect of baryonic contamination on the dynamics of the fireshell has been then considered in <cit.>, where it has been shown that even after the engulfment of a baryonic mass M_B, quantified by the baryon load B=M_Bc^2/E_e^+e^-^ tot, the fireshell remains still optically thick and continues its self-acceleration up to ultrarelativistic velocities <cit.>.The dynamics of the fireshell in the optically thick phase up to the transparency condition is fully described by E^tot_e^+e^- and B <cit.>. In the case of long bursts, it is characterized by 10^-4≲ B<10^-2 <cit.>, while for short bursts we have 10^-5≲ B≲ 10^-4 <cit.>.The fireshell continues its self-acceleration until the transparency condition is reached; then a first flash of thermal radiation, the P-GRB, is emitted <cit.>. The spectrum of the P-GRB is determined by the geometry of the fireshell which is dictated, in turn, by the geometry of the pair-creation region.In the case of the spherically symmetric dyadosphere, the P-GRB spectrum is generally described by a single thermal component in good agreement with the spectral data <cit.>.In the case of an axially symmetric dyadotorus, the resulting P-GRB spectrum is a convolution of thermal spectra of different temperatures which resembles more a power-law spectral energy distribution with an exponential cutoff <cit.>.After transparency, the accelerated baryons (and leptons) propagates through the circum-burst medium (CBM). The collisions with the CBM, assumed to occur in fully radiative regime, give rise to the prompt emission <cit.>. The spectrum of these collisions, in the comoving frame of the shell, is modeled with a modified BB spectrum, obtained by introducing an additional power-law at low energy with a phenomenological index α̅ which describes the departure from the purely thermal case <cit.>. The structures observed in the prompt emission of a GRB depend on the CBM density n_CBM and its inhomogeneities <cit.>, described by the fireshell filling factor ℛ. This parameter is defined as the ratio between the effective fireshell emitting area A_eff and the total visible area A_vis <cit.>. The n_CBM profile determines the temporal behavior (the spikes) of the light curve. The observed prompt emission spectrum results from the convolution of a large number of modified BB spectra over the surfaces of constant arrival time for photons at the detector <cit.> over the entire observation time. Each modified BB spectrum is deduced from the interaction with the CBM and it is characterized by decreasing temperatures and Lorentz and Doppler factors.The duration and, consequently, the moment at which the burst emission stops are determined by the dynamics of the e^+e^- plasma. The short duration is essentially due to the low baryon load of the plasma and the high Lorentz factor Γ≈10^4 (see Fig. 2 inand Fig. 4 in .The description of both the P-GRB and the prompt emission, requires the appropriate relative spacetime transformation paradigm introduced in <cit.>: it relates the observed GRB signal to its past light cone, defining the events on the worldline of the source that is essential for the interpretation of the data. This requires the knowledge of the correct equations relating the comoving time, the laboratory time, the arrival time, and the arrival time at the detector corrected by the cosmological effects.It is interesting to compare and contrast the masses, densities, thickness and distances from the BH of the CBM clouds, both in short and long bursts. In S-GRBs we infer CBM clouds with masses of 10^22–10^24 g and size of ≈10^15–10^16 cm, at typical distances from the BH of ≈10^16–10^17 cm (see Secs. <ref> and <ref> and ), indeed very similar to the values inferred in long GRBs <cit.>. The different durations of the spikes in the prompt emission of S-GRBs and long bursts depend, indeed, only on the different values of Γ of the accelerated baryons and not on the structure of the CBM: in long bursts we have Γ≈10^2–10^3 <cit.>, while in S-GRBs it reaches the value of Γ≈10^4 <cit.> (see Secs. <ref> and <ref>).The evolution of an optically thick baryon-loaded pair plasma, is generally described in terms of E_e^+e^-^ tot and B and it is independent of the way the pair plasma is created. This general formalism can also be applied to any optically thick e^+e^- plasma, like the one created via νν̅↔ e^+e^- mechanism in a NS merger as described in <cit.>, <cit.>, and <cit.>.Only in the case in which a BH is formed, an additional component to the fireshell emission occurs both in S-GRBs and in the binary-driven hypernovae (BdHNe, long GRBs with E_ iso>10^52 erg, details inat the end of the P-GRB phase: the GeV emission observed by Fermi-LAT and AGILE. As outlined in this article, this component has a Lorentz factor Γ>300 and, as we will show in Sec. <ref>, it appears to have an universal behavior common to S-GRBs and BdHNe. It is however important to recall that the different geometry present in S-GRBs and BdHNe leads, in the case of BdHNe, to the absorption of the GeV emission in some specific cases <cit.>. § THE S-GRB 081024B §.§ Observations and data analysis The short hard GRB 081024B was detected on 2008 October 24 at 21:22:41 (UT) by the Fermi-GBM <cit.>.It has a duration T_90≈0.8 s long and exhibits two main peaks, the first one lasting ≈0.2 s. Its location (RA, Dec)=(322^o.9, 21^o.204) (J2000) is consistent with that reported by the Fermi-LAT <cit.>. The LAT recorded 11 events with energy above 100 MeV within 15^o from the position of the burst and within 3 s from the trigger time <cit.>. Emission up to 3 GeV was seen within ∼5 s after the trigger <cit.>.GRB 081024B also triggered the Suzaku-WAM, showing a double peaked light curve with a duration of ∼0.4 s <cit.>. Swift-XRT began observing the field of the Fermi-LAT ∼70.3 ks after the trigger, in Photon Counting (PC) mode for 9.9 ks <cit.>.Three uncatalogued sources were detected within the Fermi-LAT error circle <cit.>, but a series of follow-up observations established that none of them could be the X-ray counterpart because they were not fading <cit.>.The above possible associations have been also discarded by the optical observations performed in the R_c-band <cit.>. Consequently, no host galaxy has been associated to this burst and, therefore, there no spectroscopic redshift has been determined. §.§.§ Time-integrated spectral analysis of the Fermi-GBM data We analyzed the data from the Fermi-GBM detectors, i.e., the NaI-n6 and n9 (8–900 keV) and the BGO-b1 (0.25–40 MeV), and LAT data [<http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/>] in the energy range 0.1 – 100 GeV.In order to obtain detailed Fermi-GBM light curves we analyzed the(Time-Tagged Events) files [<ftp://legacy.gsfc.nasa.gov/fermi/data/gbm/bursts>] with the package. [<http://fermi.gsfc.nasa.gov/ssc/data/analysis/rmfit/vc_rmfit_tutorial.pdf>]In Fig. <ref> we reproduced the 50 ms binned GBM light curves corresponding to the NaI-n9 (8 – 260 keV, top panel) and the BGO-b1 (0.26 – 40 MeV, second panel) detectors. We also reproduced the 100 ms binned LAT light curve (0.1 – 100 GeV, third panel) and the corresponding high energy detected photons (bottom panel), both consistent with those reported in <cit.>. All the light curves are background subtracted. The GBM light curves show one narrow spike of about 0.1 s, followed by a longer pulse lasting around ∼0.7 s.The time-integrated analysis was performed in the time interval from T_0-0.064 s to T_0+0.768 s which corresponds to T_90 duration of the burst and T_0 is the trigger time.We have fitted the corresponding spectrum with two spectral models: Comptonized (Compt, i.e., a power-law model with an exponential cutoff) and Band <cit.>, see Fig. <ref> and Tab. <ref>. The Compt and the Band models provide similar values of the C-STAT (see Tab. <ref>).Therefore, the best-fit is the Compt model because it has one parameter less than the Band one. §.§.§ Time-resolved spectral analysis of the Fermi-GBM dataWe have also performed the time-resolved analysis by using 16 ms bins. After the rebinning the GBM light curves still exhibit two pulses: the first pulse observed before the LAT emission on-set, from T_0-0.064 s to T_0+0.128 s, and the following emission, from T_0+0.128 s to T_0+0.768 s, hereafter dubbed as Δ T_1 and Δ T_2 time intervals, respectively.As proposed in <cit.>, the emission before the on-set of the LAT emission corresponds to the P-GRB emission, while the following emission is attributed to the prompt emission (see Sec. <ref>).The spectrum of the Δ T_1 time interval, which can be interpreted as the P-GRB emission, is equally best-fit, among all the possible models, by a black body (BB) and a Compt spectral models. Fig. <ref> and Table <ref> illustrate the results of this time-resolved analysis. From the difference in the C-STAT values between the BB and the Compt models (ΔC-STAT=9.88, see Tab. <ref>), we conclude that the simpler BB model can be excluded at >3σ confidence level.Therefore the best fit is the Compt model.As in the case of GRB 090510, a Compt spectrum for the P-GRB emission can be interpreted as the result of the convolution of BB spectra at different Doppler factors arising from the a spinning BH <cit.>.The spectrum of the Δ T_2 time interval, which can be interpretated as the prompt emission, is equally best-fit by a power-law (PL) and a Compt spectral models (see Fig. <ref> and Table <ref>). The PL and the Compt models are equivalent, though Compt model slightly improves the C-STAT statistic. However, because of the unconstrained value for the peak energy of the Compt model E_p, we conclude that the PL model represents an acceptable fit to the data. §.§ Theoretical interpretation within the fireshell modelWe proceed to the interpretation of the data analysis performed in Sec. <ref> within the fireshell model. §.§.§ The estimate of the redshift The identification of the P-GRB and of the prompt emission is fundamental in order to estimate the source cosmological redshift and, consequently, to determine all the physical properties of the e^+e^- plasma at the transparency point <cit.>. The method introduced in <cit.> allows to determine the source redshift from two main observational constraints: the observed P-GRB temperature kT, related to the theoretically-computed rest-frame temperature kT_blue=kT(1+z), and the ratio between the P-GRB fluence S_BB=F(Δ T_1)Δ T_1 and the total one S_tot=F(T_90)T_90, which represents a good redshift independent approximation for the ratio E_ P-GRB/E_e^+e^- (see Tab. <ref>).A trial and error procedure is then started, using various set of values for E_e^+e^-^ tot and B to reproduce the observational constraints.Each of these set of values provides various possible values for the redshift z from the relation between kT and kT_blue.The closure condition is represented by the E_ iso(z)≡ E_e^+e^-^tot, where E_ iso is computed taking into account the K-correction on S_tot <cit.>.The redshift verifying the last condition and the corresponding values of E_e^+e^-^ tot and B are the correct one for the source.The theoretical redshift z=3.12±1.82 together with all the other quantities so far determined are summarized in Tab. <ref> (for further details on the method see, e.g., ). The analogy with the prototypical source GRB 090227B (B=4.13×10^-5, ), GRB 140619B (B=5.52×10^-5, ), and GRB 090510 (B=5.54×10^-5, ) is very striking.The self-consistency of the above theoretical method to estimate the redshift has been tested in S-GRB 090510 <cit.>. In this case a theoretical redshift z_ th=0.75±0.17 has been derived, in agreement with the spectroscopic measurement z=0.903±0.003 <cit.>. §.§.§ Analysis of the prompt emission In the fireshell model, the prompt emission light curve is the result of the interaction of the accelerated baryons with the CBM <cit.>. After the determination of the initial conditions for the fireshell, i.e., E^tot_e^+e^- and B (see Tab. <ref>), to simulate the prompt emission light curve of the S-GRB 081024B (see Figs. <ref>) and its corresponding spectrum, we derived the CBM number density and the filling factor ℛ distributions and the corresponding attached errors (see Tab. <ref> and Fig. <ref>, top panel). The average CBM number density inferred from the prompt emissions of GRB 081024B is ⟨ n_ CBM⟩=(3.18±0.74)×10^-4 (see Tab. <ref>), and is larger than those of GRB 140619B, ⟨ n_CBM⟩ = (4.7±1.2)×10^-5 cm^-3 <cit.>, and GRB 090227B, ⟨ n_CBM⟩ = (1.90±0.20)×10^-5 cm^-3 <cit.>, but still typical of the S-GRB galactic halo environments.The simulation of the prompt emission light curve of the NaI-n9 (8 – 900 keV) data of GRB 081024B is shown in Fig. <ref> (middle panel).The short time scale variability observed in the S-GRB light curves is the result of the large values of the Lorentz factor (Γ≈10^4, see Tab. <ref>). Under these conditions the total transversal size of the fireshell visible area, d_v, is smaller than the thickness of the inhomogeneities (≈10^16 cm, see the values indicated in Tab. <ref>), justifying the spherical symmetry approximation <cit.> and explaining the no significant “broadening” in arrival time of the luminosity peaks.The corresponding spectrum is simulated by using the spectral model described in <cit.> with phenomenological parameters α̅=-1.99. The rebinned data within the Δ T_2 time interval agree with the simulation, as shown by the residuals around the fireshell simulated spectrum (see Fig. <ref>, bottom panel).§ THE S-GRB 140402A§.§ Observations and data analysis The short hard GRB 140402A was detected on 2014 April 2 at 00:10:07.00 (UT) by the Fermi-GBM <cit.>. The duration of this S-GRB in the 50–300 keV is T_90=0.3 s. It was also detected by the Fermi-LAT <cit.> with a best on-ground location (RA, Dec)=(207^o.47, 5^o.87) (J2000), consistent with the GBM one. More than 10 photons were detected above 100 MeV and within 10^o from the GBM location, which spatially and temporally correlates with the GBM emission with high significance <cit.>.This burst was also detected by the Swift-BAT <cit.>, with a best location (RA, Dec)=(207^o.592, 5^o.971) (J2000). No source was detected in the Swift-XRT data <cit.> after two pointings in PC mode, from 33.3 ks to 51.2 ks and from 56 ks to 107 ks, respectively. These two observation set are within the 3-sigma upper limit of the count rate of 3.6×10^-3 counts/s and 3.0×10^-3 counts/s, respectively <cit.>. Optical exposures at the full refined BAT position <cit.> took by the Swift-UVOT (during both the XRT pointings, ) and by Magellan (at 1.21 days after the burst, ) showed no optical afterglow. This allowed to set, respectively, 3-sigma upper limits of v>19.8 mag and of r>25.0 mag.Consequently, no host galaxy has been associated to this burst and, therefore, no spectroscopic redshift has been determined.§.§.§ Time-integrated spectral analysis of the Fermi-GBM dataIn Fig. <ref> we reproduced the 16 ms binned GBM light curves corresponding to detectors NaI-n3 (8 – 260 keV, top panel) and BGO-b0 (0.26 – 20 MeV, second panel), and the 0.2 s binned high-energy light curve (0.1 – 100 GeV, bottom panel).Also for this burst all the light curves are background subtracted.The NaI light curve shows a very weak and short pulse, almost at the background level, while the BGO signal exhibit two sub-structures with a total duration of ≈0.3 s. The vertical dashed line in Fig. <ref> represents the on-set of the LAT emission, soon after the first pulse seen in both the GBM light curves. The background subtracted LAT light curve within 100 s after the GBM trigger and the corresponding 20 photons with energies higher than 0.1 GeV are shown in Fig. <ref>.We performed the time-integrated spectral analysis in the time interval from T_0-0.096 s to T_0+0.288 s (hereafter T_90).To increase the poor statistics at energies ≲260 keV, we included also the data from the NaI–n0 and n1 detectors in the spectral analysis. Among all the possible models, BB and Compt equally best-fit the above data(see Fig. <ref> and the results listed in Tab. <ref>). From the value ΔC-STAT=5.99 between the above two models (see Tab. <ref>), we conclude that the Compt model is an acceptable fit to the data. Similar to the GRB 140619B <cit.>, also in the case of GRB 140402A the low-energy index of the Compt model is consistent with α∼0.From theoretical and observational considerations on the on-set of the GeV emission (see Sec. <ref> and Fig. <ref>), we investigate the presence of a spectrum consistent with a BB one, which corresponds to the signature of the P-GRB emission for moderately spinning BH <cit.>. §.§.§ Time-resolved spectral analysis of the Fermi-GBM dataThe first spike (see Fig. <ref>), observed before the on-set of the GeV, emission extends from T_0-0.096 s to T_0 (hereafter Δ T_1). Again BB and Compt spectral models equally best-fit the above data. As it is shown in Fig. <ref> and Tab. <ref>, the above two models are almost indistinguishable, with the low-energy index of the Compt model α=0.43±0.51 being consistent within almost 1-σ level with the low energy index of a BB (α=1).We conclude that the BB model is an acceptable fit to the data and identify the first pulse in the light curve with the P-GRB emission.The spectrum of the emission in the time interval from T_0 to T_0+0.288 s (hereafter Δ T_2) reveals that a Compt model fits slightly better the data points at ≈1 MeV and its low-energy index α=0.07±0.54 indicates that the energy distribution is somehow broader than that of a BB model (see Fig. <ref> and Tab. <ref>). The Compt model is consistent with the modified BB spectrum adopted in the fireshell model for the prompt emission <cit.>. Therefore we identify the Δ T_2 time interval with the prompt emission.§.§ Theoretical interpretation within the fireshell modelWe proceed to the interpretation of the data analysis performed in Sec. <ref> within the fireshell model.§.§.§ The estimate of the redshift After having identified of the P-GRB emission of the S-GRB 140402A (see Sec. <ref>), we follow the same loop procedure recalled in Sec. <ref> to infer the redshift, E_e^+e^-^ tot and B of the source.The results of this method are summarized in Tab. <ref>. In particular the theoretically derived redshift for this source is z=5.52±0.93. Again, the analogy with the S-GRBs 081024B (see Sec. <ref>), GRB 090227B <cit.>, 140619B <cit.>, and 090510 <cit.> is very striking.§.§.§ Analysis of the prompt emission Similarly to the case of the S-GRB 081024B (see Sec. <ref>), to simulate the prompt emission light curve of the S-GRB 140402A (see Fig. <ref>) and its corresponding spectrum, we derived the CBM number density and the filling factors ℛ distributions (see Tab. <ref> and Fig. <ref>, top panel).Also in this case the inferred values fully justify the adopted spherical symmetry approximation <cit.> and explain the negligible “dispersion” in arrival time of the luminosity peak. The average CBM number density in the case of GRB 140402A is ⟨ n_ CBM⟩=(1.54±0.25)×10^-3(see Tab. <ref>), which is similar to that inferred from GRB 081024B. The simulation of the prompt emission light curve of the BGO-b0 (0.26 – 40 MeV) data of GRB 140402A is shown in Fig. <ref> (middle panel).The simulation of the corresponding spectrum requires a phenomenological parameterα̅=-0.9. Fig. <ref> (bottom panel), displays the agreement between the rebinned data from the Δ T_2 time interval with the simulation. § THE GEV EMISSION IN S-GRBSBefore going into more details on the general properties of the S-GRB GeV emission, we briefly summarize the observational features and the data analysis of the high energy emission of the S-GRBs 081024B and 140402A, and then we turn back to a new analysis on the absence of the GeV emission in the S-GRBs 090227B. §.§ The GeV emission of the S-GRBs 081024B and 140402A We downloaded the LAT event and spacecraft data[<http://fermi.gsfc.nasa.gov/cgi-bin/ssc/LAT/LATDataQuery.cgi>] selecting the observational time, the energy range and the source coordinates <cit.>. We then made cuts on the dataset time and energy range, position <cit.>, region of interest (ROI) radius (typically10^o), and maximum zenith angle.[The maximum zenith angle selection excludes any portion of the ROI which is too close to the Earth's limb, resulting in elevated background levels.] Within the event selection recommendations for the analysis of LAT data using the Pass 8 Data (P8R2) we adopted the burst and transient analysis (for events lasting <200 s) with an energy selection of 0.1 – 500 GeV, a ROI-based zenith angle cut of 100^o, an event class 16, and the instrument response function .[<http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/ Cicerone_Data_Exploration/Data_preparation.html>] The additional selection of the good time intervals (GTIs) when the data quality is good () is introduced to exclude time periods when some spacecraft event has affected the quality of the data (in addition to the time selection to the maximum zenith angle cut introduced above).In the case of the S-GRB 081024B, we obtained the GeV light curve and the observed photon energies showed in Fig. <ref> (third and fourth panels), which are in agreement with those reported in <cit.>. In the case of the S-GRB 140402A, we obtained the GeV light curve showed in Fig. <ref> (upper plot).About 20 photons with energies higher than 0.1 GeV have been detected within 100 s after the GBM trigger (see Fig. <ref>, lower panel). The highest energy photon is a 3.7 GeV event which is observed at T_0+8.7 s.Then, we built up the rest-frame 0.1 – 100 GeV light curve of the S-GRBs 081024B and 140402A.For the S-GRB 081024B, we rebinned its GeV emission luminosity light curve into two bins, as displayed in <cit.>. For the S-GRB 140402A, we rebinned it into two time bins with enough photons to perform a spectral anlysis: from T_0 to T_0+0.6 s, and from T_0+0.6 s to T_0+20 s. The resulting luminosity light curves follow a common power-law trend with the rest-frame time which goes as t^-1.29±0.06 (see dashed black line in Fig. <ref>). All the light curves are shown from the burst trigger times on, while in the case of the S-GRB 090510 it starts after the precursor emission, i.e., from the P-GRB emission on <cit.>. The GeV emission of the S-GRB 140402A is the second longest in time duration after GRB 090510, which exhibits a common behavior with the light curves of the other S-GRBs after ∼1 s rest-frame time (see Fig. <ref>).Tab. <ref> lists the redshift, E_p,i, E_iso (in the rest-frame energy band 1–10000 keV), and the GeV isotropic emission energy E_LAT in the rest-frame energy band 0.1–100 GeV of the five authentic S-GRBs discussed here. These values of E_LAT are simply obtained by multiplying the average luminosity in each time bin by the corresponding rest-frame duration and, then, by summing up all the contributions for each bin. However, these estimates represent lower limits to the actual GeV isotropic emission energies, since at late times the observations of GeV emission could be prevented due to instrumental threshold of the LAT.§.§ Reanalyzing the GeV emission of the S-GRB 090227BWe performed the unbinned likelihood analysis method,[<https://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/lat_grb_analysis.html>] which is preferred when the number of events is expected to be small, for the S-GRB 090227B. We took spectra within 1 s, 10 s, 100 s, and 1000 s, after the burst trigger. The background point like sources and diffuse (galactic and extragalactic) emission within 10^ o from the GRB position are taken from LAT 4-year Point Source Catalog (3FGL). The test statistic (TS) computed from the above likelihood analysis is TS≲1 in each time interval (TS>25 corresponds to 5-σ of significance), therefore, no significant GeV emission can be associated to this GRB. A single GeV photon with energy 1.59 GeV at time 896 s after the trigger and within 1^ o from the GRB has been found. Considering the above background models, we computed the probability for this photon to belong to this GRB. The likelihood analysis gives a probability of this photon to correlate to GRB 090227B of 0.36%, while its probability of being a photon from the diffuse background is >99%.The results of this analysis are in agreement with those reported in <cit.>. There, it is also stated that an autonomous repoint request by the Fermi-GBM brought the LAT down to ≃20^ o after ∼300 s and, therefore, the source entered in the optimal LAT FoV. By using the S-GRB common power-law trend t^-1.29±0.06 (see dashed black line in Fig. <ref>), we computed the expected energy fluxes of the GeV emission of the S-GRB 090227B f_1, at the time of ∼300 s when the source entered the LAT FoV, and f_2, at 896 s when the diffuse background photon was detected. We assumed a power-law spectrum with a typical value of the photon index of -2 and obtained f_1=(1.09±0.16)×10^-9 erg cm^-2s^-1 and f_2=(2.65±0.39)×10^-10 erg cm^-2s^-1. These computed fluxes are within the Fermi-LAT sensitivity of the Pass 8 Release 2 Version 6 Instrument Response Functions,[<http://www.slac.stanford.edu/exp/glast/groups/canda/lat_Performance_files/broadband_flux_sensitivity_p8r2_source_v6_all_10yr_zmax100_n03.0_e1.50_ts25.png>] which is approximately 10^-11 erg cm^-2s^-1. Therefore, we can conclude that the GeV emission associated to the S-GRB 090227B ceased before 300 s, when the source entered the LAT FoV. §.§ Lower limits on the GeV emission Lorentz factors in S-GRBsFollowing <cit.>, it is possible to derive a lower limit on the Lorentz factor of the GeV emission Γ^ min_ GeV by requiring that the outflow must be optically thin to high energy photons, namely to the pair creation process. Using the maximum GeV photon observed energy E^ max_ GeV in Tab. <ref>, for each S-GRB various lower limits on the GeV Lorentz factors can be derived from the time resolved spectral analysis. For each S-GRB we estimate lower limits in each time interval of the GeV luminosity light curves in Fig. <ref>.Then, Γ^ min_ GeV for each S-GRB has been then determined as the largest among the inferred lower limits (see Tab. <ref>). The GeV photons are produced in ultrarelativistic outflows with Γ^ min_ GeV≳300. §.§ The energy budget of the GeV emission in S-GRBs<cit.> proposed that the 0.1–100 GeV in S-GRBs (see Fig. <ref>) is produced by the mass accretion onto the newborn KNBH.The amount of mass that remains bound to the BH is given by the conservation of energy and angular momentum from the merger moment to the BH birth. We can estimate lower limits of the needed mass to explain the energy requirements E_ LAT in Tab. <ref> by considering the above accretion process onto a maximally rotating Kerr BH. In this case, depending whether the infalling material is in co- or counter-rotating orbit with the spinning BH, the maximum efficiency of the conversion of gravitational energy into radiation is η_+=42.3% or η_-=3.8%, respectively (see Ruffini & Wheeler 1969, in problem 2 of 104 in ). Therefore, E_ LAT can be expressed asE_ LAT= f_ b^-1η_± M_ acc^η_± c^2 , where f_ b is the beaming factor which depends on the geometry of the GeV emission, and M_ acc^η_± is the amount of accreted mass corresponding to the choice of the efficiency η_±. The observational evidence that the totality of S-GRBs exhibit GeV emission and that its absence is due instrumental absence of alignment between the LAT and the source at the time of the GRB emission (see Sec. <ref>) suggest that no beaming is necessary in Eq. <ref>.Therefore, in the following we set f_ b≡1. The corresponding estimates of M_ acc^η_± in our sample of S-GRBs are listed in Tab. <ref>.§ ON THE DETECTABILITY OF THE X-RAY EMISSION OF S-GRBS GRB 090510 is the only S-GRB with a complete X-ray afterglow <cit.>. Only upper limits exist for the X-ray afterglow emission of the other S-GRBs and no special features are identifiable.As an example to evidence the difficulty of measuring the X-ray afterglow in S-GRBs, we computed the observed X-ray flux light curve of GRB 090510, actually observed at z_ in=0.903, as if it occurred at the redshifts of the other S-GRBs, i.e., z_ fin=1.61, 2.67, 3.12, and 5.52. This can be attained through four steps. (1) In each time interval of the X-ray flux light curve f_ obs^ in of GRB 090510, we assume that the best fit to the spectral energy distribution is a power-law function with photon index γ, i.e., N(E)∼ E^-γ.(2) In the rest-frame of GRB 090510, we identify the spectral energy range for a source at redshift z_ fin which corresponds to the 0.3–10 keV observed by Swift-XRT, i.e.,0.3(1+z_ fin/1+z_ in)-10(1+z_ fin/1+z_ in)keV .(3) We rescale the fluxes for the different luminosity distance d_ l. Therefore, the observed 0.3–10 keV X-ray flux light curve f_ obs^ fin for a source at redshift z_ fin is given by f_ obs^ fin= f_ obs^ in[d_ l(z_ in)/d_ l(z_ fin)]^2∫^101+z_ fin/1+z_ inkeV_0.31+z_ fin/1+z_ inkeV N(E)E dE/∫^10keV_0.3keV N(E) E dE== f_ obs^ in[d_ l(z_ in)/d_ l(z_ fin)]^2(1+z_ fin/1+z_ in)^2-γ.(4) We transform the observational time t_ in of GRB 090510 at z_ in into the observational time t_ fin for a source at z_ fin by taking into account the time dilation due to the cosmological redshift effect, i.e., t_ fin=(1+z_ fin/1+z_ in)t_ in. Fig. <ref> shows that all the computed flux light curves are well below the observational upper limits provided by the Swift-XRT repointings. - S-GRB 090227B, no repointings (see Fig. <ref>(b)). - S-GRB 140619B, a repointing from 48.7 to 71.6 ks after the GBM trigger with an upper limit of 2.9×10^3 count/s (seeand Fig. <ref>(c)). - S-GRB 081024B, two repointings within the flux light curve in Fig. <ref>(d). Each upper limit was set by using the lowest count rate among those of the uncatalogued sources within the LAT FoV, later on confirmed as not being the burst X-ray counterparts: the first one at ∼70.3 ks after the trigger for ∼9.9 ks with a count rate of1.3×10^-3 counts/s <cit.>; the second one from 1.5 to 6.1 days with an average count rate of 7.4×10^-4 counts/s <cit.>. - S-GRB 140402A, two repointings <cit.>: the first from 33.3 to 51.2 ks with a count rate upper limit of 3.6×10^-3 counts/s; the second from 56 to 107 ks with an upper limit of 3.0×10^-3 counts/s (see Fig. <ref>(d)).We converted the above count rate upper limits in fluxes by multiplying for a typical conversion factor 5×10^-11 erg/cm^2/counts <cit.>.We conclude that there is no evidence in favor or against a common behavior of the X-ray afterglows of the S-GRBs in view of the limited observations.These aspects are noteworthy since in the case of long GRBs the X-ray emission has a very crucial role <cit.>, which is not testable in the case of S-GRBs.§ ON THE SHORT BURSTS ORIGINATING IN BH–NS MERGERSAs pointed out in <cit.>, <cit.> and <cit.>, U-GRBs are expected to originate in the BH–NS binaries produced by the further evolution of the BdHNe <cit.>. We recall that BdHN progenitor systems are composed of a carbon-oxygen core (CO_ core) and a NS in a close binary system. When the CO_ core explodes as a supernova (SN) Ib/c, its ejecta starts a hypercritical accretion process onto the companion NS, pushing its mass beyond the value M_ crit^ NS, and leading to the formation of a BH. This BH, together with the new NS (νNS) produced out of the SN event, leads to the progenitor systems of the U-GRBs.The orbital velocities of the BH–NS binaries formed from BdHNe are high and even large kicks are unlike to make these systems unbound <cit.>. U-GRBs represent a new family of BH–NS binaries unaccounted for in current standard population synthesis analyses <cit.>.U-GRBs are expected to lead to harder and shorter bursts in γ-rays, which explains the lack of their observational identification <cit.>, and pose a great challenge possibly to be considered to emit fast radio bursts. They also could manifest themselves, before the merging, as pulsar-BH binaries <cit.>. § CONCLUSIONSWe have first recalled the division of short bursts into two different sub-classes <cit.>: the S-GRFs, with E_ iso≲ 10^52 erg, E_p,i≲2 MeV and no GeV emission, and the authentic S-GRBs, with E_ iso≳ 10^52 erg, E_p,i≳2 MeV and with the presence of the GeV emission, always detected by Fermi-LAT, when operative <cit.>. We then focus on two additional examples of S-GRBs: GRB 081024B, with E_iso=(2.6±1.0)×10^52 erg and E_p,i=(9.6±4.9) MeV (see Sec. <ref>), and GRB 140402A, with E_iso=(4.7±1.1)×10^52 erg and E_p,i=(6.1±1.6) MeV (see Sec. <ref>).We perform time-integrated and time-resolved spectral analyses on both these sources (see Secs. <ref>–<ref> and Secs. <ref>–<ref>) and infer their cosmological redshifts (z=3.12 for the S-GRB 081024B and z=5.52 for the S-GRB 140402A, see Secs. <ref> and <ref>, respectively). We also identify their P-GRB spectral emission. The P-GRB emission of S-GRB 081024B exhibit the convolution of BB spectra at different Doppler factors arising from a spinning BH, in total analogy with S-GRB 090510 <cit.>. The P-GRB emission of S-GRB 140402A is consistent with a single BB, expected to occur for a moderately spinning BH <cit.>.The baryon load mass M_ B, the Lorentz Γ factor and the properties of the CBM clouds are in agreement with those of the other S-GRBs: M_ B≈10^-6 M_⊙, Γ≈10^4 (see Secs <ref> and <ref>), distances of the CBM clouds r≈10^16 cm and CBM densities n_ CBM≈10^-3 cm^-3 (see Secs <ref> and <ref>), typical of galactic halos environment <cit.>.In analogy to the other S-GRBs we confirm that the turn-on of the GeV emission starts after the P-GRB emission and is coeval with the occurrence of the prompt emission (see Sec. <ref>). All these coincidences point to the fact that the GeV emission originates from the on-set of the BH formation <cit.>.Most noteworthy, the existence of a common power-law behavior in the rest-frame 0.1–100 GeV luminosities (see Fig. <ref> in Sec. <ref>), following the BH formation, points to a commonality in the mass and spin of the newly-formed BH in all these S-GRBs. This result is explainable with the expected mass of the merging NSs, each one of M≈ 1.3–1.5 M_⊙ <cit.>, and the expected range of the non-rotating NS critical mass M^ NS_ crit∼ 2.2–2.7 M_⊙ leading to a standard value of the BH mass and of its Kerr parameter a/M∼1 <cit.>.Finally, in all S-GRBs the energetic of the GeV emission implies the accretion of M≳0.03–0.08 M_⊙ or M≳0.35–0.86 M_⊙ for co- or counter-rotating orbits with a maximally rotating BH, respectively (see Sec. <ref>). This accretion process, occurring both in S-GRBs and also BdHNe <cit.>, is currently being analyzed for the occurrence of r-process <cit.>.In all the identified S-GRBs, within the Fermi-LAT FoV, GeV photons are always observed <cit.>.This implies that no intrinsic beaming is necessary for the S-GRB GeV emission. The Lorentz factor of the GeV emission is Γ^ min_ GeV≳300.From Fig. <ref> for the S-GRBs and from Fig. <ref> for S-GRFs we conclude that in both systems there is no evidence for the early X-ray flares observed in BdHNe <cit.>.Before closing, we return to the issue of the GW detectability by aLIGO from S-GRBs. We have already evidenced their non detectability in GRB 090227B <cit.> and GRB 140619B <cit.> by aLIGO by computing the signal to noise ratio S/N up to the contact point of the binary NS components. In both cases each NS has been assumed to have mass M_ NS=1.34 M_⊙=0.5M_ crit^ NS. There, it has been concluded that the GW signals emitted in such systems were well below the value S/N=8 needed for a positive detection.These considerations have been extended in <cit.> to all S-GRBs. It was there concluded that such signals might be detectable for sources located at z≲ 0.14 (i.e., at distances smaller than the GW detection horizon of 640 Mpc) for the aLIGO 2022+ run. GRB 090510, to date the closest S-GRB, is located at z=0.903 (i.e., 5842 Mpc) and, therefore, it is outside such a GW detection horizon.We can then conclude that for sources at distances larger than that of GRB 090510, like GRB 081024B (at z=3.12) and GRB 140402A (at z=5.52) analyzed in this paper, no GW emission can be detected.We thank the referee for pleasant and expert advices. M. M. and J. A. R. acknowledge the partial support of the project N 3101/GF4 IPC-11, and the target program F.06790073-6/PTsF of the Ministry of Education and Science of the Republic of Kazakhstan. Y. 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B., Ivanova, L. N., & Nadezhin, D. K. 1972, , 16, 209[Zhang et al.(2012)Zhang, Shao, Yan, & Wei]2012ApJ...750...88Z Zhang, F.-W., Shao, L., Yan, J.-Z., & Wei, D.-M. 2012, , 750, 88 | http://arxiv.org/abs/1704.08179v2 | {
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"Y. Aimuratov",
"R. Ruffini",
"M. Muccino",
"C. L. Bianco",
"A. V. Penacchioni",
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"D. Primorac",
"J. A. Rueda",
"Y. Wang"
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"published": "20170426161443",
"title": "GRB 081024B and GRB 140402A: two additional short GRBs from binary neutron star mergers"
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thmTheorem[section] cor[thm]Corollary lem[thm]Lemma prop[thm]Proposition prob[thm]Problem definition exa[thm]Example definition defi[thm]Definition remark rem[thm]Remark | http://arxiv.org/abs/1704.07979v2 | {
"authors": [
"Peter Humphries",
"Snehal M. Shekatkar",
"Tian An Wong"
],
"categories": [
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"published": "20170426055630",
"title": "Biases in prime factorizations and Liouville functions for arithmetic progressions"
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EEG-Based User Reaction Time EstimationUsing Riemannian Geometry Features Dongrui Wu1, Senior Member, IEEE, Brent J. Lance2, Senior Member, IEEE, Vernon J. Lawhern23, Member, IEEE, Stephen Gordon4,Tzyy-Ping Jung5, Fellow, IEEE, Chin-Teng Lin6, Fellow, IEEE 1DataNova, NY USA2Human Research and Engineering Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, MD USA3Department of Computer Science, University of Texas at San Antonio, San Antonio, TX USA4DCS Corp, Alexandria, VA USA5Swartz Center for Computational Neuroscience & Center for Advanced Neurological Engineering,University of California San Diego, La Jolla, CA6Centre for Artificial Intelligence, Faculty of Engineering and Information Technology, University of Technology Sydney, AustraliaE-mail: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]===================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== There is a long history of representing a quantum state using a quasi-probability distribution: a distribution allowing negative values. In this paper we extend such representations to deal with quantum channels. The result is a convex, strongly monoidal, functorial embedding of the category of trace preserving completely positive maps into the category of quasi-stochastic matrices. This establishes quantum theory as a subcategory of quasi-stochastic processes. Such an embedding is induced by a choice of minimal informationally complete POVM's. We show that any two such embeddings are naturally isomorphic. The embedding preserves the dagger structure of the categories if and only if the POVM's are symmetric, giving a new use of SIC-POVM's, objects that are of foundational interest in the QBism community. We also study general convex embeddings of quantum theory and prove a dichotomy that such an embedding is either trivial or faithful. § INTRODUCTIONIn Feynman's famous 1981 paper on quantum computation <cit.> he writes “The only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative”. In this paper we wish to make this statement exact.Of course, much work has already been done in this regard going all the way back to the Wigner quasi-probability distribution in 1932 <cit.>. The Wigner function allows you to associate a probability distribution over a phase space for a quantum particle, with the only caveat that the probability sometimes has to be negative. The negativity appearing in probabilistic representations of quantum systems is something that lies at the heart of quantum theory: Spekkens has shown <cit.> that the necessity of negativity in probabilistic representations is equivalent to the contextuality of quantum theory. It is also a necessity for a quantum speedup as states represented positively by the Wigner function can be efficiently simulated <cit.>. An operational interpretation of negative probabilities is given by Abramsky and Brandenburger <cit.>.The main contribution of this paper will be to represent all of (finite-dimensional) quantum theory as a set of quasi-stochastic processes, not just the states. In particular we will use the language of category theory to establish that the category of quantum processes is a subcategory of the category of quasi-stochastic processes. A central object for studying these representations is the informationally complete POVM (IC-POVM), this is a measurement that completely characterizes a quantum state. We are particularly interested in IC-POVMs that are minimal as these form a basis of the statespace. A particular type of such a measurement is a symmetric IC-POVM: this is a very special type of POVM that is of particular interest to the QBism community <cit.> as it allows the updating of states to be written in a particularly elegant manner. It also allows quantum states to be written down with a minimal amount of negativity <cit.>. Minimal and symmetric POVMs turn out to be essential to preserving respectively the tensor product and adjoint in the quasi-stochastic representations considered in this paper. Representing quantum processes by quasi-stochastic matrices is not a new idea. In particular it is used in <cit.> to argue the similarity of a unitary process and the Born rule, although they stop short of extending the rule to all quantum channels and of composing them. In <cit.> they also don't go into detail about the compositional nature of these representations either. As far as the author is aware, this paper is the first to consider the compositional structure of quasi-stochastic representations of quantum theory. An approach that comes close is that of the duotensor framework of Hardy <cit.>, in particular his hopping metric is similar to the transition matrices T in this paper, but because Hardy's fiducial effects don't have to form a POVM the values in the hopping metric don't form a quasi-probability distribution. In <cit.> it was shown that the negativity in the representations can be overcome by modifying the way probabilities are calculated. In this paper this modification takes the form of a category that has a modified composition rule.We will represent quantum theory using a category close to that of Selinger's <cit.> consisting of trace preserving completely positive maps. This category models the dynamics of finite dimensional quantum systems where throwing away systems is allowed. It excludes classical systems which are, for instance, present in the category of finite dimensional C^*-algebras. Most of the results carry over to this setting, a small example of which we will give at the end of section <ref>. The category consisting of quasi-stochastic matrices that compose via matrix multiplication will be denoted . The primary contribution of the paper is the following theorem.Any family of minimal informationally complete (IC) POVM's gives rise to a functor F:→. This functor is faithful (injective) and strong monoidal (preserving tensor product), it furthermore preserves the convex structure of . The functors arising from two different families of minimal IC-POVM's are naturally isomorphic. The functor preserves the adjoint of unital channels if and only if the POVM's are generalised symmetric informationally complete (SIC). Note that the fact that any two representations by minimal POVM's are naturally isomorphic in a way seems to answer the question why there doesn't seem to be a preferred way to represent quantum theory by probabilities: the category doesn't `see' this difference.This theorem establishes that can be considered a subcategory of . is not a full subcategory: for a given system not all quasi-probability distributions correspond to valid quantum states. In particular, since no orthogonal IC-POVM exists, the probability distribution that is associated to a quantum state will always be somewhat mixed. This is qualitatively similar to the concept of Spekkens' epistricted theories <cit.>. As shown in that paper, many of the characteristic features of quantum theory (no-cloning, teleportation, dense coding, entanglement) arise in classical epistemically restricted theories: theories where states of maximal knowledge are not available. Some features of quantum theory however do not occur in classical epistemically restricted theories, primarily Bell / noncontextuality inequality violations and a computational speed-up. As shown by Abramsky and Brandenburger <cit.>, any non-signalling correlations (including those that maximally violate the Bell inequality) can be represented by a hidden variable model if one allows negative probabilities. Furthermore it was shown that any tomographically local theory has a complexity bound of AWPP <cit.> which was later shown by the same authors to be achieved by a computational model based on a quasi-probabilistic Turing machine <cit.>. These results together suggest that the features of quantum theory that don't occur in classical epistricted theories can be explained by the presence of negative probabilities in quantum theory and that in fact the necessity of this negativity is the `cause' of violating Bell inequalities and achieving a computational speed-up.We also prove a converse statement to the above theorem. Defining a quasi-stochastic representation of quantum theory to be a convex functor F:→ we show the following.To each quasi-stochastic representation we can associate a family of quasi-POVM's (POVM's that don't have to consist of positive components) that determine the representation on the states. Exactly one of the following holds for all quasi-stochastic representations. * All the components of the associated quasi-POVM's are multiples of the identity, in which case the representation is trivial.* All the associated quasi-POVM's are informationally complete, in which case the representation is faithful.Furthermore, for nontrivial representations it holds that * the representation is strong monoidal if and only if the associated quasi-POVM's are minimal informationally complete.* the representation preserves the dagger if and only if the associated quasi-POVM's are symmetric informationally complete.This gives an interesting new way to look at minimal generalised SIC-POVM's: they are the only POVM's that lead to a quasi-stochastic representation of quantum theory that is both strong monoidal (preserving the tensor product) and that preserves the dagger (the adjoint). The fact that each representation comes from a family of quasi-POVM's mirrors a very similar statements about frames in <cit.>.Any representation is either trivial or faithful and if it is strong monoidal it also has to be minimal. It might even be the case that along the lines of the proof of Theorem <ref> that any nontrivial representation has to be minimal regardless, although we do not show this. If this is in fact the case then it is a strong argument that minimal (quasi-)IC-POVM's are an essential object in quantum theory: they would be the only objects inducing faithful quasi-stochastic representations.Considering the above results it is a natural question to ask whether any non-signalling theory allows an embedding into the category of quasi-stochastic processes. This turns out to be true for any causal operational-probabilistic theory <cit.> which allows coarse-graining and is nondeterministic. This follows easily from the existence of minimal IC measurements in those theories and an adaptation of the proofs in this paper. Such a theory allows a strong monoidal embedding if and only if it allows local tomography.We can also try to start withand identify suitable properties of subcategories that make them isomorphic to . Work has already been done in this direction by Appleby, Fuchs, Schack and others <cit.>. They have restricted themselves to looking at state spaces. By considering the entire compositional structure of quantum theory, finding suitable axioms might be easier. In fact, using some simple axioms and the structure ofit is possible to derive the modified composition rule q(j) = ∑_i(α p(i) - β)r(j| i) used in those papers. This will appear in later work.We assume some basic familiarity with some concepts of category theory, namely that of a category, functor and natural transformation. All other categorical concepts will be explained when necessary. In section <ref> we will establish the standard way of associating stochastic matrices and probability distributions to quantum channels and states as done in for instance <cit.>. We will extend this to a functor in section <ref>. The preservation of the tensor product and the adjoint will be studied respectively in sections <ref> and <ref>. Finally, general representations of quantum theory are considered in section <ref>. The appendix contains some of the longer proofs.Acknowledgements:This work is supported by the ERC under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant n^o 320571. The author would like to thank the anonymous referees for their valuable comments and for pointing out additional references. § PRELIMINARIESLet A∈ M^n× m(ℝ) be a n× m matrix with real entries. It is called quasi-stochastic when the values in each column sum up to 1. It is called stochastic when it is quasi-stochastic and all the entries are positive. A matrix is called doubly (quasi-)stochastic when it is (quasi-)stochastic and its transpose is also (quasi-)stochastic. A doubly quasi-stochastic matrix is necessarily square. Note that if a (doubly) quasi-stochastic matrix has an inverse, this inverse will also be (doubly) quasi-stochastic. The inverse of a stochastic matrix is not necessarily stochastic (that is, some negative components might occur). Let M_n = M^n× n(ℂ). We call ρ∈ M_n a state when ρ≥ 0 and (ρ)=1. We call E∈ M_n an effect when 0≤ E≤ 1. A set of effects {E_i}⊆ M_n is called a POVM when ∑_i E_i = I_n where I_n denotes the identity. If the set of E_i are Hermitian and satisfy ∑_i E_i = I_n while no longer being necessarily positive, then we call this set a quasi-POVM[In fact, a quasi-POVM is nothing more than a Hermitian basis where the elements happen to sum up to the identity, we use the term `quasi-POVM' to be consistent about the use of `quasi-' to refer to an object that is usually positive, but in this case not.]. If a (quasi-)POVM spans M_n we call it Informationally Complete (IC) and if the elements of a IC-POVM are also linearly independent then it forms a basis for M_n and we call it minimal informationally complete. Such a set always has n^2 elements.Measuring a state ρ using a POVM {E_i} leads to probabilities p(i) = (ρ E_i) by the Born rule. Since the E_i are positive and they sum up to I_n these probabilities indeed form a proper probability distribution: ∑_i p(i) = 1. If the POVM is minimal IC then we can write each ρ uniquely in terms of the E_i:ρ = ∑_i α_i E_i/(E_i).We have chosen to normalise E_i to trace 1 so that the α_i sum up to 1.Now we can find a relation between the α's and the p(i)'s:p(i) = (ρ E_i) = ∑_j α_j (E_j/(E_j)E_i) = ∑_j T(i| j) α_jwhere T is a matrix defined asT_ij = T(i| j) = (E_j/(E_j)E_i). T is a stochastic matrix: ∑_i T(i| j) = 1, and it is doubly stochastic if and only if all the E_i have the same trace, which then necessarily has to be 1/n because n = (I_n) = ∑_i=1^n^2(E_i). We will refer to T as a transition matrix for {E_i}.We have the relation p = Tα where we consider p and α as vectors. T has to be invertible because the E_i form a basis, so α = T^-1 p, which means we can write ρ in terms of its probabilities over E_i:ρ = ∑_i (T^-1p)_i E_i/(E_i).Note that while T is stochastic, T^-1 will in every case contain some negative elements and so will be just quasi-stochastic <cit.>. This is not too surprising because if we had a POVM that has a T^-1 that is stochastic we would have succeeded in finding a non-contextual hidden variable model for quantum theory.Now suppose we have for M_m a minimal IC-POVM {E_i^'}. And consider a completely positive trace preserving (CPTP) map Φ: M_n→ M_m. Let σ = Φ(ρ). We wish to know the probability distribution of σ over E_i^' in terms of the probability distribution of ρ over E_i. Using the expansion of ρ in terms of p: q(i) = (σ E_i^') = (Φ(ρ)E_i^') = ∑_j (T^-1p)_j (Φ(E_j/(E_j))E_i^') = ∑_j r(i| j) (T^-1p)_jwhere we have introduced a matrix r defined asr_ij = r(i| j) =(Φ(E_j/(E_j))E_i^').The matrix r is again stochastic because Φ is trace preserving. The matrix is doubly stochastic if and only if the E_i and E_j^' all have the same trace and Φ is unital.We see that we have now translated the equation σ = Φ(ρ) into the equation q = rT^-1p. We can also translate composition of maps. Let Ψ: M_m→ M_k be CPTP and {E_i^''} be a minimal IC-POVM on M_k. SetT^'(i| j) = (E_j^'/(E_j^')E_i^'),s(i| j) = (Ψ(E_j^'/(E_j^'))E_i^''). Then setting σ^' = (Ψ∘Φ)(ρ) = Ψ(Φ(ρ)) = Ψ(σ) and letting q^' be the probability distribution associated to σ^'we can derive in the same way as before thatq^' = s(T^')^-1 q = s(T^')^-1 r T^-1 p.Since the map Ψ∘Φ is completely determined by what it does on states this also completely determines the matrix that we should associate with it, namely: s(T^')^-1 r.Note that the constructions in this section work equally well with quasi-POVMs, replacing probabilities with quasi-probabilities and stochastic matrices with quasi-stochastic matrices.§ FUNCTORIAL EMBEDDING FROM POVM'S We've seen how to translate composition of quantum channels into composition of quasi-stochastic matrices. To make this transition formal we will show that this induces a functor between the relevant categories. is the category which has as objects the natural numbers n>0, and its morphisms Φ: n→ m are CPTP maps Φ: M_n→ M_m. Composition is the usual composition of linear maps. States are maps ρ̃: 1 → n.Note that the definition of states corresponds to the definition of state we have used above, because if ρ̃: M_1 = ℂ→ M_n then it is completely defined by its action on 1, and we see that ρ̃(1) = ρ∈ M_n is positive and trace 1. We will most of the time simply use the actual states and not the morphism using the abuse of notation Φ(ρ) := Φ∘ρ̃. QStoch is the category which has as objects the natural numbers n>0 and its morphisms A: n→ m are quasi-stochastic matrices A∈ M^n× m(ℝ) and composition works by regular matrix multiplication. States p: 1 → n correspond to quasi-probability distributions. We saw above that when we associate stochastic matrices s and r to Ψ and Φ, the composition Ψ∘Φ has the matrix sT^-1 r associated to it, where T is determined by the choice of POVM on the intermediate space. We can capture this in a category by modifying the notion of composition: s*r := sT^-1r. Let T = (T^(i))_i=1^∞ be a family of invertible quasi-stochastic matrices where T^(i) is a i× i square matrix. Define QStoch_T to be the category with the same objects and morphisms as QStoch but with composition of r: n→ m and s: m→ k defined as s*r = s(T^(m))^-1r.The associativity of this composition follows from the associativity of matrix multiplication and the new identity morphism for n is now T^(n) so that _T is indeed a category for any choice of invertible quasi-stochastic matrices.Fix for every finite dimension n>0 a minimal IC-POVM {E_i^(n)} and let T = (T^(i)) be the family of matrices where T^(i) = T_n is the transition matrix for {E_j^(n)} as defined above when i=n^2 and otherwise let it be the identity matrix. Then Q:→_T defined by Q(n) = n^2 and Q(Φ: n → m)_ij = (Φ(E_j^(n)/(E_j^(n)))E_i^(m))is a faithful functor. We have defined the composition in the category _T precisely so that Q preserves composition: Q(Ψ∘Φ) = Q(Ψ)*Q(Φ), and it is easy enough to realise that Q(id_n) = T_n which acts as the identity in _T, so that Q is indeed a functor.Note that M_1 = ℂ has a unique choice for a minimal IC-POVM, namely {1}. We then see that Q(ρ̃: 1→ n)_i1 = (ρ̃(1)E_i^(n)) = (ρ E_i^(n)) which is just the Born rule as expected.Since the E_i are informationally complete, all the states are sent to different probability distributions. Now suppose Q(Φ)=Q(Ψ). Then we also get Q(Φ∘ρ) = Q(Φ)*Q(ρ) = Q(Ψ)*Q(ρ) = Q(Ψ∘ρ) so that we must have Φ∘ρ = Ψ∘ρ for all states ρ. Since a map is completely defined by its action on states we must then have Φ = Ψ, so that Q is indeed faithful. Note that this functor maps all states and maps to distributions and matrices with positive entries. All the negativity is hidden in the modified composition. This is analogous to the modified probability calculus of <cit.> and the urgleichung of <cit.>. We can bring this negativity more to the forefront using the following result. Let T = (T^i)_i=1^∞ be a family of invertible quasi-stochastic matrices of size i× i.The functor F_T:_T → defined by F_T(n) = n and F_T(A: n → m) = A(T^(n))^-1 is an isomorphism of categories. The identity morphism of n in _T is T^(n), so that F_T(id_n) = F_T(T^(n)) = T^(n)(T^(n))^-1 = I_n. That it preserves composition follows easily from the definition of composition in _T, so it is indeed a functor. That it is an isomorphism of categories follows because it has an inverse functor F_T^-1(A) = AT^(n). Now we see that the composition F_T∘ Q gives for any family of minimal IC-POVM's an embedding of quantum theory into the category of quasi-stochastic maps. States are all mapped to proper probability distributions (no negative components), while effects do contain negative components. Instead of F_T(A) = A(T^(n))^-1 we could have also defined the isomorphism F_T^'(A) = (T^(m))^-1A. In that case effects would be mapped to proper probabilistic effects and states would instead contain negative components.There are a lot of ways to represent quantum theory as a quasi-stochastic theory, but one of the problems is that it is hard to find a reason to prefer one over the other. Using the functorial embedding we can make it clear why there doesn't seem to be a preferred one: Let {E_i^(n)} and {F_i^(n)} be families of minimal IC-POVM's, and let the T^1 and T^2 be the respective families of matrices as defined in Theorem <ref>, and let Q_i:→_T_i be the respective functors. Then there exists a natural isomorphism η: F_T^1∘ Q_1 ⇒ F_T^2∘ Q_2. See appendix <ref>. Any minimal embedding is naturally isomorphic to another, which means that as far as the categories are concerned there really is only one embedding. Note that the category doesn't “see” negativity, so these embeddings can still be very different in terms of which maps are represented with negative components.We've now shown that all quantum channels and all states can be represented in a coherent manner in terms of quasi-probabilities, but this is not really all of quantum theory. We should also consider as done in <cit.> the probabilities arising from measuring a state using a POVM. So lets consider a measurement with K different outcomes where the probabilities for a state ρ are given as P(k|ρ) = (A_kρ) where {A_k}_k=1^K denote a POVM on M_n. Expanding ρ in terms of its stochastic representation: P(k|ρ) = ∑_i (T_n^-1Q(ρ))_i(A_k E_i/(E_i)). It is helpful to consider the POVM {A_k} as a map Â:M_n →ℂ^K in the category of C^*-algebras where Â(B)_k = (A_kB). Abusing notation a bit we can then writeQ(Â)(k| i) = ⟨ e_k,Â(E_i/(E_i))⟩ = (A_k E_i/(E_i)) where we let Q map ℂ^K to k and we take the standard basis {e_k} as `the POVM' for ℂ^K. This makes sense when we view ℂ^K as the diagonal matrices in M_K in which case the standard basis components e_k correspond to the diagonal projections to the kth component. It is easy to check that Q(Â) is indeed a stochastic matrix and we see that P(k|ρ) = ∑_i Q(Â)(k| i)(T_n^-1Q(ρ))_i so thatP(·|ρ) = Q(Â)T_n^-1Q(ρ) = Q(Â)*Q(ρ).This looks exactly the same as applying a CPTP map to a state. In this view measuring a POVM corresponds to a `quantum-classical' channel that takes as input a quantum state and outputs a classical state (a probability distribution).§ PRESERVING THE TENSOR PRODUCT An important part of quantum theory is the possibility of parallel composition: the tensor product. This can be captured by the fact that the category of quantum processes is a (strict) monoidal category[We will work exclusively with strict monoidal categories in this paper, so we will ignore the coherence isomorphisms that usually appear.]: A category 𝔸 is called strict monoidal when it has a bifunctor ⊗: 𝔸×𝔸→𝔸, and an identity object I, such that for all objects A,B,C ∈𝔸: I⊗ A = A⊗ I = A and A⊗ (B⊗ C) = (A⊗ B)⊗ C. The functoriality condition boils down to id_A⊗ id_B = id_A⊗ B and (f_1⊗ f_2)∘ (g_1⊗ g_2) = (f_1∘ g_1)⊗ (f_2∘ g_2) for all compatible morphisms.It is easy to verify that the linear algebraic tensor product turns and into monoidal categories, where on objects it acts as n⊗ m = nm. The relevant notion of a morphism between monoidal categories is that of a strong monoidal functor.A functor between two strict monoidal categories F: 𝔸→𝔹 is called strong monoidal when the two functors F_1(A,B) = F(A)⊗ F(B) and F_2(A,B) = F(A⊗ B) are naturally isomorphic via α: F_1⇒ F_2 such that the components α_A,B: F(A)⊗ F(B) → F(A⊗ B) satisfy the coherence condition α_A⊗ B, C∘ (α_A,B⊗ id_C) = α_A,B⊗ C∘(id_A ⊗α_B,C). The naturality condition means that α_B_1,B_2∘ (F(f_1)⊗ F(f_2)) = F(f_1⊗ f_2)∘α_A_1,A_2 for all morphisms f_i:A_i→ B_i.The functor is called strict monoidal when all the α are identities and it is called lax monoidal when the α are not necessarily isomorphisms. There are multiple choices for a tensor product in _T. We will choose the tensor product such that the functor F^':_T → defined by F^'(A:n→ m) = T_m^-1 A is strict monoidal. Denote the tensor product in _T by ⊗^', then we should have F^'(A⊗^' B) = F^'(A)⊗ F^'(B). Writing this out we get T_m_1m_2^-1 (A⊗^' B) = (T_m_1^-1⊗ T_m_2^-1)(A⊗ B) so that ⊗^' should be defined as A⊗^' B := T_m_1m_2(T_m_1^-1⊗ T_m_2^-1)(A⊗ B). It is instructive to check that this indeed turns _T into a monoidal category:(A_1⊗^' A_2)*(B_1⊗^' B_2)= T_k_1k_2(T_k_1^-1⊗ T_k_2^-1)(A_1⊗ A_2) T_m_1m_2^-1T_m_1m_2(T_m_1^-1⊗ T_m_2^-1)(B_1⊗ B_2) = T_k_1k_2(T_k_1^-1⊗ T_k_2^-1) (A_1T_m_1^-1B_1)⊗ (A_2T_m_2^-1B_2) = (A_1*B_1)⊗^' (A_2*B_2) We can now check that the functor F: _T→ is strong monoidal. We have F(A⊗^' B) = T_m_1m_2(T_m_1^-1⊗ T_m_2^-1)(A⊗ B) T_n_1n_2^-1 and F(A)⊗ F(B) = (A⊗ B)(T_n_1^-1⊗ T_n_2^-1). This can be rewritten toT_m_1m_2(T_m_1^-1⊗ T_m_2^-1) (F(A)⊗ F(B)) = F(A⊗^' B)T_n_1n_2(T_n_1^-1⊗ T_n_2^-1)which means our coherence isomorphisms are α_n_1,n_2 = T_m_1m_2(T_m_1^-1⊗ T_m_2^-1). It then follows by straightforward matrix multiplication that these satisfy the coherence conditions.Note that we could have chosen the tensor product in _T such that F would be strict monoidal and F^' would be strong monoidal. The reason we have chosen this tensor product is that it makes the following easier. The functor Q: →_T as defined in Theorem <ref> for a family of minimal IC-POVM's is strong monoidal with coherence isomorphisms α_n_1,n_2=S_n_1,n_2, whereS_n_1,n_2(j| i_1i_2) = (E_i_1^n_1/(E_i_1^n_1)⊗E_i_2^n_2/(E_i_2^n_2) E_j^n_1n_2)See appendix <ref>. The composition of two strongly monoidal functors is again strongly monoidal, so F∘ Q gives a strong monoidal functor of into . The functor is strong monoidal precisely because the S_n_1,n_2 are invertible and therefore are isomorphisms. These S are invertible because the POVM's are minimal. If they weren't minimal we wouldn't get a strong monoidal functor, but at most a lax monoidal functor.§ PRESERVING THE ADJOINT Another structure that exists on quantum channels is the adjoint. If we have a CP map Φ: M_n→ M_m, its adjoint is the dual of the map with respect to the Hilbert-Schmidt inner product. That is, a map Φ^†: M_m → M_n (note that the direction of the map has changed) such that⟨Φ(A), B ⟩_HS = (Φ(A)B^†) = (A(Φ^†(B))^†) = ⟨ A,Φ^†(B) ⟩_HS.If Φ(A) = UAU^†, then Φ^†(A) = U^† A U, e.g., for unitary conjugation the adjoint is simply the Hermitian adjoint of the unitary. Hamiltonian evolution in time is represented by the unitary exp(itH) where the Hermitian adjoint is exp(-itH). The adjoint can therefore be interpreted as reversing the time direction. It is also how one transfers between the Schrödinger and Heisenberg picture of quantum mechanics. The adjoint has the property that it distributes over composition, while changing the order of the morphisms: (Ψ∘Φ)^† = Φ^†∘Ψ^†. This behaviour can be defined in a more general way:A dagger category 𝔸 is a category which has a contravariant functor †: 𝔸→𝔸 that is the identity on objects and is its own inverse. Concretely this means that id_A^† = id_A and for all morphisms (f^†)^† = f, while composition satisfies (g∘ f)^† = f^†∘ g^†. We will call a category a partial dagger category, when the dagger is only defined for a subset of the morphisms. A dagger functor is a functor between (partial) dagger categories F: 𝔸→𝔹 that preserves the dagger (when it exists) in the following way: F∘†_𝔸 = †_𝔹∘ F, e.g. F(f^†_𝔸) = (F(f))^†_𝔹. The category is not a dagger category as the adjoint of a trace preserving map Φ will only be trace preserving when Φ is unital. We could consider the category of trace preserving unital maps, but this is a bit restrictive in that it doesn't have morphisms between different objects. Therefore we will stick with as a partial dagger category and only define the dual of a map when it is unital.is also only a partial dagger category with the transpose taking the role of the dagger. The transpose of a quasi-stochastic matrix is only quasi-stochastic when it is doubly quasi-stochastic (this is the exact analogue of trace preserving unital maps). We could again `fix' this by only considering doubly quasi-stochastic maps, but instead we will only define the dagger when a map is doubly quasi-stochastic._T doesn't always have a partial dagger structure. The obvious choice is the transpose: (A*B)^T = (AT^-1B)^T = B^T (T^T)^-1A^T. We need (A*B)^T = B^T * A^T so that equality here only holds when T = T^T. Looking at the definition of the transition matrix T this is the case when all the components of the POVM have equal trace. In that case we define the partial dagger on _T to be the transpose as defined on doubly quasi-stochastic matrices. Let {E_i^(n)} be a family of minimal IC-POVM's where all the components in a single POVM have equal trace, Then the functor Q as in Theorem <ref> is a dagger functor. The dagger in is only defined when Φ: M_n→ M_n is CPTP unital, which necessitates that the input and output dimension are equal. Let {E_i} be the POVM for n (dropping the superscript). We have (E_i) = 1/n for all i, because the POVM is minimal and all the traces are equal. So nowQ(Φ^†)_ij = (Φ^†(E_j/(E_j))E_i) = 1/n(E_iΦ^†(E_j)) = 1/n(E_i(Φ^†(E_j))^†) = 1/n(Φ(E_i)E_j^†) = Q(Φ)_jiwhere E_i^† = E_i, because they are by definition Hermitian. Indeed Q(Φ^†) = Q(Φ)^T as required. Somewhat surprisingly this won't in general extend to a dagger functor from to by appending Q with F:_T →. Let A be a n× n doubly quasi-stochastic matrix and assume that T is a symmetric matrix. Recall that F(A) = AT^-1_n which gives F(A^T) = A^T T^-1_n while F(A)^T = (AT^-1_n)^T = T^-1_n A^T. Now, of course in general A^T T^-1_n ≠ T^-1_n A^T so that this functor is not a dagger functor. In fact, it will only be a dagger functor if T commutes with all quasi-stochastic matrices. Let J_n denote the n× n matrix with every component equal to 1: (J_n)_ij = 1. Suppose the n× n doubly quasi-stochastic matrix T commutes with all n× n doubly quasi-stochastic matrices, then there are real constants α,β such that T = α I_n + β J_n where α + nβ = 1. All matrices will be n× n square matrices. We note that a matrix A is quasi-stochastic if and only if JA = J. It is furthermore doubly quasi-stochastic if and only if AJ = JA = J. J is obviously a rank 1 matrix, which has a single non-zero eigenvector: 1 = (1,…,1), such that J1 = n1. We therefore see that P = 1/nJ is a 1-dimensional projection, and that the linear subspace spanned by the doubly quasi-stochastic matrices is exactly {A ; [A,P] = 0 }. This commutation essentially fixes one of the eigenvalues of the matrices, but apart from that doesn't require anything extra. This space is therefore isomorphic to ℝ× M_n-1. Now any doubly quasi-stochastic matrix A can be written as PAP + (I_n-P)A(I_n-P). Therefore for T to commute with a doubly quasi-stochastic matrix A we need (I_n-P)T(I_n-P) to commute with (I_n-P)A(I_n-P) (since it automatically commutes on the other part). But (I_n-P)A(I_n-P) is just an arbitrary (n-1)× (n-1) matrix. We know that the only matrices in the space M_k that commute with all the matrices are multiples of the identity. We therefore have (I_n-P)T(I_n-P) = α(I_n-P) (noting that I_n-P acts as the identity on this space). And so in fact T = α(I_n-P) + β P = α I_n + (β-α) P = α I_n + 1/n(β-α)Jwhich with a proper relabelling becomes T = α I_n + β J. T is automatically symmetric, and it is quasi-stochastic exactly when α + nβ = 1. So for F to be a dagger functor, the transition matrices have to be particularly simple: T_n = α_n I_n + β_n J_n. Let {E_i} be a minimal IC-POVM for M_n. Suppose its transition matrix has that form, thenT_ij = (E_j/(E_j)E_i) = αδ_ij + β.We note that the right-hand side is symmetric, so that the left-hand side has to be as well, which means that (E_j) = 1/n by minimality. This POVM has a very specific symmetry property.We say that a POVM {E_i} is generalised symmetric informationally complete (generalised SIC) for M_n when it is minimal informationally complete and(E_iE_j) = αδ_ij + βfor some constants α and β. It is called SIC (not generalised) when all the components are rank-1: E_i = 1/nΠ_i for some projections Π_i. For such a SIC the constants need to be α = 1/n^2 and β = n-1/n^4.The existence and behaviour of such POVM's has been subject of intense study especially in the case of the rank 1 SIC's where the existence in arbitrary dimension is still not proven<cit.>. The generalised SIC's have been classified<cit.> and there exist many of them for each dimension.With the previous lemma and our observations about the functor preserving the dagger we get the following theorem.Let Q be the functor from Theorem <ref>. F∘ Q: → preserves the dagger if and only if all POVM's are generalised SIC. Q always preserves the dagger, so F∘ Q only preserves the dagger when F does. We've already seen that F preserves the dagger if and only if all the transition matrices have the special symmetric form corresponding to generalised SIC-POVMs. Recalling that the adjoint of a map can be interpreted as its time reversal, this gives an interpretation of SIC-POVMs as being the only POVMs `preserving the arrow of time' in the sense that the image of the time-reversal of a map is the time-reversal of its image. § GENERAL FUNCTORIAL EMBEDDINGSSo far we have only considered minimal IC-POVM's: A POVM the components of which form a basis for the underlying matrix space. However we can also consider POVM's consisting of more components. In that case the components won't be linearly independent so that a state can be represented in multiple ways as a combination of the POVM elements:ρ = ∑_i α_i E_i/(E_i) = ∑_i α^'_i E_i/(E_i).This means that (ρ E_i) = ∑_j α_j (E_j/(E_j)E_i) = ∑_j T(i| j)α_j = ∑_j T(i| j)α^'_jor as vectors: p = Tα = Tα^'. This is the case because T is no longer a full rank matrix so that it has a nontrivial kernel. Now we can consider the generalised inverse of T defined as having the same kernel as T but otherwise acting as an inverse which gives a canonical choice of α: α = T^-1p. We can also still transfer composition: Φ(ρ) ↦ Q(Φ)T^-1Q(ρ). This is well-defined in the sense that ker(T)⊂ ker(Q(Φ)), but we can no longer define a category _T as there is no longer an identity morphism: A*T≠ A, because T^-1T ≠ I_n. The composite functor F∘ Q where we map Φ↦ Q(Φ)T^-1 also doesn't work any more as we still don't map the identity to the identity. However, we could consider that there are some smart choices which do produce a valid functor. For instance there is a priori no reason that we can't let id↦ I. To preserve the convex structure we should then also change where the other maps are sent to. Suppose we have such a functor, how much of what we previously constructed still goes through? Looking at Theorem <ref> we see that we no longer get a natural isomorphism as the matrices involved are no longer invertible. We also see that the proof of Theorem <ref> explicitly requires the minimality of the representation since otherwise the morphisms involved wouldn't be isomorphisms. The most a nonminimal representation can therefore be is lax monoidal. There are no obvious barriers to a nonminimal representation preserving the adjoint.So far we have studied embeddings of into coming from POVMs. Let's consider a bit more general view. We call F: → a quasi-stochastic representation of when F is a functor that preserves the convex structure of the categories: F(tΦ_1 + (1-t)Φ_2) = tF(Φ_1) + (1-t)F(Φ_2) and maps 1 to 1 (so that states are sent to states). We call it a positive representation when it sends all states to nonnegative distributions. A representation is called strong monoidal / faithful / dagger preserving when the functor is. We call a representation trivial when for all n∈ℕ F(ρ)=F(σ) for all states ρ,σ∈ M_n.Let F:→ be a quasi-stochastic representation. Then we can find for every n∈ℕ a quasi-POVM {E_i^(n)} such that for a state ρ∈ M_n we have F(ρ)_i = (ρ E_i^(n)). Let n∈ℕ. We see that F restricts to a convex map f^(n): DO(n) → QProb(F(n)) where DO(n)⊂ M_n represents the set of density operators and QProb(k_n) is the set of quasi-probability distributions on F(n) points. We can split this map up into its components f^(n)_i: DO(n) →ℝ. These functions are again convex and using standard methods we can extend these first to linear maps on Hermitian matrices and then to linear maps to all matrices: f^(n)_i: M_n→ℂ. This is exactly an element of the dual space of M_n and since this is finite dimensional it is isomorphic to M_n which means there exists a E_i^(n)∈ M_n such that f^(n)(A) = ⟨ A, E_i^(n)⟩_HS = (A (E_i^(n))^†). Because this function is real valued on the Hermitian matrices, E_i^(n) has to be Hermitian and because we have 1 = ∑_i f_i^(n)(ρ) = (ρ∑_i E_i^(n)) we get ∑_i E_i^(n) = I_n proving that {E_i^(n)} is indeed a quasi-POVM.We will refer to these quasi-POVM's as the representation's associated quasi-POVM's. It is easy to see that the representation is positive if and only if the associated quasi-POVM's are true POVM's (consisting of only positive components).Note: If we start out with a family of POVM's and construct the regular functor Q: →_T and then append this with the functor F^': _T→ given by F^'(A) = T^-1A then F^'∘ Q will not be a positive representation, while F ∘ Q where F(A) = AT^-1 will be. F^'∘ Q and F∘ Q are naturally isomorphic, so positivity of the representation is not a `categorical' property in the sense that it isn't preserved by natural isomorphism.It is not hard to see that a representation is faithful if and only if the associated quasi-POVM's are informationally complete. On the other hand, suppose a representation is trivial, then for a element of the quasi-POVM E∈ M_n we must have (ρ E) = (σ E) for all states ρ,σ∈ M_n. Calling this value α and realising that α = (ρα I_n) we see that (ρ (E-α I_n))=0 for all states ρ so that E= α I_n. Perhaps somewhat surprisingly these are the only options for the associated POVM's:Let F: → be a quasi-stochastic representation of . This representation is either faithful or trivial. See appendix <ref>. This means that we immediately get the following corollary using Theorems <ref> and <ref>. Let F: → be a positive nontrivial representation.* F is strong monoidal if and only if the associated POVM's are minimal IC.* F preserves the dagger if and only if the associated POVM's are generally symmetric IC.The above theorem and corollary show that any nontrivial representation must be induced by a family of informationally complete (quasi-)POVMs. It is not clear whether these representations must also be minimal. It could very well be that the proof of Theorem <ref> can be adapted to show that any representation must necessarily be minimal to preserve functoriality. If this is the case then we automatically get that the object n inis mapped to n^2 inproviding a natural way of viewing the difference between the amount of perfectly distinguishable states (n) versus the dimension of the state space (n^2). This specific relation between these values is identified as something particular to quantum theory in for instance <cit.>. Real-valued (or quaternion valued) quantum theory would have a different value here.eptcs§ PROOF OF THEOREM <REF>Let {E_i^(n)} and {F_i^(n)} be families of minimal IC-POVM's, and let the T^1 and T^2 be the respective families of matrices as defined in Theorem <ref>, and let Q_i:→_T_i be the respective functors. Then there exists a natural isomorphism η: F_T^1∘ Q_1 ⇒ F_T^2∘ Q_2. We need a family of transition matrices from the POVM {E_i^(n)} to {F_i^(n)}. So first note that there is a matrix α^(n) such thatE_j^(n)/(E_j^(n)) = ∑_k α^(n)(k| j) F_k^(n)/(F_k^(n)).Then define the matrix S^(n) byS^(n)(i| j) = (E_j^(n)/(E_j^(n))F_i^(n)) = ∑_k α^(n)(k| j) (F_k^(n)/(F_k^(n)) F_i^(n)) = ∑_k α^(n)(k| j) T^2_n(i| k)which means that S^(n) = T^2_n α^(n) or equivalently (using the fact that the POVM's are minimal so that their transition matrices are invertible) that α^(n) = (T^2_n)^-1S^(n).Now given a CPTP map Φ: M_n→ M_m we can writeQ_1(Φ)(j| i)= (Φ(E_i^(n)/(E_i^(n)))E_j^(m)) = (E_j^(m))(Φ(E_i^(n)/(E_i^(n)))E_j^(m)/(E_j^(m)))= (E_j^(m))∑_i^',j^'α^(n)(i^'| i)α^(m)(j^'| j)1/(F_j^'^(m))(Φ(F_i^'^(n)/(F_i^'^(n)))F_j^'^(m)) = ∑_i^', j^'α^(n)(i^'| i)α^(m)(j^'| j)(E_j^(m))/(F_j^'^(m))Q_2(Φ)(j^'| i^').This can be simplified by definingβ^(m)(i| j) := α^(m)(j| i) (E_i^(m))/(F_j^(m)).Note that the indexes of α and β are reversed. This is intentional as this allows us to writeQ_1(Φ)(j| i)= ∑_j^'β^(m)(j| j^') ∑_i' Q_2(Φ)(j^'| i^')α^(n)(i^'| i) = ∑_j^'β^(m)(j| j^')(Q_2(Φ)α^(n))(j^'| i) = (β^(m)Q_2(Φ)α^(n))(j| i).What is this β? If we take Φ=id_n we get Q_i(Φ) = T^i_n. By using α^(n) = (T^2_n)^-1S^(n) we then get T^1_n = Q_1(Φ) = β^(n)Q_2(Φ)α^(n) = β^(n) T^2_n (T^2_n)^-1 S^(n) = β^(n)S^(n).S^(n) has an inverse since it acts as a basis transformation between the two POVM's. This allows us to write β^(n) = (S^(n))^-1T^1_n which means we get the following expression:Q_1(Φ) = (S^(m))^-1T^1_m Q_2(Φ) (T^2_n)^-1S^(n).Now by multiplying this equation with (T^1_n)^-1 on the right and using (F_i∘ Q_i)(Φ) = Q_i(Φ)(T^i_n)^-1 this translates to(F_1∘ Q_1)(Φ) = (S^(m))^-1T^1_m(F_2∘ Q_2)(Φ) S^(n)(T^1_n)^-1.Define η_n := S^(n)(T^1_n)^-1. The equation can now be recast asη_m (F_1∘ Q_1)(Φ) = (F_2∘ Q_2)(Φ) η_nwhich means the collection of η's forms a natural transformation. For all n, S^(n) and T^(n) are both invertible, so that η^(n) is as well. Hence η is a natural isomorphism. § PROOF OF THEOREM <REF>The functor Q: →_T as defined in Theorem <ref> for a family of minimal IC-POVM's is strong monoidal with coherence isomorphisms α_n_1,n_2=S_n_1,n_2 whereS_n_1,n_2(j| i_1i_2) = (E_i_1^n_1/(E_i_1^n_1)⊗E_i_2^n_2/(E_i_2^n_2) E_j^n_1n_2)Let ρ_i be states in M_n_i, then we have ρ_i = ∑_j(T_n_i^-1Q(ρ_i))_j E_j^n_i/(E_j^n_i) which means we can writeQ(ρ_1⊗ρ_2)_j= ((ρ_1⊗ρ_2)E_j^n_1n_2) = ∑_i_1,i_2 (T_n_1^-1Q(ρ_1))_i_1(T_n_2^-1Q(ρ_2))_i_2(E_i_1^n_1/(E_i_1^n_1)⊗E_i_2^n_2/(E_i_2^n_2) E_j^n_1n_2) = ∑_i_1,i_2 S_n_1,n_2(j| i_1, i_2)(T_n_1^-1Q(ρ_1))_i_1(T_n_2^-1Q(ρ_2))_i_2.This can now be written as Q(ρ_1⊗ρ_2) = S_n_1,n_2(T_n_1^-1⊗ T_n_2^-1)(Q(ρ_1)⊗ Q(ρ_2)) and substituting the definition of the tensor product of _T ⊗^' it becomesQ(ρ_1⊗ρ_2) = S_n_1,n_2T_n_1n_2^-1 (Q(ρ_1)⊗^' Q(ρ_2)) = S_n_1,n_2*(Q(ρ_1)⊗^' Q(ρ_2)).Now fix CPTP maps Φ_i: M_n_i→ M_m_i and writeQ(Φ_1⊗Φ_2)*Q(ρ_1⊗ρ_2)= Q(Φ_1(ρ_1)⊗Φ_2(ρ_2)) = S_m_1,m_2*(Q(Φ_1(ρ_1))⊗^' Q(Φ_2(ρ_2))) = S_m_1,m_2*(Q(Φ_1)⊗^' Q(Φ_2))*(Q(ρ_1)⊗^' Q(ρ_2)) = S_m_1,m_2*(Q(Φ_1)⊗^' Q(Φ_2))S_n_1,n_2^-1T_n_1n_2*Q(ρ_1 ⊗ρ_2).Because this has to hold for all states ρ_i we can drop that term and we get the equality Q(Φ_1⊗Φ_2) = S_m_1,m_2*(Q(Φ_1)⊗^' Q(Φ_2))S_n_1,n_2^-1T_n_1n_2. Now by bringing some terms to the other side and using the definition of * again we get the desired naturality equation:S_m_1,m_2*(Q(Φ_1)⊗^' Q(Φ_2)) = Q(Φ_1⊗Φ_2)*S_n_1,n_2.We still need to check that the coherence condition holds:S_n_1n_2,n_3*(S_n_1,n_2⊗^' id_n_3) = S_n_1,n_2n_3*(id_n_1⊗^' S_n_2,n_3) S_n_1n_2,n_3T_n_1n_2n_3^-1T_n_1n_2n_3(T_n_1n_2^-1⊗ T_n_3^-1)(S_n_1,n_2⊗ T_n_3) = S_n_1,n_2n_3T_n_1n_2n_3^-1T_n_1n_2n_3(T_n_1^-1⊗ T_n_2n_3^-1)(T_n_1⊗ S_n_2,n_3) S_n_1n_2,n_3(T_n_1n_2^-1S_n_1,n_2⊗ I_n_3) = S_n_1,n_2n_3(I_n_1⊗ T_n_2n_3^-1S_n_2,n_3).To prove this equality we need to write out S_n_1n_2,n_3 and to do that we first need the following: define β such thatE_i_1^n_1n_2/(E_i_1^n_1n_2) = ∑_k_1,k_2β(k_1,k_2| i_1) E_k_1^n_1/(E_k_1^n_1)⊗E_k_2^n_2/(E_k_2^n_2)which then givesT_n_1n_2(l| i_1)= (E_i_1^n_1n_2/(E_i_1^n_1n_2) E_l^n_1n_2) = ∑_k_1,k_2β(k_1,k_2| i_1) (E_k_1^n_1/(E_k_1^n_1)⊗E_k_2^n_2/(E_k_2^n_2) E_l^n_1n_2) = ∑_k_1k_2β(k_1,k_2| i_1)S_n_1,n_2(l| k_1,k_2) = (S_n_1,n_2β)(l| i_1),so that β = S_n_1,n_2^-1T_n_1n_2. Now we can writeS_n_1n_2,n_3(j| i_1, i_2)= (E_i_1^n_1n_2/(E_i_1^n_1n_2)⊗E_i_2^n_3/(E_i_2^n_3) E_j^n_1n_2n_3) = ∑_k_1,k_2β(k_1,k_2| i_1) (E_k_1^n_1/(E_k_1^n_1)⊗E_k_2^n_2/(E_k_2^n_2)⊗E_i_2^n_3/(E_i_2^n_3) E_j^n_1n_2n_3).Defining the quantity in the trace to be S_n_1,n_2,n_3(j| k_1,k_2,i_2) (note the comma's) this becomesS_n_1n_2,n_3(j| i_1,i_2) = ∑_k_1,k_2 S_n_1,n_2,n_3(j| k_1,k_2,i_2)β(k_1,k_2| i_1) = ∑_k_1,k_2,k_3 S_n_1,n_2,n_3(j| k_1,k_2,k_3)β(k_1,k_2| i_1)δ_k_3,i_2so that we get the equality S_n_1n_2,n_3 = S_n_1,n_2,n_3(β⊗ I_n_3) = S_n_1,n_2,n_3(S_n_1,n_2^-1T_n_1n_2⊗ I_n_3).Filling this in the left-hand side of the coherence equation: S_n_1n_2,n_3 (T_n_1n_2^-1S_n_1,n_2⊗ I_n_3) = S_n_1,n_2,n_3 (S_n_1,n_2^-1T_n_1n_2⊗ I_n_3)(T_n_1n_2^-1S_n_1,n_2⊗ I_n_3) = S_n_1,n_2,n_3.By doing a similar rewriting exercise for S_n_1,n_2n_3 we get the same expression on the right-hand side, which proves the coherence equation.§ PROOF OF THEOREM <REF> We first prove a short lemma about subspaces of matrices.Let L⊂ M_n be a subspace containing the identity and at least one Hermitian matrix with at least two distinct eigenvalues, then V= span(⋃_U∈ U(n) ULU^†) = M_n, where the union is taken over all unitary matrices. We need to show that V contains all matrices. Since it contains all unitary conjugations of any matrix, it suffices to show that it contains all diagonal matrices. Let E be the matrix in L with at least two distinct eigenvalues. We can diagonalise E=UD_1U^†, so D_1 is in V. Write D_1 = diag(λ_1,…,λ_n) taking λ_1≠λ_2. Since I_n is in V we also have D_2 = 1/(λ_2-λ_1)(D_1-λ_1 I_n) in V. We see that D_2 = diag(0,1,λ_3^',…,λ_n^'). Now we can apply the unitary conjugation to D_2 that interchanges the first and second coordinate and subtract it from D_1 giving D_3 = D_2 - PD_2P = diag(-1,1,0,…,0). By considering conjugation with permutation matrices we can get the -1 and 1 at arbitrary spots along the diagonal. The linear span of these operators is the set of diagonal matrices with zero trace. Because we also have the identity we can then create arbitrary diagonal matrices.Let F: → be a quasi-stochastic representation of . This representation is either faithful or trivial. Let {E_i^(n)} denote the nth quasi-POVM associated to F. Let V = span{E_i^(n)} and let V^⊥ = {A ∈ M_n ; ⟨ A, E_i^(n)⟩_HS = 0 ∀ i}. We note that M_n = V ⊕ V^⊥. F is faithful on n if and only if V^⊥ = {0}, since otherwise we can find a state ρ = ρ_1 + ρ_2 where ρ_1∈ V, ρ_2∈ V^⊥, ρ_2≠ 0 such that F(ρ) = F(ρ_1).Because of functoriality we must have F(Φ(ρ)) = F(Φ)F(ρ) = F(Φ)F(ρ_1) = F(Φ(ρ_1)) for all CPTP maps Φ. If we can find a Φ such that Φ(ρ_2)∉V^⊥ then this leads to a contradiction because we would have F(Φ(ρ))≠ F(Φ(ρ_1)). Therefore for the POVM to fit in a valid representation we must have V^⊥ closed under application of an arbitrary CPTP map. In particular, it has to be closed under unitary conjugation: when (ρ_2 E_i^(n)) = 0 for all i we must also have (Uρ_2 U^† E_i^(n)) = (ρ_2 U^† E_i^(n) U) = 0 for all unitaries U. This means we should have (ρ_2 A) = 0 for all A∈ W = span(⋃_U∈ U(n) UV U^†).Now we distinguish two cases. Either all the E_i^(n) are multiples of the identity in which case F(ρ)_i = (ρ E_i^(n)) = (ρα_i I_n) = α_i where E_i^(n) = α_i I_n, so that the representation is trivial, or there is a E_i^(n) that isn't a multiple of the identity in which case it has at least two distinct eigenvalues. In the latter case the space W satisfies the conditions for the previous lemma which gives W=M_n. But then (ρ_2 A) = 0 for all A∈ M_n so that necessarily ρ_2 = 0, which shows that V^⊥ = {0}. The representation is then indeed faithful for n.Let us now suppose that F isn't faithful. Then there is an n such that two states in M_n are mapped to the same distribution. We then know that all states are mapped to the same distribution for this n by the argument above. Let m≥ n. Pick two orthogonal pure states |v⟩⟨v|, |w⟩⟨w|∈ M_m and let ρ and σ be orthogonal pure states in M_n, then there exists a partial isometry Φ such that Φ(ρ) = |v⟩⟨v| and Φ(σ) = |w⟩⟨w|. By functoriality we get F(|v⟩⟨v|) = F(Φ(ρ)) = F(Φ)F(ρ)=F(Φ)F(σ) = F(Φ(σ)) = F(|w⟩⟨w|). Since v and w were arbitrary, all pure states must be mapped to the same distribution and by convexity this holds for all states. This means that the representation is trivial for all m≥ n. If we now consider a CPTP surjection Φ: M_n→ M_2 we can use the same argument to show that the representation is trivial for m=2 which shows that indeed the entire representation is trivial. | http://arxiv.org/abs/1704.08525v3 | {
"authors": [
"John van de Wetering"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170427120543",
"title": "Quantum Theory is a Quasi-stochastic Process Theory"
} |
School of Physics and Astronomy, University of Glasgow,Glasgow, G12 8QQ, UKE-mail: [email protected] Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17Copenhagen, DK-2100, DenmarkE-mail: [email protected] Centre of Excellence for Particle Physics at the Terascale and CSSM,School of Physical Sciences, The University of Adelaide,Adelaide, SA 5005, Australia^*E-mail: [email protected] In N=1 supergravity the tree-level scalar potential of the hidden sector may have a minimum with broken local supersymmetry (SUSY) as well as a supersymmetric Minkowski vacuum. These vacua can be degenerate, allowing for a consistent implementation of the multiple point principle. The first minimum where SUSY is broken can be identified with the physical phase in which we live. In the second supersymmetric phase, in flat Minkowski space, SUSY may be broken dynamically either in the observable or in the hidden sectors inducing a tiny vacuum energy density. We argue that the exact degeneracy of these phases may shed light on the smallness of the cosmological constant. Other possible phenomenological implications are also discussed. In particular, we point out that the presence of such degenerate vacua may lead to small values of the quartic Higgs coupling and its beta function at the Planck scale in the physical phase. § INTRODUCTIONIt is expected that at ultra-high energies the Standard Model (SM) is embedded in an underlying theory that provides a framework for the unification of all interactions such as Grand Unified Theories (GUTs), supergravity (SUGRA), String Theory, etc. At low energies this underlying theory could lead to new physics phenomena beyond the SM. Moreover the energy scale associated with the physics beyond the SM is supposed to be somewhat close to the mass of the Higgs boson to avoid a fine-tuning problem related to the need to stabilize the scale where electroweak (EW) symmetry is broken. Despite the successful discovery of the 125 Higgs boson in 2012, no indication of any physics beyond the SM has been detected at the LHC so far. On the other hand, there are compelling reasons to believe that the SM is extremely fine-tuned. Indeed, astrophysical and cosmological observations indicate that there is a tiny energy density spread all over the Universe (the cosmological constant), i.e. ρ_Λ∼ 10^-123M_Pl^4 ∼ 10^-55 M_Z^4 <cit.>, which is responsible for its acceleration. At the same time much bigger contributions must come from the EW symmetry breaking (∼ 10^-67M_Pl^4) and QCD condensates (∼ 10^-79M_Pl^4). Because of the enormous cancellation between the contributions of different condensates to ρ_Λ, which is required to keep ρ_Λ around its measured value, the smallness of the cosmological constant can be considered as a fine-tuning problem.Here, instead of trying to alleviate fine-tuning of the SM we impose the exact degeneracy of at least two (or even more) vacua. Their presence was predicted by the so-called Multiple Point Principle (MPP) <cit.>. According to the MPP, Nature chooses values of coupling constants such that many phases of the underlying theory should coexist. This corresponds to a special (multiple) point on the phase diagram where these phases meet. At the multiple point these different phases have the same vacuum energy density.The MPP applied to the SM implies that the Higgs effective potential, which can be written asV_eff(H) = m^2(ϕ) H^† H + λ(ϕ) (H^† H)^2 ,where H is a Higgs doublet and ϕ is a norm of the Higgs field, i.e. ϕ^2=H^† H, has two degenerate minima. These minima are taken to be at the EW and Planck scales <cit.>. The corresponding vacua can have the same energy density only ifλ(M_Pl)≃ 0, β_λ(M_Pl)≃ 0 .where β_λ=d λ(ϕ)/d logϕ is the beta–function of λ(ϕ). It was shown that the MPP conditions (<ref>) can be satisfied when M_t=173± 5 and M_H=135± 9 <cit.>. The application of the MPP to the two Higgs doublet extension of the SM was also considered <cit.>.The measurement of the Higgs boson mass allows us to determine quite precisely the parameters of the Higgs potential (<ref>). Furthermore using the extrapolation of the SM parameters up to M_Plwith full 3–loop precision it was found <cit.> thatλ(M_Pl) =-0.0143-0.0066(M_t/ - 173.34 ) + 0.0018(α_3(M_Z)-0.1184/0.0007) +0.0029(M_H/ -125.15 ) .The computed value ofβ_λ(M_Pl) is also rather small, so that the MPP conditions (<ref>) are basically fulfilled.The successful MPP predictions for the Higgs and top quarks masses <cit.> suggest that we may use this idea to explain the tiny value of the cosmological constant as well. In principle, the smallness of the cosmological constant could be related to an almost exact symmetry. Nevertheless, none of the generalizations of the SM provides any satisfactory explanation for the smallness of this dark energy density. An exact global supersymmetry (SUSY) guarantees that the vacuum energy density vanishes in all global minima of the scalar potential. However the non-observation of superpartners of quarks and leptons implies that SUSY is broken. The breakdown of SUSY induces a huge and positive contribution to the dark energy density which is many orders of magnitude larger than M_Z^4. Here the MPP assumption is adapted to (N=1) SUGRA models, in order to provide an explanation for the tiny deviation of the measured dark energy density from zero.§ SUGRA MODELS INSPIRED BY DEGENERATE VACUAThe full (N=1) SUGRA Lagrangian can be specified in terms of an analytic gauge kinetic function f_a(ϕ_M) and a real gauge-invariant Kahler function G(ϕ_M,ϕ_M^*). These functions depend on the chiral superfields ϕ_M. The function f_a(ϕ_M) determine, in particular, the gauge couplings Re f_a(ϕ_M)=1/g_a^2, where the index a represents different gauge groups. The Kahler function can be presented asG(ϕ_M,ϕ_M^*)=K(ϕ_M,ϕ_M^*)+ln|W(ϕ_M)|^2 ,where K(ϕ_M,ϕ_M^*) is the Kahler potential while W(ϕ_M) is the superpotential of the SUGRA model under consideration. Here we shall use standard supergravity mass units: M_Pl/√(8π)=1.The SUGRA scalar potential can be written as a sum of F- and D-terms, i.e.[V(ϕ_M,ϕ^*_M)=∑_M, N̅ e^G(G_MG^MN̅ G_N̅-3)+ 1/2∑_a(D^a)^2,; D^a=g_a∑_i, j(G_i T^a_ijϕ_j) , G_M ≡∂ G/∂ϕ_M , G_M̅≡∂ G/∂ϕ^*_M ,; G_N̅M≡∂^2 G/∂ϕ^*_N ∂ϕ_M , G^MN̅=G_N̅M^-1 . ]In Eq. (<ref>) g_a is the gauge coupling associated with the generator T^a. In order to achieve the breakdown of local SUSY in (N=1) supergravity, a hidden sector is introduced. The superfields of the hidden sector (z_i) interact with the observable ones only by means of gravity. It is expected that at the minimum of the scalar potential (<ref>) hidden sector fields acquire vacuum expectation values (VEVs) so that at least one of their auxiliary fieldsF^M=e^G/2G^MP̅G_P̅is non-vanishing, leading to the spontaneous breakdown of local SUSY, giving rise to a non-zero gravitino mass m_3/2≃ <e^G/2>. The absolute value of the vacuum energy density at the minimum of the SUGRA scalar potential (<ref>) tends to be of order of m_3/2^2 M_Pl^2. Therefore an enormous degree of fine-tuning is required to keep the cosmological constant in SUGRA models around its observed value.The successful implementation of the MPP in (N=1) SUGRA models requires us to assume the existence of a supersymmetric Minkowski vacuum<cit.>. According to the MPP this second vacuum and the physical one must be degenerate. Since the vacuum energy density of supersymmetric states in flat Minkowski space vanishes, the cosmological constant problem is solved to first approximation. Such a second vacuum exists if the SUGRA scalar potential (<ref>) has a minimum whereW(z_m^(2))=0 ,∂ W(z_i)/∂ z_m|_z_m=z_m^(2)=0where z_m^(2) are VEVs of the hidden sector fields in the second vacuum. Eqs. (<ref>) indicate that an extra fine-tuning is needed to ensure the presence of the supersymmetric Minkowski vacuum in SUGRA models.The simplest Kahler potential and superpotential that satisfies conditions (<ref>) are given byK(z, z^*)=|z|^2 , W(z)=m_0(z+β)^2 .The hidden sector of the corresponding SUGRA model involves only one singlet superfield z. If β=β_0=-√(3)+2√(2), the corresponding SUGRA scalar potential possesses two degenerate vacua with zero energy density at the classical level. The first minimum associated with z^(2)=-β is a supersymmetric Minkowski vacuum. In the other minimum, z^(1)=√(3)-√(2), local SUSY is broken so that it can be identified with the physical vacuum. Varying β around β_0 one can get a positive or a negative contribution from the hidden sector to the total vacuum energy density of the physical phase. Thus parameter β can be always fine-tuned so that the physical and supersymmetric Minkowski vacua are degenerate.The fine-tuning associated with the realisation of the MPP in (N=1) SUGRA models can be to some extent alleviated within the no-scale inspired SUGRA model with broken dilatation invariance <cit.>. Let us consider the no–scale inspired SUGRA model that involves two hidden sector superfields (T and z) and a set of chiral supermultiplets φ_σ in the observable sector. These superfields transform differently under the dilatations (T→α^2 T,z→α z, φ_σ→α φ_σ) and imaginary translations (T→ T+iβ,z→ z, φ_σ→φ_σ), which are subgroups of the SU(1,1) group <cit.>. The full superpotential of the model can be written as a sum <cit.>:[W(z, φ_α)=W_hid + W_obs ,; W_hid=ϰ(z^3+μ_0 z^2+∑_n=4^∞c_n z^n), W_obs=∑_σ,β,γ1/6 Y_σβγφ_σφ_βφ_γ . ]The superpotential (<ref>) includes a bilinear mass term for the superfield z and higher order terms c_n z^n which explicitly break dilatation invariance. A term proportional to z is not included since it can be forbidden by a gauge symmetry of the hidden sector. Here we do not allow the breakdown of dilatation invariance in the superpotential of the observable sector to avoid potentially dangerous terms that may lead to the so–called μ–problem, etc.The full Kähler potential of the SUGRA model under consideration is given by <cit.>:[ K(ϕ_M,ϕ_M^*)= -3ln[T+T -|z|^2-∑_αζ_α|φ_α|^2] +; + ∑_α, β(η_αβ/2 φ_α φ_β+h.c.)+ ∑_βξ_β|φ_β|^2 , ]where ζ_α, η_αβ, ξ_β are some constants. If η_αβ and ξ_β have non-zero values the dilatation invariance is explicitly broken in the Kähler potential of the observable sector. Here we restrict our consideration to the simplest set of terms that break dilatation invariance. Moreover we only allow the breakdown of the dilatation invariance in the Kähler potential of the observable sector, because any variations in the part of the Kähler potential associated with the hidden sector may spoil the vanishing of the vacuum energy density in global minima. When the parameters η_αβ, ξ_β and ϰ go to zero, the dilatation invariance is restored, protecting supersymmetry and a zero value of the cosmological constant.In the SUGRA model under consideration the tree-level scalar potential of the hidden sector is positive definiteV_hid=1/3(T+T-|z|^2)^2|∂ W_hid(z)/∂ z|^2,so that the vacuum energy density vanishes near its global minima. In the simplest case when c_n=0, the SUGRA scalar potential of the hidden sector (<ref>) has two minima, at z=0 and z=-2μ_0/3. At these points V_hid attains its absolute minimal value i.e. zero. In the first vacuum, where z=-2μ_0/3, local SUSY is broken and the gravitino gains massm_3/2=<W_hid(z)/(T+T-|z|^2)^3/2> =4ϰμ_0^3/27<(T+T -4μ_0^2/9)^3/2> .All scalar particles get non–zero masses m_σ∼m_3/2ξ_σ/ζ_σ as well. This minimum can be identified with the physical vacuum. Assuming that ξ_α, ζ_α, μ_0 and <T> are all of order unity, a SUSY breaking scale M_S∼ 1 can only be obtained when ϰ is extremely small, i.e. ϰ≃ 10^-15. In the second vacuum, where z=0, the superpotential of the hidden sector vanishes and local SUSY remains intact giving rise to the supersymmetric Minkowski vacuum. If the high order terms c_n z^n are present in Eqs. (<ref>), V_hid can have many degenerate vacua, with broken and unbroken SUSY, in which the vacuum energy density vanishes. As a result the MPP conditions are fulfilled without any extra fine-tuning at the tree–level.It is worth noting that the inclusion of perturbative and non-perturbative corrections to the Lagrangian of the no–scale inspired SUGRA model should spoil the degeneracy of vacua, giving rise to a huge energy density in the minimum of the scalar potential where local SUSY is broken. Furthermore, in the SUGRA model under consideration the mechanism for the stabilization of the VEV of the hidden sector field T remains unclear. Therefore this model should be considered as a toy example only. It demonstrates that in (N=1) supergravity there might be a mechanism which ensures the vanishing of the vacuum energy density in the physical vacuum. This mechanism can also result in a set of degenerate vacua with broken and unbroken SUSY, leading to the realization of the MPP.§ IMPLICATIONS FOR COSMOLOGY AND COLLIDER PHENOMENOLOGY§.§ Model with intermediate SUSY breaking scaleNow let us assume that the MPP inspired SUGRA model of the type just discussed is realised in Nature. In other words there exist at least two exactly degenerate phases. The first phase is associated with the physical vacuum whereas the second one is identified with the supersymmetric Minkowski vacuum in which the vacuum energy density vanishes in the leading approximation. However non-perturbative effects may give rise to the breakdown of SUSY in the second phase resulting in a small vacuum energy density. This small energy density should be then transferred to our vacuum by the assumed degeneracy.If SUSY breaking takes place in the second vacuum, it can be caused by the strong interactions in the observable sector. Indeed, the SM gauge couplings g_1, g_2 and g_3, which correspond to U(1)_Y, SU(2)_W and SU(3)_C gauge interactions respectively, change with the energy scale. Their evolution obeys the renormalization group equations (RGEs) that in the one–loop approximation can be written asdlogα_i(Q)/dlogQ^2=b_iα_i(Q)/4π ,where Q is a renormalization scale, i=1,2,3 and α_i(Q)=g_i^2(Q)/(4π). In the pure MSSM b_3<0 and the gauge coupling g_3(Q) of the SU(3)_C interactions increases in the infrared region. Thus although this coupling can be rather small at high energies it becomes rather large at low energies and one can expect that the role of non–perturbative effects is enhanced.To simplify our analysis we assume that the SM gauge couplings at high energies are identical in the physical and supersymmetric Minkowski vacua. Consequently for Q>M_S, where M_S is a SUSY breaking scale in the physical vacuum, the renormalization group (RG) flow of these couplings is also the same in both vacua. When Q<M_S all superparticles in the physical vacuum decouple. Therefore the SU(3)_C beta function in the physical and supersymmetric Minkowski vacua become very different. Because of this, below the scale M_S the values of α_3(Q) in the physical and second vacua (α^(1)_3(Q) and α^(2)_3(Q)) are not the same. For Q<M_S the SU(3)_C beta function in the physical phase b̃_3 coincides with the corresponding SM beta function, i.e. b̃_3=-7. Using the value of α^(1)_3(M_Z)≈ 0.1184 and the matching condition α^(2)_3(M_S)=α^(1)_3(M_S), one can estimate the value of the strong gauge coupling in the second vacuum1/α^(2)_3(M_S)=1/α^(1)_3(M_Z)- b̃_3/4πlnM^2_S/M_Z^2. In the supersymmetric Minkowski vacuum all particles of the MSSM are massless and the EW symmetry is unbroken. So, in the second phase the SU(3)_C beta function b_3 remains the same as in the MSSM, i.e. b_3=-3. Since the MSSM SU(3)_C beta function exhibits asymptotically free behaviour, α^(2)_3(Q) increases in the infrared region. The top quark is massless in the supersymmetric phase and its Yukawa coupling also grows in the infrared with the increasing of α^(2)_3(Q). At the scaleΛ_SQCD=M_Sexp[2π/b_3α_3^(2)(M_S)] ,where the supersymmetric QCD interactions become strong in the supersymmetric Minkowski vacuum, the top quark Yukawa coupling is of the same order of magnitude as the SU(3)_C gauge coupling. So a large value of the top quark Yukawa coupling may give rise to the formation of a quark condensate. This condensate breaks SUSY, resulting in a non–zero positive value for the dark energy densityρ_Λ≃Λ_SQCD^4. The dependence of Λ_SQCD on the SUSY breaking scale M_S is shown in Fig. <ref>. Since b̃_3 < b_3 the value of the QCD gauge coupling below M_S is larger in the physical phase than in the second one. Therefore the scale Λ_SQCD is substantially lower than the QCD scale in the SM and decreases with increasing M_S. When the supersymmetry breaking scale in the physical phase is of the order of 1, we get Λ_SQCD=10^-26M_Pl≃ 100 eV <cit.>. This leads to the value of the cosmological constant ρ_Λ≃ 10^-104 M_Pl^4, which is enormously suppressed as compared with v^4 ≃ 10^-67 M_Pl^4.The measured value of the dark energy density is reproduced when Λ_SQCD=10^-31M_Pl≃ 10^-3. The appropriate values of Λ_SQCD can be obtained only if M_S≃ 10^3-10^4<cit.>. However models with such a large SUSY breaking scale do not lead to a suitable dark-matter candidate and also spoil the unification of the SM gauge couplings. §.§ Split SUSY scenarioThe problems mentioned above can be addressed within the Split SUSY scenario of superymmetry breaking <cit.>. In other words, let us now assume that in the physical vacuum SUSY is broken so that all scalar bosons gain masses of order of M_S≫ 10, except for a SM-like Higgs boson, whose mass is set to be around 125. The mass parameters of gauginos and Higgsinos are protected by a combination of an R-symmetry and Peccei Quinn symmetry so that they can be many orders of magnitude smaller than M_S. To ensure gauge coupling unification all neutralino, chargino and gluino states are chosen to lie near the TeV scale in the Split SUSY scenario <cit.>. Also a TeV-scale lightest neutralino can be an appropriate dark matter candidate <cit.>.Thus in the Split SUSY scenario supersymmetry is not used to stabilize the EW scale <cit.>. This stabilization is expected to be provided by some other mechanism, which may also explain the tiny value of the dark energy density. Therefore in the Split SUSY models M_S is taken to be much above 10 TeV. In the Split SUSY scenario some flaws inherent to the MSSM disappear. The ultra-heavy scalars, whose masses can range from hundreds of TeV up to 10^13 <cit.>, ensure the absence of large flavor changing and CP violating effects. The stringent constraints from flavour and electric dipole moment data, that require M_S>100-1000, are satisfied and the dimension-five operators, which mediate proton decay, are also suppressed within the Split SUSY models. Nevertheless, since the sfermions are ultra-heavy the Higgs sector is extremely fine-tuned, with the understanding that the solution to both the hierarchy and cosmological constant problems might not involve natural cancellations, but follow from anthropic-like selection effects <cit.>. In other words galaxy and star formation, chemistry and biology, are basically impossible without these scales having the values found in our Universe <cit.>. In this case supersymmetry may be just a necessary ingredient in a fundamental theory of Nature like in the case of String Theory.It has been argued that String Theory can have a huge number of long-lived metastable vacua <cit.> which is measured in googles (∼ 10^100) <cit.>. The space of such vacua is called the “landscape".To analyze the huge multitude of universes, associated with the “landscape" of these vacua a statistical approach is used <cit.>. The total number of vacua in String Theory is sufficiently large to fine-tune both the cosmological constant and the Higgs mass, favoring a high-scale breaking of supersymmetry <cit.>. Thus it is possible for us to live in a universe fine-tuned in the way we find it simply because of a cosmic selection rule, i.e. the anthropic principle <cit.>.The idea of the multiple point principle and the landscape paradigm have at least two things in common. Both approaches imply the presence of a large number of vacua with broken and unbroken SUSY. The landscape paradigm and MPP also imply that the parameters of the theory, which results in the SM at low energies, can be extremely fine-tuned so as to guarantee a tiny vacuum energy density and a large hierarchy between M_Pl and the EW scale. Moreover the MPP assumption might originate from the landscape of string theory vacua, if all vacua with a vacuum energy density that is too large are forbidden for some reason, so that all the allowed string vacua, with broken and unbroken supersymmetry, are degenerate to very high accuracy. If this is the case, then the breaking of supersymmetry at high scales is perhaps still favored. Although this scenario looks quite attractive it implies that only a narrow band of values around zero cosmological constant would be allowed and the surviving vacua would obey MPP to the accuracy of the width w of this remaining band. However such accuracy is not sufficient to become relevant for the main point of the present article, according to which MPP “transfers” the vacuum energy density of the second vacuum to the physical vacuum.In order to estimate the value of the cosmological constant we again assume that the physical and second phases have precisely the same vacuum energy densities and the gauge couplings at high energies are identical in both vacua. This means that the renormalization group flow of the SM gauge couplings down to the scale M_S is the same in both vacua as before. For Q<M_S all squarks and sleptons in the physical vacuum decouple and the beta functions change. At the TeV scale, the corresponding beta functions in the physical phase change once again due to the decoupling of the gluino, neutralino and chargino. Assuming that α^(2)_3(M_S)=α^(1)_3(M_S), one finds1/α^(2)_3(M_S)=1/α^(1)_3(M_Z)- b̃_3/4πlnM^2_g/M_Z^2-b'_3/4πlnM^2_S/M_g^2 ,where M_g is the mass of the gluino and b'_3=-5 is the one–loop beta function of the strong gauge coupling in the Split SUSY scenario. The values of Λ_SQCD and ρ_Λ can be estimated using Eqs. (<ref>) and (<ref>), respectively.In Fig. <ref> we explore the dependence of Λ_SQCD in the second phase on the SUSY breaking scale M_S in the physical vacuum. In our analysis we set M_g=3. As before Λ_SQCD diminishes with increasing M_S. The observed value of the dark energy density can be reproduced when M_S∼ 10^9 - 10^10 <cit.>. The value of M_S, which results in the measured cosmological constant, depends on α_3(M_Z) and the gluino mass. However this dependence is rather weak. In particular, with increasing M_g the value of M_S, which leads to an appropriate value of the cosmological constant, decreases. When α_3(M_Z)=0.116-0.121 and M_g=500-2500, the corresponding value of the SUSY breaking scale varies from 2· 10^9 up to 3· 10^10<cit.>.The obtained prediction for M_S can be tested. A striking feature of the Split SUSY model is the extremely long lifetime of the gluino. The gluino decays through a virtual squark to a quark antiquark pair and a neutralino g̃→ qq̅+χ_1^0. The large squark masses give rise to a long lifetime for the gluino. This lifetime can be estimated as <cit.>τ∼ 8(M_S/10^9 )^4 (1 /M_g)^5 s.From Eq. (<ref>) it follows that the supersymmetry breaking scale in the Split SUSY models should not exceed 10^13 <cit.>. Otherwise the gluino lifetime becomes larger than the age of the Universe. When M_S varies from 2· 10^9 (M_g=2500) to 3· 10^10 (M_g=500) the gluino lifetime changes from 1 to 2· 10^8 (1000). Thus the measurement of the mass and lifetime of gluino should allow one to estimate the value of M_S in the Split SUSY scenario.§.§ Models with low SUSY breaking scaleThe observed value of the dark energy density can be also reproduced when the SUSY breaking scale is around 1. This can be achieved if the MSSM particle content is supplemented by an extra pair of 5+5̅ supermultiplets which are fundamental and antifundamental representations of the supersymmetric SU(5) GUT. The additional bosons and fermions would not affect gauge coupling unification in the leading approximation, since they form complete representations of SU(5). In the physical phase states from 5+5̅ supermultiplets can gain masses around M_S. The corresponding mass terms in the superpotential can be induced because of the presence of the bilinear terms [η (5·5)+h.c.] in the Kahler potential of the observable sector <cit.>. In the Split SUSY scenario we assume that new bosonic states from 5+5̅ supermultiplets gain masses around the supersymmetry breaking scale, whereas their fermion partners acquire masses of the order of the gluino, chargino and neutralino masses. In our numerical studies we set the masses of extra quarks to be equal to the gluino mass, i.e. M_q≃ M_g. In the supersymmetric Minkowski vacuum new bosons and fermions from 5+5̅ supermultiplets remain massless. As a consequence they give a substantial contribution to the β functions in this vacuum. Indeed, the one–loop beta function of the strong interaction in the second phase changes from b_3=-3 (the SU(3)_C beta function in the MSSM) to b_3=-2. This leads to a further reduction of Λ_SQCD. At the same time, extra fermion states from 5+5̅ supermultiplets do not affect much the RG flow of gauge couplings in the physical phase below the scale M_S. For example, in the Split SUSY scenario the one–loop beta function that determines the running of the strong gauge coupling from the SUSY breaking scale down to the TeV scale changes from -5 to -13/3.As follows from Figs. <ref> and <ref> in the case of the SUSY model with extra 5+5̅ supermultiplets the measured value of the dark energy density can be reproduced even for M_S≃ 1 <cit.>. Nevertheless, the Split SUSY scenario which was discussed in the previous subsection has the advantage of avoiding the need for any new particles beyond those of the MSSM, provided that M_S≃ 10^9 - 10^10. On the other hand, the MPP scenario with extra 5+5̅ supermultiplets of matter and SUSY breaking scale in a fewrange is easier to verify at the LHC in the near future. §.§ The breakdown of SUSY in the hidden sectorThe non-zero value of the vacuum energy density can be also induced if supersymmetry in the second phase is broken in the hidden sector. This can happen if the SM gauge couplings are sufficiently small in the supersymmetric Minkowski vacuum and by one way or another, only vector supermultiplets associated with unbroken non-Abelian gauge symmetry remain massless in the hidden sector. Then these vector supermultiplets, that survive to low energies, can give rise to the breakdown of SUSY in the second phase. Indeed, at the scale Λ_X,where the gauge interactions that correspond to the unbroken gauge symmetry in the hidden sector become strong in the second phase, a gaugino condensate can be formed. This gaugino condensate does not break global SUSY. Nonetheless if the gauge kinetic function f_X(z_m) has a non-trivial dependence on the hidden sector superfields z_m then the corresponding auxiliary fields F^m can acquire non–zero VEVsF^z_m∝∂ f_X(z_k)/∂ z_mλ̅_aλ_a+...,which are set by <λ̅_aλ_a>≃Λ_X^3. Thus it is only via the effect of a non-renormalisable term that this condensate causes the breakdown of supersymmetry. Therefore the SUSY breaking scale in the SUSY Minkowski vacuum is many orders of magnitude lower than Λ_X, while the scale Λ_X is expected to be much lower than M_Pl. As a result a tiny vacuum energy density is inducedρ_Λ∼Λ_X^6/M_Pl^2. The postulated exact degeneracy of vacua implies then that the physical phase has the same energy density as the second phase where the breakdown of local SUSY takes place near Λ_X. From Eq. (<ref>) it follows that in order to reproduce the measured cosmological constant the scale Λ_X has to be somewhat close to Λ_QCD in the physical vacuum, i.e.Λ_X∼Λ_QCD/10 .Although there is no compelling reason to expect that Λ_X and Λ_QCD should be related, one may naively consider Λ_QCD and M_Pl as the two most natural choices for the scale of dimensional transmutation in the hidden sector.In the one–loop approximation one can estimate the value of the energy scale Λ_X using the simple analytical formulaΛ_X=M_Plexp[2π/b_X α_X(M_Pl)] ,where α_X(M_Pl)=g^2_X(M_Pl)/(4π), g_X and b_X are the gauge coupling and one–loop beta function of the gauge interactions associated with the unbroken non-Abelian gauge symmetry that survive to low energies in the hidden sector. For the SU(3) and SU(2) gauge groups b_X=-9 and -6, respectively. In Fig. <ref> we show the dependence of Λ_X on α_X(M_Pl). As one might expect, the value of the energy scale Λ_X diminishes with decreasing α_X(M_Pl). The observed value of the dark energy density is reproduced when α_X(M_Pl)≃ 0.051 in the case of the SUSY model based on the SU(2) gauge group and α_X(M_Pl)≃ 0.034 in the case of the SU(3) SUSY gluodynamics. It is worth noting that in the case of the SU(3) SUSY model the value of the gauge coupling g_X(M_Pl)≃ 0.654, that leads to α_X(M_Pl)≃ 0.034, is just slightly larger than the value of the QCD gauge coupling at the Planck scale in the SM, i.e. g_3(M_Pl)=0.487<cit.>.In this scenario SUSY can be broken at any scale in the physical vacuum. In particular, the breakdown of local supersymmetry can take place near the Planck scale. If this is the case, one can explain the small values of λ(M_Pl) and β_λ(M_Pl) by postulating the existence of a third degenerate vacuum. In this third vacuum local SUSY can be broken near the Planck scale while the EW symmetry breaking scale can be just a few orders of magnitude lower than M_Pl. Since now the Higgs VEV is somewhat close to M_Pl one must take into account the interaction of the Higgs and hidden sector fields. Thus the full scalar potential takes the formV=V_hid(z_m) + V_0(H) + V_int(H, z_m)+... ,where V_hid(z_m) is the part of the full scalar potential associated with the hidden sector, V_0(H) is the part of the scalar potential that depends on the SM Higgs field only and V_int(H, z_m) describes the interactions of the SM Higgs doublet with the fields of the hidden sector. Although in general V_int(H, z_m) should not be ignored the interactions between H and hidden sector fields can be quite weak if the VEV of the Higgs field is substantially smaller than M_Pl (say ⟨ H ⟩≲ M_Pl/10) and the couplings of the SM Higgs doublet to the hidden sector fields are suppressed.Then the VEVs of the hidden sector fields in the physical and third vacua can be almost identical. As a consequence, the gauge couplings and λ(M_Pl) in the first and third phases should be basically the same and the value of |m^2| in the Higgs effective potential can be still much smaller than M_Pl^2 and ⟨ H^† H⟩ in the third vacuum. In this limit V_hid(z^(3)_m)≪ M_Pl^4 and the requirement of the existence of the third vacuum with vanishingly small energy density again implies that λ(M_Pl) and β_λ(M_Pl) are approximately zero in the third vacuum. Because in this case the couplings in the third and physical phases are basically identical, the presence of such a third vacuum should result in the predictions (<ref>) for λ(M_Pl) and β_λ(M_Pl) in the physical vacuum.§ ACKNOWLEDGMENTSThis work was supported by the University of Adelaide and the Australian Research Council through the ARC Center of Excellence in Particle Physics at the Terascale and through grant LF0 99 2247 (AWT). HBN thanks the Niels Bohr Institute for his emeritus status. 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"published": "20170427071402",
"title": "Dark Energy Density in SUGRA models and degenerate vacua"
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http://arxiv.org/abs/1704.07993v1 | {
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"title": "Hybrid Precoder and Combiner Design with One-Bit Quantized Phase Shifters in mmWave MIMO Systems"
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Vibrational-ground-state zero-width resonances for laser filtration: An extended semiclassical analysis. Osman Atabek December 30, 2023 ======================================================================================================== Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. Recently Brendle and Hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball, which resolved a conjecture of Alexander, Hoffman and Osserman. Their proof involves asymptotic analysis of an ingeniously chosen vector field, and the divergence theorem.In this article we prove a sharp `moving-centre' monotonicity formula for minimal submanifolds, which implies the aforementioned area bound. We also describe similar moving-centre monotonicity formulae for stationary p-harmonic maps, mean curvature flow and the harmonic map heat flow.§ INTRODUCTIONFor many geometric partial differential equations, monotonicity formulae play an essential role and their discovery often leads to deep and fundamental results for those systems. Monotonicity is a particularly useful tool in the study of variational problems, and for regularity theory (see for example <cit.> and references therein). These formulae often control the evolution of energy-type quantities with respect to changes in scale, or time.An important example is the classical monotonicity formula for minimal submanifolds - critical points of the area functional - which states: Let Σ^k be a minimal submanifold in ℝ^n. Then so long as Σ∩B_r^n = ∅, we have d/dr(r^-k |Σ∩ B_r^n |) = r^-k-1∫_Σ∩ B^n_r|x^⊥|^2/|x^T|≥ 0.Here B_r^n=B^n(0,r) denotes the Euclidean ball of radius r about the origin in ℝ^n. Thus the area ratio r^-k|Σ∩ B_r^n| is monotone on balls with fixed centre, and so comparing to the limiting density as r↘ 0 yields that any minimal submanifold Σ^k⊂ B_r^n with Σ⊂ B_r^n, which passes through the origin, satisfies the sharp area bound |Σ∩ B_r^n|/r^k≥ |B_1^k| ,with equality if and only if Σ is a flat k-disk. In the case that the minimal submanifold Σ^k ⊂ B_r^n does not necessarily pass through the centre of the ball, Alexander, Hoffman and Osserman <cit.> conjectured (see also <cit.>) the following sharp area bound, which has recently been proven in full generality by Brendle and Hung <cit.> (see also Corollary <ref>).Alexander and Osserman had previously proven the conjecture only in the case of simply connected surfaces <cit.>. Let Σ^k be a minimal submanifold in the ball B_r^n with Σ⊂ B_r^n. Then|Σ∩ B_r^n|/(r^2-d^2)^k/2≥ |B^k_1|,where d=d(0,Σ) is the distance from Σ to the centre of the ball.The proof of Theorem <ref> by Brendle-Hung involves the choice of a clever, but somewhat geometrically mysterious, vector field W. They apply the divergence theorem to W away from small balls B_ϵ(y), where y∈Σ∩ B_r, and obtain the estimate in the limit as ϵ→ 0. In this paper, we show that the area bound (<ref>) in fact arises from a sharp `moving-centre' monotonicity formula, in which the centres of the extrinsic balls are allowed to move, and the scale is adjusted in a particular manner:Fix y∈ B_R^n. For s≥ 0 we letE_s = B^n((1-s)y, r(s))denote the ball with centre (1-s)y and radius r(s) := √(s(R^2-|y|^2) + s^2|y|^2). Our main theorem is then as follows (see also Theorem <ref>):Let Σ^k be a minimal submanifold in ℝ^n and y ∈ B_R^n, E_s, r(s) be as above. Then so long as Σ∩E_s =∅, we have d/ds( |Σ∩ E_s|/(r(s)^2 - d(s)^2)^k/2) =s^-k+2/2/2(R^2-|y|^2)^k/2∫_Σ∩ E_s|(x-y)^⊥|^2+s^2|y^T|^2/|(x-y+sy)^T|≥ 0 ,where d(s)=s|y| is the distance from y to the centre of the ball E_s.As s↘ 0, the monotone quantity(r(s)-d(s))^-k/2|Σ∩ E_s|again picks up the density at y. On the other hand, r(1)=R, so taking y∈Σ∩ B_1 = Σ∩ E_1 to be the point with |y|=d(0,Σ) and comparing s=0 to s=1 indeed yields the area bound (<ref>). This new proof using our monotonicity formula thus offers a new perspective on Theorem <ref>. In fact, the choice of radius r(s) above is constrained at a technical level in the proof of the monotonicity Theorem <ref> (see Lemma <ref>); with this choice in hand one may then observe that Brendle-Hung's vector field W takes a relatively simple form in terms of a function f whose sub-level sets are the balls E_s. Interestingly, this function also seems to emerge organically in the study of the exponential transform <cit.>. Moreover, the distinction between having an area bound compared to having monotonicity of an area ratio is significant (although potentially subtle). For example, for minimal submanifolds Σ^k ⊂ N^n, where N has sectional curvature bounded above by K, it holds that|Σ^k ∩B̅^N_r|/A_0(r)≥ 1,where B̅^N_r is the geodesic ball in N of radius r, and A_0(r) is the area of the geodesic ball of radius r in the k-dimensional space form of curvature K. (If K>0 then r must be less than π/2√(K).) For K≤ 0, the quantity on the left is in fact monotone non-decreasing <cit.>, but for K>0, such a monotonicity for the area ratio is not known. Instead, one can prove the area bound by proving a monotonicity for the quantity A_0(r)^-1∫_B̅_r^N |∇ρ|^2, where ρ is the distance function on N <cit.>. In <cit.> it was shown that Theorem <ref> would be a consequence of the sharp isoperimetric inequality |Σ|^k ≥ k^k |B_1^k| |Σ|^k-1 for minimal submanifolds Σ^k, although the latter is not presently known in general. In fact, isoperimetric inequalities are closely related to monotonicity properties of minimal submanifolds, particularly in the work of Choe (see <cit.> for a survey). One might therefore hope that the approaches to Theorems <ref> and <ref> could provide some insight towards the sharp isoperimetric inequality.The proof of Theorem <ref> may be found in Section <ref>. It uses only the coarea formula and the divergence theorem, except that the key is to apply the divergence theorem to a different vector field at each s. Thus, as for the classical monotonicity formula, our moving-centre monotonicity formula also holds for stationary varifolds, and also admits an almost-monotonicity for submanifolds with L^p-bounded mean curvature. We also provide a second proof using only the divergence theorem, which may clarify Brendle-Hung's vector field W. §.§ Monotonicity formulae for related geometric systemsIn Theorem <ref>, Theorem <ref> and Theorem <ref> we present moving-centre monotonicity formulae for the mean curvature flow, p-harmonic maps and harmonic map heat flow respectively. In Section <ref> we consider stationary p-harmonic maps (p>1); for the fixed-centre monotonicity in this setting one may consult <cit.>. The minimal submanifold case is morally the p=1 case; in fact it is the critical case for our moving-centre monotonicity in the sense that for p>1, there is a term of the wrong sign that cannot be fully absorbed in the naive manner. The offending term may be handled either by accepting an almost-monotonicity and multiplying the monotone quantity by a correcting factor, or instead by adjusting the scale more carefully. For p>1 the latter method applies only if the centre does not move too quickly. Interpolating between the two methods is what gives rise to our family of monotonicity formulae for these elliptic systems.In Sections <ref> and <ref> we present our results for certain parabolic systems, namely the mean curvature flow and harmonic map heat flow respectively. The respective fixed-centre monotonicity formulae are due to Huisken <cit.> and Struwe <cit.>. In both cases, the monotone quantity involves a global energy-type integral against a Gaussian weight. Thus, unlike the elliptic case, an additional factor to compensate for the motion of the centre appears to be unavoidable. However, for these geometric flows, we obtain a monotonicity for motion of the centre along any C^1 path, not just on straight lines. Finally, one should note that a type of moving-centre monotonicity was used by Colding and Minicozzi <cit.> to show that the entropy of a mean curvature flow self-shrinker is achieved by the Gaussian area at the natural centre and scale of the self-shrinker. This is an important step in their classification of entropy-stable, or generic, self-shrinkers (see also <cit.>). For the reader's convenience we briefly describe their result in Section <ref>. §.§ Notation In Euclidean space we will always use x to denote the position vector. When working with submanifolds, we will use ∇ to denote the ambient connection and ∇ for the induced connection on a submanifold, with D reserved for the Euclidean connection. We use y^T for the projection of a vector y to the tangent bundle, and y^⊥ for the projection to the normal bundle. When dealing with maps between manifolds M→ N ↪ℝ^n, we will unambiguously use ∇ for the connection on M. If M=ℝ^m we use lower, Latin indices for coordinates on ℝ^m and upper, Greek indices for coordinates on ℝ^n. Repeated indices are summed throughout, unless otherwise noted, and commas denote derivatives. In this setting we use · to distinguish contraction on M from full contraction ⟨ ,⟩. We use I to denote an open interval in ℝ. We will need the coarea formula, which states that for a proper Lipschitz function f and a locally integrable function u on a manifold M, one has ∫_{f≤ t} u|∇ f| = ∫_-∞^t τ̣∫_{f=τ} u.(See for instance <cit.>, or <cit.> for more general statements including for varifolds.)We denote by B^k(p,r) the (open) Euclidean ball in ℝ^k with centre p and radius r. For simplicity we will write B^k_r = B^k(0,r). We will often omit the dimension when it is clear from context.We typically prefer to derive our monotonicity formulae in differential form; derivatives with respect to the time or scale parameter should be interpreted in the distribution sense.Additional notation and background specific to each setting will be explained in the respective sections of this paper. §.§ Acknowledgements We are indebted to Nick Edelen, as well as Otis Chodosh, for bringing the initial problem to the author's attention and for stimulating discussions. The author would also like to thank Prof. Bill Minicozzi for numerous valuable suggestions.§ MINIMAL SUBMANIFOLDS Recall that the divergence theorem or first variation formula for submanifolds states that∫_Σ÷_Σ X = - ∫_Σ⟨ X,H⃗⟩ + ∫_Σ⟨ X,ν⟩for any smooth compactly supported ambient vector field X, and that minimal submanifolds are those that satisfy H⃗=0. Here ν is the outward unit normal of Σ with respect to Σ. The classical monotonicity formula for minimal submanifolds (see for instance <cit.>, or <cit.>) states that: Let Σ^k be a minimal submanifold in the ball B_r̅^n ⊂ℝ^n with Σ⊂ B_r̅^n. Then for 0<r<r̅ one has d/dr(r^-k |Σ∩ B_r|) = r^-k-1∫_Σ∩ B_r|x^⊥|^2/|x^T| . Equivalently, for 0<r<t<r̅, we have t^-k|Σ∩ B_t| - r^-k|Σ∩ B_r| = ∫_Σ∩ B_t∖ B_r|x^⊥|^2/|x|^k+2 In particular, the area ratio r^-k |Σ∩ B_r| is non-decreasing in r, and is constant if and only if Σ is a cone (with vertex at 0). In order to state our moving-centre monotonicity formula, we first define a family of extrinsic balls on which to view the submanifold. Note that in this section, we work with equivalent forms of Theorems <ref> and <ref>, in which the ball B_R is scaled back to the unit ball B_1. In particular y will denote a point in the unit ball. Fix y∈ B_1^n. For s≥ 0, denote the ballE_s=B^n((1-s)y, r(s))⊂ℝ^n ,wherer(s)=√(s(1-|y|^2)+s^2|y|^2)).Note that the E_s, s≥ 0 foliate the half-space defined by ⟨ x,y⟩ < 1+|y|^2/2. These balls may also be realised as sub-level sets,E_s={0≤ f < s},where explicitly f(x) = |x-y|^2/1-2⟨ x,y⟩ +|y|^2 = |x-y|^2/1-|x|^2+|x-y|^2. The moving-centre monotonicity formula is then as follows:Let Σ^k be a minimal submanifold in E_s̅⊂ℝ^n with Σ⊂ E_s̅ for some s̅. Then for 0<s<s̅, we have thatd/ds(s^-k/2|Σ∩ E_s |) = s^-k+2/2/2∫_Σ∩ E_s|(x-y)^⊥|^2+s^2|y^T|^2/|(x-y+sy)^T| .Equivalently, for 0<s<t<s̅, we have t^-k/2|Σ∩ E_t| - s^-k/2|Σ∩ E_s| = ∫_Σ∩ E_t∖ E_s f^-k/2( |(x-y)^⊥|^2+f^2|y^T|^2/|x-y|^2). In particular, the quantitys^-k/2|Σ∩ E_s|is nondecreasing, and is constant if and only if Σ is a flat disk orthogonal to y. Our proof will only require the coarea and first variation formulae, so it can be seen that Theorem <ref> also holds for stationary varifolds Σ, except that in the equality case one must allow for cones with vertex at y that are orthogonal to y. Note that the above is indeed equivalent to Theorem <ref>, since rearranging (<ref>) yields that r(s)^2 - s^2|y|^2 = s(1-|y|^2). Taking y=0 of course recovers the classical monotonicity formula for minimal submanifolds. It may be helpful to note that if Σ_0 is indeed a flat k-plane orthogonal to y, then any x∈Σ_0 satisfies |x|^2 = |y|^2 + |x-y|^2 and hence Σ_0 ∩ E_s is a flat k-disk of radius √(s(1-|y|^2)) as expected. Let Σ^k be a minimal submanifold in the unit ball B_1^n ⊂ℝ^n with Σ⊂ B_1^n and y ∈Σ. Then |Σ| ≥ |B^k_1|(1-|y|^2)^k/2, with equality if and only if Σ is a flat disk orthogonal to y. Corollary <ref> was first proven in full generality by Brendle and Hung <cit.>, using a carefully chosen vector field that we will return to later in this section.As s ↘ 0, the balls E_s are asymptotic to the balls B(y, √(s(1-|y|^2))). So the limit (1-|y|^2)^-k/2lim_s→ 0 s^-k/2|Σ∩ E_s| is equal to the density of Σ at y, which is at least 1 since y∈Σ. On the other hand E_1 = B(0,1), and thus comparing s=0 to s=1 using Theorem <ref> immediately yields Corollary <ref>. We now turn to the proof of Theorem <ref>, for which we first calculate the gradient of f:Let r(s), E_s and the function f with E_s = {f=s} be as in Definition <ref>. Then whereever f>0, we have thatDf/2f=x-y+fy/|x-y|^2.One may verify this using the explicit formula for f, but it is more illuminating to proceed using only the characterisation by level sets together with the choice of r(s). Indeed, let ρ(s) = r(s)^2 = s(1-|y|^2) + s^2|y|^2. By construction, the level sets of f are the spheres E_s with centre (1-s)y and radius r(s). So the function f satisfies|x-(1-f)y|^2 = ρ(f)or in somewhat expanded form |x-y|^2 = -2f⟨ x-y,y⟩+ρ(f) - f^2|y|^2. Moreover Df must be proportional to x-(1-f)y, so implicitly differentiating (<ref>), we find that1/2D f = x-y+fy/ρ'(f) - 2⟨ x-y, y⟩ -2f|y|^2.On the other hand, we note that ρ satisfies the differential equations(ρ'(s) - 2s|y|^2) = ρ(s) - s^2|y|^2. Therefore using (<ref>) yields 1/2Df =f(x-y+fy)/|x-y|^2. The outward unit normal ν to Σ∩{f=s} considered as the boundary of Σ∩{f< s} is given by ν = ∇ f/|∇ f|. For fixed s we let X_s be a vector field with ÷_Σ X_s ≡ k, to be chosen later. By the divergence theorem we would then have|Σ∩ E_s | = 1/k∫_Σ∩{f=s}⟨ X_s, ∇ f/|∇ f|⟩. On the other hand, by the coarea formula we have|Σ∩ E_s| = ∫_0^s τ̣∫_Σ∩{f=τ}1/|∇ f|.This allows us to compute the derivative of |Σ∩ E_s| as an integral over Σ∩ E_s, using Lemma <ref>:d/ds(s^-k/2|Σ∩ E_s|)=s^-k+2/2∫_Σ∩{f=s}1/|∇ f|( s-1/2⟨ X_s,∇ f⟩)=s^-k+2/2∫_Σ∩{f=s}s/|∇ f|( 1-⟨ X_s,∇ f/2f⟩) =s^-k/2∫_Σ∩{f=s}1/|∇ f|( 1-⟨ X_s, (x-y+sy)^T ⟩/|x-y|^2) . Choosing X_s = x-y-sy, we indeed have ÷_Σ X_s = k since s is fixed, and moreover⟨ X_s, (x-y+sy)^T ⟩ = |(x-y)^T|^2 - s^2 |y^T|^2.Thus d/ds(s^-k/2|Σ∩ E_s |) = s^-k/2∫_Σ∩{f=s}1/|∇ f|(|(x-y)^⊥|^2+s^2|y^T|^2/|x-y|^2). Using Lemma <ref> to replace |∇ f| yields the differential form (<ref>), whilst integrating (<ref>) using the coarea formula a second time gives the integral form (<ref>). It is clear from either formulation that s^-k/2|Σ∩ E_s| is constant if and only if (x-y)^⊥≡ 0 and y^T≡ 0 on Σ. The first condition implies that Σ is a cone with vertex at y (hence a plane, if Σ is smooth), and the second implies that Σ is orthogonal to y. As with the classical monotonicity formula, one still obtains an almost-monotonicity if one assumes only an L^p bound for the mean curvature H⃗, by following the proof above and bounding the vector field X_s on the set E_s to handle the extra term. For instance, if the mean curvature is bounded by |H⃗|≤ C_H, then using the bound|X_s|≤ 2s|y| + √(s(1-|y|^2)+s^2|y|^2)≤ 3s|y| + √(s(1-|y|^2))on {0≤ f≤ s}, one still obtains a monotone quantity after multiplying by the integrating factor exp(kC_Hμ), whereμ = 1/2∫(3|y| + √(1-|y|^2/s)) ṣ = 3/2s|y| + √(s(1-|y|^2)) .For completeness, we now give another proof of Theorem <ref> that is instead motivated by, and utilises, the work of Brendle-Hung <cit.>. By using the divergence theorem (twice), this proof recovers the integral formulation (<ref>). With f defined as above, the vector field utilised by Brendle and Hung may be written asW = -1/k(f^-k/2-1)(x-y) + F(f)y, F(t) := 1/k-2(t^2-k/2-1), k>2 -1/2log t, k=2.Setting W_0 := 1/k(x-y)-W, the computations of Brendle and Hung <cit.> yield that÷_Σ W_0=1-÷_Σ W = f^-k/2|(x-y)^⊥|^2 + f^-k-4/2 |y^T|^2/|x-y|^2On the other hand, for any 0<s<s̅, when restricted to E_s={f=s} we have W_0 = 1/ks^-k/2(x-y) - F(s)y. So applying the divergence theorem we find that∫_Σ∩{f=s}⟨ W_0,ν⟩ = ∫_Σ∩{f=s}⟨1/ks^-k/2(x-y) - F(s)y,ν⟩ = s^-k/2 | Σ∩{f≤ s}|,since ÷_Σ x=k and ÷_Σ y=0. Then applying the divergence theorem a second time, for any 0<s<t < s̅ we indeed havet^-k/2|Σ∩{f< t}| - s^-k/2|Σ∩{f< s}| = ∫_Σ∩{f=t}⟨ W_0,ν⟩ - ∫_Σ∩{f=s}⟨ W_0,ν⟩= ∫_Σ∩{s< f < t}÷_Σ W_0 = ∫_Σ∩{s < f < t}f^-k/2|(x-y)^⊥|^2 + f^-k-4/2 |y^T|^2/|x-y|^2 .§ MEAN CURVATURE FLOWA one-parameter family of submanifolds Σ_t^k ⊂ℝ^n flows by mean curvature if it satisfies _t x = H⃗. For submanifolds moving by mean curvature, Huisken <cit.> (see also <cit.>) discovered a monotonicity for Gaussian areas. For a submanifold Σ^k ⊂ℝ^n, define the Gaussian density with centre x_0∈ℝ^n and scale t_0>0 byF_x_0,t_0(Σ) = ∫_Σρ_x_0,t_0 ,where ρ_x_0,t_0(x)= (4π t_0)^-k/2exp(-|x-x_0|^2/4t_0) . For a mean curvature flow Σ^k_t, Huisken's monotone quantity is obtained by decreasing the scale as the flow progresses in time. Specifically, given a spacetime centre (x_0,t_0), the monotone quantity will be F_x_0, t_0-t(Σ_t) = ∫_Σ_tΦ_x_0,t_0, where we have setΦ_x_0,t_0(x,t)=ρ_x_0,t_0-t(x)= (4π (t_0-t))^-k/2exp(- |x-x_0|^2/4(t_0-t)). Suppose that {Σ^k_t}_t∈ I is a mean curvature flow in ℝ^n and fix x_0∈ℝ^n, t_0∈ℝ. Further suppose that ∫_Σ_tΦ_x_0,t_0 <∞ for all t∈ I with t<t_0. Then for all such times, one hasd/dt∫_Σ_tΦ_x_0,t_0 = -∫_Σ_t| H⃗ + (x-x_0)^⊥/2(t_0-t)|^2 Φ_x_0,t_0. In particular, ∫_Σ_tΦ_x_0,t_0 is non-increasing for t<t_0, and is constant if and only if Σ_t is a self-shrinking soliton that shrinks to (x_0,t_0).Recall that a self-shrinking soliton, which shrinks to (x_0,t_0), is a mean curvature flow that is invariant under (backwards) parabolic dilations about (x_0,t_0), or equivalently satisfies H⃗ = - (x-x_0)^⊥/2(t_0-t) on each Σ_t.Motivated by Theorem <ref>, we give a sharp monotonicity formula for mean curvature flow in which the Gaussian centre x_0 is allowed to move in time. Namely, we prove that:Suppose that {Σ^k_t}_t∈ I is a mean curvature flow in ℝ^n and fix t_0∈ℝ. Let y=y(t) be a smooth curve in ℝ^n. Further suppose that ∫_Σ_tΦ_y(t),t_0 <∞ for all t∈ I with t<t_0. Then for all such times, we have thatd/dt∫_Σ_tΦ_y(t),t_0 = -∫_Σ_t| H⃗ + (x-y-(t_0-t)y')^⊥/2(t_0-t)|^2 Φ_y,t_0 + 1/4∫_Σ_t|(y')^⊥|^2 Φ_y,t_0 . In particular, the quantity exp(-1/4∫_t^t_0 |y'(τ)|^2 τ̣) ∫_Σ_tΦ_y(t),t_0 is non-increasing for t<t_0.Of course, if the centre does not change, that is, if y(t)≡ x_0, then one recovers Huisken's monotonicity. The energy functional for curves in ℝ^n is of course minimised by straight lines; in this case the equality case is easy to interpret: Fix x_0,y_0∈ℝ^n and t_0∈ℝ and let {Σ_t}_t∈ I be as above. Then for t∈ I with t<t_0, the quantity exp(-|y_0|^2/4(t_0-t)) ∫_Σ_tΦ_x_0 + (t_0-t) y_0, t_0 is non-increasing, and is constant if and only if Σ_t is a self-shrinking soliton which shrinks to (x_0,t_0) and is orthogonal to y.For smooth, (spatially) compactly supported test functions ϕ=ϕ(x,t) the first variation formula for mean curvature flow states that (see for instance <cit.>) d/dt∫_Σ_tϕ = ∫_Σ_tϕ/ t +⟨H⃗, Dϕ⟩ - |H⃗|^2ϕ. First we compute the gradientDΦ_y,t_0/Φ_y,t_0=-x-y/2(t_0-t), Direct computation then yields that Φ_y,t_0/ t + ⟨H⃗, DΦ_y,t_0⟩ - |H⃗|^2Φ_y,t_0= Φ_y,t_0(k/2(t_0-t) -|x-y|^2/4(t_0-t)^2+⟨ x-y,y'⟩/2(t_0-t) -⟨H⃗, x-y⟩/2(t_0-t) - |H⃗|^2).By completing the square we then note that-|x-y|^2/4(t_0-t)^2+⟨ x-y,y'⟩/2(t_0-t) = -1/4(t_0-t)^2( |x-y-(t_0-t)y'|^2 - (t_0-t)^2|y'|^2).We will now make use of the divergence theorem (<ref>). In particular, for fixed t we setX = -1/2(t_0-t)(x-y-2(t_0-t)y').For this X we compute that÷_Σ_t(Φ_y,t_0 X) = Φ_y,t_0( ÷_Σ_t X -⟨ X, (x-y)^T⟩/2(t_0-t))= Φ_y,t_0(-k/2(t_0-t) +|(x-y-(t_0-t)y')^T|^2 - (t_0-t)^2|(y')^T|^2/4(t_0-t)^2),where we have used that x-y = (x-y-(t_0-t)y') + (t_0-t)y'. Combining (<ref> - <ref>), we thus have Φ_y,t_0/ t + ⟨H⃗, DΦ_y,t_0⟩ - |H⃗|^2Φ_y,t_0 + ÷_Σ_t(Φ_y,t_0X) + ⟨H⃗,Φ_y,t_0X⟩= Φ_y(t),t_0(-|(x-y-(t_0-t)y')^⊥|^2/4(t_0-t)^2 + |(y')^⊥|^2/4..- ⟨H⃗, x-y-(t_0-t)y'⟩/(t_0-t) - |H⃗|^2)= Φ_y(t),t_0(|(y')^⊥|^2/4 - | H⃗ + (x-y-(t_0-t)y')^⊥/2(t_0-t)|^2).If the Σ_t are compact, then we may immediately apply (<ref>) with ϕ=Φ_y,t_0, and (<ref>) to Φ_y,t_0X to conclude the result. If, however, the Σ_t are noncompact, then for R>0 we select a smooth cutoff function χ=χ_R on ℝ^n such that χ_R=1 on B_R, χ_R=0 outside B_2R, withR|Dχ_R| + R^2|D^2χ_R| ≤ C_0in between, where C_0 is a universal constant. Applying (<ref>) with ϕ=χΦ_y,t_0 and (<ref>) to χΦ_y,t_0X, we then haved/dt∫_Σ_tχΦ_y,t_0 = ∫_Σ_tχ( Φ_y,t_0/ t +⟨H⃗, DΦ_y,t_0⟩ - |H⃗|^2Φ_y,t_0 + ÷_Σ_t(Φ_y,t_0X) + ⟨H⃗,Φ_y,t_0X⟩)+ ∫_Σ_tΦ_y,t_0( χ/ t+ ⟨H⃗,Dχ⟩ + ⟨∇χ, X⟩).Since χ is independent of time we of course have χ/ t=0. Now using the divergence theorem again to Φ_y,t_0 Dχ, we have that ∫_Σ_tΦ_y,t_0( ⟨H⃗,Dχ⟩ + ⟨∇χ, X⟩) = ∫_Σ_tΦ_y,t_0( - ÷_Σ_t Dχ + ⟨∇χ, X-DΦ_y,t_0/Φ_y,t_0⟩).We may simplify the last term using (<ref>) and (<ref>), so ultimatelyd/dt∫_Σ_tχΦ_y,t_0 = ∫_Σ_tχΦ_y,t_0( |(y')^⊥|^2/4 - | H⃗ + (x-y-(t_0-t)y')^⊥/2(t_0-t)|^2 )+ ∫_Σ_tΦ_y,t_0(-÷_Σ_t Dχ+⟨∇χ, y'⟩). By (<ref>) we may estimate|÷_Σ_t Dχ| ≤kC_0/R^21_B_2R∖ B_R, |⟨∇χ,y'⟩| ≤ |y'| C_0/R1_B_2R∖ B_R. Since by assumption ∫_Σ_tΦ_y,t_0 <∞, as in <cit.> we may therefore let R→∞ to conclude the result. (For instance, one may use the bounds above to move the terms involving y' and Dχ to the left-hand side via an integrating factor, then apply the monotone convergence theorem to justify the remaining integral involving H⃗.)Theorem <ref> holds for Brakke flows as well, except that (<ref>) is instead an upper bound, since the first variation formula (<ref>) is also an upper bound for such flows <cit.>.§.§ Self-shrinkersIn the study of self-shrinking solitons one may make the normalisation that the soliton flow Σ_t shrinks to the origin at time 0. The flow is then determined by any negative time slice - in particular, Σ_t = √(-t)Σ_-1. The time slice Σ = Σ_-1 satisfies the elliptic equation H⃗ = -x^⊥/2, and submanifolds satisfying this equation are called self-shrinkers. Colding-Minicozzi <cit.> introduced the entropy of a submanifold, defined by λ(Σ) = sup_x_0,t_0 F_x_0,t_0(Σ),where F_x_0,t_0(Σ) = ∫_Σρ_x_0,t_0 is the Gaussian area as in (<ref>).Using a moving-centre monotonicity formula, they were able to show that if Σ is a self-shrinker, then its entropy is achieved at the natural centre x_0=0 and scale t_0=1, that is, λ(Σ) = F_0,1(Σ). Their monotonicity is as follows: Let Σ^k ⊂ℝ^n be a self-shrinker and fix a ∈ℝ. Then d/ds F_sy, 1+as^2(Σ) = -s/2(1+as^2)^2∫_Σ |(asx+y)^⊥|^2 ρ_sy,1+as^2≤0,so long as 1+as^2>0. In particular the Gaussian area F_x_0,t_0(Σ) is maximised at (x_0,t_0)=(0,1), and this maximum is strict unless Σ is invariant under either dilation or a translation. § P-HARMONIC MAPSIn this section we consider maps from a compact Riemmanian manifold M^m (possibly with boundary) to a manifold N, which we assume to be isometrically embedded in ℝ^n. (Note that n is not necessarily the dimension of N.) We fix p ∈ (1,∞). A (W^1,2) map u:M→ N ↪ℝ^n is said to be (weakly) p-harmonic if it is a (weak) solution of the elliptic equation_p(u)=÷(|∇ u|^p-2∇ u) = -A_u(∇ u,∇ u) ,where A is the second fundamental form of N in ℝ^n. Such maps are the critical points of the L^p energy functional ℰ_p(u) = ∫_M |∇ u|^p. The case p=2 corresponds to the usual case of harmonic maps, whilst the limiting case p→ 1 corresponds to the case of minimal hypersurfaces - since if a level set {u=a} is a smooth hypersurface, its mean curvature is given by ÷(∇ u/|∇ u|) (see for instance <cit.>).A p-harmonic map u is said to be stationary if it additionally satisfies∫_M |∇ u|^p ÷ X = p ∫_M|∇ u|^p-2⟨∇ X, ∇ u⊗∇ u⟩for any smooth, compactly supported vector field X on M. Explicitly, in coordinates, the expressions above are |∇ u|^2 = u^α_,i u^α_,i and ⟨∇ X, ∇ u⊗∇ u⟩ = X_i,j u^α_,i u^α_,j.For 1<p<2 we understand products of |∇ u|^p-2 with derivatives of u to be zero on the critical set {∇ u =0}. Note that A_u(∇ u,∇ u) is orthogonal to T_u N. In particular, even for weak solutions (see <cit.>, or <cit.> for the case p=2) one has ∫_M ⟨ A_u(∇ u,∇ u), ξ⟩=0for any smooth map ξ:M→ℝ^n such that ξ(x)∈ T_u(x) N for all x. It follows that any smooth harmonic map is automatically stationary, for instance by applying the divergence theorem to the vector field on M given by|∇ u|^p-2⟨ X·∇ u, ∇ u⟩ - 1/p|∇ u|^p X, and noting that derivatives of u must be tangent to N.Monotonicity formulae play an important role in the regularity theory of p-harmonic maps, the most important case being when M is a bounded domain in ℝ^m. We state the classical monotonicity formula for Euclidean balls M=B_r^m⊂ℝ^m. The statement below is found in <cit.>, but versions were proven earlier by other authors including: Schoen-Uhlenback <cit.> for minimising 2-harmonic maps, Price <cit.> for stationary 2-harmonic maps and Hardt-Lin <cit.> for minimising p-harmonic maps.Let u:B_r̅^m → N be a stationary p-harmonic map, 1<p<m. Then for 0<r<r̅, one has that d/dr( r^p-m∫_B_r |∇ u|^p ) = p r^p-m∫_ B_r |∇ u|^p-2|_r u|^2 = pr^p-m-2∫_ B_r |∇ u|^p-2 |x·∇ u|^2. Equivalently, for 0<r<t<r̅, we havet^p-m∫_B_t |∇ u|^p - r^p-m∫_B_r |∇ u|^p = p ∫_B_t∖ B_r |x|^p-m-2 |∇ u|^p-2 |x·∇ u|^2. In particular, r^p-m∫_B_r |∇ u|^p is non-decreasing in r, and is constant if and only if u is homogenous of degree zero.We use the technique of Theorem <ref> (that is, morally, the p=1 case) to provide a family of moving-centre monotonicity formulae for stationary p-harmonic maps. For p>1 there is an excess term involving derivatives of u in the y-direction, which can be handled by an explicit correction term, or by adjusting the family of balls. In the latter case, the motion of the centre must be constrained. Interpolating between these approaches gives rise to our family of monotonicity formulae. Choose q ∈ [1,p]. Then for fixed y∈ B^m_q^-1/2, we may define the nested family of ballsE^(q)_s = B^m(sy, R_q(s)) ⊂ℝ^m, where R_q(s)=√(s(1-q|y|^2)+s^2q|y|^2). Note that for q=1, this family differs from Definition <ref> only by a rigid motion; instead of starting centred near y and expanding through B(0,1), here the balls E^(1)_s will start centred near 0 and expand through E^(1)_1 = B(y,1). For q>1 the E^(q)_s foliate ℝ^m, whilst for q=1 they foliate the half-space defined by ⟨ x,y⟩ > |y|^2-1/2. As sub-level sets, we note that E^(q)_s={0≤ f_q<s} forf_q(x)= |x|^2/1+2⟨ x,y⟩ -|y|^2 , q=1 -(1-q|y|^2 +2⟨ x,y⟩) + √((1-q|y|^2 +2⟨ x,y⟩)^2 + 4(q-1)|x|^2|y|^2)/2(q-1)|y|^2 ,q>1. Our moving-centre monotonicity formulae for p-harmonic maps are then as follows: Let p∈(1,m) and fix q∈ [1,p], y∈ B^m_q^-1/2. Further, fix s̅∈ (0,∞) and let u:E^(q)_s̅→ N be a stationary p-harmonic map. Then for 0<s<s̅, one has that d/ds(s^p-m/2∫_E_s^(q) |∇ u|^p )= qs^p-m+2/2/2R_q(s)∫_ E_s^(q) |∇ u|^p-2 (|y|^2 |∇ u|^2 - |y·∇ u|^2)+s^p-m-2/2/2R_q(s)∫_ E_s^(q) |∇ u|^p-2(p|x·∇ u|^2-(p-q)s^2|y·∇ u|^2). Equivalently, for 0<s<t<s̅, we have t^p-m/2∫_E_t^(q) |∇ u|^p -s^p-m/2∫_E_s^(q) |∇ u|^p = q∫_E_t^(q)∖ E_s^(q) |∇ u|^p-2 f^p-m+4/2(|y|^2|∇ u|^2 - |y·∇ u|^2/|x|^2+(q-1)f^2|y|^2)+ ∫_E_t^(q)∖ E_s^(q) |∇ u|^p-2 f^p-m/2(p|x·∇ u|^2-(p-q)f^2|y·∇ u|^2/|x|^2+(q-1)f^2|y|^2).Bounding the y·∇ u terms using Cauchy-Schwarz yields the monotone quantities as follows:Let p∈(1,m) and fix q∈ (1,p], y∈ B^m_q^-1/2. Let u:E^(q)_s̅→ N be a stationary p-harmonic map. Then for 0<s<s̅, the quantity s^q-1/p-1p-m/2∫_E_s^(q) |∇ u|^pis non-decreasing.In particular, when q=p, we get the sharper statement:Let p∈(1,m) and fix y∈ B^m_p^-1/2. Let u:E^(p)_s̅→ N be a stationary p-harmonic map. Then for 0<s<s̅, the quantitys^p-m/2∫_E^(p)_s |∇ u|^pis non-decreasing. If y≠ 0, then this quantity is constant if and only if u is a constant map; if y=0 then it is constant if and only if u is homogenous of degree zero.Our proof will use the coarea formula, and so we will also need the version of (<ref>) for domains with boundary. Namely, let Ω⊂ℝ^m be a smooth bounded domain and let X be a smooth vector field on Ω. It follows from (<ref>) that∫_Ω |∇ u|^p-2(|∇ u|^2÷ X - p ⟨∇ X, ∇ u⊗∇ u⟩)= ∫_Ω |∇ u|^p-2(|∇ u|^2 X·ν - p⟨ X·∇ u, ν·∇ u⟩),where ν is the outward unit normal to Ω. [One way to see this is to define φ by φ(t) = t/ϵ for 0≤ t≤ϵ, ϕ(t)=1 for t≥ϵ, and let φ be a smooth approximation of φ. Apply (<ref>) to φ(d(x,Ω)) X on Ω and let ϵ→0 using the coarea formula. For regular maps one may instead directly apply the divergence theorem with boundary to (<ref>).]For the proof we will suppress the dependence on q, that is, we fix q and work with E_s = E^(q)_s, f=f_q, R=R_q. Since, by construction, the level sets of f are the spheres E_s with centre and radius as in (<ref>), the function f satisfies|x-fy|^2 =R(f)^2=f(1-q|y|^2) + f^2q|y|^2. It will be useful to record the expanded form|x|^2 + (q-1)f^2|y|^2 = 2f⟨ x,y⟩+f(1-q|y|^2) + 2f^2(q-1)|y|^2.The outward unit normal ν of E_s with respect to E_s is given byν = x-sy/|x-sy| = x-sy/R(s).But the gradient of f on each E_s must be proportional to ν, so implicitly differentiating using (<ref>) and then (<ref>), we find that1/2∇ f = x-fy/1-q|y|^2 + 2⟨ x,y⟩+2(q-1)f |y|^2 = f(x-fy)/|x|^2+(q-1)f^2|y|^2.(One may also verify this directly using the explicit form of f.)Taking the norm of both sides and using (<ref>) yields |∇ f| = 2f R(f)/|x|^2+(q-1)f^2|y|^2. The coarea formula gives thatd/ds∫_E_s |∇ u|^p = ∫_{f=s}|∇ u|^p/|∇ f| =1/2sR(s)∫_{f=s} |∇ u|^p (|x|^2+(q-1)s^2|y|^2). But for fixed s, by the stationarity (<ref>), for any vector field X on E_s which satisfies∇_i X_j = δ_i,j,using the formula (<ref>) for the normal we find that(m-p)∫_E_s |∇ u|^p =1/R(s)∫_{f=s} |∇ u|^p-2( |∇ u|^2 X· (x-sy) - p⟨ X·∇ u, (x-sy)·∇ u⟩), Choosing X=x+sy and polarising both terms on the right, we have(m-p)∫_E_s |∇ u|^p= 1/R(s)∫_{f=s} |∇ u|^p-2( |∇ u|^2 (|x|^2-s^2|y|^2) - p|x·∇ u|^2+ps^2|y·∇ u|^2 ) = 1/R(s)∫_{f=s}|∇ u|^p(|x|^2+(q-1)s^2|y|^2) + qs^2/R(s)∫_{f=s}|∇ u|^p-2(|y·∇ u|^2 - |y|^2|∇ u|^2)+1/R(s)∫_{f=s} |∇ u|^p-2((p-q)s^2|y·∇ u|^2-p|x·∇ u|^2) . Using (<ref>) for the first term on the right and rearranging gives that d/ds(s^p-m/2∫_E_s |∇ u|^p )= qs^p-m+2/2/2R(s)∫_{f=s}|∇ u|^p-2( |y|^2|∇ u|^2-|y·∇ u|^2 )+s^p-m-2/2/2R(s)∫_{f=s} |∇ u|^p-2(p|x·∇ u|^2-(p-q)s^2|y·∇ u|^2). This is the stated monotonicity formula in differential form (<ref>). On the other hand, using (<ref>) and integrating using the coarea formula again will give the integral form (<ref>). Bounding |y·∇ u|^2≤ |y|^2 |∇ u|^2 in (<ref>), and using (<ref>), we have that(m-p) ∫_E_s|∇ u|^p≤2s d/ds∫_E_s |∇ u|^p + p-q/R(s)∫_E_s |∇ u|^p s^2|y|^2 ≤ 2s d/ds∫_E_s |∇ u|^p + p-q/q-11/R(s)∫_E_s |∇ u|^p (|x|^2+(q-1)s^2|y|^2) =2s p-1/q-1d/ds∫_E_s |∇ u|^p.This implies the stated monotonicity. If p=q, then we did not lose anything in the second inequality above; therefore equality holds if and only if x·∇ u≡ 0 and |y·∇ u| = |y| |∇ u|, that is, if y is parallel to ∇ u^α for each α. The first condition implies that u is homogenous of degree zero. If y≠ 0 then the second condition implies that u is constant on lines orthogonal to y, which combined with the first forces u to be constant.§ HARMONIC MAP HEAT FLOWThe harmonic map heat flow involves deforming a map u:M^m → N ↪ℝ^n by the parabolic equation_t u =u + A_u (∇ u,∇ u).Again A is the second fundamental form of N in ℝ^n. Struwe <cit.> discovered a Gaussian-weighted monotonicity for regular solutions to (<ref>) on M=ℝ^m. In this section we use Φ to denote the kernelΦ_x_0,t_0(x,t)= (4π (t_0-t))^-m-2/2exp(- |x-x_0|^2/4(t_0-t)),and we denote ℝ^m_t = ℝ^m ×{t}⊂ℝ^m × I.Fix x_0 ∈ℝ^m, t_0∈ I.Let u:ℝ^m× I → N be a regular solution of (<ref>) with |∇ u| ≤ c<∞ and ∫_ℝ^m_t |∇ u|^2 Φ_x_0,t_0 <∞ for all t∈ I with t<t_0. Then d/dt∫_ℝ^m_t |∇ u|^2 Φ_x_0,t_0 = -2 ∫_ℝ^m_t| _t u - ∇ u ·x-x_0/2(t_0-t)|^2 Φ_x_0,t_0 .In particular, ∫_ℝ^m_t |∇ u|^2 Φ_x_0,t_0 is non-increasing for t<t_0. Allowing the centre x_0 to move in time, we are able to prove the following moving-centre monotonicity for the harmonic map heat flow:Fix t_0∈ I and a smooth curve y=y(t) in ℝ^m.Let u:ℝ^m× I→ N be a regular solution of (<ref>) with |∇ u| ≤ c<∞ and ∫_ℝ^m_t |∇ u|^2 Φ_y(t),t_0 <∞ for all t. Then d/dt∫_ℝ^m_t |∇ u|^2 Φ_y(t),t_0 = -2 ∫_ℝ^m_t| _t u - ∇ u ·x-y - (t_0-t)y'/2(t_0-t)|^2 Φ_y,t_0 + 1/2∫_ℝ^m_t |∇ u · y'|^2 Φ_y,t_0.In particular,exp(-1/2∫_t^t_0 |y'(τ)|^2 dτ)∫_ℝ^m_t |∇ u|^2 Φ_y(t),t_0 is non-increasing for t<t_0.The proof is similar to the proof of Theorem <ref>. By the exponential decay of the Gaussian weight and the assumed gradient bound, we may differentiate under the integral and integrate by parts freely: First we calculate d/dt∫_ℝ^m_t |∇ u|^2 Φ_y(t),t_0 = ∫_ℝ^m_t 2⟨∇ u,∇_t u⟩Φ_y,t_0+∫_ℝ^m_tΦ_y,t_0 |∇ u|^2( (x-y)· y'/2(t_0-t) -|x-y|^2/4(t_0-t)^2+m-2/2(t_0-t)). Integrating by parts, we have ∫_ℝ^m_t 2⟨∇ u,∇_t u⟩Φ_y,t_0 = ∫_ℝ^m_tΦ_y,t_0( -2⟨ u, _t u⟩ + 1/t_0-t⟨_t u, (x-y)·∇ u⟩)= ∫_ℝ^m_tΦ_y,t_0( -2|_t u|^2 + 1/t_0-t⟨_t u, (x-y)·∇ u⟩),where in the second line we used that A_u(∇ u,∇ u) is orthogonal to T_u N and hence to _t u. By completing the square we have that-|x-y|^2/4(t_0-t)^2+(x-y)· y'/2(t_0-t) = -1/4(t_0-t)^2( |x-y-(t_0-t)y'|^2 - (t_0-t)^2|y'|^2).Since u is regular, for a smooth compactly supported vector field Y on M, applying the divergence theorem to the vector field 2⟨ Y·∇ u, ∇ u⟩ -|∇ u|^2 Y yields that 0 = ∫_ℝ^m_t( 2⟨ Y·∇ u, u⟩ +2 ⟨∇ Y, ∇ u⊗∇ u⟩-|∇ u|^2÷ Y)= ∫_ℝ^m_t( 2⟨ Y·∇ u,_t u⟩ + 2 ⟨∇ Y, ∇ u⊗∇ u⟩ -|∇ u|^2÷ Y),where again we have used that A_u(∇ u,∇ u) is orthogonal to Y·∇ u. For fixed t we again set X= -1/2(t_0-t)(x-y-2(t_0-t)y'), so that by polarisation÷(Φ_y,t_0X) = Φ_y,t_0(-m/2(t_0-t) + |x-y-(t_0-t)y'|^2-(t_0-t)^2|y'|^2/4(t_0-t)^2),∇_i(Φ_y,t_0X)_j= Φ_y,t_0( X_i,j -(x-y)_i X_j/2(t_0-t)) = Φ_y,t_0( -δ_ij/2(t_0-t) +(x-y)_i (x-y-2(t_0-t)y')_j/4(t_0-t)^2).Contracting the last equation against the symmetric tensor ∇ u ⊗∇ u and polarising again then gives, for Y=Φ_y,t_0X,⟨∇ Y, ∇ u⊗∇ u⟩ = Φ_y,t_0( -|∇ u |^2/2(t_0-t) + |(x-y-(t_0-t)y')·∇ u|^2-(t_0-t)^2|y'·∇ u|^2/4(t_0-t)^2).Again the exponential decay means that (<ref>) in fact holds for Y=Φ_y,t_0X, so subtracting the resulting identity from (<ref>) and using the calculations above we find thatd/dt∫_ℝ^m_t |∇ u|^2 Φ_y(t),t_0 = ∫_ℝ^m_tΦ_y,t_0( -2|_t u|^2 + 1/t_0-t⟨_t u, (x-y)·∇ u⟩)+ ∫_ℝ^m_tΦ_y,t_01/t_0-t⟨_t u,(x-y-2(t_0-t)y')·∇ u⟩-∫_ℝ^m_tΦ_y,t_0( |(x-y-(t_0-t)y')·∇ u|^2-(t_0-t)^2|y'·∇ u|^2/2(t_0-t)^2) =-2 ∫_ℝ^m_tΦ_y,t_0|_t u-x-y-(t_0-t)y'/2(t_0-t)·∇ u|^2 + 1/2∫_ℝ^m_tΦ_y,t_0 |y'·∇ u|^2.This concludes the proof.plain | http://arxiv.org/abs/1704.08195v2 | {
"authors": [
"Jonathan J. Zhu"
],
"categories": [
"math.DG",
"math.AP",
"35A16, 49Q05, 53C44"
],
"primary_category": "math.DG",
"published": "20170426163333",
"title": "Moving-centre monotonicity formulae for minimal submanifolds and related equations"
} |
ms τ_syn τ_msHzmV τ_r #1#2𝔉^-1[#1](#2) E I DCisarxivtrueHow the connectivity structure of neuronal networks influences responses to oscillatory stimulitocsectionTitleHannah Bos 1,*, Jannis Schücker 1, Moritz Helias 1,21 Institute of Neuroscience and Medicine (INM-6) and Institute for Advanced Simulation (IAS-6) and JARA BRAIN Institute I, Jülich Research Centre, 52425 Jülich, Germany2 Department of Physics, Faculty 1, RWTH Aachen University, 52074 Aachen, Germany* mailto:[email protected]@fz-juelich.de § ABSTRACTPropagation of oscillatory signals through the cortex is shaped by the connectivity structure of neuronal circuits. The coherence of population activity at specific frequencies within and between cortical areas has been linked to laminar connectivity patterns. This study systematically investigates the network and stimulus properties that shape network responses. The results show how input to a cortical column model of the primary visual cortex excites dynamical modes determined by the laminar pattern. Stimulating the inhibitory neurons in the upper layer reproduces experimentally observed resonances at γ frequency whose origin can be traced back to two anatomical sub-circuits. We develop this result systematically: Initially, we highlight the effect of stimulus amplitude and filter properties of the neurons on their response to oscillatory stimuli. Subsequently, we analyze the amplification of oscillatory stimuli by the effective network structure, which is mainly determined by the anatomical network structure and the synaptic dynamics. We demonstrate that the amplification of stimuli, as well as their visibility in different populations, can be explained by specific network patterns. Inspired by experimental results we ask whether the anatomical origin of oscillations can be inferred by applying oscillatory stimuli. We find that different network motifs can generate similar responses to oscillatory input, showing that resonances in the network response cannot, straightforwardly, be assigned to the motifs they emerge from. Applying the analysis to a spiking model of a cortical column, we characterize how the dynamic mode structure, which is induced by the laminar connectivity, processes external input. In particular, we show that a stimulus applied to specific populations typically elicits responses of several interacting modes. The resulting network response is therefore composed of a multitude of contributions and can therefore neither be assigned to a single mode nor do the observed resonances necessarily coincide with the intrinsic resonances of the circuit.§ AUTHOR SUMMARYOscillations are ubiquitously generated within and propagated between biological systems, for example in ecosystems, cell biology and the brain. Recordings from neural signals often show oscillations which are either generated within the recorded brain area or imposed on it from other areas. It is an open question in neuroscience how the underlying network structure influences the interaction between internally generated and externally applied oscillations. This study systematically analyzes how these two types of oscillations are reflected in the spectra produced by neural networks and whether oscillatory input can be utilized to uncover dynamically relevant sub-circuits of the neuronal network. Previous work showed that structured neural circuits yield dynamic modes which act as filters on internally generated noise of the neuronal activity. Here we show how these filters act on external input and how their superposition can yield resonances in the network response, which cannot directly be linked to the sub-circuits generating oscillations within the network. Simulation and theoretical analysis of a column model of the primary visual cortex show that oscillatory input applied to the inhibitory population in the upper layer elicits a resonance at γ frequency in their response, which can be traced back to two distinct anatomical sub-circuits.isarxiv§ INTRODUCTIONOscillations in the γ-frequency range (30-80) are observed ubiquitously in recordings of brain activity, such as the local field potential (LFP) <cit.>. On the single cell level, these oscillations can be generated from neurons with a preferred frequency at which they transmit signals. This band-pass filtering can arise from either single cell properties, for example sub-threshold resonances of cells which are driven by fluctuations <cit.>, or from strongly driven cells whose input-output relation exhibits a peak at their firing rate <cit.>. On the network level, certain connectivity patterns have been hypothesized to facilitate very slow as well as fast oscillations in the γ-range <cit.>, namely the inter-neuron γ (ING) and pyramidal inter-neuron γ (PING) motif (, reviewed in ). Although numerous theoretical studies shed light on the emergent behavior of theses dynamical motifs in isolation, few studies considered the effect of their embedment in larger networks, such as the layered structure of the cortex <cit.>. Similarly, the dynamical interaction of the network motif with the surrounding network has been neglected when interpreting results of experimental studies gathering evidence for the ING motif <cit.> using oscillatory stimuli. In this study we analyze how the responses to oscillatory stimuli are shaped by the network alone and therefore only consider populations of neurons with non-resonant input-output relations.Neural response properties <cit.> as well as the emergence of oscillations in the γ-range <cit.> depend on the dynamical state of the network, which can be altered by externally applied stimuli. It is still a matter of debate which stimuli (natural or noise stimuli) elicit γ oscillations <cit.> and whether γ oscillations of different frequencies and peak shapes, elicited by these stimuli, are of the same anatomical origins <cit.>. Changes of the excitability of neurons, that could be induced by stimuli, have theoretically been shown to have a strong impact on the oscillations generated within the network <cit.>. Probing the anatomical origin of network oscillations generated in the cortex has become more feasible since the emergence of optogenetic experiments <cit.>, in which individual groups of neurons can be stimulated selectively. Evidence for γ oscillations being generated by the interaction of inter-neurons alone has been gathered by means of periodic light stimulation in optogenetically altered mice <cit.>. A theoretical study <cit.> reproduces the experimental results by the analytical and numerical treatment of a network composed of excitatory and inhibitory neurons. The explanation requires gap junctions and a subthreshold resonance of the inhibitory neurons. Using Hodgkin-Huxley-type model neurons, Tiesinga <cit.> showed that the results of Cardin et al. can be reproduced by a PING mechanism if the excitatory cells have an additional slow hyperpolarizing current. This result strengthened the previous statement of the author <cit.> that experimental setups using oscillatory stimuli cannot distinguish between underlying ING and PING mechanisms.We can summarize the difficulties that arise in the interpretation of these results with respect to the origin of the observed oscillations by three main points. First, it is still under debate how strongly external stimuli interfere with the dynamical state of the network. Histed et al. <cit.> pointed out that weak light impulses have a linear effect on the population responses of mice in vivo, which they found to be sufficiently predictive for changes in behavior. Second, mean-field theory of recurrent networks needs to be extended to incorporate oscillatory stimuli <cit.>. Third, the dynamical interaction of the connection pattern generating the oscillation with the surrounding network needs to be taken into account.Describing oscillations that arise on the population level from weakly synchronized neurons, Ledoux et al. <cit.> investigate how external input shapes the dynamic transfer function, which describes the response of a neuron to small rate perturbations. However, they do not discuss the implications of this alteration for the dynamical properties to the population rate spectra in high-dimensional recurrently connected populations. Employing a similar framework, Barbieri et al. <cit.> showed by comparison to experimentally measured spectra that describing an input signal as a perturbation around the stationary state suffices to predict a considerable amount of the variance of the LFP.In this work, population dynamics of spiking neurons are reduced to a rate-based description by a combined approach using mean-field theory to determine the stationary rates and linear response theory for the dynamical properties of the fluctuations. The reduction can therefore be understood as a two-step procedure. In the first step the stationary rate of the population is determined by evaluation of the nonlinear stationary transfer function <cit.>, which depends on the mean and variance of the input to the population (also referred to as the working or operating point). All fluctuations around the working point are considered linear in the second step of the reduction, yielding the dynamic transfer function of the populations <cit.>. It has been shown that this level of reduction suffices to describe oscillations in neural networks, that are visible on the population but not on the single neuron level <cit.>. This reduction effectively maps the dynamics of each population composed of numerous neurons to a single noisy rate unit, which filters its input by a dynamic transfer function. Oscillations are therefore described as filtered noise (as found in <cit.>) and the neural network is reduced to coupled units, where the connections between the units shape the correlation structure of the network. Keeping this reduction procedure in mind, we start from a rate based description to illustrate the phenomena that arise when considering oscillatory input to neural networks. In the first section, we use a negatively self-coupled population to analysize how different types of stimuli are reflected in different response measures. In particular, we consider large versus small and filtered versus non-filtered stimuli and their influence on absolute versus relative response spectra. In the second section, we study the contribution of the connectivity structure to the emergence and visibility of resonances in network responses by analyzing three characteristic network motifs composed of one excitatory and one inhibitory population each. Building on the insights gathered from low-dimensional coupled rate circuits analyzed in the first two sections, the third part is concerned with the analysis of resonances evoked by oscillatory stimuli in a microcircuit model based on primary sensory areas <cit.> comprising millions of spiking neurons.We here show that phenomenological rate models with certain connectivity patterns suffice to explain resonance in the γ range in response to oscillatory stimuli supplied to the inhibitory neurons, which is not visible when stimulating the excitatory neurons. In addition we demonstrate, that two different oscillation generating mechanisms, one involving only the inhibitory and one involving both the inhibitory and the excitatory neurons, generate similar resonances. In general terms, we show that the responses of sub-circuits in isolation are different than the responses of a system which embeds this sub-circuit. The fact that a complex system cannot be understood by the analysis of its parts in isolation, but only in its entirety has been pointed out before <cit.>. § RESULTS§.§ Dynamic responses of a self-coupled inhibitory population In this section, we analyze how input is processed in a negatively self-coupled dynamical rate unit that produces a rhythm in the γ-frequency range. The model is inspired by a population of inhibitory leaky-integrate-and-fire (LIF) neurons. We here contrast large versus small stimuli as well as stimuli that affect the input current to the unit versus the output rate of the unit. Changes induced by the input are considered in the spectrum as well as in the power ratio (the spectrum normalized by the spectrum without input).A sketch of the circuit with and without input is depicted in fig:eigenvalue_trajectoryA. The dynamics of the circuit is determined by its dynamic transfer function, which we here choose to approximate the dynamics in a corresponding population of LIF neuron models with delays (for further detail see subsec:Static-and-dynamic) as H(ω)=A e^-iω d/1+iωτ e^-σ_d^2ω^2/2.Here τ denotes the effective time constant of the dynamic transfer function and A its amplitude in response to a constant current. The multiplicative factors e^-iω d and e^-σ_d^2ω^2/2 originate from the Gaussian distributed delays and with mean d and variance σ_d. We here choose the variance equal to the mean, i.e. σ_d=d. The first factor promotes oscillations, while the second one suppresses the transfer of large frequencies. In multi-dimensional systems, the generation of each peak in the spectrum can be attributed to the dynamics of one eigenmode of the system, where the dynamic transfer function of the i-th eigenmode is given by the corresponding eigenvalue λ_i(ω) <cit.>. Each mode emerges from the interplay of several populations. The transfer function here could hence also be interpreted as the transfer function of one dynamic mode. In this case its parameters are understood as effective parameters, which are composed of the parameters of all populations contributing to this mode.The observed rate Y of the unit is given by its output rate R(ω) combined with additive white noise X(ω) with zero mean ⟨ X(ω)⟩=0 and non-zero variance ⟨ X(ω)X^T(-ω)⟩=D as Y(ω)=R(ω)+X(ω).The noise term originates from the fact that the considered rate profile actually describes a spike train. In other words, the spike train can be considered as a noisy realization of the rate profile R(ω). The internally generated noise in self-coupled populations of LIF neurons exhibits a variance of D=r_0/N <cit.>, where r_0 denotes the stationary rate of the neurons in the population and N the number of neurons. The fluctuating rate produced by the circuit (fig:eigenvalue_trajectoryA upper sketch) reads Y(ω)=-wH(ω) Y(ω)+X(ω) ⇔Y(ω)=1/1-λ(ω)X(ω),where the fed back rate is weighted by the feedback strength -w and we identify λ(ω)=-wH(ω) as the eigenvalue of the one-dimensional system. When considering LIF neuron models, the strength w is determined by the synaptic amplitude and the number of connections. The spectrum of the population without additional input is given by C(ω)=⟨ Y(ω)Y^T(-ω)=|1/1-λ(ω)|^2D. fig:eigenvalue_trajectoryB shows the Nyquist plot of the eigenvalue λ(ω), which determines the shape of the spectrum. The peak frequency is determined by the point at which the eigenvalue trajectory assumes its closest distance to unity, resulting in a large prefactor in eq:cross_spectrum_baseline (see also <cit.>). The parameters of the dynamic transfer function (eq:params_1d_model) are based on the dynamic transfer function of populations in a large scale model composed of LIF neurons <cit.> and chosen to produce a peak in the γ frequency range (fig:eigenvalue_trajectoryC). The mapping between the LIF neurons and rate models is described in the first sections of the sec:Methods.When considering the effect of external input to the spectrum in the following, we distinguish weak and strong stimuli that require different levels of description: Large input changes the stationary rate and the dynamic properties of the population. Small input can be treated as a perturbation around the stationary point which itself remains unchanged. We will show in the last part of this study, that a small oscillatory component in the input to a population is sufficient to affect its spectrum considerably. We therefore neglect the effect of the oscillatory component of the stimulus onto the stationary point and restrict this analysis to either oscillatory input, which can be treated as a perturbation, or oscillatory input with an additional constant offset that may change the stationary state. We start by applying a large constant input to the population, yielding an altered stationary rate of the system r_0→ r_0+δ r_0. This changes the spectrum for two reasons. First, the dynamic properties of the population change yielding a new dynamic transfer function H(ω)→ H_I(ω) (λ(ω)→λ_I(ω)). This occurs in systems whose dynamic transfer function depends on the statistics of the input, which is also referred to as the working point. In general, an increase in the external rate can yield large changes in the dynamic transfer function. However, reasonably sized stimuli applied to populations in the fluctuation driven regime primarily affect the offset of the transfer function and leave the shape approximately unaltered (see subsec:Approximation_of_dtf). This suggests the following approximation H_I(ω)≈(1+δ A/A) H(ω) (see fig:eigenvalue_trajectoryB for the shifted eigenvalue trajectory). Second, the input alters the stationary rate of the circuit and therefore the amplitude of the internally generated noise D→ D_I=(r_0+δ r_0)/N. The new spectrum is hence given by C_I(ω)=|1/1-λ_I(ω)|^2D_I.In the following C_I(ω) is termed the response spectrum, δ C(ω)=C_I(ω)-C(ω) the excess spectrum, and ρ(ω)=C_Iω)/C(ω) the power ratio. The latter is commonly used in experimental studies since it is insensitive to the filtering of the local field potential by the extracellular tissue <cit.> and dendritic morphology <cit.>, provided that both can be approximated as activity independent. All three measures display a peak at the frequency generated by the circuit (fig:eigenvalue_trajectoryC,D). The peak arises because the excitatory input provided to the system effectively strengthens the inhibitory loop that generates the oscillation by shifting the eigenvalue closer to the value one and therefore closer to a rate instability <cit.>. Oscillatory input current injected into a neuron is necessarily filtered by the dynamic transfer function of the neuron. It is, however, the change in population firing rate that is recurrently processed on the network level and eventually constitutes the measurable network response. To this end we compare network responses to two types of stimuli. The first type causes a modulation of the input current (“current modulation”, CM), and the second type directly modulates the output rate (“rate modulation”, RM, see also the illustrations in fig:fig1B,C). In the first case the stimulus is filtered by the dynamic transfer function before it affects the activity of the population (see subsec:spec_input), while it can be directly added to the rate in the latter case. The spectrum of the stimulated network hence reads C_I(ω) =|1/1-λ(ω)|^2(D+R_I(ω)R_I^*(ω)) R_I(ω)= I(ω)H(ω)I(ω)Here I(ω) describes the stimulus in Fourier domain. Experimental studies considered periodic stimuli to investigate the circuits underlying the generation of oscillations <cit.>. Supplying a sinusoidal stimulus with frequency ω_I (I(ω)=iπ I_0(δ(ω+ω_I)-δ(ω-ω_I))) to the one-dimensional circuit contributes an additional term to the spectrum at stimulus frequency and yields the following power ratio ρ(ω)=C_I(ω)/C(ω)=1+1/Dπ^2I_0^2, π^2I_0^2|H(ω)|^2 ω=ω_I.This expression shows in particular, that the power ratio is independent of the resonance properties of the circuit, since its contributions (which are described in eq:cross_spectrum_baseline) cancel. The power ratio is independent of the stimulus frequency for rate modulated systems (fig:fig1A), while it reflects the shape of the population filter H in current modulated systems (fig:fig1C). This tendency is also reflected in the response spectrum, which displays a constant offset in the RM system while for the CM system it approaches the spectrum without stimulus for higher frequencies due to the low-pass filter of the population.This insight can be directly transferred to experimental studies. To investigate the anatomical origin of oscillations, one seeks to analyze the dynamics of the rate fluctuations generated within the circuit. Hence, an upstream low-pass filter can give the false impression that slow rate fluctuations are generated within the circuit, even though they arise from the filter of the population. Adjusting the input strength to emphasize fast frequencies can compensate for the low-pass filtering of the populations; to this end one needs to replace the amplitude as I_0→ I_0/|H(ω)|. An exact experimental implementation of this protocol is only possible, if the dynamic transfer function of the stimulated neuronal population is known. However, the relation can still be employed in an approximate manner to counteract known influences. For example Tiesinga <cit.> modeled optogenetic stimuli as AMPA mediated currents. Since the dynamic transfer function is a convolution of the synaptic and the population filter, the synaptic filter could be counteracted in experiments by considering the underlying receptor and neurotransmitter density, which determine the time scales of the synaptic currents.The two effects described above can be combined in a stimulus that has a constant component, which increases the susceptibility and firing rate of the target population, as well as an oscillatory component. This yields the following power ratios at ω=ω_I ρ(ω) =|1-λ(ω)/1-λ_I(ω)|^2D_+|I_I(ω)|^2/D =|1-λ(ω)/1-λ_I(ω)|^2(D_/D+I_0^2/D) (D_/D+I_0^2|H_I(ω)|^2/D)From the analysis of the spectrum with constant input and from eq:cross_spectrum we know that the constant component of the stimulus shifts the eigenvalue, such that the λ-dependent prefactor in the latter equation displays a peak close to the internally generated frequency. This peak, which reflects a positive change in the excitability of the population (D_>D⇒ H_I(0),λ_(0)>H(0),λ(0)), is clearly visible in the rate modulated system (fig:fig1B). Here, the frequency independent contribution of the oscillatory stimulus is added to the internal fluctuations. It therefore amplifies the peak, which is shaped by the shift in the working point, but it cannot affect the shape of the spectrum by itself. The spectrum for current-modulated circuits experiences additional amplification at low frequencies compared to the rate modulated circuit due to the multiplications of the dynamic transfer function. If the change in excitability is large, this amplification of low frequencies can overshadow the peak caused by the shift in working point. The balance between these two effects depends on the parameters of the population filter and the rise in excitability (H_I(ω)-H(ω), for small ω) compared to the closeness of the system to an instability, characterized by the term |1-λ_I(ω)|^2(at peak frequency).In summary, the analysis of a one-dimensional self-coupled population shows that the responses of the circuit can vary widely depending on the properties of the system and the stimulus. Namely, a stimulus that affects the stationary state of the system changes its dynamic properties and therefore alters the strength of internally generated oscillations. Stimuli that can be treated as perturbations amplify the internally generated oscillation, but do not change the underlying dynamical circuits that shape the spectrum. Input that directly affects the rate of the population reveals information regarding the dynamics of the circuit, while the responses to stimuli that are added to the input current to the population run the risk of reflecting the filter properties of the populations. These effects dominate the frequency-dependence of power ratios, which become independent of the resonance properties of the circuit. The response and the excess spectrum, in contrast, both exhibit peaks at the internally generated oscillation. It is therefore advantageous to consider one of the two latter quantities in addition to the power ratio. §.§ Stimulus evoked spectra in a two dimensional network In neural circuits, recurrent loops generating characteristic oscillations do not appear in isolation, but are embedded into larger networks. To analyze the effect of the surrounding network on the responses elicited by oscillatory stimuli, we start by considering oscillation generating circuits composed of one or two populations. We first describe how the oscillations generated within the network can be understood by means of dynamical modes and how the effect of a stimulus vector can be split into components that each excite a different mode. The responses of individual modes can in principal be traced back to an anatomical circuit that generates the oscillation. However, the identification of the origin of an oscillation is usually complicated by the fact that responses to stimuli are composed of multiple modes. To isolate this phenomenon we study three exemplary circuits in which evoked responses can be treated as perturbations. Here, applied stimuli modulate the rate directly (RM), assuming a stimulation protocol which counteracts the filter properties of the population that receives the input. A two-dimensional circuit, as shown in fig:2d_ING1A, is composed of an excitatory (E) and an inhibitory (I) population, with the dynamic transfer functions of population i receiving input from population jH_ij(ω)=A_ie^-iω d_j/1+iωτ_i e^-σ_d_j^2ω^2/2, i,j∈E,I,where d_j denotes the delay of a connection starting at population j. To illustrate the phenomena analyzed here in the simplest possible setup, we assume that the neurons in the two populations have equal working points and that the delays of all synapses are identically distributed. The neurons therefore have equal stationary firing rates (r_0,,r_0,)=(r_0,r_0) as well as equal transfer functions H_ij(ω)=H(ω). The connectivity matrix is given by 𝐖=([a -b;c -d ])with all parameters being positive a, b, c, d>0. In networks of LIF-neuron models, these parameters are given by the product of in-degrees and connection strength (a=w_EEK_EE, b=w_EIK_EI, c=w_IEK_IE, and d=w_IIK_II). The effective connectivity matrix, which combines the anatomical and dynamic properties of the circuit, is given by 𝐌(ω)=H(ω) 𝐖, with the eigenvalues λ̃_0,1(ω)=H(ω)=λ_0,1(a-d/2±√((a-d)^2/4-(bc-da))).The right (u_i) and left (v_i) eigenvectors are given by 𝐮_0=𝐯_0=([ 1; 0 ]), 𝐮_1=𝐯_1=([ 0; 1 ]) b=c=0or 𝐮_i=1/y_i([ 1; x_i ]), 𝐯_0=y_0/x_1-x_0([ x_1;-1 ]), 𝐯_1=y_1/x_1-x_0([ -x_0;1 ]) x_i=a-λ_i/bc=0 c/λ_i+d y_i=√(1+|x_i|^2). They are bi-orthogonal and normalized such that |𝐮_i|^2=1 and 𝐯_i^T𝐮_j=δ_ij. Note that in a more general setting, where the populations have different stationary activities and therefore different transfer functions, the eigenvectors are frequency dependent. The spectrum produced by the circuit can be expressed as the sum of spectra produced by the eigenmodes due to their auto- (n=m) and their crosscorrelation (n≠ m) (see subsec:Composition-spec) 𝐂(ω)=∑_n,m∈{0,1}β_nm/(1-λ̃_n(ω))(1-λ̃_m^*(ω))𝐮_n𝐮_m^T*with β_nm=∑_iD_iiα_n^iα_m^i, where α_j^i=𝐯_j^T𝐞_i denotes the projection of the j-th left eigenvector onto the i-th unit vector with 𝐞_i^T∈{ ([ 1,0 ]),([ 0,1 ])}. For LIF neuron models the diagonal elements of the stationary activity matrix are given by D_ii=r_0,i/N_i. When describing neuronal populations, the auto- and crosscorrelations of the modes describe properties of groups of neurons, namely the summed correlations on the single neuron level. For example, the autocorrelation of one modes refers to the sum of all auto- and crosscorrelations between neurons that constitute that mode.The diagonal elements of 𝐂(ω) (eq:spec_2d) describe the spectra of the population activity. For frequencies ω_c at which one eigenvalue λ̃_c(ω_c) approaches the value one, a peak is visible in the spectrum. The anatomical connections that determine the amplitude and frequency of this peak can be established using the following quantities <cit.> Z_ij^amp=((Z_ij),(Z_ij))𝐤^T Z_ij^freq=((Z_ij),(Z_ij))𝐤_⊥^T,which identify the sensitivity of the eigenvalue to the connections (defined by the matrix elements M_ij) via Z_kl=v_c,kM_klu_c,l/𝐯_c^T𝐮_c,where 𝐮_c and 𝐯_c are the right and left eigenvector associated to λ_c(ω_c). The unit vectors that describe directions in the complex plane: 𝐤 points from λ_c(ω) to the one and 𝐤_⊥ perpendicular to 𝐤, are given by 𝐤=(1-(λ_c),(λ_c))/√((1-(λ_c))^2+(λ_c)^2) 𝐤_⊥=(-(λ_c),1-(λ_c))/√((1-(λ_c))^2+(λ_c)^2), where all dependencies on frequency were omitted for brevity of notation.In electrophysiological recordings, the activities of the excitatory and the inhibitory population are often indirectly observed via the local field potential (LFP), which has been related to the input of pyramidal neurons <cit.> (see subsec:LFP) C_LFP(ω)=:=C_LFP^auto(ω)a^2C_(ω)+b^2C_(ω)+:=C_LFP^cross(ω)2ab(C_ I(ω)).The LFP gets contributions from the excitatory and the inhibitory current onto the excitatory neurons as well as from their crosscorrelation. Defining the LFP in this way implicitly assumes δ-shaped synaptic currents, otherwise the contributions above would additionally be filtered by the synaptic kernels (see eq:def_CLFP_method). Stimulating the circuit with sinusoidal input F(ω)=iπ I_0[δ(ω+ω_I)-δ(ω-ω_I)] of frequency ω_I and amplitude I_0 in the direction 𝐞_I elicits the excess spectrum (see eq:spectrum_decomp_with_input) at ω=ω_I δ𝐂(ω)=𝐂_I(ω)-𝐂(ω)=∑_n,m∈{0,1}=δ𝐂^nm(ω)β_nm^I/(1-λ̃_n(ω))(1-λ̃_m^*(ω))𝐮_n𝐮_m^T*with β_nm^I=π^2I_0^2γ_nγ_m^*, where γ_j=𝐯_j^T𝐞_I marks the projection of the stimulus direction onto the left eigenvector. If the stimulus vector is parallel to the right eigenvector of one mode 𝐞_I=x𝐮_k it will excite only this mode (γ_k=x, γ_i≠ k=0). The scalar γ_i therefore measures the portion by which the i-th eigenmode is stimulated by the stimulus vector 𝐞_I. The stimulus-induced component of the LFP response that originates in the autocorrelation of the currents is given by δ C_LFP^auto(ω)=C_LFP,I^auto(ω)-C_LFP^auto(ω)=∑_n,m∈{0,1}(a^2δ C_^nm(ω)+b^2δ C_^nm(ω)).Since the contributions of individual modes are more straightforward to separate for the autocorrelations, we only consider the LFP contribution defined in eq:def_dC_LFP_auto in detail (see eq:def_CLFP_method for the full LFP spectrum). It will, however, be shown, that the crosscorrelations do not interfere with the discussed effects.In the following sub-sections, we will show on three exemplary circuits, that non-negligible connectivity between populations evokes responses of several modes when individual populations are stimulated. The interference of these mode responses can yield similar network responses for different underlying network structures.§.§.§ A self-coupled inhibitory circuit embedded in a two-dimensional network The first circuit is composed of two self-coupled populations, one excitatory and one inhibitory. The latter is coupled to the excitatory population, while the reverse connection is of negligible strength. Because the network has an approximate feedforward rather than a recurrent structure, the dynamic modes of the circuit correspond approximately to the populations in isolation. The E-E loop generates a low pass filter and the I-I loop a peak at around 140 (fig:2d_ING1B), which is visible in the LFP at around 125 (fig:2d_ING1C). This is reflected in the eigenvectors (eq:evecs_iso), which point in the direction of the populations, and in the eigenvalues λ_0≈ a, λ_1≈-d, which reflect the strength of the feedback connections of the populations. Without additional input, the signal produced by the excitatory population is dominated by the zeroth mode. Since the self-coupling of the mode is positive, but the dynamics are still stable, excitations decay slowly, reflected by enhanced slow frequencies in the spectrum (fig:2d_ING1C, blue curve). The spectrum observed in the inhibitory population is dominated by the first mode, which has a negative eigenvalue and therefore produces an oscillation similar to the isolated populations discussed in the previous section (fig:2d_ING1C, red curve). Combining these signals yields the LFP (fig:2d_ING1C, gray and black curve), which contains contributions of both modes. Since the negative feedback to the inhibitory populations is stronger than the excitatory connection, the LFP is similar to the spectrum of the inhibitory neurons. At low frequencies, the spectrum of the excitatory population is particularly large and therefore raises the LFP signal. Stimulating the excitatory population excites only the zeroth mode (γ_0=𝐯_0^T𝐞_0≈1, γ_1≈0), as sketched in the left panel of fig:2d_ING1D and reflected in the additional LFP response (fig:2d_ING1E, left panels). Similarly, stimulating the inhibitory population excites the first mode (fig:2d_ING1D, right) and therefore yields a high frequency peak in the LFP response (fig:2d_ING1E, right). In summary, in a two-dimensional network, with a feedforward structure from the inhibitory to the excitatory population, each population generates its own rhythm by self-coupling. As a result, the excitation of individual populations elicits responses which can be traced back to the original circuits generating the oscillations observable in the LFP. In particular, stimulating the inhibitory population yields a high frequency peak in the additional LFP spectrum, stimulation the excitatory population, on the other hand, yields increased low frequencies.§.§.§ Symmetric two-dimensional network A peak in the high γ-range has been shown to be generated by a network composed of LIF-neuron models with symmetric architecture <cit.>. In this setup the excitatory and the inhibitory population receive the same input (a=c=1, b=d=g) (fig:2d_ING2A) yielding one eigenvalue to be zero and one to be negative (λ_0=0, λ_1=1-g), for networks working in the inhibition dominated regime (g>1). Here the eigenvectors deviate from the vectors of the populations (𝐞_0 and 𝐞_1), revealing that the modes get contributions from both populations. Since the zeroth eigenvalue is zero, the corresponding mode has no feedback (fig:2d_ING2D, left) and the produced spectrum, which scales with 1/|1-λ_0H(ω)| (eq:spec_2d), is therefore constant. The first mode generates a peak due to the negative feedback, with a frequency of around 130 determined by the parameters of its transfer function as well as the strength of the coupling λ_1. The sensitivity analysis shows that this peak is generated by an interplay of the two populations as well as their self-coupling (fig:2d_ING2B). The LFP produced by the circuit without additional input, as well as its decomposition into the spectra observed in the populations, is shown in fig:2d_ING2C. Since the population vector of the inhibitory population points more in the direction of the right eigenvector of the first mode than the right eigenvector of the zeroth mode (𝐮_1^T𝐞_1>𝐮_0^T𝐞_1), the spectrum of the inhibitory population displays the peak generated by the first mode. The opposite is true for the excitatory population (𝐮_0^T𝐞_0>𝐮_1^T𝐞_0) which shows a spectrum dominated by the zeroth mode and the mixture of the zeroth and first mode, which is reminiscent of a low pass filter. Note, however, that this reasoning only holds for modes which are far from an instability. If one of the eigenvalues would approach the value one at a certain frequency, the prefactor in eq:spec_2d would become large and the corresponding peak would be visible in all populations. However, in the presented regime of weakly synchronized oscillations, the rhythm generated by the full circuit produces similar population rate spectra as the rhythm generated by a single population (compare fig:2d_ING2C and fig:2d_ING1C). It is notable, that in this circuit the crosscorrelation of the population rate spectra have a large impact on the LFP (fig:2d_ING2C ).Applying a stimulus to one of the two populations elicits large responses originating in the autocorrelations of the modes, as well as in their crosscorrelation (see single color and dashed curves in fig:2d_ING2E). The large crosscorrelation can be understood by considering that the same connections contribute to both modes. That was not the case in the previous circuit, where the crosscorrelation therefore remained small. In the current circuit, the sum of the auto- and crosscorrelation is small and thus the resulting spectrum is also small. An example of such a circuit is a network of neurons where the population signal displays small fluctuations, but a projection of the rates onto the direction of the modes yields strongly fluctuating signals, which are highly negatively correlated between the modes.In quantitative terms, stimulating the excitatory population elicits responses of the modes whose amplitudes scale with γ_0=√(g^2+1)/g-1 and γ_1=-√(2)/g-1 (fig:2d_ING2D, left). The signs of γ_0 and γ_1 reveal that excitation of the two modes is of opposite signs, yielding a negative crosscorrelation (which scales with γ_1γ_2, see fig:2d_ING2E, left). As the population vector of the excitatory population points more in the direction of the right eigenvector corresponding to the zeroth mode than the eigenvector of the first mode (|γ_0|>|γ_1|), the stimulus induced response visible in the LFP is dominated by contributions of the zeroth mode and the coupling of the zeroth and first mode (fig:2d_ING2E left), as already observed in the composition of the spectrum without additional stimulus. Stimulating the inhibitory population also excites both modes (γ_0=-√(g^2+1)/g-1, γ_1=g√(2)/g-1) with opposite signs (fig:2d_ING2D, right) such that the main part of the responses cancel. The contribution of the first mode is slightly larger (fig:2d_ING2E, right), imposing its peak onto the LFP spectrum. Thus the responses to stimulations obtained here are similar to the responses of the previously considered circuitry: stimulating the excitatory population elicits mainly slow frequencies, while stimulating the inhibitory population reveals a high frequency peak. These responses can therefore not distinguish between an I-I-loop and a fully connected E-I-circuit generating the high frequency peak.In principle, it is possible to selectively probe the modes of a complex circuit experimentally. This requires co-stimulation of all populations with population specific stimulus amplitudes chosen proportional to their respective entries in the right-sided eigenvectors 𝐮_0 and 𝐮_1.§.§.§ A two-dimensional network without self-coupling Recurrently connected excitatory and inhibitory populations can produce oscillations without the necessity of inhibitory self-feedback. The network motif discussed in this section can be considered the prototypical PING motif as discussed in <cit.>, which produces γ oscillations. fig:2d_PINGA shows a diagram of a circuit with connections between the populations and negligible self-couplings. In this parameter regime the LFP is determined by the inhibitory activity impinging onto the excitatory neurons. This circuit generates a peak at around 50, which, as expected, depends on the connections between the populations (fig:2d_PINGB). Since the eigenmode producing the oscillations is a mixture of the two populations, with both of them having comparably sized entries in the corresponding eigenvectors, the population spectra are similar in both populations with similar contribution to the LFP (fig:2d_PINGC, all curves lie on top of each other). The circuit is characterized by two eigenvalues, which are complex conjugates and purely imaginary (λ_0,1=± i√(bc)). Considering the eigenvalue trajectories (described by λ_0,1H(ω)) reveals that the trajectory starting at a positive imaginary part produces the 50 peak, while the other trajectory produces a peak at a very large frequency. The latter one is potentially suppressed in neural circuits by inhomogeneities in the parameters, like distributed delays, which contribute an additional multiplicative factor with low-pass characteristics (cf. eq:def_H) to the transfer function. We hence focus on the peak at lower frequency.A stimulus applied to either the excitatory or the inhibitory population excites both modes with equal strength (γ_0=γ_1=1/2√(bc+1/bc) for stimulation of the excitatory population and γ_0=γ_1^*=i/2√(bc+1) for stimulation of the inhibitory population,illustrated in fig:2d_PINGD). Since γ_0γ_1^*>0 when stimulating the excitatory and γ_0γ_1^*<0 when stimulating the inhibitory population, the contribution of the crosscorrelations to the response spectrum is of opposite sign for the two stimuli, as seen in fig:2d_PINGE (dashed curves). The cancellations of the cross- and autocorrelation when stimulating the excitatory population yields an LFP response, which is reminiscent of a low-pass filter with a small peak at very low frequencies (fig:2d_PINGE, left upper panel). Since no cancellation occurs when stimulating the inhibitory population, the spectrum shows amplifications at the frequency that is generated by the circuit autonomously (fig:2d_PINGE, right upper panel). Hence, also with this circuit motif, stimulation of the inhibitory population yields a peak in the LFP which is missing when stimulating the excitatory population, even though the excitatory population is involved in generating the peak.In order to isolate the response of one mode, the stimulus vector needs to point in the direction of its right eigenvector. Since the eigenvectors are complex, with real entries for the excitatory and imaginary entries for the inhibitory population (𝐮_0,1=√(1/bc+1)(√(bc),e^∓ iπ/2)), adjusting the amplitude of the sinusoidal signal applied to the population is not sufficient to segregate the mode responses. Since the Fourier transform of a phase-shifted sine wave is given by ℱ[sin(ω_0t+ϕ)]=e^iω/ω_0ϕℱ[sin(ω_0t)], complex entries in the stimulus vector can be achieved by adjusting the relative phase of the input to the populations. The mode generating the peak around 50 is excited in isolation if the stimulus applied to the inhibitory population lags the stimulus to the excitatory population by π/2. Reversely, if the excitation of the inhibitory population precedes the stimulation of the excitatory population by π/2, the LFP response is determined by the first mode, which amplifies fast oscillations. §.§ Stimulus evoked spectra in a model of a microcircuit In this section we analyze the responses observed in a model of a microcircuit of the primary sensory cortex to oscillatory input, discuss the results in comparison with experimental data and point out potential pitfalls when utilizing these results to identify the anatomical sources of oscillations produced by the circuit. First, a previously introduced theoretical framework, which enables the prediction of population rate spectra as well as the location of their origin, is extended by oscillatory input. Subsequently we demonstrate that the theoretical prediction reproduces the responses observed in simulations and additionally offers insight into the anatomical origin of the components contributing to these responses. Analyzing the responses of the populations in layer 2/3, we demonstrate that ad-hoc interpretations can yield misleading conclusions.§.§.§ The microcircuit with oscillatory input The microcircuit model has been introduced by Potjans et al. <cit.> to represent a layered circuit typical for the primary sensory cortex. The model is composed of around 10^5 LIF model neurons, which are divided into four layers (L2/3, L4, L5, L6) with one excitatory and one inhibitory population each. Connection probabilities between theses eight populations are gathered from 50 anatomical and physiological studies. The model has been shown to reproduce typical rate profiles <cit.>. In agreement with experimental evidence it supports the emergence of slow rate fluctuations in layer 5, as well as low- and high-γ oscillations in the upper layers <cit.>. As yet, the microcircuit has only been analyzed in the resting state, when each population receives uncorrelated Poisson input, which mimics the input of remote areas. Oscillatory stimuli are introduced by modulating a ratio a (0<a<1) of the external input rate to the populations with a sinusoid of a given frequency f_ext ν_ext^k(t)=ν_ext^0N_ext^k(1+a sin(2π f_extt)).Here ν_ext^k(t) denotes the total external input applied to the k-th population, ν_ext^0 the firing rate associated with one incoming connection and N_ext^k the external indegree to population k. fig:Population-rate-responses shows the instantaneous firing rate, as well as the spectra observed in populations 2/3E, 2/3I, 4E, and 4I in the resting state and with oscillatory input of 10 and 50 to the populations of strength a=0.05. The spectra show that this modulation of 5 percent of the external static inputsuffices to reproduce population responses of strength comparable to LFP responses measured in experiments applying oscillatory light stimuli to optogenetically altered mice (see Fig. 3c in <cit.>). The rate of population 4E adapts strongly to both rhythms (see left panel of fig:Population-rate-responsesC), which is interesting since layer 4 is considered the main recipient of thalamic input <cit.>. In the microcircuit, excitatory populations show a stronger amplification of the 10 stimulus, while the inhibitory populations show higher amplitudes when stimulated with 50 (fig:Population-rate-responses). These results agree with measurements conducted by Cardin et al. (Supplementary Fig. 5 in <cit.>) by trend: Low frequency stimulation of excitatory neurons show stronger responses in the LFP than stimulation of inhibitory neurons and vice versa for high frequency stimulations. However, both excitatory and inhibitory currents contribute to the LFP <cit.>, such that the measured responses in experiments should be compared with a superposition of the individual population rate spectra. The effect that low frequency stimulation of excitatory populations evokes strong responses can be understood intuitively: Our previous examples show that excitatory populations more prominently participate in modes that resemble low-pass filters. In other words, excitatory activity and, in particular, excitatory self-feedback drive the circuit towards a rate instability which facilitates slowly decaying modes. Responses of these modes are also elicited when stimulating the excitatory population at low frequencies, resulting in amplified low frequency responses.§.§.§ Theoretical description of oscillatory input Here we describe how the amplification of oscillatory stimuli can be understood theoretically. Previous work <cit.> shows that the population rate spectra produced by this model are sufficiently described by a theoretical two-step reduction composed of mean-field theory, which yields the stationary firing rates <cit.>, as well as linear response theory, which characterizes the response properties of the neurons <cit.>, formally described by the dynamic transfer function. Since already small modulations of the external input have considerable impact on the population rate spectra, but negligible effect on the stationary firing rates (fig:Population-rate-responses), we constrain our analysis in this section to purely oscillatory stimuli which do not alter the working point of the populations. This assumption can be justified by the observation that the width of the response peaks shown in fig:Population-rate-responses is narrow (in particular for high frequencies) and the stimulated frequency therefore approximately does not couple to other frequencies. The network can therefore be analyzed analogously to the two-dimensional circuits discussed in the previous section, after mapping the dynamics in the microcircuit model to interacting linear rate models <cit.>. Extensions to stimuli that affect the stationary state of the system are discussed in subsec:Approximation_of_dtf. The spectrum as well as the excess spectrum δ𝐂(ω) of the microcircuit with sinusoidal input are hence described by the same equation as the 2d-circuit (eq:spec_2d, eq:excess_spec_2d) extended to eight populations (for detail see subsec:spec_input) δ𝐂(ω)=𝐂_I(ω)-𝐂(ω) =∑_n,m=1^8β_ext,nm(ω)/(1-λ_n(ω))(1-λ_m^*(ω))𝐮_n(ω)𝐮_m^T*(ω),with (eq:beta_ext)β_ext,nm(ω)=w_ext^2N_ext,k^2ν_ext^2a^2T/4|H_k(ω_I)|^2α_n^k(ω)α_m^k*(ω).The latter factor, which quantifies the amplitude of the excited auto and cross-correlations of the modes, depends on the modulated external firing rates N_ext,kν_exta, the external synaptic weights w_ext, the transfer function of the population that receives the input H_k(ω_), the projection of the modes on the direction of the stimulus α_n^k(ω)=𝐯_n^T(ω)𝐞_k as well as the measurement time T.Since the populations are set at different working points, which is reflected in differentpopulation specific firing rates and transfer functions, the left and right eigenvectors are frequency dependent, in contrast to the models considered in the previous sections. The power ratio, which describes the relative size of the evoked spectrum at stimulation frequency, is given by ρ(ω)=𝐂_I(ω)/𝐂(ω)=1+δ𝐂(ω)/𝐂(ω)and evaluated at stimulus frequency ω=ω_I. To show that the theoretical description suffices to predict the impact of oscillatory input to the population rate spectra, we modulate 5percent of the static external input to each population at frequencies between 0 and 200 and compare the power ratios at stimulation frequency observed in each population with the theoretically predicted power ratios (fig:Power-ratios-sim). As expected, the response to a stimulus is strongest in the population the stimulus is applied to (see the diagonal panels in fig:Power-ratios-sim).Considering only the diagonal panels, it is evident that stimuli applied to excitatory populations amplify mostly low frequencies, which confirms the previously found tendencies. The inhibitory populations, except for 4I, display resonances at frequencies larger than zero. The most prominent peak is visible in population 2/3I, which, due to its location at just above 40, could be interpreted as a resonance phenomenon related to the low-γ peak produced by the circuit. The responses of 2/3E and 2/3I (fig:Composition-of-powerA0, B0) show similar tendencies as the LFP power ratio measured in experiments (see Fig. 3 in <cit.>), namely 2/3E supports mainly low frequencies, while 2/3I displays a resonance at around 41. However, it remains to be investigated whether the peaks in the population rate spectra would also be visible in the LFP, which is given as a weighted sum of the spectra of one row in fig:Power-ratios-sim in addition to the crosscorrelations of the currents (as outlined for an examplary 2d-circuit in subsec:LFP). §.§.§ The origin of the peak visible in the population rate spectrum of 2/3IIn the following, the results of the last sections are exploited to analyze whether and how the peak in the power ratio of population 2/3I can shed light on the circuit producing the low-γ oscillation. Since power ratios can give misleading results, especially if the resting spectra are not entirely flat (as discussed in subsec:I-I-loop), the resting state spectra as well as the response and excess spectra are shown in fig:Composition-of-powerA1 and fig:Composition-of-powerB1. The response spectra are between one to two orders of magnitude larger than the resting state spectra and are therefore approximately equivalent to the excess spectra. The excess spectrum for population 2/3E shows a peak in the low-γ frequency range, as well as a large offset for low frequencies. Since the stimulus amplifies low frequencies stronger than the low-γ peak, the γ peak is not visible in the power ratio. The excess spectrum visible in 2/3I has a contrary tendency: it is weak for low frequencies and saturates at a higher value for frequencies above 40. The peak in the power ratios originates in the prompt increase of the excess spectrum between 20 and 40.In the following we decompose the excess spectrum into contributions from the respective dynamic modes (see eq:spectrum_decomp_with_input_microcircuit) and identify the dominant contributions at peak frequency 41. Subsequently we employ the recently derived sensitivity measure <cit.> to trace the dominant modes back to their anatomical origin.We recall from the previous chapters, that the spectrum visible in one population is composed of contributions arising from the autocorrelation as well as the crosscorrelation of the modes. If the peak in the power ratio of population 2/3I would reflect the resonance of one sub-circuit, one would expect the excess spectrum to be composed mainly of the contribution of the corresponding mode. The diagonal contributions to the excess spectrum are given by δ𝐂_auto(ω)=∑_n=1^8β_ext,nn(ω)/|1-λ_n(ω)|^2 𝐮_n(ω)𝐮_n^T*(ω).The dominant contributions originating in the autocorrelations of the modes as well as the sum of all contributions, are shown in fig:Composition-of-powerA2 and fig:Composition-of-powerB2. The main contribution to the low-γ peak in the excess spectrum of population 2/3E is indeed given by the trajectory corresponding to the origin of the low-γ peak (compare dashed to purple curve), which is determined by connections located in layer 2/3 and 4 (fig:Anatomical-origins-ofB). The eigenvalue trajectory, that gives rise to the low-γ peak, originates in a pair of complex conjugate eigenvalues (see fig:Anatomical-origins-ofA at frequency zero, where their absolute values agree), similar to the eigenvalues in the exemplary circuit where the rhythm was entirely determined by the coupling between the excitatory and the inhibitory population rather than their self-coupling (see subsec:An-two-dimensional-network). Hence the modes corresponding to the complex conjugated pair of eigenvalues are shaped by the same connections and a stimulus applied to population 2/3E also elicits a response of the mode associated to the counterpart of the low-γ eigenvalue (see pink trajectory, its origin fig:Anatomical-origins-ofC and its contribution to the excess spectrum fig:Composition-of-powerA2). However, even though the peak shape appears to be formed by these modes, the offset of the excess spectrum cannot be explained by the diagonal contributions alone (compare dashed to gray curve). Population 2/3I contributes to the generation of two different peaks, the low-γ oscillation, as well as a high-frequency oscillation originating in the self-coupled population alone (see green eigenvalue trajectory in fig:Anatomical-origins-ofA and its origin in fig:Anatomical-origins-ofD). Hence, stimulating population 2/3I elicits responses of modes with different anatomical origins (fig:Composition-of-powerB2). The sum of contributions originating in the autocorrelations, however, deviates considerably from the total excess spectrum. This shows that crosscorrelations between the modes play a role. The total excess spectrum can be reproduced by adding the largest terms of the cross-contributions between the modes, which are given by δ𝐂^cross(ω)=∑_n,m=1,m<n^8(β_ext,nm(ω)/(1-λ_n(ω))(1-λ_m^*(ω)) 𝐮_n(ω)𝐮_m^T*(ω)).The contributions to the spectrum due to the crosscorrelations between the dominant modes are shown in fig:Composition-of-powerA3 and fig:Composition-of-powerB3. For population 2/3E the crosscorrelation between the two modes is positive and provides the missing offset to the excess spectrum. For population 2/3I, the crosscorrelation between the mode associated to the low-γ peak and the self-coupling of population 2/3I is negative below 40.In other words, providing oscillatory input between 0 and 40 to population 2/3I elicits large responses of two modes. The responses of the modes are given by weighted linear combinations of the population rates corresponding to a multiplication of the rate vector containing all rates with the left eigenvector of the modes. However, the anti-correlation between the modes at low frequencies, which is strongest at 20, lowers the fluctuations in the total response of population 2/3I. Since the amplitude of the negative correlation decreases for larger frequencies, a peak appears in the power ratios. This peak could be misinterpreted as the self-coupling of population 2/3I to generate an oscillation in the low-γ frequency range, even though the peak is generated by a more complex sub-circuit in the upper layers. § DISCUSSION In this manuscript we analyze the responses of networks, that generate oscillations internally by means of embedded sub-circuits, to oscillatory stimuli and demonstrate how the stimulus is reflected in different measures of the network response. We first employ a sequence of phenomenological rate models, each of which illustrating one dynamical effect that crucially shapes the response. Finally, we apply these insight to circuits composed of populations of LIF model neurons that can be mapped to the former class of models analytically. Our results shed light on the amount of information regarding the anatomical origin of produced oscillations that can be inferred from the responses to oscillatory stimuli. The insights outlined here can be exploited to design stimulation protocols that uncover dynamically relevant sub-circuits.Theoretical advances have been made to explain the results of Cardin et al. <cit.>, who have shown that inhibitory neurons exhibit a resonance in the γ frequency regime when stimulated with oscillatory light pulses, while excitatory neurons amplify low frequencies. Theoretical studies reproduced these results by considering certain neuronal or synaptic dynamics. Tchumatchenko et al. <cit.> assume sub-threshold resonances of inhibitory neurons as well as gap junctions between them and Tiesinga <cit.> considers slow synaptic currents for the connections from the excitatory to the inhibitory neurons. Here we investigate the effect of the network architecture on the measured responses. Using three characteristic connectivity motifs of an E-I-circuit, we identify connectivity motifs that yield responses resembling those shown in Cardin et al. We also demonstrate how these results crucially depend on the size of the stimulus as well as on the measure of the network response. §.§ The effect of small versus large stimulus amplitudes The effect of stimuli on the dynamics of a circuit can be classified into three regimes. In the simplest case, additional input can be treated as a small perturbation, which does not influence the dynamical state of the circuit (as done in <cit.>). In this case, the stimulus can be treated analogously to the internally generated noise of the population rates, which is, for example, produced within networks of LIF neurons of finite size <cit.>. In this setting, external input to populations of LIF neurons can be described as a modulation of the synaptic input, which is filtered by the dynamic transfer function of the populations the stimulus is applied to. The dynamic transfer function is typically a low-pass filter yielding the emphasis of low frequencies by the network response. This explanation is similar to the results in <cit.>, where the externally applied signal is described by an Ornstein-Uhlenbeck process and the low-pass filter is therefore explicitly introduced. Since the elevated low-frequency components in the network response contain the filter properties of individual populations rather than network dynamics, we suggest a modified stimulation protocol which eliminates the effect of the initial filtering and therefore enables the analysis of a signal which emerges from internal network dynamics alone. An input that affects the stationary properties of the circuit may change the working point within the linear regime of the static transfer function. The dynamical properties of the circuit then stay approximately unaltered and only the change of the stationary rates needs to be accounted for. In contrast, an input that changes the stationary rates in the nonlinear regime also changes the dynamic transfer functions of the populations, which potentially alters the dynamic behavior of the entire network. We show that this change can often be approximated by adjusting the prefactor of the dynamic transfer function (see subsec:Approximation_of_dtf). Applying this approximation to describe constant positive input to a self-coupled inhibitory population, which generates a distinct frequency, shows that the peak in the spectrum increases, while its peak frequency remains unaltered. In other words, the eigenvalue trajectory which determines the dynamics of the circuit is shifted towards instability by the constant input, while its shape remains unaltered. Since this effect dominates the response spectrum compared to changes evoked by a purely oscillatory stimulus, this finding supports the statement by Tiesinga et al. <cit.>, who suggested to employ constant stimuli as an alternative to oscillatory stimuli to investigate the origin of oscillations experimentally. The result is in line with the high sensitivity of the low-γ power to the external input to population 4E: the microcircuit model used in this paper is derived from the model employed in <cit.> by reducing the external input to population 4E and shows strongly reduced oscillations in the low-γ range.The approximation of the modified dynamic transfer function by a change in the prefactor needs to be applied with caution, when the population is embedded in a network. Here the static transfer function can show higher degrees of nonlinearity due to the recurrent feedback. As a result a stimulus is more likely to change the dynamic structure of the network. These effects can only be captured by linearizing the system around the new working point. An analytical description of the transition between the resting state and the stimulus induced state remains to be investigated. Recent results on the nonlinear transfer function of the LIF model will be useful in this approach <cit.>. In summary, the results derived here show that the way in which a stimulus affects the dynamical regime of a network, as well as the filtering properties of the populations, should be taken into account when probing a network for the origin of internally generated oscillations. §.§ Power ratios versus response spectra Experimental studies <cit.> demonstrate the responses to oscillatory stimuli by means of power ratios, the LFP power at stimulus frequency normalized by the LFP power at that frequency at rest. Theoretical studies, on the contrary, consider normalized network responses <cit.> as well as absolute responses at stimulus frequency <cit.>.We analyze the effectiveness of detecting the anatomical origin of oscillations when using power ratios compared to response spectra evoked by oscillatory stimuli. Power ratios can yield misleading results if the spectrum at rest is not entirely flat. In these cases it appears advantageous to consider the difference of the spectra with and without additional stimulus. Small stimuli that do not affect the stationary dynamics of the circuit evoke responses of oscillatory modes that are also responsible for the oscillations in the resting condition. The peaks may be canceled in the relative spectrum and therefore information can get lost when considering power ratios. In particular, in one-population systems the power ratio becomes independent of the intrinsic resonance phenomena of the network and reflects the filter properties of the population. In higher dimensional systems, a stimulation protocol which allows for the reconstruction of all circuit internal variables (eq:spectrum_decomp_with_input) can, in principle, be designed from the population rate spectra and cross-spectra obtained by separately stimulating each population. Applications of stimuli that affect the stationary dynamics of the circuit in the nonlinear regime, however, change the dynamic response properties of the circuit. If the response properties are changed such that the resonance of an oscillatory mode is strengthened, these effects can dominate and also show up in power ratios. Therefore, we propose to consider both, absolute and relative spectra. §.§ Connecting network responses to dynamic network architecture Oscillations induced by the network structure can either be generated by the self-coupling of one population and imposed onto other populations or by the interplay of several populations. We analyze here whether the involvement of one population in the generation of an oscillation can be investigated by means of oscillatory stimuli to that population. We show how the network response to oscillatory stimuli can be decomposed into responses generated by the auto- as well as crosscorrelation of dynamic modes. Each mode can, individually, be traced back to its anatomical origin, namely the sub-circuit that generates the associated oscillation. However, in the analysis of experimental or simulated data, such decomposition into modes is inaccessible. The anatomical origin of the oscillation could therefore only be inferred from responses generated by a single mode. It turns out that the stimulation of an individual population elicits the response of a single mode only in trivially connected circuits, in which the connections between populations are negligible. In more complicated circuits, populations typically participate in multiple dynamical modes, which are activated together when stimulating that population.The mode composition of the network response depends on the considered measure of the network response. Experiments typically measure LFP responses <cit.>, which have been linked to the input onto excitatory neurons <cit.>.Tchumachenko et al. <cit.> defined the network response as the response of the population that is stimulated and therefore measured different responses depending on the stimulus. Tiesinga <cit.> considers the activity of the excitatory cells. These two theoretical studies referred to the findings presented by Cardin et al. who showed that that the γ-resonance was present in the LFP ratio when stimulating the inhibitory cells at γ frequency, but was missing when stimulating the excitatory cells. Given that the connectivity plays a role in the composition of the LFP, it is possible that a resonance is visible in the population spectrum, but not in the LFP response (see for example the spectra of the two-dimensional network without self-coupling fig:2d_PINGB). In the presented example, the feedback of the excitatory population response is missing and the γ rhythm is therefore not relayed back to the pyramidal neurons where it would contribute to the LFP. Even though an E-I network with a missing E-E-loop might not be biologically realistic, the same effect could be caused by a large amount of NMDA receptors at the synapses of the excitatory neurons: The slow synapses then act as a low-pass filter which the γ oscillation cannot pass (the same mechanism was investigated in <cit.>). In other words, the connection would not be present dynamically at γ frequency.To test the hypothesis that the findings of Cardin et al. suggest an oscillation generating mechanism which solely involves the inhibitory neurons, we compare the responses of two exemplary circuits (see subsec:ING1 and subsec:ING2). In the first circuit, the γ oscillation is generated by the I-I-loop and subsequently imposed onto the excitatory population. The second circuit, in contrast, requires all connections for the generation of γ. We show that the LFP response to oscillatory stimulation of the inhibitory neurons shows a resonance at γ, while the response to stimulated excitatory neurons resembles a low pass filter, regardless of the origin of the oscillation. We therefore conclude, similarly to <cit.>, that oscillatory stimuli cannot exclude the involvement of excitatory neurons in the oscillation generating mechanism.We discuss the design of a stimulation protocol that isolates the responses of individual dynamic modes by exciting populations in the same ratio in which they contribute to the oscillation generating sub-circuit. However, it remains an open question how these single mode responses can be distinguished from mixtures of mode responses without the knowledge of the dynamic transfer of the mode. It is also debatable whether this protocol is experimentally feasible, given that, if the structure of the mode were unknown, numerous runs in which several populations are stimulated with various strength and time lags would be required.§.§ The emergence of ambiguous resonances in the network response of a microcircuit model Applying oscillatory stimuli to the populations in a multi-layered model of a column composed of LIF model neurons demonstrates that the response spectra in large spiking networks can be predicted theoretically. The results (fig:Power-ratios-sim) show that stimuli evoking firing rate fluctuations as large as the firing rate itself (see left panel in fig:Population-rate-responsesC) as well as response spectra of amplitudes comparable to those evoked in experiments <cit.> (see right panels in fig:Population-rate-responses), can still be sufficiently well described by the employed theoretical framework, which is based on mean-field and linear response theory.The power ratios of population rates with oscillatory stimuli applied to the respective population reveal a resonance in the low-γ frequency range when stimulating population 2/3I. However, it has been shown that the low-γ peak is generated within a sub-circuit which is located in the upper layers and involves several populations <cit.>. Decomposing the network response at γ frequency into contributions of the dynamic modes that shape the oscillations in the microcircuit model, reveals that the response is mainly shaped by two modes including their crosscorrelation; in addition to the mode that is responsible for the low-γ peak, population 2/3I contributs strongly to the mode which is composed of the 2/3I-2/3I loop and which is responsible for the generation of a peak at very high frequencies <cit.>. Stimulating population 2/3I therefore elicits responses of both modes, which are anti-correlated for low frequencies and therefore cancel the contributions of the autocorrelations of the modes. This cancellation for low frequencies, but not for high frequencies, gives rise to a peak in the power ratio that could be misinterpreted as the signature of an underlying I-I loop that generates the low-γ peak.In summary, we demonstrate the importance of correctly identifying the dynamic influence of the stimulus on the system as well as the considered output measure when interpreting experimental results. By analyzing reduced circuits as well as a model of a column in the primary sensory area, we demonstrate that the entire underlying network needs to be taken into account when interpreting emerging signals with respect to the origin of oscillations.§ METHODS§.§ Static and dynamic transfer function of LIF-neurons The description of the population dynamics discussed here follows the outline in <cit.> and the terminology has been introduced in <cit.>. Activity entering one population can be regarded as first passing a linear filter g(t) (dynamic transfer function) and subsequently being sent through a static nonlinear function (static transfer function) (see Fig. 1B in <cit.>). Hence the rate of one population of unconnected neurons receiving white noise input x(t) with strength I_0 can be described as r(t)=ν([g∗ I_0x](t))≈ r_0+I_0ν'(0) [g∗ x](t),where ∗ denotes the convolution of two signals. In the second step, the nonlinear function was linearized around the static point (also referred to as the working point) with ν(0)=r_0. The linearized version of the linear-nonlinear model above can be mapped to the dynamics of LIF neuron models. Here, we consider LIF model neurons with exponentially decaying synaptic currents, i.e. with synaptic filtering. The dynamics of the membrane potential V and synaptic current I_s are given by <cit.>dV/dt= -V+I_s(t) dI_s/dt= -I_s+∑_j=1^NJ_j s_j(t-d_j) ,whereis the membrane time constant andthe synaptic time constant. The membrane resistance /C_m has been absorbed into the definition of the current. Input is provided by the presynaptic spike trains s_j(t)=∑_kδ(t-t_k^j), where the t_k^j mark the time points at which neuron j emits an action potential. The synaptic efficacy is denoted as J_j= w_j/C_m, with w in Ampere. Whenever the membrane potential V crosses the threshold V_θ, the neuron emits a spike and V is reset to the potential V_r, where it is clamped during . In the diffusion approximation the dynamics reads <cit.>dV/dt= -V+I_s(t) dI_s/dt= -I_s+μ+σ√() ξ(t),where the input to the neuron is characterized by its mean μ and a variance proportional to σ, and ξ is a centered Gaussian white process satisfying ⟨ξ(t)⟩=0 and ⟨ξ(t)ξ(t^')⟩=δ(t-t^'). The static transfer function ν can be obtained for white noise (originating from δ-synapses, i.e. =0) <cit.> or colored noise (originating from filtered synapses, i.e. 0) <cit.>. The stationary rate is then given by r_0=ν(μ,σ). The dynamic transfer function h(t)=ν'(0)g(t) has been derived in the Fourier domain using linear response theory to systems exposed to white <cit.> and colored noise <cit.>. To employ linear response theory, the system has to be linearized around the static point, yielding a dynamic transfer function that also depends on the working point H(ω):=H(ω,μ,σ). The dynamic transfer function of the LIF model with δ-synapses is given by <cit.> H_WN(ω,μ,σ)=G 1/1+iωΦ_ω^'|_x_θ^x_R/Φ_ω|_x_θ^x_R,where G=√(2)r_0/σ and we introduced Φ_ω(x)=u^-1(x) U(iωτ-1/2,x) as well as Φ_ω^'=∂_xΦ_ω. Here, U(iωτ-1/2,x) is the parabolic cylinder function <cit.> and the boundaries are x_{R,θ}=√(2) {V_R,V_θ}-μ/σ. The effect of the synaptic filtering ∼ is twofold: First, input is low-pass filtered by the factor 1/1+iω appearing in the transfer function. Second, it causes a shift of the boundaries <cit.>, i.e. x_{R̃,θ̃}=√(2){V_R,V_θ}-μ/σ+√(/2), which is correct up to linear order in k=√(/) and valid up to moderate frequencies. Finally the dynamical transfer function is given by H(ω,μ,σ)=G 1/1+iω1/1+iωΦ_ω^'|_x_θ̃^x_R̃/Φ_ω|_x_θ̃^x_R̃.Note that we only consider the dominant part of the dynamical transfer function, i.e. the modulation of the output rate caused by a modulation of the mean input. The part of the transfer function corresponding to a modulation of the variance of the neurons' input <cit.> is one order of magnitude smaller and neglected here. The formalism for rate fluctuations of a single unconnected population can be extended to an N-dimensional recurrent network of populations with the connectivity matrix 𝐌^A and delays d, where each population receives input from other populations, each of which described as a rate r_i(t) with additional noise x_i(t) which is subsequently filtered by the population specific dynamic transfer function h_i(t)r_i(t)=h_i∗∑_j=1^NM_ij^A(r_j(∘-d_ij)+x_j(∘-d_ij)).Here, h is obtained from the Fourier transform of H. The connectivity matrix 𝐌^A follows from the LIF-network parameters, i.e. M_ij^A≡ I_ijJ_ij, with I_ij being the indegree from population j on population i. §.§ Approximation of the dynamic transfer function The dynamical response to a constant current, in the following termed the DC limit, can be obtained by evaluating the dynamic transfer function at frequency zero, i.e.A(μ,σ):=H(0,μ,σ) =∂ν(μ,σ)/∂μ .The equal sign follows from the fact that the integral over the impulse response h(t)=H(ω)t is given by the response to a constant input <cit.>. We now investigate how the dynamic transfer function behaves for an isolated population which receives external input defined by its mean μ and variance ∝σ^2 μ= K_extJ_extν_ext,σ^2= K_extJ_ext^2ν_ext.Here K_ext denotes the external number of synapses weighted by J_ext, with the external firing rate ν_ext. Perturbing the external rate typically yields a larger change of the mean than the variance (dμ/dν_ext∝ K_𝕖𝕩𝕥 ,dσ/dν_ext∝√(K_ext), with K_ext≈10^3). We therefore neglect the variation of σ and restrict the analysis to a perturbation of the mean, i.e. μ→μ^'=μ+δμ. fig:transfer_function_lifA shows that the DC limit of H significantly changes while its shape stays approximately constant. This suggests that we can approximate H(ω,μ^',σ) by a change of the DC-limit by altering the prefactor in the following way H(ω,μ^',σ)≈ A(μ^',σ)/A(μ,σ) H(ω,μ,σ)=:H^approx(ω,μ,μ^',σ).In the approximation above a change in the input yields the following dynamic transfer function H_I(ω) =H(ω,μ',σ)=(1+δ A(μ,μ',σ)/A(μ,σ))H(ω,μ,σ)δ A(μ,μ',σ)=∂^2ν/∂μ^2 δμ. This approximation can be evaluated defining the relative errorR(μ,σ,μ^') =∫_0^ω_max(|H^approx(ω,μ,μ^',σ)|-|H(ω,μ^',σ)|)dω/∫_0^ω_max(|H(ω,μ^',σ)|)dω,which is shown in fig:transfer_function_lifC. In the fluctuation driven regime, which corresponds to low values of μ and high values of σ (bottom part of the figure), H^approx constitutes a good approximation. In the regime with large μ and low σ the approximation H^approx is less accurate since the change in the shape of H is not negligible (fig:transfer_function_lifB), in line with the finding of a resonance at the firing rate in the mean driven regime <cit.>. In conclusion, in the fluctuation driven regime the perturbation can be approximately absorbed into the prefactor of the dynamic transfer function. Note that the DC-limit does not change for variations of μ that affect the static transfer function ν(μ,σ) in the linear regime, where ∂ν(μ,σ)/∂μ is constant by definition. However, it turns out that the static transfer functions of the populations with working points equal to those in the microcircuit model (shown in <ref>C) are affected nonlinearly by a perturbation in μ (∂^2ν(μ,σ)/∂μ^2≠0).So far, populations were considered in isolation. To investigate how a perturbation effects the dynamic transfer function of a population embedded into a network, we now treat a perturbation of the mean external input μ_ext to a population in the microcircuit model. Since the stationary rates of the populations in the network depend on each other, the static transfer function needs to be solved self-consistently when introducing a perturbation to one population. This yields a new stationary rate r_0^'=r_0+δ r_0 and working point for each population. We first investigate the induced changes in the rates (fig:Perturbation-of-microcircuitA). A perturbation to one population has an effect on all eight populations, where by trend a positive perturbation to an excitatory population causes an increase in the rates while the opposite is true for a perturbation of an inhibitory population (compare pop_pert=4E/4I). However, increased input can also yield higher rates in some populations and lower ones in others (see pop_pert=6E and pop_pert=6I). Another tendency is that excitatory populations are more strongly affected by perturbations than inhibitory ones. In particular, population 5E is very sensitive to perturbations of populations in L4 or L5, while populations in L4 are very sensitive to perturbations in L4.We further investigate the corresponding changes in the DC-limit of the dynamic transfer function (fig:Perturbation-of-microcircuitB). In general the DC-limit follows the changes of the rates. However, some differences can be observed: for example when perturbing population 4E the rate of population 5I is sensitive to the perturbation, but the DC-limit stays almost constant, which hints on the perturbation acting on the linear regime of the static transfer function of population 5I. In summary, comparing the response of a population in isolation and embedded in a network to a perturbation in its input shows that the network structure can amplify or decrease as well as reverse the sign of the response. The responses of the other populations to this perturbation can be uniform as well as diverse.The relative error, which here needs to account for both, the change in μ as well as the change in σ induced by rate changes of the other populations (compared to eq:rel_error which does not depend on changes in σ), reads R(μ,σ,μ^') =∫_0^ω_max(|H^approx(ω,μ,μ^',σ,σ^')|-|H(ω,μ^',σ^')|)dω/∫_0^ω_max(|H(ω,μ^',σ^')|)dωand is shown in fig:Perturbation-of-microcircuitC. The error follows the behavior of the rate and the DC-limit and therefore shows that the higher the changes in the working point of the populations the higher the error. The error of the approximation of the dynamic transfer functions is large compared to the error for the same populations in isolation. However, it stays within the limits of 10% given an alteration of the input of the same order. How these changes effect the prediction of the spectrum remains to be investigated. §.§ Mapping changes in the stationary rate to changes in the eigenvalue We identify the eigenvalue trajectory of the one-dimensional circuit (discussed in subsec:I-I-loop) as the weighted dynamic transfer function λ(ω)=-wH(ω),where λ(ω) denotes the Fourier transformation of the time dependent eigenvalue defined as λ̃(t)=1/2π∫λ(ω) e^iω t dω. The eigenvalue trajectory of the circuit with an additional large constant input reads λ_I(ω)=-wH_I(ω)≈-w(1+δ A/A)H(ω),where we inserted the approximation of the dynamical transfer function, which is discussed in the previous section. Changes in the eigenvalue can therefore be parameterized as λ_I=(1+α_λ)λ,with α_λ=δ A/A being the ratio by which the prefactor of the dynamic transfer function is shifted and which is related to the excitability of the circuit (fig:eigenvalue_trajectory). The frequency dependence of the eigenvalues was omitted for clarity of notation. The following considerations show how the shift in the eigenvalue relates to the change of the stationary rate.A constant stimulus is applied by an increase in the external rate (ν̃_ext=(1+α_μ)ν_ext), yielding a change in the mean value of the input current (δμ_ext=α_μμ_ext, see eq:working_point). Following the argument in the previous section, we neglect changes in the variance. The change in the external input yields the following change in the stationary rate r_0 α_rr_0=.∂ν/∂μ|_μ=μ_extδμ+1/2.∂^2ν/∂μ^2|_μ=μ_extδμ^2,where higher orders in the derivative of the static transfer function ν(μ,σ) were neglected. In this recurrent network δμ is composed of the perturbation in the external input δμ_ext in addition to a contribution from the feedback connection δμ_rec=-wr_0. We now identify that A=.∂ν/∂μ|_μ=μ_ext, abbreviate m=.∂^2ν/∂μ^2|_μ=μ_extand recall that δ A=mδμ (eq:H_I). Inserting this in the equation above yields the following relation between the ratio of the eigenvalue shift and the rate change (α_λ+1)^2-1/2α_r=r_0m/A^2.This shows that in this approximation the eigenvalue does not shift, if the working point sets the static transfer function in the linear regime, i.e. m=0. However, we demonstrated that, in particular, in recurrent networks the nonlinear effect can play a role (fig:Perturbation-of-microcircuit).For the self-coupled inhibitory units (discussed in subsec:I-I-loop), we chose the following parameters:α_λ=0.15 ,r_0/N=1, α_r=0.8, w=-4 mV A=0.05 mV^-1, A_I=0.058 mV^-1τ=2, d=3.6, I_0=0.5/π.The parameters of the dynamic transfer function for both populations in the two-dimensional circuits (discussed in subsec:2d) are given by:r_0,/N_=r_0,I/N_=1, A=0.5 mV^-1, w=1 τ=2, d=1.5, I_0=0.5/π. §.§ Composition of the spectrumThe systems considered in this work are given by, or can be reduced to, N-dimensional rate models with noise, while N denotes the number of populations. The spectrum of the populations is hence identical to the diagonal of 𝐂(ω)=⟨𝐘(ω)𝐘^T(-ω)⟩,where 𝐘(ω) is the rate vector in Fourier space, composed of the rate 𝐑(ω) and a noise term 𝐗(ω) as 𝐘(ω)=𝐑(ω)+𝐗(ω). Inserting the effective connectivity matrix, which is composed of the connection strengths M_ij and the dynamical transfer function of the populations incorporating the connection delays H_i(ω)e^-iω d_ij(for further detail see <cit.>) into the self-consistent equation for the population rates (eq:rate_convolution), yields 𝐂(ω)=(𝕀-𝐌̃_d(ω))^-1 𝐃 (𝕀-𝐌̃_d(-ω))^-1,T,with the diagonal matrix 𝐃=⟨𝐗(ω)𝐗^T(-ω)⟩ describing the power spectrum of the noise. In the reduction of spiking networks, the noise is a finite-size effect with the autocovariance D_ii=r̅_i/N_i≡ r_i , where r̅_i is the stationary firing rate of the i-th population and N_i the number of neurons within population i.The eigenvectors and eigenvalues of the effective connectivity matrix are defined as 𝐌̃_d(ω)𝐮_i(ω) =λ_i(ω)𝐮_i(ω) 𝐯_i^T(ω)𝐌̃_d(ω) =λ_i(ω)𝐯_i^T(ω).The eigenvectors 𝐮_i(ω) and 𝐯_i(ω) with the convention |𝐮_i(ω)|^2=1 and 𝐯_i^T(ω)𝐮_j(ω)=δ_ij constitute a bi-orthogonal basis. The eigenvectors and eigenvalues of the term (𝕀-𝐌̃_d(ω))^-1 appearing in eq:spectrum are given by (𝕀-𝐌̃_d(ω))^-1𝐮_i(ω) =1/1-λ_i(ω)𝐮_i(ω) 𝐯_i^T(ω)(𝕀-𝐌̃_d(ω))^-1=1/1-λ_i(ω)𝐯_i^T(ω).The diagonal noise correlation matrix 𝐃 can be written as the sum of outer products of the unit vectors 𝐞_i, which have the entry one at position i and zero everywhere else, weighted by the diagonal entries r_i 𝐃=∑_ir_i𝐞_i𝐞_i^T.The unit vectors can be rewritten in the basis spanned by the eigenvectors of the effective connectivity matrix 𝐞_i=∑_jα_j^i(ω)𝐮_j(ω)α_j^i(ω)=𝐯_j^T(ω)𝐞_i.Here α_j^i(ω) describes the projection of the i-th unit vector onto the j-th eigenmode. Inserting the decomposition of the unit vectors (eq:unit_vector_decomp) into eq:D_old_basis yields the noise correlation matrix in the new basis 𝐃=∑_n,mβ_nm(ω)𝐮_n(ω)𝐮_m^T*(ω),with β_nm(ω)=∑_ir_iα_n^i(ω)α_m^i*(ω) weighing the i-th component of the n-th and m-th left eigenvector with the rate of the i-th population. In other words, the i-th population has a large contribution to β_nm(ω) if the unit vector of population i points in a similar direction as the right eigenvector of the n-th and m-th mode and population i has a large rate r_i. Here we also used that the effective connectivity matrix is real valued in the time domain and therefore has the property M̃(-ω)=M̃(ω)^∗ in Fourier domain. Inserting eq:D_new_basis and the effective connectivity matrix in the new basis (eq:decomp_propagator) into the expression for the spectrum (eq:spectrum) results in𝐂(ω) =(∑_n1/1-λ_n(ω)𝐮_n(ω)𝐯_n^T(ω))(∑_j,kβ_jk(ω)𝐮_j(ω)𝐮_k^T*(ω)) (∑_m1/1-λ_m^*(ω)𝐯_m^*(ω)𝐮_m^T*(ω)) =∑_n,mβ_nm(ω)/(1-λ_n(ω))(1-λ_m^*(ω))𝐮_n(ω)𝐮_m^T*(ω).The spectrum of the eight dimensional microcircuit is hence composed of 64 contributions. Eight contributions arise from individual modes (n=m) and the remaining terms arise from contributions of pairs of different modes (n≠ m). §.§ Composition of the spectrum with input So far the contribution of external input on the spectrum produced by the circuit has been neglected. In the microcircuit, all model neurons receive external Poisson input from a population-specific number of independent sources that each fire with the rate ν_ext=8. Given that all N_ext,i external sources are independent, the total Poisson input received by the jth neuron in the ith population can be approximated by Gaussian white noise described byy_ext,i^j(t) =∑_k=1^N_ext,i(ν_ext+√(ν_ext)χ_ext,i^jk(t))=N_ext,iν_ext+∑_k=1^N_ext,i√(ν_ext)χ_ext,i^jk(t) =N_ext,iν_ext+√(N_ext,iν_ext)χ_ext,i^j(t).Here χ_ext,i^jk(t) is a random variable with zero mean (⟨χ_ext,i^jk(t)⟩=0) and variance ⟨χ_ext,i^jk(t)χ_ext,i'^j'k'(t')⟩=δ_ii^' δ_kk^' δ_jj^' δ(t-t'). The last equal sign in eq:ExtInputToNeuron follows from the fact that two Gaussian white noise processes are equal if they have equal first and second moments. The population averaged input of the ith population is given by y_ext,i(t) =1/M_i∑_j=1^M_i(N_ext,iν_ext+√(N_ext,iν_ext)χ_ext,i^j(t)) =N_ext,iν_ext+1/M_i∑_j=1^M_i√(N_ext,iν_ext)χ_ext,i^j(t) =N_ext,iν_ext+√(N_ext,iν_ext/M_i)χ_ext,i(t),where M_i is the number of neurons in population i. The statistics of the input therefore does not depend on whether each population receives input form several Poisson sources with low firing rates or one Poisson source with high firing rate. When the firing rate of the external input is modulated in time ν_ext(t)=ν_extf(t), a neuron in population i receives Gaussian white noise with time dependent mean and variance y_ext,i(t)=μ_ext,i(t)+σ_ext,i(t)χ_ext,i(t),with μ_ext,i(t)=N_ext,iν_ext(t) and σ_ext,i(t)=√(N_ext,iν_ext(t)/M_i). The description introduced here is valid around one working point of the populations and therefore for fluctuations around one stationary firing rate. Motivated by this, we choose f(t) to describe a modulation around the external rate, which does not change its original value when averaged over long time series, i.e. T→∞lim1/2T∫_-T^Tν_ext(t)dt=ν_ext.Incorporating the external input into the self-consistency of the rate fluctuations (eq:rate_convolution) yields r_i(t)=h_i∗∑_j=1^NM_ij^Ay_j(∘-d_ij)+τ_mJ_ext[h_i∗ y_ext,i(∘)](t),with the strength of the external current w_ext (J_ext= w_ext/C_m) in Ampere, which is chosen to equal the synaptic strength within the network as defined in eq:diffeq_iaf. Defining the effective connectivity matrix 𝐌̃_d(ω) to incorporate the convolution of the dynamic transfer function h_i and the anatomical connectivity M_ij^A, as well as the delays and delay distribution <cit.>, the fluctuating rate of the circuit reads in Fourier space 𝐘(ω)=𝐌̃_d(ω)𝐘(ω)+𝐃_ext(ω)𝐘_ext(ω)+𝐗(ω) ⇔ 𝐘(ω)=𝐏(ω)(𝐃_ext(ω)𝐘_ext(ω)+𝐗(ω)),where the eigenvalues and eigenvectors of 𝐏(ω)=(𝕀-𝐌̃_d(ω))^-1 are defined in eq:decomp_propagator and 𝐃_ext(ω) is a diagonal matrix with elements D_ext,ii(ω)=τ_mJ_extH_i(ω). The population rate spectra read𝐂_I(ω) =⟨𝐘(ω)𝐘^T(-ω)⟩ ==𝐂_0(ω)𝐏(ω)⟨𝐗(ω)𝐗^T(-ω)⟩𝐏^T(-ω)+=𝐂_ext(ω)𝐏(ω)𝐃_ext(ω)⟨𝐘_ext(ω)𝐘_ext^T(-ω)⟩𝐃_ext^T(-ω)𝐏^T(-ω),where we used that the internal and external noise sources have zero mean and we assumed them to be uncorrelated. The spectrum observed in the circuit is thus given by the spectrum generated within the circuit (eq:spectrum) and the spectrum imposed from the outside 𝐂_ext(ω). Before calculating the expectation value of the external fluctuations ⟨𝐘_ext(ω)𝐘_ext^T(-ω)⟩, we notice that the autocorrelation of the signal defined in eq:ExtInputTimeDep depends on two time arguments, namely the time-lag τ as well as the global time t, which is induced by the modulation of the rate⟨ y_ext,i(t)y_ext,j(t+τ)⟩=N_ext,iN_ext,jν_ext^2f_i(t)f_j(t+τ)+δ_ijδ(τ) N_ext,iν_ext/M_if_i(t),where we used that ⟨χ_ext,i(t)χ_ext,j(t+τ)⟩=δ_ijδ(τ). The autocorrelation is modulated in time and the process is therefore not stationary. The time dependent spectral density, also known as the Wigner-Ville spectrum (GWVS) <cit.> is given by ⟨𝐘_ext(ω,t)𝐘_ext^T(-ω,t)⟩_ij=∫_-∞^∞e^-iωτ⟨ y_ext,i(t)y_ext,j(t+τ)⟩ dτt'=t+τ=N_ext,iN_ext,jν_ext^2f_i(t)e^iω tT→∞lim∫_-T^Te^-iω t'f_j(t') dt' +δ_ijδ(τ) N_ext,iν_ext/M_i f_i(t) =N_ext,iN_ext,jν_ext^2f_i(t)e^iω tF_j(ω)+δ_ijδ(τ)N_ext,iν_ext/M_if_i(t),where F(ω) denotes the Foureir transform of f(t). The GWVS describes the time-frequency distribution of the mean energy of the signal y_ext(t). The normalized marginal distribution can be obtained by taking the average over time ⟨𝐘_ext(ω)𝐘_ext^T(-ω)⟩_ij=T→∞lim1/2T∫_-T^T⟨𝐘_ext(ω,t)𝐘_ext^T(-ω,t)⟩_ijdt =μ^2F_i(ω)T→∞lim1/2T∫_-T^Tf_j(t)e^iω tdt+σ^2T→∞lim1/2T∫_-T^Tf_i(t)dt =N_ext,iN_ext,jν_ext^2T→∞lim1/2TF_i^*(ω)F_j(ω)+δ_ijN_ext,iν_ext/M_iT→∞lim1/2T∫_-T^Tf_i(t)dt =N_ext,iN_ext,jν_ext^2T→∞lim1/2TF_i^*(ω)F_j(ω)+δ_ijN_ext,iν_ext/M_i,where the last integral could be discarded since we required that the time dependent modulation of the firing rate f_i(t) averages out over long time series. The terms above show that a modulation of the external firing rates yields two contributions to the spectrum. The first term describes the contribution that arises from the modulation of the mean, which can give rise to new peaks. In the original microcircuit model, the external rate is constant (f_i(t)=1). The first term thus solely gives a contribution at zero frequency (F_i(ω)=2πδ(ω)). The second term describes the effect of the modulation of the variance. This term cannot introduce new peaks in the spectrum, since it does not depends on ω. It, however, gives a contribution at all frequencies and can therefore influence the amplification of the dynamical modes of the systems. The contribution of the external variance is large compared to the variance of the noise generated within the network, which is of order ν_ext/M_i. However, its contribution is negligible, since it is small for in the microcircuit model and additionally filtered out by the transfer function of the population for larger frequencies (see definition of 𝐃_ext(ω) and eq:spec_network). In the following we will therefore focus on the dynamical contribution of the first term.Let us now assume that the fraction a of the external input to the kth population is modulated by a sinusoid of frequency ω_ such that f(t)=1+a sin(ω_t) yielding the modulated mean and variance μ_ext,k(t) =N_ext,kν_ext(1+a sin(ω_t)) σ_ext,k^2(t) =N_ext,k/M_kν_ext(1+a sin(ω_t)),while all other populations receive unmodulated external input. The contribution of the mean modulation to the spectrum of the external signal is then, for ω>0, given by ⟨𝐘_ext(ω)𝐘_ext^T(-ω)⟩_ij= N_ext,k^2ν_ext^2T→∞lim1/2Ta^2π^2(δ^2(ω-ω_)+δ^2(ω+ω_)) i=j=k0 Thus the spectrum exhibits a δ-peak at the frequency of the modulating signal. To determine the height of the peak, we consider that the measurement or the modulation lasts only a finite duration. Inserting the definition of the δ-function 2πδ(ω-ω_)=T→∞lim∫_-T^Te^-i(ω-ω_)τdτ, the peak height at ω_, when measured in the interval [0,T] (which changes the normalization constant in eq:spec_ext_inf_time from 1/2T to 1/T), is given by ω→ω_limδ_T(ω-ω_)=1/2πω→ω_lim∫_0^Te^-i(ω-ω_)τdτ=1/2πω→ω_lime^-i(ω-ω_)T-1/-i(ω-ω_)=T/2π,yielding the following spectrum of the external signal ⟨𝐘_ext(ω)𝐘_ext^T(-ω)⟩_ij= N_ext,k^2ν_ext^2a^2T/4i=j=kω=ω_0The additional contribution to the spectrum visible in the network due to the external input is therefore given byC_ext,ij(ω) =∑_n,mP_in(ω)D_ext,nn(ω)⟨ Y_ext(ω)Y_ext^T(-ω)⟩_nmD_ext,mm(-ω)P_jm(-ω)yielding 𝐂_ext(ω)= w_ext^2N_ext,k^2ν_ext^2a^2T/4|H_k(ω)|^2𝐏(ω)𝐏^T(-ω) ω=ω_0The expression above shows that the peak imposed on one population from the outside is propagated through the network and is therefore visible in all populations. The height of the peaks scales linearly with time.As opposed to the above described current modulation, rate modulating input is directly added on top of the population rate and not filtered by the transfer function of the population, yielding the following additive term in the spectrum𝐂_ext^RM(ω)= w_ext^2N_ext,k^2ν_ext^2a^2T/4𝐏(ω)𝐏^T(-ω) ω=ω_0Analyzing the current modulating input further by decomposing the externally imposed spectrum, when population k is stimulated in the eigenbasis of the propagator matrix, as already done for the internally generated spectrum eq:spectrum_decomp, yields 𝐃_ext(ω)⟨𝐘_ext(ω)𝐘_ext^T(-ω)⟩𝐃_ext(-ω) =∑_n,kmβ_ext,nm(ω)𝐮_n(ω)𝐮_m^T*(ω) with β_ext,nm(ω)=w_ext^2N_ext,k^2ν_ext^2a^2T/4|H_k(ω)|^2α_n^k(ω)α_m^k*(ω) and α_j^k(ω)=𝐯_j^T(ω)𝐞_k. Inserting this into eq:spec_ext_final yields the decomposition of the population rate spectra at ω_ 𝐂_I(ω) =∑_n,mβ_nm(ω)+β_ext,nm(ω)/(1-λ_n(ω))(1-λ_m^*(ω))𝐮_n(ω)𝐮_m^T*(ω).The expression above is referred to as the response spectrum. The excess spectrum is defined as the additional power due to the input δ𝐂(ω)=𝐂_I(ω)-𝐂_0(ω). The power ratio, which describes the response spectrum at stimulus frequency ω_ normalized by the original spectrum at that frequency, is evaluated at ω=ω_ and given by ρ(ω)=𝐂_I(ω)/𝐂_0(ω)=1+δ𝐂(ω)/𝐂_0(ω). §.§ Approximation of the LFP The LFP is described as the input to pyramidal cells. The rate fluctuations received by pyramidal cells are given by Y_AMPA(ω) =W_H_AMPA(ω)Y_(ω)=ae^-iω d/1+iωτ_AMPAY_(ω)Y_GABA(ω) =W_H_GABA(ω)Y_(ω)=-be^-iω d/1+iωτ_GABAY_(ω),where 𝐘(ω)=(Y_E(ω),Y_I(ω)) denotes the vector of fluctuating rates of the excitatory and the inhibitory population. Here we assumed exponentially decaying synaptic currents with time constants τ_AMPA and τ_GABA. The synaptic weights a and -b are defined in eq:conn_matrix-1. Mazzoni et al. <cit.> showed that the LFP is well approximated by the sum of absolute values of the currents received by the pyramidal neurons and is therefore given by C_LFP(ω) =⟨(Y_AMPA(ω)-Y_GABA(ω))(Y_AMPA(-ω)-Y_GABA(-ω))⟩ =a^2/1+ω^2τ_AMPA^2⟨|Y_(ω)|^2⟩+b^2/1+ω^2τ_GABA^2⟨|Y_(ω)|^2⟩+2ab(⟨ Y_(ω)Y_(ω)⟩/(1+iωτ_AMPA)(1-iωτ_GABA)) = C_LFP^auto(ω)a^2/1+ω^2τ_AMPA^2C_(ω)+b^2/1+ω^2τ_GABA^2C_(ω)+ C_LFP^cross(ω)2ab(C_E(ω)/(1+iωτ_AMPA)(1-iωτ_GABA)).The LFP is thus determined by the autocorrelation of the rate fluctuations of each population as well as their crosscorrelation. The synaptic dynamics induces an additional low-pass filtering of the contributions. 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New York: Dover Publications; 1974. Oppenheim96 Oppenheim A, Wilsky A. Systems and signals. Prentice Hall; 1996. Matz03 Matz G, Hlawatsch F. Time-varying power spectra of nonstationary random processes. In: Time- Frequency Signal Analysis and Processing: A Comprehensive Reference. Elsevier; 2003. p. 400–409.> | http://arxiv.org/abs/1704.08372v1 | {
"authors": [
"Hannah Bos",
"Jannis Schücker",
"Moritz Helias"
],
"categories": [
"q-bio.NC"
],
"primary_category": "q-bio.NC",
"published": "20170426225224",
"title": "How the connectivity structure of neuronal networks influences responses to oscillatory stimuli"
} |
firstpage–lastpage 2018A Massive Prestellar Clump Hosting no High-Mass Cores Ken'ichi Tatematsu Accepted 2018 February 28. Received 2018 February 23; in original form 2017 April 26 ======================================================================================== We quantify the gas-phase abundance of deuterium and fractional contribution of stellar mass loss to the gas in cosmological zoom-in simulations from the Feedback In Realistic Environments project. At low metallicity, our simulations confirm that the deuterium abundance is very close to the primordial value. The chemical evolution of the deuterium abundance that we derive here agrees quantitatively with analytical chemical evolution models. We furthermore find that the relation between the deuterium and oxygen abundance exhibits very little scatter. We compare our simulations to existing high-redshift observations in order to determine a primordial deuterium fraction of (2.549±0.033)×10^-5 and stress that future observations at higher metallicity can also be used to constrain this value. At fixed metallicity, the deuterium fraction decreases slightly with decreasing redshift, due to the increased importance of mass loss from intermediate-mass stars. We find that the evolution of the average deuterium fraction in a galaxy correlates with its star formation history. Our simulations are consistent with observations of the Milky Way's interstellar medium: the deuterium fraction at the solar circle is 85-92 per cent of the primordial deuterium fraction. We use our simulations to make predictions for future observations. In particular, the deuterium abundance is lower at smaller galactocentric radii and in higher mass galaxies, showing that stellar mass loss is more important for fuelling star formation in these regimes (and can even dominate). Gas accreting onto galaxies has a deuterium fraction above that of the galaxies' interstellar medium, but below the primordial fraction, because it is a mix of gas accreting from the intergalactic medium and gas previously ejected or stripped from galaxies. nuclear reactions, nucleosynthesis, abundances – stars: mass loss – ISM: abundances – galaxies: star formation – intergalactic medium – cosmology: theory § INTRODUCTIONDeuterium is one of the few stable isotopes produced in astrophysically interesting amounts during Big Bang nucleosynthesis, together with helium and lithium (seefor a review). Helium and lithium can be produced after this initial phase, in stars and via collisions of cosmic ray nuclei, potentially increasing their gas-phase abundances. However, the gas-phase deuterium abundance can only decrease. All primordial deuterium is burned during the collapse of a protostar and deuterium synthesized in stellar interiors is immediately destroyed, because deuterium fuses at relatively low temperatures, T≈10^6 K, easily reached in the interiors of stars and even brown dwarfs <cit.>. Therefore, mass lost from stars (also referred to as `recycled' gas) is deuterium-free, i.e. (D/H)_recycled=0. The primordial deuterium fraction, (D/H)_prim, is sensitive to cosmological parameters and, in particular, to the baryon–photon ratio and thus to the baryonic density of the Universe. Measurements of the cosmic microwave background (CMB) radiation have pinned down the ratio of the mean density of baryons to the critical density and the Hubble parameter <cit.>. The most recent theoretical models of Big Bang nucleosynthesis have incorporated these and derived, for example, (D/H)_prim=(2.45±0.05)×10^-5 <cit.> and (D/H)_prim=(2.58±0.13)×10^-5 <cit.>, where the quoted errors are 1σ. An accurate determination of the primordial deuterium fraction, in conjunction with Big Bang nucleosynthesis reaction rates, gives an independent constraint on the cosmic baryon density. If there is disagreement with the value derived from CMB measurements, this could point to a deviation in the expansion rate of the early universe and to non-standard models of big bang nucleosynthesis. Low-metallicity gas likely has a deuterium fraction close to the primordial value, because it has not been substantially enriched by stellar mass loss. Absorption lines in spectra of background quasars have been used to determine the primordial deuterium fraction observationally, finding e.g. (D/H)_prim=(2.547±0.033)×10^-5 <cit.>. Modern estimates are thus consistent with each other and there is currently no conflict with the standard model of cosmology <cit.>. Intermediate-mass and massive stars return material to the interstellar medium (ISM) via stellar winds before and during the asymptotic giant branch (AGB) phase and via supernova explosions, respectively. One well-known effect of this recycling process of baryons that become part of a star and are later returned into space is the release of metals into the ISM and the intergalactic medium (IGM). However, it is also important for the destruction of light elements, such as deuterium. If there is no fresh infall of gas onto galaxies and the ISM of these objects is replenished by stellar mass loss, both the metallicity of the gas and young stars increases and the deuterium fraction in the ISM decreases. The ratio of the deuterium fraction in the ISM or IGM and the primordial deuterium fraction, (D/H)/(D/H)_prim, is therefore a measure of the fraction of the gas that has not been processed in stars. The inverse of this, i.e. (D/H)_prim/(D/H), is known as the astration factor. Measurements of the evolution of the deuterium abundance have been used to constrain galactic chemical evolution models <cit.>. These models predict astration factors higher than observed in the local ISM when they only take into account cosmological inflow <cit.>. Models that additionally allow for galactic outflows predict lower astration factors <cit.>. In this way, measurements of the deuterium fraction (and thus the astration factor) can shed light on the balance between primordial inflow, metal-enriched outflow, and recycling through stellar mass loss, which are all related to the star formation and accretion history of a galaxy <cit.>. The fuelling of star formation by stellar mass loss is likely more important in high-mass galaxies and high-density environments. Massive early-type galaxies and satellite galaxies have specific star formation rates (SFRs) far below those of central late-type galaxies. It is not known which process(es) quench(es) galaxies, but galactic outflows are at least partially responsible for quenching massive galaxies and preventing subsequent gas accretion <cit.>. However, a substantial fraction of local early-type galaxies still have a detectable molecular or atomic gas reservoir <cit.>.Some of these exhibit gas kinematics indicating a predominantly external gas supply, such as through minor mergers, whereas others (especially those located in a cluster environment) are consistent with their ISM being fed through stellar mass loss <cit.>. Furthermore, some massive galaxies in the centres of clusters are forming stars at a substantial rate (1-100 M_ yr^-1) and contain a considerable amount of dust <cit.>. Dust is produced by stars and destroyed by sputtering in hot gas. Therefore, the gas supply is unlikely to have cooled out of the hot halo gas. This also indicates that stellar mass loss may be an important contributor to the fuel for the observed star formation <cit.>. The amount of mass supplied to the ISM through stellar mass loss could also be sufficient to fuel most of the star formation in present-day star-forming galaxies, including the Milky Way <cit.>. However, galactic outflows were not included, but are likely required to produce correct stellar masses and metallicities. Hydrodynamical simulations that included feedback from stars and/or black holes have found that stellar mass loss becomes more important for fuelling star formation towards lower redshift, although, in general, it does not become the dominant fuel source for star formation <cit.>. The predicted deuterium fraction and the importance of stellar mass loss are the focus of this paper. We present results from a suite of high-resolution, cosmological `zoom-in' simulations from the `Feedback In Realistic Environments' (FIRE) project,[http://fire.northwestern.edu/] which spans a large range in halo and galaxy mass. The FIRE simulation suite has been shown to successfully reproduce a variety of observations, which is linked to the strong stellar feedback implemented. These galactic winds efficiently redistribute gas from galaxies out to large galactocentric distances (see ). For the purposes of this paper, we highlight the fact that the simulations match the derived stellar–to–halo mass relationship <cit.>, the galaxy mass-metallicity relation and gas-phase metallicity gradients at z=0-3 <cit.>, and the dense neutral hydrogen, H i, content of galaxy haloes <cit.>.This is the first time cosmological, hydrodynamical simulations are used to study the deuterium abundance in the ISM and IGM. Our simulations self-consistently follow the time-dependent assembly of dark matter haloes, the accretion of gas onto galaxies, the formation of stars, the return of mass in stellar winds, and the generation of large-scale galactic outflows, whereas chemical evolution models - which are often used for similar studies - are based on analytic prescriptions. Specifically, we do not assume instantaneous recycling of stellar mass loss nor instantaneous mixing of metals nor a specific parametrized gas accretion or star formation history. However, our simulations do not consider mixing of the gas between resolution elements.In Section <ref> we describe the suite of simulations used, as well as the way we compute the deuterium abundance and the fractional contribution of stellar mass loss to the gas, i.e. the `recycled gas fraction' or f_recycled (Section <ref>). The deuterium retention fraction and recycled gas fraction are related via (D/D_prim)=1-f_recycled. In Section <ref> we present our results, including comparisons to existing observations. Section <ref> describes the evolution of the deuterium fraction (and hence of the recycled gas fraction), while Section <ref> focuses on high redshift and Section <ref> on low redshift. We discuss our results and conclude in Section <ref>.§ METHODThe simulations used are part of the FIRE-1 sample. These were run with gizmo[http://www.tapir.caltech.edu/.17ex∼phopkins/Site/GIZMO.html] <cit.> in `P-SPH' mode, which adopts the Lagrangian `pressure-energy' formulation of the smoothed particle hydrodynamics (SPH) equations <cit.>. The gravity solver is a heavily modified version of gadget-2 <cit.>, with adaptive gravitational softening following <cit.>. Our implementation of P-SPH also includes substantial improvements in the artificial viscosity, entropy diffusion, adaptive timestepping, smoothing kernel, and gravitational softening algorithm. The FIRE project consists of a suite of cosmological `zoom-in' simulations of galaxies with a wide range of masses, simulated to z=0 (; Feldmann et al. in preparation), to z=1.7 <cit.>, and to z=2 <cit.>. The simulation sample used is identical to the one used in <cit.> and the simulation details are fully described in <cit.> and references therein. The three Milky Way-mass galaxies that are the focus of Figure <ref> and <ref> are simulations `m12i', `m12v', and `m11.9a' (from highest to lowest stellar mass) from <cit.> and <cit.>. A ΛCDM cosmology is assumed with parameters consistent with the 9-yr Wilkinson Microwave Anisotropy Probe (WMAP) results <cit.>. The initial particle masses for baryons (dark matter) vary from 2.6×10^2-4.5×10^5 M_ (1.3×10^3-2.3×10^6 M_) for the 16 simulations that were run to z=0 (see alsofor further details). The 23 simulations that were run to z≈2 are described in <cit.> and <cit.> and their initial baryonic (dark matter) masses are (3.3-5.9)×10^4 M_ ((1.7-2.9)×10^5 M_). Star formation is restricted to molecular, self-gravitating gas above a hydrogen number density of n_H≈5-50 cm^-3, where the molecular fraction is calculated following <cit.> and the self-gravitating criterion following <cit.>. The majority of stars form at gas densities significantly higher than this imposed threshold. Stars are formed from gas satisfying these criteria at the rate ρ̇_⋆=ρ_molecular/t_ff, where t_ff is the free-fall time. When selected to undergo star formation, the entire gas particle is converted into a star particle.We obtain stellar evolution results from STARBURST99 <cit.> and assume an initial stellar mass function (IMF) from <cit.>. Radiative cooling and heating are computed in the presence of the CMB radiation and the ultraviolet (UV)/X-ray background from <cit.>. Self-shielding is accounted for with a local Sobolev/Jeans length approximation. We impose a temperature floor of 10 K or the CMB temperature. The primordial abundances are X_prim=0.76 and Y_prim=0.24, where X_prim and Y_prim are the mass fractions of hydrogen and helium, respectively. The simulations include a metallicity floor at metal mass fraction Z_prim≈10^-4 Z_ or Z_prim≈10^-3 Z_, because yields are very uncertain at lower metallicities and we do not resolve the formation of individual first-generation stars. The abundances of 11 elements (H, He, C, N, O, Ne, Mg, Si, S, Ca and Fe) produced by massive and intermediate-mass stars are computed following <cit.>, <cit.>, and <cit.>. The amount of mass and metals ejected in a computational time-step depends on the age of the star particle and our simulations therefore self-consistently follow time-dependent chemical enrichment. Mass ejected through supernovae and stellar winds are modelled by transferring a fraction of the mass of a star particle to its neighbouring gas particles, j, within its SPH smoothing kernel as follows: f_j = m_j/ρ_j W(r_j,h_sml)Σ_i m_i/ρ_i W(r_i,h_sml), where h_sml is the smoothing length of the star particle (determined in the same manner as for gas particles), r_i is the distance from the star particle to neighbour i, W is the quintic SPH kernel, and the summation is over all SPH neighbours of the star particle, 62 on average. There is no sub-resolution metal diffusion in these simulations.The FIRE simulations include an explicit implementation of stellar feedback by supernovae, radiation pressure, stellar winds, and photo-ionization and photo-electric heating (seeand references therein for details). Feedback from active galactic nuclei (AGN) is not included. For star-forming galaxies, which constitute the majority of our simulated galaxies, AGN are thought to be unimportant. However, AGN-driven outflows are potentially important for the high-mass end of our simulated mass range. We measure a galaxy's stellar mass, M_star, within 20 proper kpc of its centre. The deuterium fraction of a galaxy's ISM is measured within 20 proper kpc of its centre for gas with a temperature below 10^4 K, which selects the warm ionized and cold neutral gas in the ISM. These choices have a mild effect on the normalization of some of our results, but not on the trends or on our conclusions. §.§ Deuterium fraction in hydrodynamical simulationsDetermining (D/H)/(D/H)_prim in our simulations is straightforward. The mass of a gas particle can only increase during the simulation by receiving mass lost from nearby stars (no particle splitting is implemented). Therefore, any mass above the initial particle mass, m_initial, is deuterium-free. This is mixed with the initial particle mass, which has the primordial fraction of deuterium. Therefore, for each gas particle, we calculate (D/H)/(D/H)_prim= D/D_primH_prim/H = m_initial/m_initial+m_recycledX_prim/X_gas = m_initial/m_gasX_prim/X_gas, where m_recycled is the amount of mass received from evolving stars, i.e. the amount of gas that has been `recycled', X_gas is the mass fraction of hydrogen, and m_gas is the mass of the particle at the redshift of interest. We refer to this quantity as the deuterium retention fraction, because it is the fraction of deuterium, produced during Big Bang nucleosynthesis, that is not destroyed. The inverse of Equation <ref> is the astration factor. The value of (D/H)_prim is well-constrained, both directly from absorption-line observations of low-metallicity gas and indirectly from CMB measurements coupled with Big Bang nucleosynthesis reaction rates <cit.>. Another way to constrain the primordial value is by comparing observations to cosmological simulations, as done in Section <ref>.The deuterium retention fraction in Equation <ref> is directly related to the fractional contribution of stellar mass loss to the gas, i.e. the recycled gas fraction, f_recycled= m_recycled/m_gas = m_gas-m_initial/m_gas = 1-D/D_prim = 1-(D/H)/(D/H)_primX_gas/X_prim, which is used to study the importance of stellar mass loss in fuelling the ISM and star formation. Besides destroying all deuterium, a fraction of the hydrogen is fused into helium and metals before the gas is recycled into the ISM, X_gas=1-Y_gas-Z_gas. This can be approximated well by X_gas≈ X_prim-3Z_gas. The factor X_gas/X_prim is close to unity for subsolar metallicities, but becomes more important at supersolar metallicities. Even though the differences do not change our conclusions, we will show and discuss both (D/H)/(D/H)_prim and D/D_prim or f_recycled when relevant. § RESULTSObservations of the deuterium fraction exist at both high and low redshift. We will first discuss the evolution of (D/H) and then discuss predictions and observational comparisons at z=3 and z=0 separately. Throughout the paper, we use oxygen abundance ratios of gas as compared to those of the Sun, i.e. [O/H]= log_10(n_O/n_H) - log_10(n_O/n_H)_, where n_O is the oxygen number density, n_H the hydrogen number density, and log_10(n_O/n_H)_=-3.31 is the solar oxygen abundance taken from <cit.>. §.§ Evolution of the deuterium fractionWhile the total deuterium content of the Universe decreases with time, its total metallicity increases, leading to an inverse correlation between the deuterium and oxygen abundance <cit.>. Figure <ref> shows the median deuterium retention fraction (black curves) for all gas particles in our simulations as a function of oxygen metallicity at z=0 (top panels) and z=3 (bottom panels). The left panels only take into account deuterium and oxygen, whereas the right panels fold in the hydrogen abundance for both axes. Z_O (Z_O,) is the oxygen mass fraction of the gas (for solar abundances). The grey, shaded regions show the 1σ (dark) and 3σ (light) scatter around the median. Some of our simulations implemented a relatively high metallicity floor of [O/H]_initial=-2.8. Here, in order to not be affected by the imposed metallicity floor, we excluded gas with a metallicity within a factor of 2 from its initial oxygen abundance, but this choice does not affect our conclusions. At solar oxygen metallicity, about 90 per cent of the primordial deuterium is not destroyed at z=3 and 88 per cent at z=0. (D/H)/(D/H)_prim=0.91 (z=0) and 0.93 (z=3) at [O/H]=0, slightly higher because of the small decrease of the hydrogen fraction. The 1σ scatter in this relation is very small, which shows that the destruction of deuterium and the enrichment with oxygen are tightly correlated. However, the scatter increases at z=0 at the highest metallicities ([O/H]≳0.5). Additionally, we find large non-Gaussian tails at all metallicities, which means that even at low metallicity, a small fraction of gas particles have substantially reduced deuterium abundances. Our calculations likely underestimate the mixing of gas, because elements in our simulation are stuck to gas particles and do not diffuse to neighbouring gas particles. Adding turbulent diffusion to our simulations would only decrease the scatter in the correlation between deuterium and oxygen, because it smoothes out variations, and would thus strengthen our conclusions. The dependence of (D/H) on [O/H] is very steep at high metallicity, because [O/H] is a logarithmic quantity.A small fraction of the gas (0.5 per cent) reaches extremely high metallicities ([O/H]>0.5), which have not been observed. It is possible that such rare systems exist, outside the Milky Way, but are beyond current observational capabilities. Note, however, that the (average) metallicity in sightlines through our simulation is always [O/H]≤0.5 (see Figure <ref>). Another possibility is that there is not enough mixing in our simulations, since the metals are stuck to particles and cannot diffuse, resulting in small metal-rich pockets. Additionally, the yields are very uncertain at such high metallicities. The real uncertainty is therefore larger than the scatter in this regime. Note that although D/D_prim≤1, (D/H)/(D/H)_prim can be larger than unity in rare cases at high metallicity, because H/H_prim can become very small due to hydrogen fusion. For comparison, the red, dashed curves (identical in top and bottom left panels) show the relation between the oxygen and deuterium abundances obtained from a one-zone chemical evolution model <cit.>. This model assumes that chemical equilibrium is reached in the ISM due to the balance between gas inflow and outflow, enrichment though stellar mass loss, and gas consumption due to star formation. The only parameters in the relation are the recycling fraction, r, i.e. the fraction of mass returned to the ISM by a simple stellar population, and the oxygen yield, m_O, i.e. the mass fraction of a simple stellar population released into the ISM in oxygen,DD_prim=11+rZ_O/m_O, where Z_O is the oxygen mass fraction of the gas. We also compare our findings to analytic results derived from a closed box model, i.e. no gas inflow or outflow <cit.>. The resulting relation between the deuterium retention fraction and oxygen abundance is[Note that in the literature, this equation is usually expressed with a yield defined with respect to the final stellar remnant mass (after mass loss) rather than the initial stellar mass as is the case in our definition of m_O.]DD_prim=e^-r Z_Om_O as shown by the purple, dotted curves (identical in top and bottom panels). Both chemical evolution models assume instantaneous stellar mass loss and enrichment, with no time dependence (whereas our simulations consistently follow time-dependent mass loss and enrichment as the stellar population ages). The ratio r/m_O=26.7 (using r=0.4, and m_O=0.015, the fiducial values from ) is thus the only free parameter. The models match the relative abundances at z=0 surprisingly well. As can be seen by comparing the two panels of Figure <ref>, there is relatively little evolution in the correlation between deuterium and oxygen. At fixed oxygen metallicity, the deuterium abundance is slightly higher at z=3 than at z=0. This is because most of the oxygen is produced in core-collapse supernovae, which also dominate the stellar mass loss at early times. At late times, AGB stars are responsible for most of the mass loss, adding deuterium-free material, but not substantially enriching the gas with oxygen. This can be tested by dividing the cumulative amount of mass loss added to the gas in the simulations by the total amount of gas-phase oxygen at different redshifts. As mentioned before, the fiducial ratio used is r/m_O=26.7, which is close to, though slightly higher than, the value we find at z=0, r/m_O=24.4 (directly computed from and averaged over all our simulations). At z=3, however, the average ratio in our simulations is substantially different, r/m_O=19.4. We therefore added extra model curves (cyan, dashed using Equation <ref> and orange, dotted using Equation <ref>) to each panel of Figure <ref>, where we changed the value of r/m_O to match the average value in the simulations. The level of agreement between these simple models, especially the closed box model from <cit.>, and our cosmological simulation results is remarkable given the very different approaches. This lends credence to both methods and shows that the most important factor in this correlation is the ratio r/m_O, which can be calculated from stellar population synthesis models. The complex processes involved in the formation of galaxies, such as galaxy mergers, time-variable star formation and galactic outflows, as well as the lack of mixing in these simulations are thus likely unimportant where these relative abundances are concerned. The improvement from the small variation in r/m_O with redshift supports our claim that the evolution is due to the extra (almost oxygen-free) mass loss from AGB stars at late times.Accurate observations of (D/H) at [O/H]>-1 in combination with an accurate determination of (D/H)_prim, either from observations at low metallicity or derived from CMB measurements, would be able to determine r/m_O. The recycling fraction is governed by intermediate-mass stars as well as massive stars, whereas the oxygen yield depends only on the latter. Therefore, the relation between (D/H)/(D/H)_prim and [O/H] can potentially be used to constrain stellar evolution models and/or the variation of the IMF at the high-mass end. Although all our simulations were run with the same stellar evolution model and IMF, numerical chemical evolution models have already demonstrated that the deuterium fraction depends on these choices <cit.>. In the top right panel, observational constraints independently derived by <cit.> (blue cross with error bars) and <cit.> (red cross with error bars) for the local ISM have been included, slightly offset from [O/H]=0 for clarity and using (D/H)_prim=2.547×10^-5, as recently obtained by <cit.>. Our solar deuterium value is in excellent agreement with that of <cit.>, but higher than that of <cit.>. The latter, however, is interpreted by the authors as a lower limit on the true value, in which case it is also in agreement with the result from our simulations (see Section <ref> for further discussion on these results). Additionally, it is of interest to compare our results to those from numerical chemical evolution models. We therefore reproduced one of the models, based on aIMF andstellar lifetimes, from <cit.>, shown in the top right panel as the purple, dotted curve. Although these results agree qualitatively, there is a clear 2σ discrepancy at high metallicity. This is potentially caused by the different IMF and stellar evolution models used, by the different star formation histories (and thus different importance of AGB stars), or by the inclusion of galactic outflows. Given the excellent match between our simulations and the simple closed box model, we believe the former explanation is the most plausible. Although the 1σ scatter in the relation between deuterium and oxygen abundances is small, it is nonzero and slightly larger at lower redshift. As mentioned before, almost all oxygen is produced by massive stars and released in core-collapse supernovae, whereas this is only the case for about half of the iron. The other half is synthesized in intermediate-mass stars and released in type Ia supernova explosions and winds from AGB stars <cit.>. The iron abundance at fixed oxygen abundance therefore enables us to trace the relative importance of massive (younger) stars and intermediate-mass (older) stars and check whether variations in the contribution of stellar mass loss by AGB stars is responsible for the scatter seen in Figure <ref>. Figure <ref> shows the residual dependence of the normalized deuterium fraction on the iron metallicity at fixed oxygen metallicity at z=0 (top panel) and z=3 (bottom panel). Z_Fe/Z_Fe, is the iron mass fraction of the gas, normalized by the solar value. The coloured curves show the median relation between deuterium and iron in four bins, 0.1 dex wide, with (from top to bottom) Log_10 Z_O/Z_O, ≈-1 (blue), ≈-0.5 (cyan), ≈0 (black), and ≈0.5 (red). Error bars show the 16th and 84th percentiles of the distribution and only bins containing at least 100 gas particles are included. A larger iron abundance at fixed Z_O/Z_O, means that older, intermediate-mass stars have been relatively more important for enriching the ISM. The deuterium fraction is thus expected to decrease with increasing iron abundance. Figure <ref> proves that this is indeed the case, although there is significant scatter in the residual relation between deuterium and iron. We therefore conclude that the (small) scatter in the relation between deuterium and oxygen is at least in part due to the varying importance of AGB stars and thus to the varying age of the stellar population responsible for enriching the gas. Comparing the z=0 and z=3 results, it is clear that the residual dependence of the deuterium fraction on the iron abundance is stronger at z=0 than at z=3. This is likely because the variation in stellar population ages, and thus in the importance of AGB stars, is smaller at higher redshift, when the Universe was much younger. This is also consistent with the fact that the 1σ scatter in the correlation between the deuterium fraction and oxygen abundance is smaller at higher redshift. Measuring [Fe/H] besides [O/H] and (D/H) will provide even better constraints on the stellar IMF and stellar evolution models.§.§.§ Milky Way-mass galaxies To understand the chemical evolution of galaxies like the Milky Way, Figure <ref> shows the deuterium evolution for three of our simulated galaxies with stellar masses close to that of the Milky Way at z=0, as indicated in the legend. The mean deuterium retention fraction is calculated for the gas within 20 proper kpc of the galaxies' centres, with a temperature below 10^4 K, which selects the warm ionized and cold neutral gas in the ISM. The black curves include the evolution of the hydrogen fraction, as it would be measured in observations. The orange curves show the recycled gas fraction, which is lower, but show the same trends with look-back time. The final z=0 values vary between the three galaxies, because they have different stellar masses. The mass dependence will be discussed in Section <ref>. Here, we are interested in the evolution of (D/H), that is, in the shape of the curves. Initially, the deuterium fraction is equal to its primordial value, after which it decreases. Two of the galaxies show an approximately linear decrease towards z=0 (solid and dashed curves), whereas for the galaxy with M_star=10^10.4 M_ (dotted curve; `m12v') the deuterium fraction levels off in the last ≈5 Gyr. The former have therefore not reached an equilibrium between the inflow of deuterium-rich gas from the IGM, the addition of deuterium-free gas through stellar mass loss, and the outflow of deuterium-poor gas. The latter galaxy has potentially reached chemical equilibrium in its ISM.Our three galaxies have different stellar masses and gas masses, on top of different star formation histories, and we lack the statistical power to control for this. Despite this limitation, we checked whether or not the low-redshift behaviour of the deuterium fraction is related to the star formation history. The galaxy which has reached deuterium equilibrium (M_star=10^10.4 M_; `m12v') has already formed half of its stars by z≈1.1, whereas the other two reach half of their present-day stellar mass only at z≈0.4 and have thus experienced much more low-redshift star formation and thus more low-redshift stellar mass loss. Therefore, the reason for the different low-redshift behaviour may indeed lie in the different star formation histories of our simulated galaxies.We also checked for a dependence of the deuterium evolution on the mass loading factor, i.e. the gas outflow rate from a galaxy divided by the galaxy's SFR, as suggested by <cit.>. The average mass loading factor at z_1>z>z_2 is calculated in the following way. We select all the gas particles within 20 proper kpc of the galaxies' centres and a temperature below 10^4 K. We then divide the total mass of those selected particles that have been turned into stars by z=z_2 by the total mass of the selected particles that are still gaseous, but located beyond 20 proper kpc of the galaxies' centres at z=z_2. We take z_1 and z_2 to be approximately 1.5 Gyr apart, which is similar to the gas consumption time-scale. We find that the mass loading factor is relatively constant in the last 5 Gyr. From the most to least massive of our simulated Milky Way-mass galaxies, their average late-time mass loading factors over the last 5 Gyr are 0.2 (`m12i'), 0.4 (`m12v'), and 2.1 (`m11.9a'). The values for `m12i' and `m12v' are consistent with the upper limits from <cit.>, who argue that these low mass loading factors are not driven by galactic winds, but caused by random gas motions and/or close passages of satellite galaxies. We conclude that there is no clear correlation of the mass loading factor with the late-time deuterium evolution.Knowing the evolution of (D/H) can thus potentially help us understand a galaxy's star formation history. This could be achieved for the Milky Way with an accurate determination of the deuterium fraction in giant planets in the Solar System, such as Jupiter, in combination with present-day measurements in the local ISM <cit.>. The deuterium fraction in the giant planets provides a fossil record of the deuterium fraction in the local ISM during the time the Solar System was formed, about 4.5 Gyr ago. Using (D/H)_prim=2.547×10^-5 from <cit.>, the measurement by <cit.> implies (D/H)/(D/H)_prim=0.82^+0.12_-0.15 in Jupiter, which is consistent with all three of our simulated Milky Way-mass galaxies and is not precise enough to distinguish between a declining or constant deuterium fraction. Future observations with higher accuracy would be well-suited for this purpose. §.§ Deuterium fraction at high redshiftThere has been a large observational effort to measure the deuterium fraction in metal-poor gas through absorption lines in spectra of background quasars. Lyman Limit Systems (LLSs; 10^17.2<N_H i<10^20.3 cm^-2, where N_H i is the H i column density) and Damped Lyman-α Systems (DLAs; N_H i>10^20.3 cm^-2) are optically thick to Lyman limit photons. To make a fair comparison between our simulations and these observations, we calculate column densities based only on the neutral gas. Because the gas comprising these strong absorbers is partially shielded from the ambient UV radiation, it is more neutral than if it were optically thin. This is taken into account in our simulations by using the fitting formula from <cit.>, which has been shown to capture the effect of self-shielding well.<cit.> argue that the most precise measurements can be made in absorbers with N_H i>10^19 cm^-2. In order to compare to these systems, we also restrict ourselves to sightlines with column densities above this limit. Additionally, we discard the rare systems with N_H i>10^21 cm^-2 in order to not be dominated by molecular gas. We note that neither this selection nor the self-shielding correction affects our results. We do not find a dependence of (D/H) on column density at fixed metallicity, so absorption line systems at any column density could be used. The vast majority of the selected high column density absorbers are located in the haloes around galaxies <cit.>. We therefore use a simulated region of 300 by 300 proper kpc centred on the main galaxy in each of our zoom-in simulations. We grid this volume into 1 by 1 proper kpc pixels to calculate the column density of H i, D i, and O i. We assume that the neutral fraction is the same for all three atoms, because their ionization potentials are very similar, as is also done in observations.The black curve in Figure <ref> shows the median fraction of deuterium in neutral gas, divided by its primordial value, in the selected LLSs and DLAs at z=3 as a function of their metallicity. The different grey scales show the 1σ (dark) and 3σ (light) scatter around the median. Observations of (D i/H i) compiled by <cit.> and their associated 1σ errors are shown as red error bars, where we assumed that (D/H)_prim=2.547×10^-5, the weighted mean of their measurements. Our simulations confirm that at [O/H]≲-2 the deuterium abundance is very close to the primordial value (within 0.1 per cent), as seen before in Figure <ref>. These low-metallicity systems are therefore appropriate to use to determine (D/H)_prim. At [O/H]=-1 the median deuterium abundance is still only 1 per cent below primordial, similar to the 1σ error in the weighted mean of the observational values from <cit.>. The scatter in the relation is even smaller than in Figure <ref>, because we are including all (neutral) gas along a particular line-of-sight (rather than individual gas particles), decreasing the importance of small fluctuations. Observations of N_H i, N_D i, and N_O i are therefore well-suited to determine (D/H)_prim and the relation between the deuterium and oxygen abundances. Instead of assuming no variation as a function of metallicity for the 6 observed systems shown in Figure <ref>, we can test how well they match our simulations, which show a slight downward trend and minor additional scatter. We select those sightlines in our simulations that have the same metallicity as one of the observed absorbers, within 1σ errors. We then use least square fitting and calculate χ^2 between our simulated sightlines and the observations as a function of (D/H)_prim, which sets the relative normalization. The minimum χ^2 is reached for (D/H)_prim=(2.549±0.033)×10^-5, where the errors are 1σ and calculated from the difference in χ^2. This is consistent with theoretical models of Big Bang nucleosynthesis, based on cosmological parameters <cit.>. Our best estimate is very similar to, though slightly higher than, the weighted mean calculated by <cit.>, who assumed no metallicity dependence. For this low-metallicity sample, we do not gain much accuracy from comparing the data to our simulations.However, given that the scatter in the simulations is much lower than the observational measurement error at all metallicities, more metal-rich absorption-line systems can be used to determine the primordial deuterium fraction. This would allow for the expansion of the observational sample, which would improve the accuracy of (D/H)_prim. Even absorbers with [O/H]≳-1 can be used when taking into account the relation between the deuterium and oxygen abundance based on hydrodynamical simulations or on Equation <ref> combined with a prescription for the change of X_gas with metallicity. For the latter, one should use a slowly evolving ratio of recycling fraction to oxygen yield, r/m_O, increasing with time as the contribution of mass lost by intermediate-mass stars increases. Vice versa, observations of metal-rich absorbers can set constraints on the ratio of the recycling fraction and the oxygen yield, assuming that the primordial abundance of deuterium is known from either CMB measurements or from absorption-line observations at [O/H]≲-2. r/m_O depends on the relative number of intermediate- and high-mass stars and on their stellar yields and can thus potentially help constrain the high-mass end of the stellar IMF and/or stellar evolution models. It is important to note that the depletion of deuterium onto dust and preferential incorporation into molecules could cause large scatter in (D/H) between quasar sightlines at fixed metallicity, which are not due to variations in the recycled gas fraction. This is probably seen in the local ISM at solar metallicity <cit.> and briefly discussed in Section <ref>. Unfortunately we cannot address this issue with our current simulations. A relatively large sample of (D/H) measurements in absorption-line systems could quantify the scatter in (D/H) between sightlines at fixed metallicity. This will tell us whether the depletion of deuterium onto dust is important in the intergalactic medium at [O/H]>-2, because our simulations have shown that the scatter due to variations in stellar mass loss at fixed metallicity is negligibly small. If dust depletion turns out to be dominant, these systems cannot be used for determining (D/H)_prim or r/m_O. However, there is some evidence that LLSs tend to reside in dust-poor environments <cit.>. Additionally, dust depletion seems to be less important in lower column density absorbers <cit.>. It is therefore possible that these systems are well-suited for determining (D/H)_prim or r/m_O even at [O/H]>-2. §.§ Deuterium fraction at low redshiftTo compare with observations of (D/H) at z=0, we focus on the deuterium fraction in the ISM of our simulated Milky Way-like galaxies. The black curves in Figure <ref> show how the ratio of the present-day abundance of deuterium to the primordial abundance varies with 3D distance from the galactic centre, R_GC, for the same three galaxies as shown in Figure <ref>. For completeness, we show the recycled gas fraction as orange curves. (D/D_prim) is similar to, though slightly lower than,(D/H)/(D/H)_prim, because of the decrease of the hydrogen fraction. One galaxy (dashed curve; `m11.9a') has a central hole in its ISM, created by galactic winds, consistent with its relatively large average mass loading factor (see Section <ref>). It therefore has no deuterium measurement at R_GC<4 kpc. This galaxy has the lowest deuterium abundance at large radii, because the gas that was originally in its centre has been moved to larger radii. The deuterium retention fraction is low in the centres of the other two galaxies, where the density of stars is high and most of the star formation takes place. The deuterium fraction for all three galaxies increases with galactocentric radius, as previously shown by chemical evolution models <cit.>. The importance of stellar mass loss therefore increases towards the galaxy centre and recycled gas accounts for about half of the gas in the central kpc. The steepness of the deuterium abundance gradient could also reveal information on the assembly history of a galaxy <cit.>. In our sample, the galaxy with the flattest deuterium profile has the highest outflow rate. It may therefore depend more strongly on (bursty) galactic outflows than on (smooth) gas accretion. Large scatter exists between measurements of the deuterium abundance in the local ISM via absorption-line observations. This scatter could be explained by localized infall of pristine gas, with very little mixing. In this case, the average astration factor is relatively high (and mass lost from stars dominates the ISM). However, if this is the case, the oxygen abundance is also expected to decrease locally as it becomes diluted with the metal-free, infalling gas, resulting in large scatter. The fact that oxygen shows much smaller abundance variation than deuterium argues against such localized infall <cit.>. Another, more likely, explanation for the large (D/H) sightline variations is that some of the deuterium is depleted onto dust. The probability of deuterium depletion onto dust grains and incorporation into molecules is high, since the zero-point energies of deuterium-metal bonds are lower than those of the corresponding hydrogen-metal bonds <cit.>. When the ISM is heated, dust grains and molecules can be destroyed, returning deuterium to the atomic gas phase. Metals, such as iron, silicon, and titanium, are also depleted onto dust grains and the correlation of their abundances with deuterium supports this theory <cit.>. Based on the assumption that the observational scatter is caused by deuterium depletion onto dust, relatively high deuterium abundances, and low astration factors, are derived for the local Milky Way ISM by <cit.> and <cit.>. The deuterium retention percentages in the solar neighbourhood, here defined as 7<R_GC<9 kpc, lie between 85 and 92 per cent for our three simulations of star-forming galaxies with masses similar to that of the Milky Way[Although our calculations are done using 3D distance, the results are unchanged when we restrict ourselves to the ISM, because most of the gas mass lies within the star-forming disc.]. This is consistent with Figure <ref>, where we found that 91 per cent of its primordial value is recovered for [O/H]=0 at z=0. Using the value from <cit.> for the primordial deuterium fraction as in Section <ref>, the deuterium abundance derived by <cit.> implies that the local ISM still contains 91^+9_-10 per cent of the primordial deuterium abundance. This is consistent with our simulations within 1σ. <cit.> use the same data compilation, but a different method, to derive a deuterium retention percentage of 79±4 per cent (again assuming (D/H)_prim=2.547×10^-5). This is consistent (within 2σ) with our most massive Milky Way-like galaxy, with M_star=10^10.8 M_. However, <cit.> stress that their measurement can also be interpreted as a lower limit in the event that all available sightlines are affected by dust depletion. In this case, our other galaxies are also consistent with their model. Our simulations exhibit low astration factors and therefore agree with the explanation that the large scatter in local ISM observations is due to dust depletion rather than due to poor mixing of freshly accreted gas.No known galaxy besides the Milky Way has a measurement of the deuterium fraction in their ISM. Such observations would be interesting, because our simulations predict a strong dependence on stellar mass. Figure <ref> shows the mean deuterium retention fraction (top panel) and recycled gas fraction (bottom panel) within 20 kpc of the centre of the galaxy for gas with a temperature below 10^4 K as a function of stellar mass at z=0. Due to the depletion of hydrogen at higher metallicity, the differences between the deuterium retention fraction and recycled gas fraction increase with stellar mass, but the trends with mass remain the same. The black crosses show the mass-weighted mean, while the red diamonds show the (instantaneous) SFR-weighted mean. The latter is therefore a better indicator of how important stellar mass loss is for the fuelling of star formation, whereas the former is the value that would be measured, for example, in sightlines through the ISM. The galaxy shown in grey with the highest stellar mass has large uncertainties for its deuterium abundance, because it contains only 12 star-forming gas particles (but more than 1000 gas particles in total). Two other massive galaxies are not included, because they contained no star-forming gas and very little non-star-forming gas. Galaxies with M_star<10^8 M_ are also excluded, because they contained no or very little star-forming gas. The other galaxies (shown in black, red, and blue) have 100 star-forming gas particles or more. There is a definite trend with stellar mass, where more massive galaxies generally have lower deuterium fractions and thus a larger contribution of stellar mass loss. This is a clear prediction from our simulations. The exception is the lowest mass galaxy which has a low (D/H), as well as a high [O/H] (as expected from Figure <ref>). This reflects the variation in star formation, inflow, and outflow history. Our sample of galaxies is limited, but quantifying the scatter in the deuterium fraction between galaxies of similar stellar mass would help us to understand how this correlates with their formation history. Due to their tight relation, the oxygen abundance could provide very similar information as the deuterium abundance. We expect this to work well at subsolar metallicities. At solar and supersolar metallicities, the dependence of (D/H) on [O/H] is very steep, which means that to determine the recycled gas fraction, very precise measurements of [O/H] are needed. Additionally, the scatter in the relation increases towards high metallicity at z=0, due to varying contributions of AGB stars, impeding the determination of the recycled gas fraction through [O/H]. The deuterium fraction is always lower when weighted by star formation rate (red diamonds) rather than by mass (black crosses), because this dense, star-forming gas is close to newly formed stars with high mass loss rates. This means that mass loss is more important for fuelling star formation than it is for replenishing the more diffuse ISM. A measurement of the deuterium fraction (such as those in the local ISM) generally provides information on the latter, but not the former. In our simulations, the star formation in low-mass galaxies is fuelled predominantly by gas accretion, although there is a small contribution by stellar mass loss. For two of the galaxies with M_star≈10^10.5 M_, mass loss fuels about half of the current star formation, while one other is still dominated by gas accretion. For the massive galaxy (or galaxies, if we include the one with large uncertainties) with M_star≈10^11 M_, mass loss fuels the majority of star formation. As mentioned in Section <ref>, stellar mass loss has been suggested as fuel for the observed star formation in local, massive, early-type galaxies and in central cluster galaxies. Our simulations are consistent with this interpretation. Notably, there are also detections of neutral deuterium in quasar sightlines through clouds outside the galactic disc of the Milky Way, in the lower galactic halo <cit.>. These clouds could provide the fuel to sustain the Milky Way's steady star formation. However, the errors associated with these measurements are large and therefore not very constraining. More precise determinations of (D/H) could help constrain the nature of this gas. If the deuterium fraction of the gas around the Milky Way is high, it is likely pristine gas falling in from the IGM. If the deuterium fraction is similar to that of the Milky Way, it is likely part of a galactic fountain <cit.>. In intermediate cases, the gas could be a mix of both high and low (D/H) gas or the gas could be ejected or stripped from lower mass satellites <cit.>. These different theories can potentially be tested via estimates of the scatter in (D/H) in the halo, which would be large if the fountain gas is not fully mixed with the gas falling in from the IGM, but small if the gas was ejected or stripped from satellites.In order to compare the deuterium fraction in the galaxy to the deuterium fraction of accreting gas, the blue triangles in Figure <ref> show (D/H)/(D/H)_prim at z=0.1 of the gas that is accreted at 0<z<0.1, which means it is located at R_GC<20 kpc and has T<10^4 K at z=0, but is located at larger radii and/or has higher temperatures at z=0.1. Our radial boundary is somewhat arbitrary, but our results are not very sensitive to this (except for the normalization, because the deuterium fraction is generally higher at larger radius). The accreting gas has a higher deuterium retention fraction than the ISM gas. This is consistent with a substantial part of the material accreting for the first time onto a galaxy. The fact that (D/H)/(D/H)_prim<1, however, shows that another part of the accreting gas has previously been ejected or stripped from a galaxy. Such a combination of different accretion channels has already been found in these and other simulations <cit.>. The difference in (D/H) between accreting gas and the ISM is especially large for the more massive galaxies in our sample. As mentioned above, this balance between accretion from the IGM and reaccretion could potentially be shown observationally by measuring (D/H) in clouds that are accreting onto the Milky Way <cit.>. § DISCUSSION AND CONCLUSIONSWe have quantified the evolution of the deuterium fraction and its dependence on the oxygen abundance, galactocentric radius, and stellar mass in cosmological zoom-in simulations with strong stellar feedback. Because deuterium is only synthesized in the early Universe, it provides an interesting constraint on cosmology and on galaxy evolution. The normalized deuterium fraction is a measure of the recycled gas fraction, i.e. the fractional contribution of stellar mass loss to the gas, because deuterium is completely destroyed in stars and therefore f_recycled=1-(D/D_prim). Observations, however, measure (D/H)/(D/H)_prim, which is higher than (D/D_prim), especially at high metallicity (because X_gas/X_prim≈1-3Z_gas). Our simulations self-consistently follow gas flows into and out of galaxies and the (metal-rich and deuterium-free) mass loss by supernovae and AGB stars. We have compared our predictions to available observations at low and high redshift and found them to be consistent.Our main conclusions can be summarized as follows: * The deuterium fraction exhibits a tight correlation with the oxygen abundance, evolving slowly with redshift (Figure <ref>). This is captured well by simple chemical evolution models <cit.>, which depend only on the ratio of the recycling fraction and the oxygen yield. We find a small increase in r/m_O with time, because of the increased importance of AGB stars. The variation in the importance of AGB stars at fixed oxygen abundance can be traced by the iron abundance and is responsible for some of the (small) scatter in the deuterium fraction (Figure <ref>).* The three Milky Way-mass galaxies in our sample exhibit different evolution at low redshift (Figure <ref>). The galaxies that form many of their stars at late times have continually decreasing deuterium fractions in their ISM. The galaxy which forms most of its stars before z=1 shows a constant deuterium fraction in the last ≈5 Gyr, indicating that it may have reached chemical equilibrium. The evolution of the deuterium fraction may therefore be directly connected to the galaxy's star formation history.* The deuterium fraction is very close to primordial at [O/H]≲-2 (within 0.1 per cent). These are the metallicities of LLSs and DLAs typically used to measure the primordial deuterium abundance. Because of the tight correlation with metallicity, deuterium measurements in more metal-rich systems can also be used to constrain the primordial deuterium fraction (Figure <ref>) if dust depletion is unimportant.* We compared our simulations to the observational sample of <cit.> to determine a primordial deuterium fraction of (D/H)_prim=(2.549±0.033)×10^-5, very close to, though slightly higher than, their original estimate, which assumed no dependence of the measured (D/H) on metallicity. Our result is also in agreement with cosmological parameters and Big Bang nucleosynthesis <cit.>.* The deuterium fraction increases with galactocentric radius. Our simulations are consistent with the available estimates from the local Milky Way ISM where the observed scatter between sightlines is assumed to be caused by depletion onto dust (Figure <ref>).* The deuterium fraction decreases with increasing stellar mass, which means that the importance of stellar mass loss in our simulations increases with stellar mass (Figure <ref>). Mass loss is more important for fuelling star formation than for replenishing the general ISM (which has, on average, a lower gas density). Accreting gas has a higher deuterium fraction than the ISM of galaxies, but lower than primordial. This is consistent with previous findings that some gas accretes directly from the IGM, but some has been previously been ejected or stripped from a galaxy <cit.>. Due to the tight correlation of (D/H) and [O/H], shown in Figure <ref>, measurements of the oxygen abundance could provide the same information as the deuterium abundance if this relation is accurately calibrated by observations at [O/H]≳-1. The relation evolves slowly, because of the increased importance of stellar mass loss from AGB stars, which should be taken into account (see Section <ref>). If the importance of mass loss for fuelling the ISM increases for massive galaxies, as in our simulations, (D/H) will decrease with mass and [O/H] (and other metal abundances) will increase. There is observational evidence from gas-phase and stellar metallicities that this is indeed the case, although [O/H] may saturate at the highest stellar masses <cit.>. Using the median relation derived from our simulations (Figure <ref>) one can immediately estimate the contribution of stellar mass loss given the gas-phase oxygen metallicity. However, this only works in the situation where galactic winds remove mass loss from supernovae and AGB stars (approximately) equally, but may not if ejecta from young stars are removed from a galaxy (through supernovae or AGN, quenching star formation) after which its ISM is replenished by mass loss from old stars alone. This is likely the reason that the scatter in (D/H) increases at supersolar metallicity in our simulations. In observations, however, the scatter at high metallicity is probably dominated by the depletion of deuterium onto dust, an effect which is not included in our calculations. Our cosmological, hydrodynamical simulations follow time-dependent chemical enrichment and the assembly of galaxies self-consistently, whereas simple, analytical chemical evolution models assume instantaneous recycling and specify a specific star formation history <cit.>. We nevertheless compare our results to these models and find a remarkable agreement (especially with ) when considering the relation between the deuterium and oxygen abundance, despite the very different methods used. Although other galaxy properties are not necessarily well reproduced in these simplified models <cit.>, these do not play a dominant role when relative abundances are concerned. The bursty star formation and strong galactic outflows present in our simulations thus have no major impact on the correlation between (D/H) and [O/H]. Although <cit.> find that a relatively high mass loading factor is necessary to match the z=0 deuterium abundance in the local ISM, two of our simulated galaxies show low mass loading factors at late times (but high mass loading factors at early times, see ).<cit.> use zoom-in simulations with and without stellar mass loss to show that mass loss dominates the fuelling of star formation at late times for galaxies in haloes of similar mass to that of the Milky Way. However, their simulation did not include outflows from either star formation or AGN, which results in the galaxies being too massive at z=0. This means that too much baryonic mass is locked up in stars and therefore less gas is available for accretion at late times. Furthermore, the majority of the mass lost by stars is retained by the galaxy and not ejected by a galactic wind. Our simulations show that for Milky-Way mass galaxies with strong stellar feedback, cosmological inflow either dominates over or rivals stellar mass loss for the fuelling of star formation. However, recycled gas dominates the SFR at M_star≈10^11 M_ in our simulations. <cit.> use large-volume simulations with stellar and AGN feedback that match the stellar–to–halo mass relation and find that the contribution of mass loss to the fuelling of star formation is largest for galaxies with M_star≈10^10.5 M_. The deuterium fraction in the ISM is therefore the lowest around this mass and increases for more massive galaxies. Our simulations (without AGN feedback) find a different behaviour at the high-mass end, where mass loss becomes increasingly important and the deuterium fraction decreases. <cit.> also show results from simulations without AGN feedback, which do not match the stellar–to–halo mass relation, but agree qualitatively with our simulations. In the absence of AGN feedback, there are no strong galactic outflows, which means that most of the stellar mass loss is retained in the ISM. Different implementations of AGN feedback (ejective or preventive) could also result in different deuterium abundances. Future observations of (D/H) in massive galaxies therefore have the potential to discriminate between these different models. Numerical chemical evolution models generally find somewhat lower deuterium fractions (higher astration factors) than we do in our cosmological simulations <cit.>. This mild difference could be due to the different assumptions made for the stellar IMF, metal yields, and stellar mass loss. A detailed quantitative comparison between the two different numerical approaches requires using the same IMF and stellar evolution models, which is left for future work. The star formation history of a galaxy determines the relative importance of AGB stars with respect to core-collapse supernovae, which may explain any remaining discrepancy. This could potentially be tested by comparing the oxygen and iron abundances (see Figure <ref>). Another possibility is that the inclusion of galactic outflows in our simulations reduces the importance of stellar mass loss, because a substantial fraction of the recycled gas is ejected from the ISM. The large observed scatter in the deuterium fraction between different sightlines through the local ISM could be interpreted as evidence for an inhomogeneous ISM due to localized gas accretion, giving rise to a low average deuterium fraction <cit.>. Our work instead favours the interpretation that the ISM is well-mixed and the observational scatter is caused by the depletion of deuterium onto dust, which leads to a local deuterium fraction not much lower than the primordial value <cit.>. It is also possible that both dust depletion and localized infall play a roll and the true value is intermediate between those cases <cit.>. More precise observations of the deuterium and metal abundances would help clarify which of these interpretations is correct.In summary, we have quantified the deuterium fraction in a suite of zoom-in simulations and found it to be tightly correlated with the oxygen metallicity and consistent with current observational constraints. We conclude that the primordial deuterium fraction (and thus early cosmological expansion and Big Bang nucleosynthesis) can also be constrained by using observations at medium to high metallicity in combination with our simulations. Or, vice versa, if the primordial deuterium fraction is known, these measurements can inform us about the ratio of the recycling fraction to the oxygen yield and thus about the high-mass end of the stellar IMF and stellar evolution models. Our simulations predict that the deuterium fraction is lower at smaller galactocentric radii and for higher mass galaxies. This means that stellar mass loss could provide most of the fuel for star formation in massive early-type galaxies and in the centres of less massive, star-forming galaxies. Grid-based calculations or SPH simulations with explicit diffusion would be useful to determine whether or not small-scale mixing modifies the deuterium and oxygen abundances. Accurate observations of the deuterium fraction provide us with the possibility to understand the fuelling of star formation through stellar mass loss in galaxies in general and the Milky Way in particular. § ACKNOWLEDGEMENTS We would like to thank the Simons Foundation and the organizers and participants of the Simons Symposium `Galactic Superwinds: Beyond Phenomenology', in particular David Weinberg, for interesting discussions and inspiration for this work. We also thank Thomas Guillet and Joop Schaye for helpful discussions and Tim Davis for useful comments on an earlier version of the manuscript. We would like to thank the referees for valuable comments that helped clarify our results and put them into context. Support for FvdV was provided by the Klaus Tschira Foundation. EQ was supported by NASA ATP grant 12-APT12-0183, a Simons Investigator award from the Simons Foundation, and the David and Lucile Packard Foundation. CAFG was supported by NSF through grants AST-1412836 and AST- 1517491 and by NASA through grant NNX15AB22G. DK was supported by the NSF through grant AST-1412153 and by the Cottrell Scholar Award from the Research Corporation for Science Advancement. Support for PFH was provided by an Alfred P. Sloan Research Fellowship, NASA ATP Grant NNX14AH35G, and NSF Collaborative Research Grant #1411920 and CAREER grant #1455342. Numerical calculations were run on the Caltech compute cluster “Zwicky” (NSF MRI award #PHY-0960291), through allocation TG-AST120025, TG-AST130039 and TG-AST150045 granted by the Extreme Science and Engineering Discovery Environment (XSEDE) supported by the NSF, and through NASA High-End Computing (HEC) allocation SMD-14-5189, SMD-14-5492, SMD-15-5950, and SMD-16-7592 provided by the NASA Advanced Supercomputing (NAS) Division at Ames Research Center.mnras | http://arxiv.org/abs/1704.08254v2 | {
"authors": [
"Freeke van de Voort",
"Eliot Quataert",
"Claude-André Faucher-Giguère",
"Dušan Kereš",
"Philip F. Hopkins",
"T. K. Chan",
"Robert Feldmann",
"Zachary Hafen"
],
"categories": [
"astro-ph.GA",
"astro-ph.CO"
],
"primary_category": "astro-ph.GA",
"published": "20170426180001",
"title": "On the deuterium abundance and the importance of stellar mass loss in the interstellar and intergalactic medium"
} |
CTP-SCU/2017008 Institut de Physique Nucléaire, CNRS/IN2P3, Univ. Paris-Sud, Université Paris-Saclay, 91406 Orsay, FranceInstitut de Physique Nucléaire, CNRS/IN2P3, Univ. Paris-Sud, Université Paris-Saclay, 91406 Orsay, FranceCenter for Theoretical Physics, Department of Physics,Sichuan University, 29 Wang-Jiang Road, Chengdu, Sichuan 610064, ChinaInstitut de Physique Nucléaire, CNRS/IN2P3, Univ. Paris-Sud, Université Paris-Saclay, 91406 Orsay, France Department of Physics, University of Arizona, Tucson, AZ 85721,USAWe present a new rearrangement of short-range interactions in the ^1S_0 nucleon-nucleon channel within Chiral Effective Field Theory. This is intended to reproduce the amplitude zero (scattering momentum ≃ 340 MeV) at leading order, and it includes subleading corrections perturbatively in a way that is consistent with renormalization-group invariance. Systematic improvement is shown at next-to-leading order, and we obtain results that fit empirical phase shifts remarkably well all the way up to the pion-production threshold. Anapproach in which pions have been integrated out is included, which allows us to derive analytic results that also fit phenomenology surprisingly well. The Two-Nucleon ^1S_0 Amplitude Zeroin Chiral Effective Field Theory U. van Kolck December 30, 2023 ====================================================================== § INTRODUCTION The nuclear effective field theory (EFT) program <cit.> conceives nuclear physics as the renormalization-group (RG) evolution of Quantum Chromodynamics (QCD) at low energies, formulated in terms of effective degrees of freedom (nucleons, pions, etc.). The link with QCD written in terms of more fundamental objects (quarks and gluons) is ensured by imposing QCD symmetries (particularly approximate chiral symmetry) as the only constraints on the otherwise most general EFT Lagrangian. Power counting (PC) rules tell which terms in this Lagrangian (out of an infinite number) should be taken into account when computing observables at a given order in an expansion in powers of the small parameter Q/M_ hi, where Q is the characteristic external momentum of a process and M_ hi M_ QCD∼ 1 GeV is the EFT breakdown scale. Thanks to the recent development of ab initio methods, which bridge the gap between nuclear forces and currents on one hand and nuclear structure and reactions on the other, Chiral EFT (χEFT) <cit.> is now better exploited than ever. However, problems remain in the formulation of this EFT, some of which we address here in the simplest, yet surprisingly challenging, two-nucleon (NN) channel — the spin-singlet, isospin-triplet S wave, ^1S_0.The initial applications of χEFT followed a scheme suggested by Weinberg <cit.> and Rho <cit.>, where a PC dictated by naive dimensional analysis (NDA) <cit.> was assumed to apply to the nuclear potential and currents. The truncated potential is inserted into a dynamical equation — Lippmann-Schwinger (LS), Schrödinger, or one of their variants for the many-body system — from whose exact solution nuclear wave functions are obtained. Averages of the appropriate, truncated currents give rise to scattering amplitudes when the system is probed by external particles such as photons or pions. To deal with the singular nature of the potential and currents, an arbitrary regularization procedure must be introduced. Unfortunately, already at leading order (LO) NDA does not yield all the short-range interactions necessary for the NN amplitude to be approximately independent of the regulator choice <cit.>. Similar issues appear at higher orders <cit.>and also affect electromagnetic currents<cit.>. Given that non-perturbative renormalization can differ significantly from the perturbative renormalization used to infer NDA, it is perhaps unsurprising that a scheme based solely on NDA fails to produce nuclear amplitudes consistent with RG invariance.This problem appears even in NN scattering in the ^1S_0 channel, where one-pion exchange (OPE) has a delta-function singularity in coordinate space. While NDA prescribes that the contact term, which supplements OPE in the LO potential, is chiral-invariant, renormalization demands that a chiral-symmetry-breaking short-range interaction also be present <cit.>. According to NDA, such a chiral-breaking interaction, being proportional to two powers of the pion mass, should not appear before two more orders (next-to-next-to-leading order, or N^2LO) in the Q/M_ hi expansion. This “chiral inconsistency” motivated Kaplan, Savage, and Wise <cit.> to propose a PC where pion exchanges are treated as perturbative corrections starting at next-to-leading order (NLO). However, higher-order calculations soon made clear that such an approach is not valid at low momenta in certain partial waves <cit.>. The alternative is to treat OPE as LO only in the lower waves <cit.>, where suppression by the centrifugal barrier is not effective. The angular-momentum suppression factor has been studied recently in peripheral spin-singlet channels <cit.>.The ^1S_0 partial wave was excluded from the analysis in Ref. <cit.> because this particular channel presents, in addition to the above renormalization issue, other features that are not completely understood. The situation has not improved greatly since the late 90s, despite considerable effort <cit.>.Some of this work has been reviewed recently in Refs. <cit.>.A unique feature of this channel, which was recognized early on, is fine tuning in the form of a very shallow virtual bound state. OPE is characterized by two scales, its inverse range given by the pion mass m_π and its inverse strength given by M_NN≡ 16π f_π^2/(g_A^2 m_N) = 𝒪(f_π), where m_N = 𝒪(M_ QCD) is the nucleon mass, f_π= 𝒪(M_ QCD/(4π)) is the pion decay constant, and g_A = 𝒪(1) is the axial-vector coupling constant. At the physical pion mass m_π≈ 140 MeV, the virtual state's binding momentum ℵ∼ 10 MeV is much smaller than the pion scales, and can only be reproduced at LO through a fine tuning of the short-range interaction. Physics of this state can be described simply by another successful, renormalizable EFT, Pionless (or Contact) EFT (πEFT). In the very-low-energy regime of nuclear physics, Q≪ m_π, pion exchange cannot be resolved, the EFT Lagrangian contains only contact interactions, and the two-body amplitude reduces <cit.> to the effective range expansion (ERE). To simultaneously capture physics at Q∼ m_π, however, pion exchange needs to be retained. The perturbative expansion in Q/M_NN prescribed by Refs. <cit.> converges very slowly, if at all, in the low-energy region <cit.>, which leads to the identification of M_NN as a low-energy scale M_lo, just as suggested by NDA.Yet, it is disturbing that the NDA-prescribed LO potential produces ^1S_0 phase shifts that show large discrepancies with the Nijmegen partial-wave analysis (PWA) <cit.> even at moderate scattering energies. In Ref. <cit.> it was shown that — again at variance with NDA — the first correction in this channel appears already at NLO, in the form of a contact interaction with two derivatives. Still, only about half of the energy dependence of the amplitude near threshold is accounted for at LO, so Ref. <cit.> went a step further by suggesting the promotion to LO of an energy-dependent short-range interaction that reproduces the effective range — a generalization of the same suggestion for πEFT <cit.>. Even this promotion leaves significant room for improvement when compared to the Nijmegen PWA. In particular, the empirical ^1S_0 phase shift, thus theamplitude, vanishes at a center-of-mass momentum k = k_0≃ 340 MeV. Since k_0 is significantly below the expected breakdown scale M_ QCD, we should consider it as a soft scale where the EFT converges. In contrast, we find that the LO phase shift of Ref. <cit.> is around 25^∘ at k=k_0 and does not vanish until k reaches a few GeV. Since higher orders need to overcome LO,convergence at momenta k∼ k_0 will be at best very slow. This can only be remedied if LO contains the amplitude zero. As pointed out in Ref. <cit.>, a low-energy zero requires a different kind of fine tuning than the one that gives rise to a shallow bound state. When the zero appears at very low energies, a contact EFT can be devised (the “other unnatural EFT” of Ref. <cit.>) which gives rise to a perturbative expansion of the amplitude. Such an expansion around k=k_0 in the presence of pions was developed in Ref. <cit.>.Here we propose a rearrangement of the short-range part of χEFT that leads to the existence of the amplitude zero at LO, in addition to the shallow virtual state. The PC of Ref. <cit.> is generalized with the purpose of including the non-perturbative region that contains the virtual state. This is patterned on an idea originally developed for doublet neutron-deuteron (nd) scattering at very low energies <cit.>, where the amplitude has a zero at small imaginary momentum, in addition to a shallow virtual state. We develop an expansion in Q/M_ hi for Q∼ M_ lo, which gives a renormalizable amplitude order by order. Following a successful approach to πEFT <cit.>, the virtual state is assumed to be located right at threshold at LO and is moved to a binding momentum ∼M_lo^2/M_hi at NLO. We calculate NLO corrections and show a systematic improvement in the description of the phase shift.A challenging feature of χEFT is that it usually does not yield analytical expressions for amplitudes. In order to facilitate an understanding of the properties of the NN amplitude, we also consider a version of our PC for the theory without explicit pions, where we retain k_0∼ M_ lo but take M_NN→∞. To our surprise, even though k_0 > m_π, this new version of πEFT also produces a good description of the empirical phase shifts.Our approach is in line with Refs. <cit.>, which argued that short-range forces in the spin-singlet S wave must produce rapid energy dependence. It is a systematic extension of the potential proposed in Ref. <cit.>, and it resembles the unitarized approach of Ref. <cit.>. More generally, it can be seen as the EFT realization of Castillejo-Dalitz-Dyson (CDD) poles <cit.> in S-matrix theory. Traditional S-matrix tools, such as the N/D method, have recently received renewed attention in the NN system (e.g. Ref. <cit.>). The D function is determined modulo the addition of CDD poles, which result in zeros of the scattering amplitude. In particular, the momentum k_0 may be identified with the position of a CDD pole in the ^1S_0 channel <cit.>. An EFT provides a systematic description of the two-body CDD pole that can be naturally extended to more-body systems.This article is structured as follows. In Sec. <ref> we present an initial approach (“warm-up”) to the problem on the basis of a modified organization of πEFT up to NLO. The proposed PC is discussed in detail, and RG invariance is demonstrated explicitly. In Sec. <ref>we bring OPE into LO; also, we compare with theresults <cit.> of the high-quality Nijm93 potential <cit.> before and after the inclusion ofthe NLO potential in this χEFT. Conclusions and outlook are presented in Sec. <ref>.§ PIONLESS THEORYOur first approach to the problem will omit explicit pion exchange (and also electromagnetic interactions, which are small for k≳10 MeV anyway, as well as other small isospin-breaking effects <cit.>). This will allow us to find analytical results, which cannot be reached if one includes OPE in (fully iterated) LO.In the absence of explicit pions and nucleon excitations, all interactions among nucleons are of contact type. The part of the “standard” πEFT Lagrangian relevant for the NN ^1S_0 channel isℒ_π^( ct) = N^†(i∂_0 + ∇^2/2m_N)N - C_0(N^T P⃗_^1S_0 N)^†·(N^T P⃗_^1S_0 N) + ⋯,where N is the isodoublet, bispinor nucleon field and the NN ^1S_0 projector is expressed in terms of the Pauli matrices σ (τ⃗) acting on spin (isospin) space as P⃗_^1S_0 = σ_2 τ⃗τ_2/√(8), while “⋯” means more complicated interactions and relativistic corrections suppressed by negative powers of the breakdown scale of the theory. Now, the interaction term in Eq. (<ref>) may be rewritten if, following Ref. <cit.>, an auxiliary “dibaryon” field ϕ⃗ with quantum numbers of an isovector pair of nucleons is introduced,-C_0 (N^T P⃗_^1S_0 N)^†·(N^T P⃗_^1S_0 N)↔ ϕ⃗^ †·Δϕ⃗ - g (ϕ⃗^ †· N^T P⃗_^1S_0N + H.c.).The dibaryon residual mass Δ and the dibaryon-NN coupling gare such that C_0 = g^2/Δ, as can be straightforwardly checked if one performs the corresponding Gaussian path integral. This parameter redundancy permits the convenient choice <cit.>g^2 ≡4π/m_N.Higher-order contact interactions can be reproduced by the inclusion of the dibaryon's kinetic term and derivative dibaryon-NN couplings.The standard PC of πEFT <cit.> accounts for the presence of a shallow virtual state at LO, but does not produce as much energy dependence as the phenomenological phase shifts. A promotion of the dibaryon kinetic term to LO <cit.> allows for the reproduction of the derivative of the amplitude with respect to the energy around threshold, but it is unable to generate the amplitude zero by itself. This is not a problem in the context of standard πEFT, since k_0 — numerically larger than m_π — is presumably outside the scope of this theory. But here we aim at reformulating the theory in a way such that k_0 is considered below the breakdown scale, so as to illustrate the proposed reformulation of the χEFT PC in Sec. <ref>.Inspired by an EFT for nd scattering at very low energies <cit.>, we consider here a generalization with two such dibaryon fields, ϕ⃗_1,2,ℒ_π^(2ϕ) =N^†(i∂_0 + ∇^2/2m_N)N + ∑_j=1,2ϕ⃗_j^ †·[Δ_j+c_j(i∂_0+∇^2/4m_N)] ϕ⃗_j- √(4π/m_N)∑_j=1,2(ϕ⃗_j^ †· N^T P⃗_^1S_0N + H.c.) +⋯,where we have made use of Eq. (<ref>) and displayed explicitly the kinetic dibaryon terms with dimensionless factors c_j. As we will see, such an extension naturally allows us to reproduce the amplitude zero already at LO, greatly improving the description of the empirical phase shifts. To illustrate the effects of the two dibaryons, we neglect for now the interactions represented by “⋯” in Eq. (<ref>). At momentum k=√(m_NE), where E is the center-of-mass energy, the on-shell T matrix is written in terms of the S matrix and the phase shift δ asT(k) = 2π i/m_N k[S(k) -1 ] =4π/m_N[-kδ(k)+ik]^-1.Loops are regularized by a momentum cutoff Λ in the range Λ≳ M_hi≫ k and a regulator function f_R(q/Λ), with q the magnitude of the off-shell nucleon momentum, that satisfiesf_R(0) = 1,f_R(∞) = 0.Computing the two-dibaryon self-energy, i.e. dressing up the bare two-dibaryon propagatorℬ(k;Λ) = ∑_j(Δ_j(Λ)+c_j(Λ) k^2/m_N)^-1≡m_N/4π V(k;Λ)with nucleon loops (see Fig. <ref>), yields𝒟(k;Λ) = (1/ℬ(k;Λ)+ℐ_0(k;Λ))^-1≡m_N/4π T(k;Λ) .In this equation we introduced the regularized integralℐ_0(k;Λ) = 4π∫d^3q/(2π)^3 f_R(q/Λ)/q^2-k^2-iϵ = ik + θ_1Λ + k^2/Λ∑_n=0^∞θ_-1-2n(k/Λ)^2n,where the dimensionless coefficients θ_n depend on the specific regularization employed. For example, for a sharp-cutoff prescription with a step function it turns out that θ_n=2/(nπ), while in dimensional regularization with minimal subtraction we have simply θ_n=0. We thus arrive at[m_N/4πT(k;Λ)]^-1 = [Δ_1(Λ)+c_1(Λ)k^2/m_N] [Δ_2(Λ)+c_2(Λ)k^2/m_N]/Δ_1(Λ)+Δ_2(Λ) +[c_1(Λ)+c_2(Λ)]k^2/m_N+ik+ θ_1Λ+θ_-1k^2/Λ +𝒪(k^4/Λ^3). When k is much smaller than any other scale, this inverse amplitude reduces at large cutoff to the ERE,[m_N/4πT(k)]^-1 = 1/a + ik - r_0/2 k^2 - P_0/4k^4 +⋯,where, for neutron-proton (np) scattering, a≃ -23.7 fm≃ -(8 MeV)^-1 <cit.> is the scattering length, r_0≃ 2.7 fm≃(73 MeV)^-1 <cit.> is the effective range, P_0≃ 2.0 fm^3≃(158 MeV)^-3 <cit.> is the shape parameter, and so on. In addition, Eq. (<ref>) allows for a pole at a momentum k_0≃ 340 MeV <cit.>, around which the amplitude can be expanded as <cit.>m_N/4πT(k) = k^2-k_0^2/k_0^3 [z_1 +z_2k^2-k_0^2/k_0^2 + O((k-k_0)^2/k_0^2) ]in terms of dimensionless parameters z_n, with |z_n|= O(1) in the absence of further fine tuning. One can easily check that δ(k) behaves linearly around k=k_0, with a slope proportional to z_1,δ(k∼ k_0)=-2z_1/k_0(k-k_0)+⋯.From the Nijm93 phase shifts <cit.> we find z_1 ≃ 0.6.It has long been recognized that the anomalously large value of |a| is a consequence of a fine tuning that places a virtual bound state very close to threshold, and introduces an accidental, small scale ℵ∼ 10 MeV corresponding to its binding momentum. In the standard version of πEFT, higher ERE parameters are assumed to depend on a single higher-energy scale M̃_hi, 1/r_0 ∼ 1/P_0^1/3∼⋯= 𝒪(M̃_hi). The PC then organizes the contributions to an observable characterized by a momentum Q∼ℵ in an expansion in powers of Q/M̃_hi, i.e. M̃_hi becomes the breakdown scale of the theory. Naively one expects M̃_hi≲ m_π, but there is some evidence that πEFT works also at larger momenta. For example, the binding momenta for the ground states of systems with A=3,4,6,16 nucleons are near 100 MeV, and yet their physics is well described by the lowest orders of πEFT (see, for example, Refs. <cit.>). In fact, it has been suggested that the characteristic scale of πEFT is set by these binding momenta through the LO three-nucleon force, so that ℵ appears only at NLO or higher <cit.>.Here we propose to accommodate an enlarged range of validity of πEFT and the smallness of 1/a by changing the standard PC of πEFT in the ^1S_0 channel on the basis of the replacements M̃_hi→ M_lo and ℵ→ M_lo^2/M_hi. The phenomenological parameters of the theory are assumed to scale as1/a = 𝒪(M_lo^2/M_hi),k_0 ∼ 1/r_0 ∼ 1/P_0^1/3∼⋯ =𝒪(M_lo),with M_hi≫ M_lo. This assumption will allow us to develop an expansion for an observable at typical momentum Q∼ M_lo in powers of Q/M_hi. The usefulness of such an expansion is far from obvious, but as we show below it seems to give good results. Our prescription includes the correct position of the amplitude zero at LO, and moves the virtual state at NLO very close to its empirical position. For Q∼ℵ the NLO amplitude is similar to that of standard πEFT with M̃_hi=𝒪(M_lo). The assignment ℵ→ M_lo^2/M_hi is somewhat arbitrary but motivated by the expectation that M_lo∼ 100 MeV and M_hi∼ 500 MeV, when it holds within a factor of 2 or so. If ℵ were taken to be smaller, a reasonable description of observables at momenta Q∼ℵ would only emerge at higher orders. Conversely, had we decided to treat ℵ as M_lo, the very-low-energy region would be well reproduced already at LO, but it would be more difficult to see improvements at NLO.Quantities in the theory can be organized in powers of the small expansion parameter M_lo/M_hi. For a generic coupling constant 𝗀, we expand formally𝗀(Λ) = 𝗀^[0](Λ)+𝗀^[1](Λ)+⋯,where the superscript ^[ν] indicates that the coupling appears at N^νLO. The “renormalized” coupling 𝗀̅^[ν] — i.e. the regulator-independent contribution to the bare (running) coupling 𝗀^[ν](Λ) — is nominally suppressed by O(M_lo^ν/M_hi^ν) with respect to 𝗀̅^[0].Likewise, the amplitude is writtenT(k;Λ) = T^[0](k;Λ) + T^[1](k;Λ) +⋯,whereT^[0](k;Λ)=V^[0](k;Λ)[1+m_N/4πV^[0](k;Λ) (ik+θ_1Λ + k^2/Λ∑_n=0^∞θ_-1-2nk^2n/Λ^2n) ]^-1, T^[1](k;Λ)= (T^[0](k;Λ)/V^[0](k;Λ))^2 V^[1](k;Λ),etc., in terms ofV^[0](k;Λ)= 4π/m_N∑_j(Δ_j^[0](Λ) +c_j^[0](Λ)k^2/m_N)^-1, V^[1](k;Λ)= - 4π/m_N∑_j(Δ_j^[0](Λ)+c_j^[0](Λ) k^2/m_N)^-2(Δ_j^[1](Λ)+c_j^[1](Λ)k^2/m_N),etc. Neglecting higher-order terms, the phase shifts at LO, LO+NLO and so on can be written asδ^[0](k;Λ)=-^-1(4π/m_Nk Re( T^[0](k;Λ))^-1),δ^[0+1](k;Λ) =-^-1(4π/m_Nk Re[ (T^[0](k;Λ))^-1(1-T^[1](k;Λ)/T^[0](k;Λ)) ]),etc. At higher orders interactions in the “⋯” of Eq. (<ref>) appear. We now consider the first two orders of the expansion in detail.§.§ Leading OrderFrom Eq. (<ref>) we see that reproducing the amplitude zero at LO with a shallow pole requires a minimum of three bare parameters. Both residual masses, Δ_1(Λ) and Δ_2(Λ), must be non-vanishing, otherwise the resulting inverse amplitude at threshold would be proportional to Λ, i.e. not properly renormalized. At the same time, at least one of the kinetic factors, which we choose to be c_2(Λ), needs to appear at LO, otherwise the amplitude zero could not be reproduced.Since we attribute in Eq. (<ref>) the smallness of the inverse scattering length to a suppression by one power of the breakdown scale M_hi, we take1/a^[0]=0.In other words, we perform an expansion of the NN ^1S_0 amplitude around the unitarity limit, as in Refs. <cit.>. One of the dibaryon parameters, which turns out to be Δ_2(Λ), carries such an effect, so that its observable contribution vanishes at LO. The regulator-independent parts of the remaining LO parameters, Δ_1 and c_2, are assumed to be governed by the scale M_lo [NDA <cit.> gives for a dibaryon-NN coupling g=𝒪(4π/√(m_N)), which differs from our convention (<ref>) by a factor of √(4π). Since it is the combination g^2/Δ that enters the amplitude, Δ is expected to be 𝒪(M_hi/(4π))=𝒪(M_lo) instead of 𝒪(M_hi).]. In a nutshell,Δ̅_1^[0] = 𝒪(M_lo), c̅_1^[0]/m_N = 0, Δ̅_2^[0] = 0, c̅_2^[0]/m_N =𝒪(1/M_lo).Because of the vanishing of c_1^[0], eliminating dibaryon-1 via Eq. (<ref>) generates a momentum-independent contact interaction. Thus, at LO we obtain — except for our additional requirement (<ref>) — the M_NN→∞ version of the model considered in Ref. <cit.>, where a dibaryon (our dibaryon-2) is added to a series of nucleon contact interactions.In order to relate Δ_1^[0](Λ), Δ_2^[0](Λ), and c_2^[0](Λ) — our three non-vanishing LO bare parameters — to observables, we impose onF(z;Λ) ≡Re{[m_N/4πT^[0](√(z);Λ)]^-1}three renormalization conditions,F(0;Λ) =0, . ∂ F(z;Λ)∂ z|_z=0 = -r_0/2, F^-1(k_0^2;Λ)=0.The dependence of loops on positive powers of Λ is canceled by that of the bare couplings,Δ_1^[0](Λ) = Δ̅_1^[0] - θ_1Λ + ⋯,Δ_2^[0](Λ) = 2θ_1/r_0^3k_0^2[θ_1 (r_0Λ)^2 -(r_0^2k_0^2/2+2θ_1θ_-1)r_0Λ + 4θ_1θ_-1^2 + ⋯],c_2^[0](Λ)/m_N = c̅_2^[0]/m_N - 2θ_1 /r_0^3k_0^4[θ_1 (r_0Λ)^2 -(r_0^2k_0^2+2θ_1θ_-1)r_0Λ + 4θ_1θ_-1^2 + ⋯],where “⋯” stands for terms that become arbitrarily small for an arbitrarily large cutoff. Equation (<ref>) ensures that the non-vanishing renormalized couplings,Δ̅_1^[0] = r_0k_0^2/2, c̅_2^[0]/m_N = -r_0/2,are indeed consistent with Eq. (<ref>). Apart from a residual cutoff dependence that can be made arbitrarily small by increasing the cutoff, the amplitude can now be expressed in terms of the renormalized couplings or, using Eq. (<ref>), in terms of r_0 and k_0:[m_N/4πT^[0](k;Λ)]^-1 = ik -r_0/2k^2/1-k^2/k_0^2(1+2θ_-1/r_0Λ k^2/k_0^2) +𝒪(k^4/Λ^3).Although the scales and the zero location are different, Eq. (<ref>) is similar to the one <cit.> for nd scattering at very low energies [DefiningA ≡r_0/2k_0^2≡ -R,Eq. (<ref>) may be rewritten as[m_N/4πT^[0](k;Λ)]^-1= A+R/1-k^2/k_0^2+ik+𝒪(k^2/Λ),which is a form used in early work on nd scattering, such as Ref. <cit.>.].Many interesting consequences can be extracted from Eq. (<ref>). For momenta below the amplitude zero, our expression reduces to the unitarity-limit version of the ERE (<ref>) but with predictions for the higher ERE parameters, starting with the shape parameterP_0^[0](Λ) = 2r_0/k_0^2[1+ 2θ_-1/r_0 Λ +𝒪(k_0^2/r_0 Λ^3)].Using the cutoff dependence to estimate the error under the assumption M_hi∼ 500MeV, the LO prediction is P_0^[0] k_0^2/(2r_0)=1.0 ± 0.3. These high ERE parameters are difficult to extract from data. A careful analysis in Ref. <cit.> obtains P_0k_0^2/(2r_0)=1.1, which is well within our expected truncation error. Yet, values obtained for P_0 from the phenomenological np potentials NijmII and Reid93 <cit.> are of the same order of magnitude as the value from Ref. <cit.>, but with a negative sign <cit.>.We conjecture that, in contrast to standard πEFT, Eq. (<ref>) also applies at momenta around the amplitude zero, with terms which are 𝒪(M_lo) and corrections of 𝒪(M_lo^2/M_hi). Around the amplitude zero, the amplitude is perturbative <cit.>. Indeed, a simple Taylor expansion of Eq. (<ref>) gives a perturbative expansion in the region |k-k_0| k_0, i.e. an equation of the form (<ref>) with LO predictions for the coefficients,z_1^[0](Λ) = 2/r_0k_0(1-2θ_-1/r_0Λ+⋯), z_2^[0](Λ) = -2/r_0k_0[1 +2i/r_0k_0(1-4θ_-1/r_0Λ)+⋯],where the “⋯” account for 𝒪(M_lo^2/Λ^2). Numerically, these coefficients are z_1^[0]=0.4± 0.1 and z_2^[0]= - (0.4 ± 0.1) -i (0.2± 0.1), which are indeed of O(1). The former is in fact reasonably close to z_1≃ 0.6 extracted from the phenomenological data. Note that we could have imposed as a renormalization condition that z_1 had a fixed value — the one that best fits the empirical value — at any Λ, thus trading the information about energy dependence carried by r_0 for that contained in the derivative of the phase shift at its zero, see Eq. (<ref>).Equation (<ref>) interpolates between the two regions, k≪ k_0 where the amplitude is non-perturbative and |k-k_0|≪ k_0 where it is perturbative. Compared to standard πEFT, it resums not only range corrections as in Ref. <cit.>, but also corrections that give rise to the pole at k=k_0. Compared to the expansion around the amplitude zero <cit.>, it resums the terms that become large at low energies and give rise to a resonant state at zero energy. The pole structure of the LO amplitude can be made explicit by rewriting Eq. (<ref>) as[m_N/4πT^[0](k;Λ)]^-1 = (k - iκ_1^[0])(k - iκ_2^[0])(k - iκ_3^[0])/i (k_0^2 - k^2) +𝒪(k^2/Λ),withκ_1^[0]=0, κ_2^[0]=r_0k_0^2/4(1 - √(1-(4/(r_0k_0))^2)), κ_3^[0]=r_0k_0^2/4(1 + √(1-(4/(r_0k_0))^2)).In addition to the amplitude zero, T^[0](k_0;Λ)=0, it is apparent that there are three simple poles, T^[0](iκ_j^[0];∞)→∞, the nature of which is linked to the sign of i Res S^[0](iκ_j^[0]): * The pole at k=0 represents a resonant state at threshold, as it induces the vanishing of δ(0). Such a pole can be reproduced even with a momentum-independent contact potential, just as it is done at LO in standard πEFT (<ref>) in the unitarity limit. (Since i ResS^[0](iκ_1^[0])=0, this state has a non-normalizable wavefunction.) * The pole at k=iκ_2^[0], κ_2^[0]≃ 190 MeV, lies on the positive imaginary semiaxis. However, since i ResS^[0](iκ_2^[0])<0, the condition to produce a normalizable wavefunction is not satisfied. The pole at k=iκ_2^[0] cannot correspond to a bound state, whose wavefunction has finite support in coordinate space. It is a redundant pole <cit.>. * The pole at k=iκ_3^[0], κ_3^[0]≃ 600 MeV, lies deep on the positive imaginary semiaxis. It represents a bound state because i ResS^[0](iκ_3^[0])>0. Since no such state exists experimentally, it sets an upper bound on the regime of validity of the EFT, M_hiκ_3^[0]. In Fig. <ref>, we plot the ^1S_0 phase shifts (<ref>) from the LO amplitude (<ref>) in comparison with the Nijm93 results <cit.>. As input, we use the empirical values of the effective range and the position of the amplitude zero. We display the cutoff band for a generic regulator by taking θ_-1=± 1 and varying Λ from around the breakdown scale (500 MeV) to infinity — as the cutoff increases, our results converge, as evident in Eq. (<ref>). This cutoff band provides an estimate of the LO error, except at low momentum where there is an error that scales with 1/|a| instead of k. The LO phase shifts are in good agreement with empirical values for most of the low-energy momentum range, except at the very low momenta where the small but non-vanishing virtual-state binding energy is noticeable. A plot of k δ shows that differences at the amplitude level are indeed small. We plot phase shifts to better display the region around the amplitude zero, which our PC is designed to capture. There, while the phase shifts themselves are not too far off empirical values, the curvature is not well reproduced. Nevertheless, the agreement is surprisingly good given the absence of explicit pion fields. In the next section we examine how robust this agreement is. §.§ Next-to-Leading OrderAs pointed out in Ref. <cit.>, the leading residual cutoff dependence of an amplitude, together with the assumption of naturalness, gives an upper bound on the order of the next correction to that amplitude. In standard πEFT, for example, the LO amplitude has an effective range r_0 ∼ 1/Λ, indicating that there is an interaction at order no higher than NLO <cit.> which will produce a physical effective range r_0∼ 1/M̃_hi. The leading residual cutoff dependence in Eq. (<ref>) is proportional to k^4 and of relative order O(M_lo/Λ). Thus, the NLO interaction must give rise to a contributionP_0^[1](Λ) ≡ P_0-P_0^[0](Λ) =O(1/M_lo^2M_hi)to the LO shape parameter (<ref>). This correction requires a higher-derivative operator. Although we could add a momentum-dependent contact operator, a simpler, energy-dependent strategy will be implemented here: we allow for a non-vanishing c_1^[1].In addition, since we are interpreting ℵ→ M_lo^2/M_hi, one combination of parameters including Δ_2^[1] enforces1/a^[1]= 1/a =O(M_lo^2/M_hi).We also introduce corrections in the other two parameters, c_2^[1] and Δ_1^[1], in order to keep r_0 and k_0 unchanged. Since NLO interactions must all be suppressed by M_hi^-1,Δ̅_1^[1] = 𝒪(M_lo^2/M_hi), c̅_1^[1]/m_N= 𝒪(1/M_hi), Δ̅_2^[1] = 𝒪(M_lo^2/M_hi), c̅_2^[1]/m_N= 𝒪(1/M_hi).This scaling — together with what was learned at LO — is consistent with the imposition of four renormalization conditions onG(z;Λ) ≡ -Re{[m_N/4π T^[1](√(z);Λ)][m_N/4π T^[0](√(z);Λ)]^-2},which ensure that a, r_0, P_0, and k_0 are fully Λ independent at NLO:G(0;Λ)=1/a, . ∂ G(z;Λ)/∂ z|_z=0= 0, . ∂^2G(z;Λ)/∂ z^2|_z=0 = -P_0^[1](Λ)/2,G(k_0^2;Λ) =0. Defining the renormalized parametersΔ̅_1^[1] = Δ̅_2^[1]+3k_0^2m_N c̅_1^[1], c̅_1^[1]m_N = -r_02(1-P_0k_0^22r_0),Δ̅_2^[1] = 1a +r_0k_0^2(1-P_0k_0^22r_0),c̅_2^[1]m_N = -4(c̅_1^[1]m_N+Δ̅_2^[1]2k_0^2),which with Eq. (<ref>) give Eqs. (<ref>) and (<ref>), the cutoff dependence of the bare parameters that guarantees Eq. (<ref>) isΔ_1^[1](Λ)= Δ̅_1^[1]+⋯,c_1^[1](Λ)/m_N = c̅_1^[1]/m_N+⋯,Δ_2^[1](Λ)= Δ̅_2^[1] - θ_1/r_0^4P_0^[1](Λ)[θ_1(r_0Λ)^2 + (r_0^2k_0^2-4θ_1 θ_-1)r_0Λ - 2θ_-1(r_0^2k_0^2-6θ_1θ_-1)] - 4θ_1/ar_0^2k_0^2(r_0Λ-2θ_-1) + ⋯,c_2^[1](Λ)/m_N = c̅_2^[1]/m_N +1/k_0^2(Δ̅_2^[1] - Δ_2^[1](Λ) ) + ⋯,where the ellipsis account for terms that disappear when we take Λ→∞.The NLO contribution to the amplitude, Eq. (<ref>), then satisfiesT^[1](k;Λ)/T^[0]2(k;Λ) = -m_N/4π[ 1/a + r_0/2k_0^2k^4/1-k^2/k_0^2(1-P_0k_0^2/2r_0+2θ_-1/r_0Λ) +𝒪(k^4/Λ^3)],which is indeed suppressed by one negative power of M_hi. If we resum T^[1](k;Λ) while neglecting N^2LO corrections,then[m_N/4π(T^[0](k;Λ) +T^[1](k;Λ))]^-1 = 1/a+ik-r_0/2k^2 - P_0/4k^4/1-k^2/k_0^2 +𝒪(k^6/k_0^2Λ^3). Now the ERE (<ref>) is reproduced for k< k_0 with the experimental scattering length and shape parameter. Additionally, there are predictions for the higher ERE parameters which are hard to test directly since they are difficult to extract from data. The zero at k_0 remains unchanged due to our choice of renormalization condition. Once expanded around k=k_0, the distorted amplitude (<ref>)yields NLO coefficients such asz_1^[1](Λ) =z_1^[0](∞) (1-P_0k_0^2/2r_0)+⋯, z_2^[1](Λ) =z_2^[0](∞) (1-i r_0k_0/2)^-1[2(1-P_0k_0^2/2r_0)- i/ak_0] + ⋯,where “⋯” stands for 𝒪(M_lo^3/Λ^3). NLO contributions are of relative 𝒪(M_lo/M_hi) with respect to their LO predictions z_1^[0] and z_2^[0], consistently with the residual cutoff dependence displayed in Eqs. (<ref>) and (<ref>). Since z_1^[0](∞) underestimates the slope of the phenomenological phase shifts around the amplitude zero, a better description of data requires z_1^[1](∞)> 0 and thus, according to Eqs. (<ref>) and (<ref>), P_0≲ P_0^[0](∞). The value given in Ref. <cit.> leads to a small change |z_1,2^[1](∞)/z_1,2^[0](∞)|≲ 1/10, but unfortunately it is ∼10% larger than P_0^[0](∞). Since Ref. <cit.> provides no error bars it is difficult to decide whether this is a real problem. We can reproduce the phenomenological value for z_1 with P_0^[1](∞)≃ -0.6 P_0^[0](∞), which is still compatible with convergence but not so small a change with respect to LO. Of course, not all the discrepancy between LO and phenomenology should come from NLO, but this might be indicative that something is missing. We will return to the shape parameter in the next section. NLO also shifts the LO position of the poles (<ref>) of the S matrix. One can obtain these shifts reliably by means of perturbative tools only for the two shallowest LO poles, finding in the large-cutoff limitκ_1^[1]=1/a, κ_2^[1]= -k_0^2+κ_2^[0]2/k_0^2-κ_2^[0]2[1/a+ 1/2r_0κ_2^[0]4/k_0^2+κ_2^[0]2(1-P_0k_0^2/2r_0)].We see that, as expected, |κ_1^[1]|∼ |κ_2^[1]| =𝒪(M_lo^2/M_hi), as long as κ_2^[0]=𝒪(M_lo). As a consequence:* The shallowest pole is moved from threshold to k ≃ - 8i MeV, and represents the well-known virtual state. Its new location almost coincides with the physical one. * The redundant pole is moved from k≃190i MeV to k≃215i MeV, when the value of P_0 given in Ref. <cit.> is used. This represents a shift of relative size ∼15% with respect to LO. Roughly two thirds of this shift are due to the finiteness of the scattering length, while the other third corresponds to the NLO correction to the shape parameter. If we take the value of P_0 that gives the slope of the phenomenological phase shifts at k_0, then the shape correction overcomes the scattering length and the pole moves to k≃ 155 i MeV, still a modest shift. The LO+NLO ^1S_0 phase shift can now be obtained from Eqs. (<ref>) and (<ref>), see Fig. <ref>. Now, in addition to the empirical values of r_0 and k_0, also the values of the scattering length and the shape parameter from Ref. <cit.> are input. We show a band corresponding to a variation of ±30% around the P_0 value of Ref. <cit.> to account for its (unspecified) error. Since the cutoff dependence of theNLO result (<ref>)is very quickly convergent (∼ 1/Λ^3), it has been neglected in Fig. <ref>. The band thus does not reflect the uncertainty of the NLO truncation, but of the input.As expected, the physical value of a greatly improves the description of the phase shifts at low energies (k≲50 MeV). However, at middle energies (k∼ 100 MeV) this improvement is much less clear. In particular, as anticipated above, only for a shape parameter ∼ 30% smaller than in Ref. <cit.> does δ^[0+1](k;∞) get slightly closer to empirical values than δ^[0](k;∞) (see Fig. <ref>). Such a change is within the LO error and, overall, the reproduction of the phase shifts is very good at NLO. Agreement could be further improved, particularly around k_0, by taking an even smaller value for the shape parameter — in particular, the value that reproduces the phenomenological value for z_1. However, even in that case the curvature of the resulting phase shifts would remain different from empirical at middle energies, which suggests that our expansion is lacking terms at either LO or NLO. §.§ Resummation and Higher OrdersThe choice of identifying the fine-tuning scale ℵ with M_lo^2/M_hi led to a non-zero scattering length only at NLO. This assignment is motivated by the numerical values estimated for these scales. Alternative choices are possible, leading to slightly different amplitudes at various orders. When plotting phase shifts, these differences are amplified. For example, taking ℵ as M_lo leads to a renormalization condition where 1/a is non-zero already at LO. In this case our running and renormalized parameters given above all change by 1/a terms. The amplitude itself (or equivalently its pole positions) changes only slightly, but in terms of phase shifts there appears to be a large improvement.Given our previous identification of ℵ with M_lo^2/M_hi, the alternative procedure just described amounts to a resummation of higher-order corrections. Because a bare parameter (Δ_2(Λ)) exists already at LO to ensure proper renormalization, this resummation can be done without harm. However, because some NLO contributions are shifted to LO, we see less improvement when going from LO to NLO. Provided that one has a PC that converges, this is just one of many ways in which we can make results at one order closer to phenomenology while remaining within the error of that order.Regardless of such resummation, corrections at higher orders are expected to improve the situation further. The cutoff dependence of Eq. (<ref>) suggests that there are no new interactions at next order, N^2LO, which would solely consist of one iteration of the NLO potential. However, the fact that our pionless phase shifts look too low in the middle range represents a significant, systematic lack of attraction between nucleons at k∼ m_π. This could be a reminder to include pions explicitly. We now consider our expansion with additional pion exchange. § PIONFUL THEORY We now modify the theory developed in the previous section to include pion exchange. This is done under the assumption that the pion mass, the characteristic inverse strength of OPE, and the magnitude of the relevant momenta have similar sizes, not being enhanced or suppressed by powers of the hard scale:m_π∼ M_NN∼ Q = 𝒪(M_lo).Such an assumption has been standard in χEFT since its beginnings <cit.>. In the NN sector, it underlies the (non-perturbative) LO character of the OPE interaction, as well as the suppression of multiple pion exchanges by powers of (M_lo/M_QCD)^2. Moreover, the Coulomb interaction between protons — the dominant electromagnetic effect — contributes an expansion in α m_N/M_lo∼ℵ/M_lo, where α≃ 1/137 is the fine-structure constant. As we took ℵ = 𝒪(M_lo^2/M_hi), we should account for the Coulomb interaction at NLO. (Other isospin-breaking effects, such as the nucleon mass splitting, are to be accounted for perturbatively, too.) Within the πEFT framework, the (subleading) Coulomb effects were included in an expansion around the unitarity limit (without consideration of the amplitude zero) in Ref. <cit.>. Since we anticipate no new features here, in this first approach we neglect isospin breaking.We also ignore the explicit dependence on quark mass, because theexpansion is already quite complicated at a fixed value of m_π^2.Pions are introduced in the usual way, by demanding that the most general effective Lagrangian transforms under chiral symmetry as does the QCD Lagrangian written in terms of quarks and gluons. (For reviews and references, see Refs. <cit.>.) In the particular case of one dibaryon field, this was done in Ref. <cit.>. The extension to the two dibaryons of the previous section is straightforward. If π⃗ is the pion isotriplet, the effective Lagrangian readsℒ_χ^(2ϕ) =1/2(∂_μπ⃗·∂^μπ⃗ - m_π^2π⃗^2) + N^†[i∂_0 + ∇^2/2m_N -g_A/2f_πτ⃗·(σ·∇π⃗) ]N+ ∑_j=1,2{ϕ⃗_j^ †·[Δ_j+c_j (i∂_0 + ∇^2/4m_N) ] ϕ⃗_j - √(4π/m_N)(ϕ⃗_j^ †· N^T P⃗_^1S_0N + H.c.)} +⋯,in the same notation as Eq. (<ref>). The omitted terms, which include chiral partners of the terms shown explicitly, are not needed up to NLO.Inspired by the pionless theory, weuse for the pionful case thesame dibaryon arrangement of short-range potentials as inSec. <ref>.Adding the long-range,spin-singlet projection of OPE, the LO potential is m_N/4πV^[0](p',p,k;Λ)=-m_π^2/M_NN1/(p'-p)^2+m_π^2 +1/Δ_1^[0](Λ) + 1/Δ_2^[0](Λ)+c_2^[0](Λ)k^2/m_N≡ m_N/4π(V^[0]_L(p',p) +V^[0]_S(k;Λ)),where p (p') is the relative incoming (outgoing) momentum and the inverse OPE strength is defined as <cit.>M_NN≡16π f_π^2/g_A^2m_N≈ 290 MeV.The momentum-independent, contact piece of OPE has been absorbed in the short-range potential V^[0]_S through the redefinition(1/Δ_1^[0](Λ)+1/M_NN)^-1→Δ_1^[0](Λ).The long-range part of OPE is the Yukawa potential represented by V^[0]_L. Integrating out dibaryon-1 we obtain the potential considered previously in Ref. <cit.>. Since two-pion exchange (TPE) enters only at N^2LO and higher <cit.>, at NLO the interaction is entirely short-ranged,m_N/4πV^[1](k;Λ)=- Δ_1^[1](Λ)+c_1^[1](Λ) k^2/m_N/Δ_1^[0]2(Λ) -Δ_2^[1](Λ)+c_2^[1](Λ)k^2/m_N/(Δ_2^[0](Λ)+c_2^[0](Λ)k^2/m_N)^2.In the limit Δ_2^[0]→∞ the potential is an energy-dependent version of the momentum-dependent LO+NLO interaction of Ref. <cit.>, while the interaction of Ref. <cit.> emerges in the limit Δ_1^[0]→∞.Because OPE cannot be iterated analytically to all orders, we can no longershow explicitly that the amplitude has a zero at LO or that the amplitude isRG invariant. However, these two important features of the pionless theory areexpected to be retained by the pionful theory on the basis that the strengthof OPE is known to be numerically moderate in spin-singlet channels andthat V^[0]_L is non-singular. Moreover, we continue to usethe scalings (<ref>) and (<ref>).Below we confirm through numerical calculations that the EFT obeying such aPC indeed has an amplitude zero and preserves RG invariance.§.§ Leading OrderThe off-shell LO amplitude is found from the LO potential (<ref>) by solving the LS equationT^[0](p',p, k;Λ) = V^[0](p',p, k;Λ) - m_N∫d^3q/(2π)^3f_R(q/Λ)/q^2-k^2-iϵV^[0](p',q,k;Λ) T^[0](q,p,k;Λ),with f_R(q/Λ) a non-local regulator function (<ref>). Defining the Yukawa amplitude,T_L^[0](p',p,k;Λ) = V_L^[0](p',p) - m_N∫d^3q/(2π)^3f_R(q/Λ)/q^2-k^2-iϵ V_L^[0](p',q)T_L^[0](q,p, k;Λ),the Yukawa-dressing of the incoming/outgoing NN states,χ_L^[0](p, k;Λ) = 1 - m_N∫d^3q/(2π)^3f_R(q/Λ)/q^2-k^2-iϵT_L^[0](p,q,k;Λ),and the resummation of NN bubbles with iterated OPE in the middle,ℐ_L^[0](k;Λ) = 4π∫d^3q/(2π)^3f_R(q/Λ)/q^2-k^2-iϵ χ_L^[0](q,k;Λ),Eq. (<ref>) can be rewritten as <cit.>[m_N/4π(T^[0](p',p,k;Λ) - T_L^[0](p',p,k;Λ))]^-1 =[m_N V_S^[0](k;Λ)/(4π)]^-1 + ℐ_L^[0](k;Λ)/χ_L^[0](p',k;Λ) χ_L^[0](p,k;Λ).This is the generalization of Eq. (<ref>) for LO in the presence of pions. Because V_L^[0] is regular, the cutoff dependence of the integrals T_L^[0] and χ_L^[0] is only residual, i.e. suppressed by powers of Λ. In contrast, just as the ℐ_0 inEq. (<ref>), ℐ_L^[0] has a linear cutoff dependence due to the singularity of V_S^[0]. Additionally, it exhibits a logarithmic dependence ∼ (m_π^2/M_NN)lnΛ <cit.> arising from the interference between V_L^[0] and V_S^[0]. This cutoff dependence is at the root of one of the shortcomings of NDA in the NN system.Compared to Refs. <cit.>, our V_S^[0] has a different k dependence. As in the previous section, two dibaryon parameters are needed to describe the zero of the amplitude and its energy dependence near threshold, while the third parameter ensures the fine tuning that leads to a large scattering length. These three parameters are sufficient for renormalization, leaving behind only residual cutoff dependence. Our LO amplitude is analogous to that of Ref. <cit.>, which results from the unitarization of an expansion around the amplitude zero.Taking the sharp-cutoff function f_R(x)=θ(1-x), we solve numerically the S-wave projection of Eq. (<ref>), as done in, e.g., Refs. <cit.>. In order to determine the values of the three bare parameters at a given cutoff, three cutoff-independent conditions on the amplitude are needed. We choose them to be the same as in the previous section, i.e. * unitarity limit, 1/a^[0]=0;* physical effective range, r_0=2.7 fm;* physical amplitude zero, k_0=340.4 MeV.The values of Δ_1^[0](Λ), Δ_2^[0](Λ), and c_2^[0](Λ) in our numerical calculations must be very well tuned in order to reproduce the required values of 1/a^[0], r_0, and k_0 within a given accuracy. The need for such a tuning becomes more and more noticeable as Λ is increased <cit.>. But the resulting phase shift changes dramatically depending on whether 1/a^[0] is very small and negative (for a shallow virtual state) or very small and positive (as it would correspond to a bound state close to threshold). Thus, in order to facilitate the numerical solution of the LS equation, we kept the scattering length large and negative, a^[0]=-600 fm. The difference with the unitarity limit cannot be seen in the results presented below.The LO pionful phase shift is obtained from the on-shell, S-wave-projected T matrix in the usual way (<ref>). The result, presented in Fig. <ref>, shows little cutoff dependence, even though the cutoff parameter is varied from 600 MeV to 2 GeV. It is likely that a more realistic estimate of the systematic error coming from the EFT truncation is obtained via the variation of the input inverse scattering length between its physical value and zero. We will come back to such an estimate later, when we resum finite-a effects. In any case, comparing with Fig. <ref> we confirm that pions help us achieve a better description of phase shifts between threshold and the amplitude zero. From the results in Fig. <ref> we can obtain numerical predictions for parameters appearing in the ERE and in the expansion around the amplitude zero. As an example, we extract the LO shape parameter P_0^[0](Λ) using our low-energy results and the unitarity-limit version of the ERE (<ref>) truncated at the level of the shape parameter. Results are shown in Fig. <ref>. For Λ large enough, we findP_0^[0](Λ) ≈ P_0^[0](∞) (1+ Q_P/Λ),with P_0^[0](∞) ≈ -1.0 fm^3 and Q_P ≈ 100MeV. Unlike the result for the shape parameter given in Ref. <cit.>, P_0^[0](∞) is negative, being reasonably close to P_0=-1.9 fm^3 — the value extracted in Ref. <cit.> from the NijmII fit <cit.>. The large change in the prediction for P_0^[0](∞) compared to the corresponding pionless result (<ref>) is confirmation of the importance of pions at LO.§.§ Next-to-Leading OrderAs before, we can infer the short-range contributions at NLO from the residual cutoff dependence of the amplitude. Figure <ref> shows that the cutoff dependence of P_0^[0](Λ) is proportional to 1/Λ, with Q_P= O(M_lo) as expected. Just as in the pionless case, this behavior implies that at least one extra short-range parameter needs to be included at NLO. This is represented by the NLO potential V^[1], Eq. (<ref>).Treating V^[1] in distorted-wave perturbation theory, we obtain a separable NLO amplitude,T^[1](p',p, k;Λ) = χ^[0](p', k;Λ) V^[1](k;Λ) χ^[0](p, k;Λ),whereχ^[0](p, k;Λ) = 1 - m_N∫d^3q/(2π)^3f_R(q/Λ)/q^2-k^2-iϵT^[0](p,q,k;Λ),is defined in terms of the full LO amplitude in analogy with Eq. (<ref>) for the long-range LO amplitude. As in the pionless case, we obtain the pionful LO+NLO phase shift from Eq. (<ref>).The dibaryon parameters are fixed in virtue of four cutoff-independent conditions, which we choose to be the values of the Nijm93 phase shifts <cit.> at four different momenta:* δ^[0+1](20.0 MeV;Λ)=61.1^∘;* δ^[0+1](40.5 MeV;Λ)=64.5^∘;* δ^[0+1](237.4 MeV;Λ)=21.7^∘;* δ^[0+1](340.4 MeV;Λ)=0^∘. The LO+NLO phase shifts are shown in Fig. <ref>.The narrow band when the cutoff is varied from 600 MeV to 2 GeV confirms that, as in Fig. <ref>, very quick cutoff convergence takes place. The LO+NLO prediction almost lies on the Nijm93 curve, which means that now the description of the empirical phase shifts throughout the whole elastic range 0≲ k≲√(m_Nm_π) is much better than at LO.Indeed, the improvement is clear not only in the very-low momentum regime (which had been expected considering that now we relaxed the unitarity-limit condition), but — more importantly from the χEFT point of view — also for momenta k∼ m_π. Comparison with the pionless result at NLO (Fig. <ref>) confirms that adding OPE significantly improves predictions in this momentum range. §.§ Resummation and Higher OrdersDespite the systematic improvement and good description of data at NLO, one might be distressed by the unusual appearance of our LO phase shift (Fig. <ref>) at low momentum. Within potential models — whether purely phenomenological or based on Weinberg's prescription — it is traditional to attempt to describe all regions below some arbitrary momentum on the same footing.As emphasized earlier, plotting phase shifts is misleading when it comes to errors in the amplitude, which is the observable the PC is designed for. A plot of kδ shows that only a small amount of physics is missed at LO even at low energies. Our strategy is a consequence of the fact that the PC assumes momenta Q ∼ M_lo, and it is in principle only in this region that we expect systematic improvement order by order. The higher the momentum, the smaller the relative improvement with order, till we reach M_hi and the EFT stops working. In the other direction, that of smaller momenta, the χEFT PC may no longer capture the relative importance of interactions properly. A simple example is pion-nucleon scattering in Chiral Perturbation Theory, where sufficiently close to threshold the LO P-wave interaction (stemming from the axial-vector coupling in Eq. (<ref>)) is smaller than NLO corrections to the S wave. Therefore the region of momenta much below the pion mass is not where one wants to judge the convergence of χEFT. However, it might be of practical interest to improve the description near threshold already at LO. As in πEFT, we can choose to reproduce the empirical value of a phase shift in the very low-momentum region — thus accounting for non-vanishing 1/a already at LO — without doing damage to renormalization. As is the case with any other choice of data to fit, the difference with respect to what we have done earlier in this section is of NLO: we are just resumming some higher-order contributions into LO. As an example, in Fig. (<ref>) we show LO and LO+NLO results with an alternative fitting protocol. In the renormalization conditions at LO we replace the unitarity limit of our original fit with the physical scattering length, that is, we impose the following cutoff-independent conditions: * a=-23.7 fm;* r_0=2.7 fm;* k_0=340.4 MeV.Likewise, at NLO we substitute the lowest Nijm93 phase shift of our earlier fit with the physical scattering length: * a=-23.7 fm;* δ^[0+1](40.5 MeV)=64.5^∘;* δ^[0+1](237.4 MeV)=21.7^∘;* δ^[0+1](340.4 MeV)=0^∘.As before we vary the cutoff from 600 MeV to 2 GeV, but the Λ convergence of the phase shifts is so quick that the cutoff bands cannot be resolved in our plot. The improved description of the very low-energy region at LO compared to that seen in Fig. <ref>, which is entirely due to the resummation of the finite scattering length, is evident. The predicted LO+NLO phase shifts virtually lie on the the Nijm93 curve, and this fit is even more phenomenologically successful than the original LO+NLO shown in Fig. <ref>. The relatively small improvement over the alternative LO curve is consequence of the resummation of higher-order contributions into LO. The small difference between alternative and original LO+NLO curves attests to the fine-tuning of the ^1S_0 channel, i.e. to the smallness of 1/a effects.Given the importance of OPE, one expects potentially large changes in theposition of the poles of T^[0] in χEFT with respect to the πEFT result (<ref>). Yet, the virtual state near threshold (at k≃ i/a) is guaranteed by construction, since m_N/4πT^[0](k;Λ) k→0≃(1/a+ik)^-1.Using the technique described in Ref. <cit.>, one may obtain numerically the positions of the other two poles. The redundant pole seems to become deeper and deeper when the cutoff Λ is increased. This is consistent with the point of view that the redundant pole accounts in πEFT for the neglected left-hand cut due to OPE. In contrast, the binding energy of the deep bound state oscillates with Λ, but we always find it to be ≳ 200 MeV, which corresponds to a binding momentum ≳ 450 MeV. This is, again, an estimate for the breakdown scale M_hi.One might worry that the LO+NLO result shown in Fig. <ref> is so good that higher orders could destroy agreement with the empirical phase shifts and undermine the consistency of our EFT expansion. At N^2LO and N^3LO there are several contributions to account for: TPE and the associated N^2LO counterterms <cit.> in first-order distorted-wave perturbation theory, as well as NLO interactions in second- and third-order distorted-wave perturbation theory. At these higher orders it might be convenient to use the perturbation techniques of Ref. <cit.> or to devise further resummation of NLO interactions.To investigate the potential effects of higher-order corrections we have performed an incomplete N^2LO calculation where the long-range component of the N^2LO TPE potential was included in first-order distorted-wave perturbation theory, following the analogous calculation in Ref. <cit.>. Since the short-range component of this potential can be absorbed in Eq. (<ref>), there are no new short-range parameters and we impose the same four renormalization conditions as in NLO. We have repeated the extraction of the phase shifts and found a negligible effect on the final result, so that this incomplete N^2LO phase shift is at least as good as the one plotted in Fig. <ref>. This indicates that in the ^1S_0 channel the effects of the N^2LO TPE potential can be compensated by a change in the strengths of our LO and NLO short-range interactions. Of course, this is not a full calculation of the amplitude up to N^2LO, but since the change from LO to LO+NLO is small, we might expect the iteration of NLO interactions to also produce small effects. We intend to pursue full higher-order calculations in the future.§ CONCLUSIONS AND OUTLOOKDespite its simplicity from the computational perspective, the two-nucleon ^1S_0 channel has proven remarkably resistant to a systematic expansion. In this work we have developed a rearrangement of Chiral EFT in this channel based on specific assumptions about the scaling of effective-range parameters and the amplitude zero with a single low-energy scale M_lo∼ 100 MeV. Through the introduction of two dibaryon fields, we were able to reproduce empirical phase shifts very well already at NLO — that is, including interactions of up to relative O(M_lo/M_hi) — from threshold to beyond the zero of the amplitude at k_0≃ 340 MeV. The existence of a deep bound state at LO indicates that the expansion in powers of M_lo/M_hi breaks down at a scale M_hi∼ 500 MeV.The new power counting is particularly transparent when pions are decoupled by an artificial decrease of their interaction strength, in which case a version of Pionless EFT is produced. Even in this case LO and NLO fits to empirical phase shifts look reasonable, although the lack of pion exchange is noticeable in the form of the energy dependence.The apparent convergence of our LO and NLO results towards the empirical phase shifts suggests that our PC might be the basis for a new chiral expansion in this channel. Our new expansion relies only on the identification of the NN amplitude zero as a low-energy scale. The ^1S_0 is unique in having such a zero and a low-energy S-matrix pole — in the ^3S_1 channel, the amplitude zero lies beyond the pion-production threshold, while the ^3P_0 phase shift crosses zero at a lower energy but displays no low-energy pole. Moreover, both ^3S_1 and ^3P_0 channels are well described alreadyat LO in a power counting consistent with RG invariance <cit.>.Before a claim of convergence in the ^1S_0 channel can be made, however, one or two higher orders should be calculated, where additional long-range interactions appear in the form of multi-pion exchange. Indications already exist <cit.> that two-pion exchange and its counterterms, which enter first at N^2LO, are amenable to perturbation theory in this channel. However, it is yet to be checked whether their contributions are small enough not to destroy the excellent agreement obtained at NLO. This calculation is demanding because it requires treating the NLO interaction beyond first order in distorted-wave perturbation theory. An incomplete N^2LO calculation which omits these demanding terms suggests that higher orders might provide only very small corrections.If this approach succeeds, then it raises new questions. As one example, can we find an equivalent momentum-dependent approach, which would be better suited to many-body calculations? As another, what is the role of the quark masses in this power counting? We have worked at physical pion mass, but it remains to be seen how this new proposal can be implemented for arbitrary m_π in a renormalization-consistent manner. We intend to address these issues in future work. § ACKNOWLEDGMENTS We thank J.A. Oller and M. Pavón Valderrama for useful discussions. UvK is grateful to R. Higa, G. Rupak, and A. Vaghani for insightful comments on the role of low-energy amplitude zeros, which have inspired this manuscript. 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"C. -J. Yang",
"Bingwei Long",
"U. van Kolck"
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"published": "20170427120510",
"title": "The Two-Nucleon 1S0 Amplitude Zero in Chiral Effective Field Theory"
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In this paper we study a semilinear problem for the fractional laplacian that are the counterpart of the Neumann problems in the classical setting. We show uniqueness of minimal energy solutions for small domains.LHC as an Axion Factory: Probing an Axion Explanation for (𝐠-2)_μ with Exotic Higgs Decays Andrea Thamm^b December 30, 2023 ============================================================================================§ INTRODUCTIONIn recent years there has been an increasing amount of attention to problems involving nonlocal diffusion operators. These problems are so vast that is impossible to give a comprehensive list of references, just to cite a few, see<cit.> for some physical models, <cit.> for some applications in finances, <cit.> for appications in fluid dynamics, <cit.> for application in ecology and <cit.> for some applications in image processing.Among these applications, one operator that is of particular importance is the fractional laplacian that is defined (up to some normalization constant) as(-Δ)^s u(x) = p.v.∫_Ωu(x)-u(y)/|x-y|^n+2sdy,where p.v. stands for in principal value and Ω⊂^n is a bounded domain.This operator is classical and have been studied by several authors. See for instance <cit.>, etc.Recall that this operator is to the so called regional fractional laplacian that corresponds to Levy processes where the long jumps are restricted to be inside Ω. See, for instance, <cit.> for a discussion on this.This operator is commonly used as a fractional version of the Neumann laplacian. In the literature, there exists several forms of fractional version of the Neumann laplacian. We refer the interested reader again to the article <cit.> for more on this. In this paper, we address the following semilinear problem associated to (-Δ)^s(-Δ)^s u + u = λ |u|^q-2u,in Ω.This is the fractional counterpart of the classical Neumann problem-Δ u + u = λ |u|^q-2uin Ω ∂ u/∂ν = 0on ∂Ω.Here, q is a subcritical exponent in the sense of the Sobolev embeddings. That is,1≤ q< 2^*_s := 2n/n-2s. The problem can be separated into three different types of behavior: sublinear (1≤ q<2); linear (q=2) and superlinear (q>2).The linear case is by now well understood as an eigenvalue problem and will not be considered here.For the sublinear and superlinear cases, the parameter λ is superfluous since if one gets a solution for some particular value of λ, then taking a suitable multiple of the solution the value of λ can be taken to be 1.It is fairly easy to see that any solution to (<ref>) is a critical point of the functional(u) := 1/2 [u]_s;Ω^2 + u_2;Ω^2/u_q;Ω^2,where [u]_s;Ω := (∬_Ω×Ω(u(x)-u(y))^2/|x-y|^n+2sdxdy)^1/2is the so-called Gagliardo seminorm of u and, as usual, u_r;Ω denotes the L^r(Ω)-norm.By standard variational methods, one can see that there exists minimal energy solutions to (<ref>). That is, functions u∈ H^s(Ω) such that(u) = inf_v∈ H^s(Ω)(v).Moreover, by a direct application of the Ljusternik – Schnirelmann method, one can construct a sequence λ_k↑∞ of critical energy levels and a sequence of critical points {u_k}_k∈ ofassociated to {λ_k}_k∈.Therefore, there exist infinitely many solutions to problem (<ref>).In this paper, we focus on minimal energy solutions to (<ref>). In particular to the multiplicity problem of such solutions.To this end, inspired by the results of <cit.>, given a domain Ω we consider the family of contracted domainsΩ_ := ·Ω = { x x∈Ω}as ↓ 0and look for the asymptotic behavior of minimal energy solutions in Ω_ as ↓ 0.We first show that the asymptotic behavior of every minimal energy solution is the same and, using this asymptotic behavior, we are able to conclude the uniqueness of minimal energy solution for contracted domains.Finally, we give an estimate on the contraction parameter in order to obtain the uniqueness result. To end this introduction, we want to remark that the same ideas can be used to deal with the Neumann problem (<ref>). The changes needed are easy and are left to the interested reader.We want to recall that uniqueness results for problem (<ref>) has been considered in the literature before. See for instance <cit.>.This result is a first step in an investigation which might be pursued in various directions: for instance, taking different Neumann fractional operators, as the one considered in <cit.>, or show if the known uniqueness results for the classical Neumann laplacian hold in this situation. We leave these questions for further investigation. § PRELIMINARIES.Let Ω⊂^n be a bounded smooth domain and 0<s<1. The fractional order Sobolev space is defined asH^s(Ω) := {u∈ L^2(Ω)u(x)-u(y)/|x-y|^n/2+s∈ L^2(Ω×Ω)}.This space is endowed with the normu_s; Ω := ([u]_s;Ω^2 + u_2;Ω^2)^1/2. It is well known (see, for instance, <cit.>) that there exists a critical exponent2^*_s := 2n/n-2s if2s<n, ∞ otherwisesuch that for any 1≤ q<2^*_s the embedding H^s(Ω)⊂ L^q(Ω) is compact.We define the Sobolev constant as the numberS(Ω) = S_s,q(Ω) := inf_u∈ H^s(Ω)u_s;Ω^2/u_q;Ω^2. It is easy to see, as a consequence of the compactness of the embedding, that there exists an extremal for S(Ω). That is a function u∈ H^s(Ω) where the above infimum is attained.Also, any extremal for S(Ω) is a minimal energy (weak) solution to (<ref>).The constant λ in (<ref>) depends on the normalization of the extremal. For instance, if the extremal u is normalized as u_q;Ω = 1 then λ = S(Ω).Recall that the operator (-Δ)^s is a bounded operator between the Sobolev space H^s(Ω) and its dual (H^s(Ω))' and can be computed by⟨ (-Δ)^s u, v⟩ = 1/2∬_Ω×Ω(u(x)-u(y))(v(x)-v(y))/|x-y|^n+2sdxdy. Therefore, for a solution to (<ref>) we mean a function u∈ H^s(Ω) such that1/2∬_Ω×Ω(u(x)-u(y))(v(x)-v(y))/|x-y|^n+2sdxdy + ∫_Ω uvdx = λ∫_Ω |u|^q-2uvdx,for every v∈ H^s(Ω).§ ASYMPTOTIC BEHAVIOR IN THIN DOMAINS Throughout this section we fix the exponent q∈ [1,2^*_s), q≠ 2.Our objective in this section is to study the asymptotic behavior of the constant S(Ω_) as ↓ 0, where the contracted domains Ω_ are given by (<ref>). We begin with a simple estimate.Under the above notations, we have thatS(Ω_) ≤ |Ω_|^1-2/q = ^n(1-2/q) |Ω|^1-2/q. The lemma follows just by taking u=1 as a test function in the definition of S(Ω_). Now we want to be more precise. We need to show that the asymptotic behavior of S(Ω_) is given precisely by ^n(1-2/q) and also to study the behavior of the associated extremals.Under the above notations, we have thatlim_↓ 0S(Ω_)/^n(1-2/q) = |Ω|^1-2/q.Moreover if u_∈ H^s(Ω_) is an extremal for S(Ω_), the rescaled extremals u̅_(x) := u_( x) normalized such that u̅__q;Ω = 1 verify thatu̅_→ |Ω|^-1/qstrongly inH^s(Ω). First, observe that for v∈ H^s(Ω_), if we denote v̅(x) = v( x), then v̅∈ H^s(Ω). Moreover, [v]_s; Ω_ = ^n/2-s [v̅]_s;Ω and v_r;Ω_ = ^n/rv̅_r;Ω for 1≤ r<2^*_s. Thereforev_s;Ω_^2/v_q;Ω_^2 = ^n(1-2/q)^-2s[v̅]_s;Ω^2 + v̅_2;Ω^2/v̅_q;Ω^2. Now, let u_∈ H^s(Ω_) be an extremal for S(Ω_) and let u̅_(x) = u_( x). Then,S(Ω_) =^n(1-2/q)^-2s[u̅_]_s;Ω^2 + u̅__2;Ω^2/u̅__q;Ω^2. Now, by Lemma <ref>, it follows that^-2s[u̅_]_s;Ω^2 + u̅__2;Ω^2/u̅__q;Ω^2≤ |Ω|^1-2/q.Let us now fix the normalization of the extremal u_ such that u̅__q;Ω=1, and by (<ref>), we obtain that u̅_ is bounded in H^s(Ω) uniformly on >0. So, there exists u̅∈ H^s(Ω) such that (up to some sequence _k→ 0),u̅_⇀u̅weakly inH^s(Ω),u̅_→u̅strongly inL^r(Ω)for any1≤ r < 2^*_s.Also, from (<ref>) and (<ref>), we have that[u̅]_s;Ω≤lim inf_↓ 0[u̅_]_s;Ω = 0,therefore u̅ is constant and, since u̅__q;Ω=1, from (<ref>) we obtain that u̅_q;Ω=1.All these together imply that u̅ = |Ω|^-1/q.From these estimates, one easily concludes that|Ω|^1-2/q ≤lim inf_↓ 0^-2s[u̅_]_s;Ω^2 + u̅__2;Ω^2 = lim inf_↓ 0S(Ω_)/^n(1-2/q)≤lim sup_↓ 0S(Ω_)/^n(1-2/q)≤ |Ω|^1-2/q.The proof is complete.§ UNIQUENESS OF EXTREMALS FOR SMALL DOMAINS In this section we show the uniqueness of extremals if the domain is contracted enough.For that purpose, observe that if u_ is an extremal for S(Ω_) and u̅_ is the rescaled extremal normalized as u̅__q;Ω=1, then u̅_ is a weak solution of the problem(-Δ)^s u + ^2s u = ^2sλ_ |u|^q-2u in Ω,where λ_ = S(Ω_) ^-n(1-2/q). Recall also, that 0≤λ_≤ |Ω|^1-2/q (c.f. Lemma <ref>).So, we define the space:= {u∈ H^s(Ω)u_q;Ω = 1}.It is easy to see thatis a C^1 manifold.We then define F× [0,1)→ (H^s(Ω))' as⟨ F(u,), v⟩ :=1/2∬_Ω×Ω(u(x)-u(y))(v(x)-v(y))/|x-y|^n+2sdxdy + ^2s∫_Ω uvdx - ^2sλ_∫_Ω |u|^q-2uvdx. Denote u_0 = |Ω|^-1/q∈ and observe that F(u_0,0)=0.Following the ideas of <cit.> we want to use the Implicit Function Theorem (IFT) to show the existence of a small number δ>0 and a curve ϕ [0,δ)→ such thatϕ(0)=u_0and F(ϕ(),) = 0for every0≤<δ,and if (u,)∈× [0,δ) is such that F(u,)=0 and u is close to u_0 then u=ϕ().Observe that if we can apply the IFT then, combining this with Lemma <ref>, automatically we obtain the uniqueness of extremals of S(Ω_) forsmall.In order to be able to apply the IFT we need to check that d_u F|_(u_0,0) is invertible (see <cit.> or <cit.>). Recall that since F is define on a manifold, the derivative is defined on the tangent space ofat the point u_0.Let us begin with a couple of lemmas.The tangent space ofat u_0, that we denote by T_u_0 is given byT_u_0 = {v∈ H^s(Ω)∫_Ω vdx = 0}. Let v∈ T_u_0. Then, there exists a differentiable curve, α (-1,1)→ such that α(0)=u_0 and α̇(0)=v.But, since α(t)∈ for every t∈ (-1,1) it follows that∫_Ω |α(t)|^qdx = 1 for everyt∈ (-1,1). Differentiating both sides of the equality gives∫_Ω q |α(t)|^q-2α(t)α̇(t)dx = 0.So, if we evaluate at t=0 and recall that u_0 is constant, we obtain that ∫_Ω vdx = 0. On the other hand, if v∈ H^s(Ω) verifies (<ref>), we construct the curve α (-1,1)→ asα(t) = u_0 + tv/u_0+tv_q.Straightforward computations show that α(0)=u_0 and α̇(0)=v. Now, we denote = (span{1})^⊥ = {f∈ (H^s(Ω))'⟨ f, 1⟩ = 0}. We have that d_u F|_(u_0,0) T_u_0→.Moreover, the following expression holds⟨ d_u F|_(u_0,0)(u), v⟩ = 1/2∬_Ω×Ω(u(x)-u(y))(v(x)-v(y))/|x-y|^n+2sdxdy. To prove the Lemma, first observe that⟨ F(u,0), v⟩ =1/2∬_Ω×Ω(u(x)-u(y))(v(x)-v(y))/|x-y|^n+2sdxdy,for every u∈.From this expression the lemma follows. It remains to see that d_u F|_(u_0,0) has a continuous inverse.The derivative d_u F|_(u_0,0) T_u_0→ has a continuous inverse.First observe that T_u_0 is a Hilbert space with inner product given by(u,v) = ⟨ d_u F|_(u_0,0)(u), v⟩ =1/2∬_Ω×Ω(u(x)-u(y))(v(x)-v(y))/|x-y|^n+2sdxdy.So, the Lemma follows from the Riesz representation Theorem. Combining Lemmas <ref>, <ref>, <ref> we are in position to apply the IFT and conclude the main result of the sectionGiven Ω⊂^n smooth and of finite measure and 1< q <2^*_s, there exists δ>0 such that S(Ω_) has a unique extremal for 0<<δ.At this point the proof is a direct consequence of the IFT and the remarks made at the beginning of the section. § ESTIMATES FOR THE CONTRACTION PARAMETER Let us first define _0 = sup{δ>0∃ !normalized extremal forS(Ω_)∀<δ}.From the results of the previous section, we know that _0>0. We now want to find a lower bound for _0. There exists u_0∈ H^s(Ω__0) an extremal for S(Ω__0), such that the rescaled function u̅_0(x) = u_0(_0 x) normalized as u̅_0_q;Ω=1 verifies that d_u F|_(u̅_0,_0) is not invertible.Assume the oposite. We first claim that there is a unique extremal for S(Ω__0). Otherwise, if u_0≠ u_1 are extremals such that the rescaled functions u̅_i(x) = u_i(_0 x), i=0,1 normalized as u̅_i_q;Ω=1, i=0,1, verify that d_u F|_(u̅_i,_0) is invertible for i=0,1. But then, by the IFT, there exists δ>0 and two curves ϕ_i (_0-δ, _0+δ)→ such thatF(ϕ_i(),)=0,for every ∈ (_0-δ, _0+δ), i=0,1.But this contradicts the uniqueness of extremals for <_0.Now, let _k>_0 be such that _k→_0 as k→∞ and let u_k be an extremal for S(Ω__k). We normalized these extremals so that the rescaled functions u̅_k(x) = u_k(_k x) verify u̅_k_q; Ω=1.We want to see that {u̅_k}_k∈ converges to u̅_0 that is the rescaled function of the unique extremal u_0 for S(Ω__0).In fact, it is immediate to see that sup_k∈u̅_k_s;Ω<∞, and so, up to a subsequence, there exists w̅∈ H^s(Ω) such thatu̅_k ⇀w̅weakly inH^s(Ω)u̅_k →w̅strongly inL^r(Ω),for every1≤ r<2^*_s.From these convergence results it follows that1=u̅_k_q;Ω→w̅_q;Ωand_0^-2s [w̅]_s,2;Ω^2 + w̅_2;Ω^2 ≤lim inf_k→∞_k^-2s [u̅_k]_s,2;Ω^2 + u̅_k_2;Ω^2 =lim inf_k→∞S(Ω__k)/λ__k. Now, let u_0∈ H^s(Ω__0) be the unique extremal for S(Ω__0) normalized such that the rescaled function u̅_0 satisfies u̅_0_q;Ω=1.Then,lim sup_k→∞S(Ω__k)/λ__k≤lim sup_k→∞_k^-2s [u̅_0]_s,2;Ω^2 + u̅_0_2;Ω^2 = _0^-2s [u̅_0]_s,2;Ω^2 + u̅_0_2;Ω^2 =S(Ω__0)/λ__0 These two inequalities combined imply that w(x) = w̅(_0^-1x) is an extremal for S(Ω__0) and so w=u_0.Now, since we are assuming that d_u F|_(u_0,_0) is invertible, we can apply the IFT as in the proof of Theorem <ref> to conclude that for some δ>0 there is a unique extremal for S(Ω_) for <_0+δ. But this contradicts the definition of _0.By a simple application of the Fredholm's alternative, it follows that d_u F|_(u_0,_0) is not invertible if and only if it has a nontrivial kernel. The following Poincaré-type inequality plays an important role in the bound of _0Let Ω⊂^n be an open, smooth and of finite measure. Let 0<s<1 and 1≤ q<2^*_s. Then, there exists c>0, that depends on q,s and Ω, such thatc w_q;Ω^2 ≤1/2 [w]_s;Ω^2,for every w∈ H^s(Ω) such that ∫_Ω wdx = 0.The proof follows by a standard compactness argument and is omitted. We are ready to prove the main result of the section.Under the notations and assumptions of the section, we have that_0≥(c/(q-1) |Ω|^1-2/q)^1/2s,where c>0 is the constant in the Poincaré-type inequality of Lemma <ref>.By Lemma <ref>, there exists u_0 an extremal for S(Ω__0) such that the rescaled function u̅_0 normalized such that u̅_0_q;Ω=1 verifies that d_u F|_(u̅_0,_0) is not invertible.Moreover, by Remark <ref>, d_u F|_(u̅_0,_0) has a nontrivial kernel.Let 0≠ z∈(d_u F|_(u̅_0,_0)), then z is a nontrivial weak solution to the problem(-Δ)^s z + _0^2s z = _0^2sλ__0 (q-1) |u̅_0|^q-2zin Ω ∫_Ω zdx = 0. Using z as a test function in the weak formulation of (<ref>) gives1/2 [z]_s,2;Ω^2 + _0^2sz_2;Ω^2 = _0^2sλ__0 (q-1)∫_Ω |u̅_0|^q-2 z^2dx. Now, we use the Poincaré-type inequality of Lemma <ref>, Lemma <ref> and Hölder's inequality to deduce from (<ref>) thatcz_q;Ω^2 ≤_0^2s |Ω|^1-2/q (q-1) u̅_0_q;Ω^q-2z_q;Ω^2.Finally, recalling that u̅_0_q;Ω=1 and that z≠ 0 we arrive that the desired result.§ ACKNOWLEDGEMENTSThis paper was partially supported by grants UBACyT 20020130100283BA, CONICET PIP 11220150100032CO and ANPCyT PICT 2012-0153. J. Fernández Bonder and A. Silva are members of CONICET. This work started while JFB was visiting the National University of San Luis. He wants to thank the Math Department and the IMASL for the hospitality that make the visit so enjoyable.amsplain | http://arxiv.org/abs/1704.08203v2 | {
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"Juan Spedaletti"
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"title": "Uniqueness of minimal energy solutions for a semilinear problem involving the fractional laplacian"
} |
XXVIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions(Quark Matter 2017)^1 Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA ^2 Physik Department, Technische Universität München, D-85748 Garching, Germany We report on the lattice calculations of the heavy quark potential at T>0 in 2+1 flavor QCD at physical quark masses using the Highly Improved Staggered Quark discretization.We study in detail the systematic effects in the determination of the real and imaginary parts of the potential when using the moment method. § INTRODUCTION The problem of quarkonium properties at high temperatures received a lot of theoretical attention since the famous paper by Matsui and Satz <cit.> (see e.g. Ref. <cit.> for a recent review). A lattice calculation of the quarkonium spectral functions would solve the problem. However, the reconstruction of meson spectral functions from a discrete set of data points in Euclidean time proved to be an extremely challenging task (see e.g. Refs. <cit.>.) Another method of obtaining quarkonium spectral functions relies on effective field theory approaches. One effective field theory approach is called NRQCD and studies based on this approach have been reported in Refs. <cit.>. Another effective field theory approach for calculating quarkonium spectral functions is pNRQCD<cit.>. The basic ingredients of the latter approach are the heavy quark anti-quark potentials. In the weakly-coupled regime these potentials can be calculated perturbatively <cit.>, while in the strong-coupling regime a lattice calculation is needed <cit.>. Attempts to calculate the heavy quark anti-quark potential on the lattice using a Bayesian approach have been presented <cit.>. A different solution to the problem, which isbased on using a simple Ansatz for the spectral function and fitting the lattice data, has been presented in Ref. <cit.>. In this contribution we will follow on the latter approach extending it using the method of moments of correlation functions.§ MOMENTS METHOD AND THE NUMERICAL RESULTS To obtain the potential on the lattice we study the correlation function of an infinitely heavy (static) quark anti-quark (Q Q̅) pair in Euclidean time τ. The quark and anti-quark are separated by thedistance r and we use Coulomb gauge to define the correlation functions. We also considered HYP smeared Wilson loops and found very similar results. The QQ̅ correlation functions have been calculated on 48^3 × 12 lattice generated by HotQCD collaboration <cit.>. In our study we used the lattice gauge couplings β=10/g^2=7.03, 7.28, 7.373, 7.596 and 7.825 corresponding to temperatures T=199, 251, 273, 333 and 407 MeV, respectively. The correlation function has the following spectral representation for τ < 1/TW_r(τ,T)=∫_-∞^∞ d ωσ_r(ω,T) e^-ωτThe real part of the potential corresponds to the lowest peak position in σ_r(ω,T), while the imaginary part of the potential corresponds to the width of the peak. The moments of the correlation functions are defined asm_1(τ,r,T)=- ∂_τ W_r(τ,T)/W_r(τ,T), m_n=-∂_τ m_n-1(τ,r,T), n>1.The first moment is just the so-called effective mass used in lattice calculations at zero temperature. For large τ it approaches the energy of the ground state. In some limiting cases the above moments are related to themoments of the spectral function around the peak. For example, in the case of a Gaussian spectral function we have m_1(τ)=∫_-∞^∞ d ωσ(ω,T) e^-ωτ/∫_-∞^∞ d ωσ(ω,T)e^-ωτ = ⟨ω⟩≃ Re V-( Im V)^2 τ,which for small Im V or τ is just the real part of the potential. Similarly we find that m_2 ≃ ( Im V)^2. As we will see below the moments m_n provide an efficient way to summarize the information contained in the static Q Q̅ correlator.On the lattice the moments are defined asm_1(τ-a s/2,r,T)=-1/sln(W_r(τ-a s,T)/W_r(τ,T))m_n(τ-a s/2,r,T)=1/s(m_n-1(τ,r,T)-m_n-1(τ-s/a,r,T)), n=2,3…, s=1,2 …In our investigations we use a Lorentzian form of the spectral function:σ_r(ω,T)=1/πΓ/(ω-μ)^2+Γ^2Θ(ω-η), μ= Re V(r,T), Γ = Im V(r,T)The Lorentzian form alone is not expected to describe the spectral function accurately away from the peak. For this reason we included a regulator parameter η that cuts off the low ω tail of the Lorentzian. For large ω no regulator is needed as the exponential kernel in the Laplace transform cuts off the tail of the Lorentzian. To check the viability of this simple form we use the spectral functions calculated in HTL perturbation theory <cit.> and fit the corresponding Euclidean correlation function with the above Lorentzian form. The results of these fits are shown in Fig. <ref>. Except for very small τ the fits can describe the HTL perturbative results well, even-thoughthe HTL spectral functions are not of Lorentzian shape <cit.>.The input values of the peak position (μ/T=4.72) and width (Γ/T=0.122) are reproduced well. Thus, we conclude that a modified Lorentzian form can be used forparametrizing the spectral functions in our study.Our numerical results for the first and second moments are shown in Fig. <ref> for T=407 MeV and rT=0.85. The results for other temperature and r T values are similar. In the figure we show the zero temperature results for m_1 as well. At zero temperature m_1 approaches a plateau for τ T>0.6, but this is not the case at finite temperature. The decrease of m_1 at finite temperature at large τ is qualitatively similar to the one observed in HTL perturbation theory and thus could be associated with the imaginary part of the potential. Since at small τ the high ω part of the spectral function, which corresponds to the excited statesis important we need to isolate the corresponding contribution in the correlator. It is customary to parametrize the τ dependence of the correlator in terms of two exponentials. The first exponential corresponds to the ground state, while the second exponential corresponds to the higher excited states and can parametrize the high ω part of the spectral function. We performed a double exponential fit of the zero temperature correlator and subtracted thecontribution due to the second exponent. The value of m_1 calculated from the subtracted correlator is also shown in Fig. <ref>. As one can see from the figure the zero temperature m_1 from the subtracted correlator shows a plateau for all τ values. Since at very large ω we do not expect the spectral function to be temperature dependent we perform the same subtraction for the finite temperature correlator, which makes the value of m_1 smaller for small τ but has no effect on its large τ behavior.The subtracted correlator then is fitted with the Lorentizan form. We performed different fits, varying the fit interval in τ and using different values of the η parameter. The fits work very well and the values of μ and Γ are not very sensitive to the choice of the fit interval and the value of η. The results of the fits are also shown in Fig. <ref>.The horizontal lines in Fig. <ref> (left) show the results for μ. In addition we calculated m_2 from the subtracted correlator and the results are shown in Fig. <ref> (right). At small τ the second moment decreases almost by a factor of three as the result of subtraction. Our fits describe the subtracted m_2 well even-though the corresponding data do not enter the fits.Performing the above fits for all values of rT we get the real and imaginary parts of the potential as functions of rT. The results for T=407 MeV are shown in Fig. <ref>. We compare the real part of the potential with the corresponding zero temperature result as well as with the singlet free energy defined as F_S(r,T)=-T ln W_r(τ=1/T,T). We see that at short distances the potential agrees with the singlet free energy, while at larger distances it is significantly above the singlet free energy. While the potential at finite temperature is always smaller than the zero temperature result we do no see any indications for the expected screened behavior at large distances. The imaginary part of the potential on the other hand behaves as expected: it is very small at small rT and seems to saturate for r T ≃ 1. However, the imaginary part is larger than in HTL perturbation theory. § CONCLUSIONSWe discussed a lattice calculation of the static QQ̅ potential using the moments of the correlation function. We used a Lorentzian form of the spectral function and fitted the lattice results on the moments to obtain the real and imaginary parts of the potential. The contribution from the high energy part of the spectral function has been subtracted using the fits of zero temperature correlators. We find that our fit procedure works very well but the real part of the potential is much larger than the singlet free energy and does not show a screening behavior. We think this is due to the fact that the subtraction of the high energy contribution from the correlator is too simplistic. In order to make our strategy for calculating the potential at non-zero temperature viable a better modeling of the high energy part of the spectral function will be needed. The work on this is in progress.Acknowledgmentsnoindent This work was supported by U.S. Department of Energy under Contract No. DE-SC0012704.We acknowledge the support by the DFG Cluster of Excellence “Origin and Structure of the Universe”. The calculation have been carried out on the computing facilities of the Computational Center for Particle and Astrophysics (C2PAP). elsarticle-num | http://arxiv.org/abs/1704.08573v1 | {
"authors": [
"P. Petreczky",
"J. Weber"
],
"categories": [
"hep-lat",
"hep-ph"
],
"primary_category": "hep-lat",
"published": "20170427135216",
"title": "Lattice Calculations of Heavy Quark Potential at Finite Temperature"
} |
theoremTheorem[section] *theorem*Theorem thm[theorem]Theorem proposition[theorem]Proposition *prop*Proposition *lemma*Lemma prop[theorem]Proposition corollary[theorem]Corollary cor[theorem]Corollary lemma[theorem]Lemma claim[theorem]Claim lem[theorem]Lemma conjectureConjecture def-theorem[theorem]Theorem-Definition *statement*Statementnotation[theorem]NotationProof:Proof of Claim: definition remark[theorem]Remark *thm*Theorem questionQuestion comment[theorem] definition[theorem]Definition defn[theorem]Definition innercustomthmTheorem Strong density of definable types and closed ordered differential fields]Strong density of definable types and closed ordered differential fieldsQ. Brouette]Quentin Brouette -Quentin BrouetteDépartement de Mathématique (Le Pentagone)Université de Mons20 place du ParcB-7000 MonsBelgium, Belgium. [email protected]. Cubides Kovascics]Pablo Cubides Kovacsics^∗ -Pablo Cubides Kovacsics Laboratoire de mathématiques Nicolas OresmeUniversité de Caen CNRS UMR 6139Université de Caen BP 518614032 Caen cedex, [email protected] ^∗Supported by the ERC project TOSSIBERG (Grant Agreement 637027)F. Point]Françoise Point^†-Françoise PointDépartement de Mathématique (Le Pentagone)Université de Mons20 place du ParcB-7000 MonsBelgium, Belgium. [email protected] ^† Research Director at the FRS-FNRS,this material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017.[ [===== The following strong form of density of definable types is introduced for theories T admitting a fibered dimension function d: given a model M of T and a definable set X⊆ M^n, there is a definable type p in X, definable over a code for X and of the same d-dimension as X. Both o-minimal theories and the theory of closed ordered differential fields (CODF) are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.[2000 Mathematics Subject Classification. 03C64, 12H05, 12L12, 03C45. Key words and phrases. Ordered differential fields, density of definable types, closed ordered differential fields.] § INTRODUCTION After Hrushovski's abstract criterion of elimination of imaginaries was introduced in <cit.>, density properties of definable types have drawn more and more attention (see <cit.>, <cit.> and <cit.>). Given a complete theory T and a model M of T, the property which often links imaginaries and definable types is the following: for every definable set X⊆ M^n, there is a definable type p∈ S_n^T(M) in X (i.e.,p contains a formula defining X) which is moreover definable over acl^eq(e) for e a code for X. Here, a code for X is an element of M^eq which is fixed by all automorphisms (of a sufficiently saturated extension) fixing X setwise. We say in this case that definable types are dense over a-codes, and if the type p can be taken to be definable over a code of X without passing to the algebraic closure, we say that definable types are dense over codes.For theories T admitting a fibered dimension function d, we strengthen the above density properties as follows: for every definable set X⊆ M^n, there is a definable type p in X which is definable over a code of X (resp. over the algebraic closure of a code of X) and has the same d-dimension of X. We say in this case that definable types are d-dense over codes (resp. over a-codes). Very much inspired by Johnson's presentation of Hrushovski's abstract criterion for elimination of imaginaries <cit.>, we obtain the following Proposition (later Proposition <ref>): Let T be a complete theory. If definable 1-types are dense over codes (resp. over a-codes) then definable n-types are dense over codes (resp. a-codes). Moreover, assuming T admits a code-definable fibered dimension function d, if 1-types are d-dense over codes (resp. a-codes) then so are all definable n-types. The first part of the previous proposition is contained in <cit.>. The result for d-density is new.We obtain as a corollary that in any o-minimal theory, definable types are -dense over codes (Proposition <ref>), wheredenotes the usual topological dimension. We use this for the particular case of real closed fields in order to show that definable types are dense over codes for closed ordered differential fields (CODF), the main motivation of this paper. A Let K be a model of CODF, X⊆ K^n be a non-empty definable set and C be a code for X. Then there is a C-definable type p∈ S_n^CODF(K) in X.We apply then Theorem A to give a new proof of elimination of imaginaries in CODF (see Theorem <ref>), very much inspired by the recent proofs given for algebraically closed valued fields by Hrushovski <cit.> and further simplified by Johnson <cit.>. Finally, using that models of CODF can be endowed with a fibered dimension function δ-, we strengthen Theorem <ref> by showing that definable types are δ--dense over codes for CODF.B Let K be a model of CODF, X⊆ K^n be a non-empty definable set and C be a code for X. Then there is a C-definable type p∈ S_n^CODF(K) in X such that δ-(p)=δ-(X).The reason why we state Theorems <ref> and <ref> separately is simply to stress that the new proof of elimination of imaginaries for CODF does not use the δ-dimension (and the stronger density property of definable types). Motivated (in part) by the fact that in CODF, isolated types are not dense, the first author had shown in his thesis that definable types are dense (see <cit.> and <cit.>). A key difference with the present work is that we carefully take into account the parameters over which types are defined. As already pointed out, this is necessary if, for instance, one is aiming to obtain elimination of imaginaries.Even though CODF is NIP, it is not strongly dependent (hence not dp-minimal) <cit.>. We would like to point out a connection of the above proposition (Proposition <ref>) with a result of Simon and Starchenko on definable types in dp-minimal theories <cit.>. Let us briefly state their result. Let T be a dp-minimal theory, M be a model of T and A be a subset of M. Simon and Starchenko show that, under the hypothesis that every unary A-definable set X contains an A-definable type p∈ S_1^T(A), every non-forking formula φ(x) (in possibly many variables) can be extended to a definable type over a model of T. Notice that assuming elimination of imaginaries, their hypothesis is equivalent to the assumption of the first part of Proposition <ref>. Nevertheless, their result states the existence of a definable type extending a non-forking formula, without specifying the parameters over which such a type is definable. It is a natural question to ask whether a direct analogue of Proposition <ref> holds for dp-minimal theories. Finally, we expect the strategy followed in this paper can be applied to other theories. In particular, we would like to apply similar techniques in future work to other topological structures endowed with a generic derivation, as defined in <cit.>.The article is laid out as follows. In Section <ref>, we set the background and fix some terminology.In particular, we introduce what we call a code-definable fibered dimension function. Section 2 is divided in two parts. Subsection 2.1 is devoted to the proof of Proposition <ref> and subsection 2.2 to the proof of Theorem <ref>. The alternative proof of elimination of imaginaries for CODF is presented in Section <ref>.In Section <ref>, we first recall the fibered dimension function δ-dim that was defined in models of CODF and show that δ-dim is code-definable. We end by proving that definable types in CODF are δ-dim-dense over codes, that is, Theorem <ref>.§ PRELIMINARIESLetbe a language, T be a complete -theory andbe a monster model of T, namely a model of T,κ-saturated and homogeneous for a sufficiently large cardinal κ.A subset ofis small if it is of cardinality strictly less than κ. Throughout, all substructures and subsets ofunder consideration will be assume to be small. Let A⊆. Given an -formula φ(x), we let ℓ(x) denote the length of the tuple x. By an -definable set, we mean a set defined by an -formula with parameters in . Whenis clear from the context, we also use `definable' instead of `-definable' and `A-definable' to specify that the parameters come from A.We denote by () the set of all -definable sets in . For integers n≥ k≥ 1, we denote by π_k:^n→^k the projection onto the first k coordinates. Given a subset X⊆^n+m and a∈π_n(X), we let X_a denote the fiber of X over a, that is, X_a:={x∈^m: (a,x)∈ X}. By S_n^T(A), we denote the Stone space of complete n-types consistent with T with parameters in A. Let B⊆. Recall that a type p∈ S_n^T(A) is B-definable if for every -formula φ(x,y) with x=(x_1,…, x_n), y=(y_1,…, y_m) and without parameters, there is an -formula ψ(y) with parameters in B such that for all a∈ A^m,φ(x,a)∈ p if and only if ψ(a). That formula ψ(y) is classically denoted by d_p xφ(x,y). §.§ Imaginaries and codes Let X⊆^n be a definable set.A finite tuple e∈^eq is a code for Xif for every σ∈ Aut_(), it holds that σ(X)=X if and only if σ(e)=e.Every non-empty -definable set X has at least one code and any two codes for X are interdefinable.We let c(X) denote some code for X.Let A be a subset ofand p∈ S^T_n(A),a code for p is a (possibly infinite) tuple e in ^eq such that for every σ∈ Aut_(), it holds that σ(p)=p if and only if σ(e)=e.Every definable type p has at least one code, namely, a tuple consisting of the elements{c(d_p xφ(x,)): φ(x,y)an -formula}. Classically (see for instance <cit.> or <cit.>),one sometimes calls canonical parameter what we call here a code for a definable setand canonical base what we call here a code for a type.Notice that any definable set X is c(X)-definable. When T has elimination of imaginaries, each code e is interdefinable with a finite tuple ofand, abusing of notation, we also denote such a finite tuple by c(X).§.§ Definable typesWe gather three well-known properties on definable types whose proofs are left to the reader. Let A⊆ M≺ and a∈^n be such that tp(a/M) is A-definable. If b∈ dcl(A,a), then tp(a,b/M) is also A-definable. Let C⊂ B⊂ and let a,b be two (possibly infinite) tuples of elements in . If tp(a/B) is C-definable and tp(b/Ba) is Ca-definable, then tp(a, b/B) is C-definable. We define ternary relations a[C] B and a[C] B bya[C] B ⇔there exists a model M⊇ BC such that tp(a/M) is C-definable,and a[C] B if and only if a[D] B where D=acl^eq(C) (see also <cit.>). The following is a standard consequence of Lemma <ref> (see <cit.> and <cit.>).Letdenote eitheror . Assume that a[C]B and b[Ca]B, then ab[C]B. §.§ Dimension functionsWe recall the definition of a fibered dimension function from <cit.>. A fibered dimension function onis a function d:()→ℕ∪{-∞} satisfying the following conditions for X, Y∈(): * d(X) = -∞ iff X = ∅, d({a})=0 for each a∈ and d()= 1; * d(X∪ Y)=max{d(X), d(Y)}; * if X⊆^n, then d(X)=d(X^σ) for each permutation σ of {1,…,n}; * if X⊆^n+1 and l∈{0,1}, then the set X(l):={a∈π_n(X): d(X_a)=l} belongs to () and d({(x,y)∈ X : x∈ X(l)})=d(X(l))+l. If in addition, d satisfies the property (Dim 5) below, we will say that d is code-definable. * If X⊆^n+1, the sets X(0) and X(1) are c(X)-definable.Let d be a dimension function on . Given a set C⊆, we extend the function d to the spaces of types S_n^T(C) by setting, for any p∈ S_n^T(C), D d(p):=inf{d(φ(,c)) : φ(x,c) ∈ p}. Note that there always exists a formula φ(x,c)∈ p such that d(p)=d(φ(,c)). The converse is shown in Proposition <ref>. For a tuple a∈^n, we set d(a/C) as d(tp(a/C)).Here are some standard consequences of Definition <ref> whose proofs are left to the reader.Let X and Y be definable sets.* if X⊆ Y, then d(X)⩽ d(Y); * if X is non-empty and X⊆^n+1, then π_n(X)=X(1)∪ X(0);* for any C⊂, a∈^n and b∈, we have that d(a/C)⩽ d(a,b/C);* for any C⊆ D⊂ and a∈^n, we have that d(a/D)⩽ d(a/C). For any C⊂, a∈^n and b∈, we have that d(a,b/C)=d(a/C)+d(b/Ca).Let X⊆^n+1 be a C-definable set such that (a,b)∈ X and d(X)=d(a,b/C).Let Z⊆^n be such that a∈ Z and d(a/C)=d(Z).After possibly replacing X by the subset X':={(x,y)∈ X: x∈ Z},we may further assume that d(π_n(X))=d(a/C), keeping the property that d(X)=d(a,b/C). We split in cases depending on the value d(b/Ca). Case 1: Suppose that d(b/Ca)=0. By possibly restricting X further (adding a formula witnessing that d(b/Ca)=0), we may assume that a∈ X(0).So d(a/C)= d(X(0))=d(π_n(X)). For Y:={(x,y)∈ X: x∈ X(0)}, we have that (a,b)∈ Y and that d(a,b/C)⩽ d(Y)=d(X(0))=d(a/C).Part (3) of Lemma <ref> implies the equality. Case 2: Suppose that d(b/Ca)=1. By the case assumption, a∈ X(1), and since d(a/C)=d(π_n(X)), we have d(a/C)=d(X(1)).Therefore, d(a,b/C)=d(X)=d(X(1))+1=d(a/C)+1. Let X⊂^n be a definable set with parameters in C. Then there exists a tuple a∈^n such that d(a/C)=d(X). We proceed by induction on n and use the fact that the dimension is fibered. Suppose n=1.If d(X)=0, then X≠∅, so take a∈ X. By definition d(tp(a/C))=0.If d(X)= 1, observe that for any formula φ over C, we have X=(φ∧ X)∨ (φ∧ X). By (Dim 2), one of (or both) φ or φ is of dimension 1. Consider the following partial type over C,p:={φ: φ()⊂ X, d(φ())<d(X),φ with parameters in C}. By (Dim 2), p is finitely consistent with X.Let a be a realisation of p, then d(a/C)=d(X). Suppose that d(a/C)=0, then there would exist ψ a formula with parameters in C such that d(ψ)=0 and ψ(a) holds. This implies that d(ψ∧ X)=1, and so ψ∧ X∈ p, a contradiction. Suppose now X⊂^n with n>1 and d(X)=d. We have that d(X)=d(X(i))+i for some i∈{0,1}. By induction hypothesis there is an tuple b∈ X(i) such that d(b/C)=d(X(i)). Since d(X_b)=i, by the case n=1, there is b_0∈ X_b with d(b_0/C, b)=i. Therefore, by Lemma <ref>, d(X)=d(X(i))+i=d(b/C)+d(b_0/C, b)=d(b, b_0/C). §.§ O-minimal theories and dimension Letbe a language containing the order relation < and let T be any o-minimal theory (we will always assume that < is dense). Our main example will be the theory RCF of real-closed fields, which we regard as a theory in the language of ordered rings _or:={<,+,·,0,1}. Letbe a monster model of T. The topological closure of a subset X⊆^n is written X and its interior Int(X).Notice that both X and Int(X) are c(X)-definable. For a definable set X⊆^n, we denote by (X) the dimension of X.It corresponds to the biggest integer k≤ n for which there is a coordinate projection π:^n→^k such that Int(π(X))≠∅. This dimension function onis fibered<cit.>, namely it satisfies the first four properties of Definition <ref>.Moreover since one can express that a subset ofis of dimension 1, the functionsatisfies (Dim 5).Indeed, for a definable set X⊆^n+1, we have thatX(1)={x∈π_n(X): ∃ a∃ b [a<b ∧ (a,b)⊆ X_x]}.So, X(1) is c(X)-definable.Since X(0)=π_n(X)∖ X(1), we have that X(0) is c(X)-definable as well. Let A⊂ and a∈. Since T is o-minimal, tp(a/A) is determined by the quantifier-free order type of a over A.We say that z realizes the type a^+ over A whenever z is a realization of the a-definable type {(a<x<c) :c∈ A and c>a}.Analogously, we say that z realizes the type -∞ over A whenever z is a realization of the ∅-definable type {(x<b) : b∈ A}.Let K be a real closed field and Y⊆ K^n be a semi-algebraic set. Suppose that (Y)<n. Then there is a non-trivial polynomial P∈ K[x] with x=(x_1,…,x_n) such that Y⊆{x∈ K^n: P(x)=0}.Since Y is semi-algebraic, it is a finite union of non-empty sets Y_1∪…∪ Y_m of the form Y_i:={a∈ K^n: ⋀_j=1^n_i Q_ij(a)>0 ∧ P_i(a)=0},with Q_ij, P_i∈ K[x].Suppose that for some i , the polynomial P_i is the zero polynomial. Then Y_i would be open and so (Y_i)=n (and so (Y)=n) a contradiction. So we take P(x)=∏_i=1^ℓ P_i(x), and the set Y is contained in the zero locus of P.§.§ Closed ordered differential fields Hereafter we let _δ:=_or∪{δ} be the language of ordered differential rings, where δ is a unary function symbol. Let _or^+:=_or∪{ ^-1} be the language of ordered fields and _δ^+:=_or^+∪{δ} the language of ordered differential fields. The _δ^+-theory of ordered differential fields is simply the _or^+-theory of ordered fields together with the axioms for a derivation δ and CODF is its model completion <cit.>. Since CODF is the model completion of a universal theory, it admits quantifier elimination in _δ^+ and it is easy to check that it also admits quantifier elimination in _δ. Letbe a monster model of CODF.In particular,is also a monster model of RCF.We equipwith the order topology. Given a,b∈^n and ϵ∈ with ϵ>0, we abbreviate by | a-b|<ϵ the formula ⋀_i=1^n | a_i-b_i| <ϵ.For a subfield F⊆ and a subset A⊆, the smallest real closed field incontaining both F and A is denoted by F(A)^rc. When F is a differential field, the smallest differential subfield ofcontaining both F and A is denoted by F⟨ A⟩. Given a=(a_1,…,a_n)∈^n we let F(a)^rc (resp. F⟨ a⟩) denote F({a_1,…,a_n})^rc (resp. F⟨{a_1,…,a_n}⟩). For n⩾ 0 and a∈, we define δ^n(a):=δ∘⋯∘δ_ntimes(a),with δ^0(a):=a,and δ̅^n(a) as the finite sequence (δ^0(a),δ(a),…,δ^n(a))∈^n+1. We let δ̅(a) denote the infinite sequence (δ^n(a))_n⩾ 0.We will denote by K{x} the differential ring of differential polynomials (in one differential indeterminate x) over K. Abusing of notation, we identify K{x} with the ordinary polynomial ring K[δ^j(x): j∈] in indeterminates δ^j(x),endowed with the natural derivation extending the one of K with the convention that δ^0(x)=x and δ(δ^j(x))=δ^j+1(x).The order of a differential polynomial f(x)∈ K{x}, denoted by (f), is the smallest integer n⩾ 0 such that there is a polynomial F∈ K[δ̅^n(x)] such that f(x)=F(δ̅^n(x)).The (algebraic) polynomial F is unique and will be denoted hereafter by f^*.Below, ∂/∂δ^n(x) denotes the usual derivative of polynomialsin K[δ^j(x): j∈] with respect to the variable δ^n(x).Let us recall the axiomatisation of CODF given in <cit.>. An ordered differential field K is a model of CODF if it is real-closed and given any f, g_1,⋯,g_m∈ K{x}∖{0}, with n:=(f)⩾(g_i) for all 1⩽ i ⩽ m, if there is c∈ K^n+1 such that f^*(c)=0 ∧(∂/∂δ^n(x)f^*)(c)≠ 0 ∧⋀_i=1^m g_i^*(c)>0,then there exists a∈ K such that f(a)=0∧⋀_i^m g_i(a)>0. Equivalently <cit.>, K is a model of CODF if it is real-closed and given f∈ K{x}∖{0} with n:=(f) and ϵ >0, if there exists c∈ K^n+1 such that f^*(c)=0∧(∂/∂δ^n(x)f^*)(c)≠ 0, then there exists a such that f(a)=0∧|δ̅^n(a)-c|<ϵ.Let _δ^-={<,+,-,·,δ} be the reduct of _δ where the constants have been removed. Let φ(x_1,⋯,x_m) be a quantifier-free _δ^--formula with ℓ(x_i)=1 for all i∈{1,…,m}. For each i∈{1,…,m}, let n_i⩾ 0 be the largest integer such that δ^n_i(x_i) is a term of φ. For x_i a tuple of variables with ℓ(x_i)=n_i+1, we write φ^∗ for the _or-formula φ^*(x_1,⋯,x_n) such that the _δ^--formula φ(x_1,⋯,x_m) is φ^*(δ̅^n_1(x_1),⋯,δ̅^n_m(x_m)). We extend this functor to quantifier-free _δ-formulas with parameters (constants treated as such) as follows. Let ψ(x_1,⋯,x_m) be a quantifier-free _δ-formula with parameters c_1,…,c_k in . Then ψ is of the form φ(x_1,⋯,x_m,c_1,…,c_k) with φ(x_1,⋯,x_m,z_1,…,z_k) a quantifier-free _δ^--formula without parameters. Define ψ^* as the _or-formula φ^*(x_1,⋯,x_m,c̅_1,…, c_k) with c̅_i=δ̅^ℓ(z̅_i)(c_i), where z̅_i is the tuple of variables appearing in φ^*(x_1,⋯,x_m, z̅_1,⋯,z_k). For any B⊆, the _δ-type of a tuple a=(a_1,…,a_n)∈^n over A is denoted by tp_δ(a/B),whereas tp(a/B) denotes its restriction to _or. To distinguish between codes in _or and _δ, we use the following notational convention: given an _δ-definable set X, we let c_δ(X) denote a code for X in _δ and, for an _or-definable set X, we let c(X) denote a code for X in _or.Throughout K will always denote a submodel of CODF. A useful observation is the following lemma. Let a∈ and B⊆ K^eq. If for every integer n⩾ 0 the type tp(δ̅^n(a)/K) is B-definable, then the type tp_δ(a/K) is B-definable. By quantifier elimination it suffices to show that every quantifier free _δ-formula has a definition. Let y=(y_1,…,y_m) and φ(x,y) be a quantifier-free _δ-formula without parameters. Let ℓ be a sufficiently large integer such that the formula φ^∗ may be expressed as a formula φ^∗(x_0,…,x_ℓ,y̅) where y̅=(y̅_1,…,y̅_m) with y̅_i:=(y_i1,…,y_iℓ). Since tp(δ̅^ℓ(a)/K) is B-definable, let ψ(y̅) be an _or-formula with parameters over B such that for all b̅∈ K^mℓφ^*(δ̅^ℓ(a),b̅) ⇔ψ(b̅).Define now the _δ-formula (with parameters in B) θ(y_1,…,y_m) as ψ(δ̅^ℓ(y_1), …, δ̅^ℓ(y_m)). We show that θ is a definition for φ. Indeed for every u=(u_1,…,u_m)∈ K^m[ φ(a,u)⇔ φ^*(δ̅^ℓ(a),δ̅^ℓ(u_1), …, δ̅^ℓ(u_m)); ⇔ ψ(δ̅^ℓ(u_1), …, δ̅^ℓ(u_m)); ⇔θ(u_1,…,u_m). ]In models of CODF, we have the following folklore topological property. For the reader convenience, we give a proof below. It is similar to the proof of <cit.> and <cit.>.For any b∈ K, any non-empty open subset O of K^n and any n∈^*, there is a∈ K such that δ̅^n-1(a)∈ O and δ^n(a)=b. Consider the differential polynomial f(x):=δ^n(x)-b and let c∈ O. We have thatf^*(c,b)=0 ∧(∂/∂δ^n(x)f^*)(c,b)=1≠ 0.Let ϵ>0 in K be such that the ball {x∈ K^n : | x-c|<ϵ}⊆ O. By the axiomatization of CODF, there is an element a∈ K such that f(a)=0 and |δ̅^n(a)-c|<ϵ. Thus δ^n(a)=b and δ̅^n-1(a)∈ O. Let us recall the following lemma about differential fields. Let a∈. Suppose δ^n+1(a) is algebraic over K(δ̅^n(a)). Then for all k⩾ n+1, δ^k(a) is algebraic over K(δ̅^n(a)) and also belongs to K(δ̅^n+1(a)).We will also need the following lemma about definable sets in CODF, which is essentially contained in <cit.>. The proof in <cit.> uses semi-algebraic cell decomposition and that RCF has finite Skolem functions.That last property means that for every definable set X⊆ K^n+1, if for every a∈π_n(X), X_a is of finite cardinality d, then it holds that there are definable functions f_i:π_n(X)→ K (where 1≤ i ≤ n), such that X_a=⋃_i=1^d f_i(a). A similar result also holds in the broader context of topological differential fields given in <cit.>. Both for completeness and the reader's convenience, we include here a proof for CODF (from which the more general proof can be easily extracted).We will use that RCF has a fibered dimension (which can be shown directly, without using cell decomposition)and finite Skolem functions.We also need the fact that for a definable set X, it holds that (X∖ Int(X))< X andthe fact that on an open definable set, definable functions are discontinuous on a subset of smaller dimension.Let A⊂, then A^∇_n:={δ̅^n-1(z):z∈ A} (if the context is clear we drop the index n and simply use A^∇). Let X ⊆ K be a non-empty _δ-definable set defined by a quantifier-free formula φ and let n be the number offree variables of the formula φ^*. Then there is an _or-definable set X^⊛⊆ K^n such that X^∇ is contained and dense in X^⊛. In particular, X is dense and contained in π_1(X^⊛).Let X^* denote the set φ^*(K). Given any finite partition {X^*_i}_i∈ I of X^* such that for each i∈ I the set X^*_i is defined by an _or-formula χ_i, it is enough to show the result for each _δ-definable set X∩π_1(X^*_i). Indeed, since each set X∩π_1(X^*_i) is defined by the formula θ_i(x):=φ(x)∧χ_i(δ̅^n-1(x)), notice that θ_i^*=φ^*∧χ_i is equivalent to χ_i since χ_i→φ^*. So assuming the result for each θ_i(K), there are sets θ_i(K)^⊛ such that θ(K)^∇ is contained and dense in θ_i(K)^⊛. Taking X^⊛=⋃_i∈ Iθ_i(K)^⊛ shows the result for X. We will use this observation throughout the proof.For any _δ-definable subset Y⊆ X, define an integer e(Y^*)⩾ 1 by e(Y^*):= min{m : (π_m(Y^*))<m}if such an m⩽ n exists n+1otherwise. Let 1≤ k ≤ n. Note that e(Y^*)=k+1 implies that (π_m(Y^*))=k.We will show the result by induction on e(X^*), for all _δ-definable sets X simultaneously. To show the base case of the induction, suppose that e(X^*)=1.This implies that π_1(X^*) is finite and therefore that X (and X^∇) are finite too. Setting X^⊛:=X^∇ satisfies all the requirements (as a finite set, it is certainly _or-definable). Suppose that we have shown the result for all integers smaller than m⩽ n and that we have e(X^*)=m+1. We split in cases depending on whether m=n or m<n.Case 1: Suppose that e(X^*)=n+1. This implies that (X^*)=n. Set X_1^*=Int(X^*) and X_2^*:=X^*∖ X_1^*. Consider for i=1,2 the set X_i:=X∩π_1(X_i^*). By the starting observation, it suffices to show the result for X_1 and X_2. Since (X_2^*)<n, e(X_2^*)⩽ n, and the result follows for X_2 by induction. We show that for X_1 the set X_1^⊛:=X_1^* satisfies the result.The fact that X_1^∇ is contained in X_1^* follows by assumption, so it suffices to show that X_1^∇ is dense in X_1^*. Pick a∈ X_1^* and ϵ>0 in K. Since X_1^* is open, let ϵ_0⩽ϵ be such that {x∈ K^n: |x-a|<ϵ_0} is contained in X_1^*. Then by Lemma <ref>, there is b∈ K such that |δ̅^n-1(b)-a|<ϵ_0. This shows that δ^n-1(b)∈ X_1^*, which implies that b∈ X_1 and completes the proof of this case. Case 2: Suppose that e(X^*)=m+1⩽ n. By assumption we have that (π_m(X^*))=m so Int(π_m(X^*))≠∅. By induction we may assume that π_m(X^*) is open. Indeed, partition X^* into X_1^* and X_2^*, with X_1^*:{y∈ X^*: π_m(y)∈ Int(π_m(X^*))} and X_2^*:= X^*∖ X_1^*. Set again X_i to be X∩π_1(X_i^*). We have that π_m(X_1^*) is open, and the result follows by induction for X_2 since, as π_m(X_2^*) has dimension strictly less than m, hence e(X_2^*)⩽ m. By Lemma <ref>, there is a non-trivial polynomial P(z,w) with coefficients in K and ℓ(z)=m and ℓ(w)=1, such that π_m+1(X^*) is contained in the zero locus of P, which we denote by Z. For every x∈π_m(X^*), there are at most s elements in Z_x. By possibly partitioning π_m(X^*) (and X) we may assume Z_x has exactly s elements. By the existence of finite Skolem functions, let h_1,…,h_s be definable functions such that for all x∈π_m(X^*), Z_x={h_1(x),…, h_s(x)}. Therefore, by decomposing X^* and X, we may suppose that π_m+1(X^*)=⋃_i=1^s{(x,y)∈π_m(X^*)× K: h_i(x)=y}. Moreover, we may suppose that h_i is continuous. This follows by induction and further partitioning X^* and X since the set {x∈π_m(X^*): h_iis discontinuous as x} is of strictly lower dimension than π_m(X^*). Writing P as a polynomial in the variable w, we obtain polynomials P_k(z) such that P(z,w)=∑_k=0^d P_k(z)w^k. We will show the result by induction on d; we will call this inductive step, the “case induction”, as opposed to the “ambient induction”. Note that the case d=0 never arises, since it would imply that π_m(X^*) is included in the zero locus of P_0(z), which contradicts that π_m(X^*) is open.Express δ(P)=∑_k=0^d δ(P_k).w^k+∑_k=1^d P_k.k.w^k-1.δ(w). So for ℓ≥ 1 and for a tuple of elements (z,w)∈ Z, there is a polynomial g_ℓ in z,δ̅^ℓ-1(w) such that δ^ℓ(w).(∑_k=0^d-1 P_k+1(z).(k+1).w^k)=g_ℓ(z,δ̅^ℓ-1(w)). We suppose the result for all X^* such that π_m+1(X^*) is included in the locus of a non zero polynomial of degree <d in the last variable. So by the case induction, we may assume that for all (z,w)∈π_m+1(X^*), we have ∑_k=0^d-1 P_k+1(x).(k+1).w^k≠ 0. Define f_0,i(x):=h_i(x) and by induction on ℓ⩾ 1, define the function f_ℓ,i:π_m(X^*)→ K by f_ℓ,i(x)=g_ℓ(x,f̅_ℓ-1,i(x))/∑_k=0^d P_k(x).k.h_i(x)^k-1 where f̅_ℓ-1,i(x)=(f_0,i(x),…,f_ℓ-1,i(x)). Each function f_ℓ,i is well-defined by the case induction hypothesis and continuous everywhere except on a set of dimension strictly less than m. Therefore, by possibly partitioning X^* and X and applying our ambient induction, we may assume each function f_ℓ,i is well-defined and continuous at every point in π_m(X^*).Define Y_i^⊛ as follows:Y_i^⊛:={(x,y_0,…,y_n-m-1)∈ X^*: ⋀_ℓ=0^n-m-1 f_ℓ,i(x)=y_ℓ}.Either π_m(Y_i^⊛) is an open subset of π_m(X^*) or we consider on one hand Int(π_m(Y_i^⊛)) and on the other hand π_m(Y_i^⊛)∖ Int(π_m(Y_i^⊛)). We apply the induction hypothesis to that last subset.Set X_i^⊛:={(x,y_0,…,y_n-m-1)∈ K^n: x∈ Int(π_m(Y_i^⊛))∧⋀_ℓ=0^n-m-1 f_ℓ,i(x)=y_ℓ}Finally define X^⊛:=⋃_i=1^s X_i^⊛. Let us first show that X^∇⊆ X^⊛. Pick δ̅^n-1(b)∈ X^∇. Since δ̅^m(b) ∈π_m+1(X^*), this implies that for some i, δ^m+1(b)∈ X_m,i.Therefore, by our construction of f_ℓ,i, δ^m+ℓ(b)=f_ℓ,i(δ̅^m+ℓ-1(b)), for all ℓ⩾ 1. This shows that δ̅^n-1(b)∈ X^⊛. To show the density of X^∇ in X^⊛, let a=(a_0,…,a_n-1)∈ X^⊛ and ϵ>0 in K. Notice that π_m(X^⊛) is open. Let ϵ_0⩽ϵ be such that {x∈ K^m: |x-(a_0,…,a_m-1)|<ϵ_0} is contained in π_m(X^⊛). Since the functions f_0,i,…,f_n-m-1,i are continuous, let δ⩽ϵ_0 be such that for all ℓ∈{0,…,n-m-1} and all x∈π_m(X^*)|x-(a_0,…,a_m-1)|<δ⇒ |f_ℓ,i(x)-f_ℓ,i((a_0,…,a_m-1))|<ϵ_0.By the axiomatisation of CODF, there is a point b∈ K such that P(δ̅^m-1(b),δ^m(b))=0 and |δ̅^m(b)-(a_0,…,a_m)|<δ. So there is 1≤ i≤ s such that δ^m(b)=h_i(δ̅^m-1(b)) and so by continuity using (<ref>),f_ℓ,i(δ̅^m-1(b)) (which is equal to δ^m+ℓ(b)) will be close to a_m+ℓ+1 for 0≤ℓ≤ n-m-1. This shows in particular that δ̅^m-1(b)∈π_m(X^*) and that δ̅^n(b)∈ X_i^⊛. Therefore b∈ X and|δ̅^n(b)-a|<ϵ. § DEFINABLE TYPES AND IMAGINARIES In this section, we show that in the Stone space of the theory CODF, definable types are dense over codes. We start by proving that for any complete theory,it suffices to show that density result for definable 1-types (instead of definable n-types for arbitrary n).§.§ Reducing to the case n=1Let T be a complete -theory,be a monster model of T and d be a fibered dimension function on . Let K be a model of T. We say that definable n-types are dense over codes in T (resp. dense over a-codes) if for every non-empty definable subset X⊆ K^n there is a∈^n such that* a∈ X(),* tp(a/K) is c(X)-definable (resp. acl^eq(c(X))-definable).If moreover we impose that d(a/K)=d(X), we say that definable n-types are d-dense over codes in T (resp. d-dense over a-codes in T). Notice that with this terminology, Theorem <ref> states that all definable n-types in CODF are dense over codes, and Theorem <ref> that all definable n-types in CODF are -dense over codes, whereis the δ-dimension (see later Section <ref>). The following Proposition corresponds to the reduction to n=1 and is very much inspired by <cit.>. Let T be a complete theory. If definable 1-types are dense over codes (resp. over a-codes) then definable n-types are dense over codes (resp. a-codes). Moreover, assuming T admits a code-definable fibered dimension function d, if 1-types are d-dense over codes (resp. a-codes) then so are all definable n-types.A proof for the first part of the proposition can be found in <cit.>. We include the proof for the reader's convenience. Let us first show the argument for types without the dimension assumption. We proceed by induction on n≥ 1, the base case being given by assumption. Let X⊆ K^n+1 be a definable set. By induction, let a∈π_n(X)() be such that tp(a/K) is c(X)-definable i.e. a[c(X)]K. Let K' be an elementary extension of K containing a and c(X). By the case n=1, let b∈ X_a() be such that tp(b/K') is c(X_a)-definable, so in particular (c(X)∪{a})-definable. Then we have both a[c(X)]K and b[c(X)a]K, so by Lemma <ref>, we have (a,b)[c(X)]K, which shows that tp(a,b/K) is c(X)-definable. By assumption (a,b)∈ X() which completes the proof of the first statement. Note that exactly the same proof of the first statement of Proposition <ref> works when replacing density over codes by density over a-codes. One simply replacesbyand notices that if b∈ X_a() and tp(b/K') is acl^eq(c(X_a))-definable, then it is also acl^eq(c(X)∪{a})-definable given that acl^eq(c(X_a))⊆ acl^eq(c(X)∪{a}). This shows that b[c(X)a]K and the proof is completed using Lemma <ref> exactly as before.Let us now show the second statement. We only show the result for d-density over codes since, as in the previous case, the proof for d-density over a-codes is completely analogous. We proceed by induction on n≥ 1, the base case being given by assumption. Fix i∈{0,1} such that d(X)=d(X(i))+i. Notice that this implies that X(i)≠∅. By induction, let a∈ X(i)() be such that tp(a/K) is c(X(i))-definable and d(a/K)=d(X(i)). Since X(i) is c(X)-definable by condition (Dim 5), tp(a/K) is also c(X)-definable. Let K' be an elementary extension of K containing a. By the case n=1, let b∈ X_a() be such that tp(b/K') is c(X_a)-definable and d(b/K')=d(X_a)=i. By Lemma <ref>, tp(a,b/K) is c(X)-definable. It suffices to show that d(a,b/K)=d(X(i))+i. We split in two cases: Case 1: If i=1, then by part (4) of Lemma <ref>, d(b/Ka)=1. This implies, by Lemma <ref>, that d(a,b/K)=d(a/K)+1=d(X(1))+1=d(X). Case 2: If i=0, since d(X_a)=0 and X_a is (K∪{a})-definable, d(b/Ka)=0. Again by Lemma <ref>, we have that d(a,b/K)=d(a/K)+0=d(X(0))+0=d(X).Let T be an o-minimal theory and letbe its dimension function. Then all definable types are -dense over codes.In view of Proposition <ref>, it suffices to show that definable 1-types -dense over codes. Let K be a model of T and let X⊆ K be a definable set. By o-minimality, X is either finite or there is an interval (b_0,b_1) with b_0∈ K∪{±∞} which is maximally contained in X with respect to inclusion. If X is finite, let a be its minimal element. Then tp(a/K) is c(X)-definable and (a/K)=0=(X). Otherwise, if b_0=-∞, let a∈ be any element realizing the type -∞ over K. Then we have that a∈ X(), tp(a/K) is ∅-definable. Finally, if b_0∈ K, let a∈ realize the type b_0^+, which is a c(X)-definable typeand again a∈ X(). In both cases, one easily cheks that (a/K)=1=(X). §.§ The case n=1In this sectiondenotes a monster model of CODF. We will show that definable 1-types are dense over codes. Recall that for B⊆ and c∈, tp(a/B) stands for the type of a over B in the language _or. Let a=(a_1,…, a_n)∈^n and (a/K)=n. Let ϵ_0 realise the type 0^+ over K(a)^rc. Then for every b=(b_1,…,b_n)∈^n such that |a-b|<ϵ_0, we have that tp(a/K)=tp(b/K).Let x=(x_1,…,x_n), φ(x,y) an _or-formula and c∈ K^|y|. Suppose that φ(a,c) holds.By the dimension assumption, a belongs to Int(φ(,c)) (which is not empty).Therefore, we have that ∃ϵ>0∀ x (|a-x|<ϵ→φ(x,c)).Since K(a)^rc is an _or-elementary substructure of ,let ϵ∈ K(a)^rc satisfy the above condition.By assumption |a-b|<ϵ_0<ϵ, hence φ(b,c). Let a∈ be such that (δ̅^n-1(a)/K)=n. Then there is b∈ such that tp(δ̅^n-1(b)/K)=tp(δ̅^n-1(a)/K) and for all k⩾ n, δ^k(b)=0. Let ϵ_0∈ realise 0^+ over K(δ̅^n-1(a))^rc. By Lemma <ref>, there is b_n∈ such that δ(b_n)=0and |b_n-δ^n-1(a)|<ϵ_0. Iterating this argument, applying again Lemma <ref>, we get b_i∈, 1≤ i<n, such that δ(b_i)=b_i+1 and|b_i-δ^i(a)|<ϵ_0. Therefore by Lemma <ref>, tp(b_1,…, b_n/K)=tp(δ̅^n-1(a)/K). Since (b_1,…, b_n)=δ̅^n-1(b_1), the proof is completed. In CODF, definable 1-types are dense over codes.Let X⊆ K be a definable set. Without loss of generality, we may assume that X is infinite. Let X^⊛⊆ K^n be the _or-definable set constructed in Lemma <ref> and which is such that X^∇ is contained and dense in X^⊛. It follows that X^∇ is also dense in X^⊛and this property can be expressed as follows: for x=(x_0,…,x_n-1) E1 x∈X^⊛↔ (∀ϵ>0)( ∃ z∈ X)[|δ̅^n-1(z)-x|<ϵ].This shows that X^⊛ is c_δ(X)-definable. Let (X^⊛)=ℓ≤ n. Notice that ℓ⩾ 1 as X is infinite. Define Y:=Int(π_ℓ(X^⊛)).By Lemma <ref>, Y≠∅. Moreover, Y is c_δ(X)-definable. By Proposition <ref>,let u=(u_0,u_1…,u_n-1)∈X^⊛ be such that for û=(u_0,…,u_ℓ-1)we have that û∈ Y, (û/K)=l and tp(û/K) is c(Y)-definable.Let ϵ_0 realise the type 0^+ over K(u)^rc. Since u∈X^⊛,by (<ref>) applied to ϵ_0, let a∈ X be such that |δ̅^n(a)-u|<ϵ_0.By Lemma <ref>, we have that tp(δ̅^ℓ-1(a)/K)=tp(û/K).Therefore, tp(δ̅^ℓ-1(a)/K) is c(Y)-definable, so in particular c_δ(X)-definable. We split in cases. If ℓ<n, then since (Y)=ℓ we have that for all x∈ X, E2δ^k(x)∈ dcl^eq(c(Y),δ̅^ℓ-1(x)) for all k≥ℓ.Therefore, by Lemma <ref> and (<ref>) we have that tp(δ̅^k(a)/K) is c_δ(X)-definable for every k⩾ 0. This implies that tp_δ(a/K) is c_δ(X)-definable by Lemma <ref>. Now suppose that ℓ=n. Then (δ̅^n-1(a)/K)=n, so by Lemma <ref> there is b∈ such that tp(δ̅^n-1(b)/K)=tp(δ̅^n-1(a)/K) and δ^k(b)=0 for all k⩾ n. This implies that δ̅^n-1(b)∈ X^⊛ and therefore b∈ X (by Lemma <ref>). Since tp(δ̅^n-1(a)/K) is c(Y)-definable, by Lemma <ref>, tp(δ̅^k(b)/K) is also c(Y)-definable for all k⩾ 0. Therefore by Lemma <ref>, tp_δ(b/K) is c(Y)-definable and hence, in particular, c_δ(X)-definable.The theorem follows from Proposition <ref> and Proposition <ref>.§ ELIMINATION OF IMAGINARIES The former proof of elimination of imaginaries for CODF, given by the third author <cit.>,uses the démontage of semialgebraic sets in real closed fields (a fine decomposition of semialgebraic sets),as well as elimination of imaginaries for RCF.We will also use that last result together with the following general criterion due to E. Hrushovski <cit.>. The formulation below is taken from <cit.>.Let T be an -theory, with home sort(meaning we work in a monster modelsuch that ^eq= dcl^eq()). Letbe some collection of sorts. Then T has elimination of imaginaries in the sorts , if the following conditions all hold: * For every non-empty definable set X⊂^1, there is an acl^eq(c(X))-definable type in X. * Every definable type in ^n has a code in , that is, there is some (possibly infinite) tuple fromwhich is interdefinable with a code for p. * Every finite set of finite tuples fromhas a code in G. That is, if S is a finite set of finite tuples from G, then c(S) is interdefinable with a finite tuple from . We will need the following characterizations of definable types over models in RCF and CODF. The characterization for RCF follows from a more general characterization of definable types in o-minimal structures due to Marker and Steinhorn <cit.>. Letbe a monster model of RCF, K be a model and a∈^n. Then the type tp(a/K) is K-definable if and only if K is Dedekind complete in K(a)^rc. Letbe a monster model of CODF, K be a model and a=(a_1,…,a_n)∈^n.Then, the type tp_δ(a/K) is K-definable if and only if K is Dedekind complete in K(δ̅(a_1),…,δ̅(a_n))^rc.The following lemma ensures condition (2) of the previous criterion for CODF. Letbe a monster model of CODF and q∈ S_n^CODF() be a definable type. Then q has a code in . Suppose that q is definable over a model K. It suffices to show that p:=q|K (the restriction of q to K) has a code in K. Let (a_1,⋯,a_n) be a realization of p. By Theorem <ref>, we have that K is Dedekind complete in K(δ̅(a_1),…, δ̅(a_n))^rc. For each m⩾1, let p_m:= tp(δ̅^m(a_1),…, δ̅^m(a_n)/K).By Theorem <ref>, each type p_m is definable. By elimination of imaginaries in RCF, let b_m be a code for p_m in K. We show that (b_m)_m⩾ 1 is a code for p. Let σ∈ Aut__δ(). Since p⊢ p_m for all m, one direction is straighforward (namely if σ(p)=p, then σ(p_m)=p_m and so σ(b_m)=b_m). For the converse, suppose that σ(b_m)=b_m for all m⩾ 1. Then σ(p_m)=p_m for all m⩾ 1. Let us show that σ(p)=p.Let φ(x_1,…, x_n,y_1,⋯,y_m) be an _δ-formula (without parameters). By quantifier elimination in CODF, we may assume φ is quantifier-free. Let φ^∗(x_1, …, x_n,y_1,⋯,y_m) be its associated _or-formula. Without loss of generality, and to ease notation, let k be an integer such that ℓ(x_i)=ℓ(y_j)=k for all i∈{1,…,n} and all j∈{1,…,m}. For c=(c_1,…,c_m)∈^m, let δ̅^k(c) denote the tuple (δ̅^k(c_1),…,δ̅^k(c_m)). For any tuple c∈ K^m we have that: φ(a_1,…, a_n,c) ⇔φ^∗(δ̅^k(a_1),…, δ̅^k(a_n),δ̅^k(c))⇔φ^∗(δ̅^k(σ(a_1)),…, δ̅^k(σ(a_n)),δ̅^k(c))⇔φ(σ(a_1),…, σ(a_n), c)Indeed, the first and third equivalences hold by definition of φ^* while the second equivalence holds since σ(p_k)=p_k implies that (φ^∗∈ p_k φ^∗∈σ(p_k)). CODF has elimination of imaginaries.This follows by Theorem <ref>. Condition (1) is implied by Theorem <ref>, condition (2) corresponds to Lemma <ref> and condition (3) is straightforward since the main sort is a field (so weak elimination of imaginaries implies elimination of imaginaries <cit.>).It is not clear how to extend the previous theorem to general topological fields with generic derivations as defined in <cit.> even assuming that definable types are dense over codes (or over a-codes). The main obstacle is to show an analog of Lemma <ref> in the general context. If one tries to mimic the proof given for CODF, one runs into the problem of showing that given a type tp_δ(a/K), the types tp(δ̅^n(a)/K) (in the ring language) are definable. Here we used the characterization of definable types in CODF given in Theorem <ref>. A substitute to Theorem <ref> can be found in a recent paper of Rideau and Simon <cit.>. Let us recall the precise setting they are working in. Let T be a NIP -theory admitting elimination of imaginaries and let T̃ be a complete L̃-enrichment of T. Let NT̃ and let A=dcl_^eq(A)⊂ N_L̃^eq. In <cit.>, the authors provide a sufficient condition on T̃ implying that the underlying -type of any ^eq(A)-definable type over N is A∩ℛ-definable, where ℛ denotes the set of all -sorts. In the case of T=RCF and T̃=CODF, we get that such -type is A∩ N-definable. However, their sufficient condition requires the existence of a model M of T̃ whose -reduct is uniformly stably embedded in every elementary extension. Somehow surprisingly, in the case of T̃=CODF and T=RCF, it has been shown that there is such a model. Indeed, building on a previous proof that CODF has archimedean models, the first author showed in <cit.> thatcan be endowed with a derivation in such a way that its expansion becomes a model of CODF. By the above theorem of Marker and Steinhorn (Theorem <ref>),(viewed as an ordered field) is uniformly stably embedded in every elementary extension.§ Δ-DIMENSION AND DEFINABLE TYPES IN CODFThe fact that closed ordered differential fields may be endowed with a fibered dimension was first proven in <cit.>. Their definition relies on a cell decomposition theorem in CODF. Alternatively, one can obtain the same dimension function proceeding as in <cit.>, as was done in <cit.> working in a broader differential setting (see <cit.>). Here we will call such a dimension function the δ-dimension and denote it by(in place of t-, the notation used in <cit.>). Throughout this section letbe a monster model of CODF and K⊂ be a small model. Instead of providing the original definition given in <cit.> of , we will use a characterization in terms of the following closure operator, also proven to be equivalent in <cit.>. For B⊆ and a∈, say that a∈ cl(B) if and only if there is a differential polynomial q∈⟨ B⟩{x}∖{0} such that q(a)=0. This closure operator cl defines a pre-geometry on(see <cit.>). The natural notion of relative independence induced by cl is defined as follows: a set A is cl-independent over B if and only if for every a∈ A one has that a∉ cl(B∪ (A∖{a})). Finally, defineE3cl-(a/B) := max{ |C| : C⊆⟨ Ba⟩, C is cl-independent over B}. Let a∈^n, b∈^m and B⊆. Notice that (a/B)≤ n and that (a/B)≤(ab/B). Moreover,is additive, that is,(ab/B)=(a/B)+(b/Ba). The following proposition characterizes the δ-dimension in terms of cl and gathers a second property that we will later need.Let X⊆ K^n be a definable set. Then * (X)=max{(a/K) : a∈ X()}, * if n=1, φ(x) is a quantifier-free _δ-formula that defines X, then (X)=1 if and only φ^*() contains a non-empty open set.Part (2) corresponds to <cit.> and part (1) is a special case of <cit.>.The following lemma addresses the somehow classical issue of showing that the function δ- induced on types and tuples as in Subsection 1.3 (equation (<ref>)), coincides, as expected, with . Let a=(a_1,…,a_n)∈ and p:=tp_δ(a/K). Then (a/K)=inf{(φ()): φ(x)∈ p}. Set m:=inf{δ-dim(φ()): φ(x)∈ p}⩽ n and let φ∈ p be such that δ-(φ())=m. Since φ(a), by Proposition <ref> we have that cl-(a/K)⩽max{cl-(b/K) : b∈φ()}= δ-(φ())=m. We show that m⩽cl-(a/K). Suppose that cl-dim(a/K)=d≤ n. Let b∈^d belong to K⟨ a⟩ such that cl-dim(b/K)=d=cl-dim(a/K). So there are q_i∈ K⟨ b⟩{x_i}∖{0}, 1≤ i≤ n, be such that q_i(a_i)=0. Rewrite q_i as differential polynomials p_i∈ K⟨ y,x_i⟩. The formula ψ(x_1,…,x_n):=∃ y ⋀_i=1 ^n (p_i(y,x_i)=0∧∃ u p_i(y,u)≠ 0) is in p. Let c∈^n be any tuple such that ψ(c) holds. Then there exists z∈^d such that ⋀_i=1^n(p_i(z,c_i)=0∧∃ u p_i(z,u)≠ 0). This shows that (c/Kz)=0. Notice that since z has length d, we have that (z/K)≤ d. Therefore, we have that (c/K)⩽(cz/K)= (z/K)+(c/Kz)⩽ d. By Proposition <ref>, (ψ())≤ d, which shows m⩽ d. Now let us show that the dimension function δ- is code-definable. The δ-dimension is code-definable. Let X⊆^n+1 be a definable set. It suffices to show that X(1) is c_δ(X)-definable. Suppose that the quantifier-free formula φ(x,y,c) defines X where x=(x_1,…,x_n), y is a single variable and c=(c_1,…, c_s) ∈^s are all the parameters in the formula. Let σ∈ Aut__δ() be such that σ(X)=X. We show that σ(X(1))=X(1). Pick a∈ X(1), so by definition a∈π_m(X) and (X_a)=1. Since σ(π_m(X))=π_m(X), we have that σ(a)∈π_m(X). It remains to show that (X_σ(a))=1. Consider the _or-formula φ^*(x_1,…,x_n, y,δ̅^t_1(c_1),…,δ̅^t_s(c_s)) and let m_i:=ℓ(x_i) for i∈{1,…,n} and ℓ:=ℓ(y). Recall that we have that φ(x,y,c) is equivalent to φ^*(δ̅^m_1(x_1),…,δ̅^m_n(x_n),δ̅^ℓ(y),δ̅^t_1(c_1),…,δ̅^t_s(c_s)). To ease notations in this proof, we will denote by δ̅(a) the tuple (δ̅^m_1(a_1),⋯,δ̅^m_n(a_n)), and by δ̅(c) the tuple (δ̅^t_1(c_1),…,δ̅^t_s(c_s)). By Proposition <ref> (part 2), we have that (X_a)=1 ⇔φ^*(δ̅(a),,δ̅(c)) contains a non-empty open set⇔∃z ∃ϵ>0 ∀w [ |w-z|<ϵ→φ^∗(δ̅(a),w,δ̅(c))]⇔∃z ∃ϵ>0 ∀w [ |w-z|<ϵ→φ^∗(δ̅(σ(a)),w,δ̅(σ(c)))] ⇔φ^*(δ̅(σ(a)),,δ̅(σ(c))) contains a non-empty open set⇔(X_σ(a))=1. Before proving Theorem <ref>,we need the following technical lemma which will ensure that we can find an element of the required dimension in a definable subset. Let a∈ be such that dim(δ̅^n-1(a)/K)=n and x a variable of length 1. Let Σ be the set of _δ-formulas Σ:=Σ_1∪Σ_2∪Σ_3 with parameters in K where[Σ_1:={q(x)≠ 0q∈ K{x}∖{0}};Σ_2:={φ(δ̅^n-1(x)) : φ∈ tp(δ̅^n-1(a)/K) }; Σ_3:=⋃_i⩾ 1{δ^n+i(x)< c : c∈ K(δ̅^n+i-1(x))};]Then Σ is consistent and can be completed into a unique type p∈ S_1^CODF(K).By quantifier elimination, we may assume that all formulas in Σ are quantifier-free. By compactness, it suffices to show that every finite subset Θ of Σ(x) is consistent (that Σ can be completed into a unique type will follow from the fact that CODF admits quantifier elimination). Let m⩾ 0 be an integer such that[ Θ∩Σ_1⊆{q(δ̅^m(x))≠ 0q∈ K[δ̅^m(x)]∖{0}}; Θ∩Σ_3⊆⋃_1<i⩽ m{δ^n+i(x)< c : c∈ K(δ̅^n+i-1(x))}.;]By multiplying all polynomials appearing in Θ∩Σ_1, we may assume Θ∩Σ_1={q(x)≠ 0} for some q∈ K[δ̅^m(x)]∖{0}.Similarly, by replacing all formulas in Θ∩Σ_2(x) by their conjunction,we may suppose Θ∩Σ_2={φ}. Therefore, Θ is consistent if and only if the following formula θ(x) is consistent θ(x):=q(δ̅^m(x))≠ 0 ∧φ(δ̅^n-1(x)) ∧ψ(δ̅^n+m(x))where ψ(δ̅^n+m(x)) corresponds to the conjunction of all formulas in Θ∩Σ_3. Suppose that ψ(δ̅^n+m(x)):=⋀_i=1^m ⋀_j=1^m_iδ^n+i(x)<f_ij(δ̅^n+i-1(x)), with m_i a positive integer for i∈{1,…,m} and f_ij a rational function over K in n+i-1 variables. Since dim(δ̅^n-1(a)/K)=n, we may further suppose that φ defines an open subset of K^n.Consider the _or-definable set θ^*(K)⊆ K^n+m.By Lemma <ref>, to show that θ is consistent it suffices to show that θ^*(K) is open and non-empty. For x̅=(x_0,…,x_m) we have that θ^*(x̅):=q^*(x̅)≠ 0 ∧φ(x_0,…,x_n-1) ∧ψ^*(x̅),where q^* is a polynomial over K in m+1 variables. The formula φ, seen as a formula over the variables x̅, defines an open subset of the form U× K^m where U is a non-empty open subset of K^n. The formula ψ^*(x̅) defines an open subset of the form K^n× V for V a non-empty open subset of K^m. Therefore φ∧ψ^* defines the non-empty open subset U× V of K^n+m. Finally, the intersection of a Zariski open subset of K^n+m with any non-empty open subset of K^n+m (in particular with U× V) is non-empty and remains open. This shows that θ^*(K) is a non-empty open subset of K^n+m. We have all the elements to show Theorem <ref>.By Lemma <ref> and Proposition <ref>, it suffices to prove the property for non-empty definable subsets X⊆ K. By Proposition <ref>, there is a∈ X() such that tp(a/K) is c_δ(X) definable. We split in cases depending on the value of δ-(X). Suppose first that δ-(X)=0. Since X is K-definable, by Lemma <ref>, we must already have that δ-(a/K)=0. Suppose now that δ-(X)=1. By Lemma <ref>, there is an _or-definable subset X^⊛⊂ K^nsuch thatE1 x∈X^⊛↔ (∀ϵ>0)( ∃ z∈ X)[|δ̅^n-1(z)-x|<ϵ].Since a∈ X, we have that δ̅^n-1(a)∈ X^⊛.Moreover, since δ-(X)=1, we must have that (δ̅^n-1(a)/K)=n. We need to find an element b∈ X() such that tp(δ^n-1(a)/K)=tp(δ^n-1(b)/K), δ-(b/K)=1 and tp_δ(b/K) is still c_δ(X)-definable.We will use Lemma <ref> to find such an element. Note that the element a which we found in Proposition <ref> is always such that δ-(a/K)=0. Let[ Σ_1:={q(x)≠ 0q∈ K{x}}; Σ_2:={φ(δ̅^n-1(x)) : φ∈ tp(δ̅^n-1(a)/K) }; Σ_3:=⋃_i⩾ 1{δ^n+i(x)< c : c∈ K(δ̅^n+i-1(x))},; ] by Lemma <ref>, Σ_1∪Σ_2∪Σ_3 determines a unique type pand we will show that it has all the required properties. Let b be a realization of p. Since tp(δ̅^n-1(b)/K)=tp(δ̅^n-1(a)/K), by (<ref>) we have that b∈ X(). The fact that it satisfies Σ_1 ensures that δ-(b/K)=1. It remains to show that tp_δ(b/K) is c_δ(X)-definable. We will show that for every m, tp(δ̅^m(b)/K) is c_δ(X)-definable. For m<n, it follows from the fact that tp(δ̅^n-1(a)/K) is c_δ(X)-definable. By induction on i≥ 0, let us show that tp(δ̅^n+i(b)/K) is c_δ(X)-definable. By Lemma <ref>, it suffices to show that tp(δ^n+i(b)/Kδ̅^n-1+i(b)) is c_δ(X)-definable.Actually, tp(δ^n+i(b)/Kδ̅^n-1+i(b)) is even ∅-definable because it is the type -∞ over K(δ̅^n-1+i(b)). Since tp(δ̅^n-1(a)/K) is c_δ(X)-definable, by Lemma <ref>, tp(δ̅^k(b)/K) is also c_δ(X)-definable for all k⩾ 0. Therefore, tp_δ(b/K) is c_δ(X)-definable by Lemma <ref>.alpha | http://arxiv.org/abs/1704.08396v3 | {
"authors": [
"Quentin Brouette",
"Pablo Cubides Kovacsics",
"Francoise Point"
],
"categories": [
"math.LO",
"03C64, 12H05"
],
"primary_category": "math.LO",
"published": "20170427010719",
"title": "Strong density of definable types and closed ordered differential fields"
} |
Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USADepartment of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USADepartment of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA We study the thermalization, injection, and acceleration of ions with different mass/charge ratios, A/Z, in non-relativistic collisionless shocks via hybrid (kinetic ions–fluid electrons) simulations. In general, ions thermalize to a post-shock temperature proportional to A.When diffusive shock acceleration is efficient, ions develop a non-thermal tail whose extent scales with Z and whose normalization is enhanced as (A/Z)^2, so that incompletely-ionized heavy ions are preferentially accelerated. We discuss how these findings can explain observed heavy-ion enhancements in Galactic cosmic rays.Chemical Enhancements in Shock-accelerated Particles: Ab-initio Simulations Anatoly Spitkovsky December 30, 2023 ===========================================================================Introduction.—Non-relativistic shocks are well-known as sources of energetic particles.Prominent examples of such shocks are the blast waves of supernova remnants (SNRs), which are thought to be the sources of Galactic cosmic rays (GCRs) <cit.>, and heliospheric shocks, where solar energetic particles (SEPs) are measured in situ <cit.>.Chemical abundances in GCRs and SEPs provide crucial information about their sources and the processes responsible for their acceleration.At trans-relativistic energies, the chemical composition of GCRs roughly resembles the composition of the solar system <cit.>, the most evident deviation being the enhancement of secondaries produced by spallation of primary GCRs during their propagation in the Milky Way. A more careful analysis, however, reveals that the GCR composition is controlled by volatility and mass/charge ratios: refractory elements show larger enhancements than volatile ones, and heavier volatile elements are more abundant than lighter ones <cit.>. Moreover, elemetns with low first ionization potential tend to be overrepresented in GCRs <cit.>. At TeV energies, where spallation is negligible, the fluxes of H, He, C-N-O, and Fe do not differ by more than one order of magnitude <cit.>. Since their typical solar number abundances relative to H are χ_He=0.0963, χ_CNO=9.54× 10^-4, χ_Fe=8.31× 10^-5 <cit.>, the abundances observed in GCRs suggest that heavy ions must be preferentially injected and accelerated compared to protons. Diffusive shock acceleration (DSA) <cit.> at SNR shocks is likely the mechanism responsible for ion acceleration up to ∼ 10^17 eV <cit.>. DSA produces universal power-law momentum spectra f(p)∝ p^-3r/(r-1), where r is the shock compression ratio; for strong shocks r→ 4 and f(p)∝ p^-4. For relativistic particles the energy spectrum is then f(E)=4π p^2 f(p)dp/dE∝ E^-2, while at non-relativistic energies one gets f(E)∝ E^-3/2 <cit.>. Hybrid simulations.—In order to study ab initio how DSA of ions with different mass/charge ratio works, we performed 2D kinetic simulations with dHybrid, a massively parallel hybrid code, in which ions are treated kinetically and electrons as a neutralizing fluid<cit.>. Hybrid simulations of non-relativistic shocks have been extensively used for assessing the efficiency of proton DSA <cit.>, the generation of magnetic turbulence due to plasma instabilities driven by accelerated particles <cit.>, the diffusion of energetic particles in such self-generated magnetic fields <cit.>, and the injection of protons into the DSA process <cit.>. In the literature there are few examples of kinetic simulations with heavy ions, e.g., the pioneering 1D hybrid simulations of weak shocks including α-particles <cit.> and the recent hybrid study of the thermalization of weakly-charged ions at shocks <cit.>. However, a self-consistent kinetic characterization of ion enhancement in DSA has never been performed before <cit.>. In the hybrid simulations presented in this Letterwe include additional ion species characterized by number abundances χ_i, atomic mass A_i, and charge Z_i (in proton units), initially in thermal equilibrium with protons and electrons. We fix χ_i≠ H=10^-5 to effectively make ions other than protons dynamically unimportant. Lengths are measured in units of c/ω_p, where c is the speed of light and ω_p≡√(4π n e^2/m), with m,e and n the proton mass, charge and number density; time is measured in units of ω_c^-1≡ mc/eB_0, B_0 being the strength of the initial magnetic field; velocities are normalized to the Alfvén speed v_A≡ B/√(4π m n), and energies to E_sh≡ mv_sh^2/2, with v_sh the velocity of the upstream fluid in the downstream frame. We account for the three spatial components of the particle momentum and of the electric and magnetic fields.Shocks are produced by sending a supersonic flow against a reflecting wall and are characterized by their sonic and Alfvénic Mach numbers M_s≡ v_sh/c_s, M_A≡ v_sh/v_A, with c_s the sound speed;in this work we consider M_s≃ M_A≡ M. The shock inclination is defined by the angle þ between the direction of B_0 and the shock normal, such that þ≲ 45 corresponds to quasi-parallel shocks.The time-step is chosen as Δ t=0.01/M ω_c^-1 and the computational box measures 2.5× 10^4c/ω_p by 2Mc/ω_p, with two cells per ion skin depth. In order to suppress the numerical heating that can arise in long-term simulations with species of disparate densities, we use 100 protons per cell, and 4 particles per cell for all the other species. We have checked the convergence of our results against 3D simulations, time and space resolution, number of particles per cell, and transverse size of the simulation box <cit.>. The electron pressure is a polytrope with an effective adiabatic index chosen to satisfy the shock jump conditions with thermal equilibration between downstream protons and electrons <cit.>.Our benchmark case comprises ion species with A={1,2,4,8} and Z={1,2} and a quasi-parallel (þ=20) shock with M=10, which exhibits efficient proton DSA and magnetic field amplification <cit.>. In our case we find that ∼ 10% of the shock kinetic energy is converted into accelerated protons, and the field is amplified by a factor of ≳ 2 in the upstream.The downstream spectra of different ion species are shown in Fig. <ref>, as a function of E/Z and normalized to their abundances χ_i.The color code gathers species with the same A/Z, while solid and dashed lines correspond toZ=1 and 2, respectively. Each of the species shows a thermal peak plus a power-law tail with the universal DSA slope γ≃ 3/2; non-thermal spectra roll over at a maximum energy E_max,i, which increases linearly with time <cit.>. For strong shocks, Rankine–Hugoniot conditions return a downstream thermal energy ℰ≃ 0.6E_sh <cit.>.Since half of the post-shock proton energy goes into electron heating by construction, we expect ℰ_H≃ℰ/2. Then, since heavier ions have more kinetic energy to convert into thermal energy, their temperature is expected to scale with their masses, i.e.,ℰ_i≠ H=A_iℰ.Dotted lines in Fig. <ref> correspond to Maxwellian distributions with such expected temperatures: they provide a good fit for the positions of thermal peaks, but only a rough one for the shape of the thermal distributions of heavy ions, whose relaxation is still ongoing [We checked that the scaling ℰ_i∝ A_i is recovered also for protons if the electron pressure is set to zero.]. When comparing different ion curves in Fig. <ref>, we notice three important scalings: * At fixed Z, the thermal peaks are shifted to the right linearly in A, i.e, each species thermalizes at a temperature proportional to its mass <cit.>;* All the ion spectra rollover at the same E_max/Z, consistent with the fact that DSA is a rigidity-dependent process [Rigidity is defined as p/Z, not E/Z: the two definitions are equivalent only for relativistic particles. However, we showed in <cit.> that the self-generated diffusion coefficient and, in turn, the acceleration time do scale as E/Z in the non-relativistic case <cit.>.];* The normalization of the non-thermal spectra at given E/Z is an increasing function of the mass/charge ratio, which implies that the efficiency of injection into DSA depends on A/Z.The first two results validate the theoretical expectations, while the last one represents the first self-consistent characterization of the parameter that regulates the injection of ions into the DSA process.Injection enhancement in DSA.— In this section we discuss how the observed boost in ion injection depends on A/Z. The ion non-thermal spectra, neglecting the cutoffs, are power laws that can be written as f_i(E)=(γ-1) nχ_iη_i /E_inj,i(E/E_inj,i)^-γ,where η_i is the fraction of ions that enter DSA above the injection energy E_inj,i.We then introduce the ratio≡f_i(E/Z_i)/χ_i f_p(E)= η_i/χ_iη_p(E_inj,i/E_inj,p)^γ-1as a measure of the enhancement in energetic ions with respect to protons at fixed E/Z. is promptly read from Fig. <ref> by taking the ratio of the power-law spectra at any E/Z between 10 and 100E_sh.Note that the enhancement has two contributions: one straightforward, η_i/η_p, which depends on the fraction of particles that enter DSA for each species, and one more subtle that depends on E_inj,i, which cannot be predicted analytically.Fig. <ref> shows the enhancements obtained for shocks with þ=20 and M={5,10,20,40};injection fractions and enhancements are calculated at time t=10^3ω_c^-1, when DSA spectra have been established, by considering the post-shock spectra of species with A/Z up to 8, integrated over 10^3 c/ω_p.For shocks with M≳ 10, where accelerated protons generate non-linear upstream magnetic turbulence with δ B/B_0≳ 1, the fraction of injected particles is η_p≈ 1% for protons and increases linearly with A/Z (top panel); at the same time, ∝ (A/Z)^2, attesting to a very effective enhancement of particles with large charge/mass (bottom panel). The scaling with A/Z is weaker for the lowest-M shock, for which δ B/B_0≈ 0.2: η_i is roughly constant at the percent level and ∝ A/Z. Chemical enhancements.— The high-M case is relevant, e.g., for SNR shocks propagating into the warm interstellar medium (ISM), where atoms are typically singly ionized. Ions that are injected into DSA will then be stripped of their electrons while being accelerated up to ∼PV rigidities <cit.>.In the popular scenario in which GCRs are produced at SNR shocks via DSA <cit.>, we can compare our findings with the chemical enrichment measured in GCRs <cit.>. In order to compare observations at Earth and shock injection simulations,we take the observed GCR flux ratios at 1 TeV, ϕ_i(E) <cit.>, weigh them with the fiducial solar abundances, χ_i <cit.>, and write the enhancement at a given E as Z_i^1-γ (see Eq. <ref>). We also account for the rigidity-dependent residence time in the Galaxy ∝(E/Z)^-δ, with δ≃ 1/3 above a few GV <cit.>, and extrapolate the enhancements down to the non-relativistic injection energies. Such an extrapolation introduces an additional factor A_i^-1/2, because DSA spectra are power laws in momentum and hence energy spectra flatten by E^1/2 at ∼ A_iGeV. Finally, we obtain that ion injection into DSA must be enhanced at SNR shocks according to^ GCRs=.ϕ_i/χ_iϕ_p|_ TeVZ_i^γ-1-δ/A_i^1/2≃.ϕ_i/χ_iϕ_p|_ TeVZ_i^1/6/A_i^1/2in order to explain theabundances observed in GCRs.We consider a strong quasi-parallel shock with M=20 and singly-ionized He, CNO, and Fe atoms with effective A/Z={4,14,56} and calculatein the upstream, sinceat t=10^3ω_c^-1 ions A/Z≳ 14 have already been over-injected but have not yet developed the universal downstream DSA spectrum. The enhancements found in simulations and those in GCR data (Eq. <ref>) are compared in Fig. <ref>: the scaling ≃ (A/Z)^2 found for strong shocks provides a very good fit, with singly-ionized He, CNO, and Fe particles enhanced by a factor of about ten, hundred, and a few thousand, respectively. It is remarkable that such a Fe enhancement requires a very large fraction η_Fe≲ 50% of the pre-shock Fe ions to enter DSA;this may have implications for the overall ISM chemical composition, since regions processed by shocks may become depleted in heavy elements. Nevertheless, in the ISM many elements are typically trapped in molecules (C,O) and dust grains (Fe, refractory elements), so that fragmentation and sputtering may represent crucial steps in the injection of heavy elements <cit.>. Our results suggest that dust grains with very large A/Z≫1 should also have no problem of being efficiently energized via DSA-like processes, thereby sputtering pre-energized ions that can be easily injected <cit.>. In the low-M regime relevant to heliospheric shocks, our simulations show that DSA can account for enhancements by factors of a few to ten, which are commonly observed in SEP events <cit.>. However, chemical enhancements in SEPs may be time dependent <cit.> and greatly vary from event to event; in addition to shock strength and inclination, they seem to depend on the presence of pre-existing magnetic turbulence and energetic seed particles (produced, e.g., in solar flares) <cit.>, which makes it nontrivial to compare individual SEP events with our simulations where ion injection only occurs from the thermal pool. Dependence on shock inclination.— Oblique shocks with þ≳ 50 cannot inject thermal protons and drive self-generated magnetic turbulence <cit.>. We find that in such shocks ions with large A/Z do thermalize at a temperature ∝ A, but progressively further in the downstream with respect to the quasi-parallel case. Neither protons nor heavier ions are injected into DSA or develop a non-thermal tail, which confirms that having a large gyroradius (∝ A/Z) is not a sufficient condition for being injected into DSA.The injection mechanism.— Ion injection occurs in a qualitatively different way than proton injection, which is due to specular reflection off the time-dependent potential barrier at the shock and energization via shock-drift acceleration <cit.>. Unlike protons, heavy ions are not halted by the shock barrier and always penetrate in the downstream for at least one gyroradius (∼ MA/Z c/ω_p); here, their distribution tends to become more isotropic due to the presence of rapidly-varying fields, an analog of the violent relaxation in stellar dynamics. If isotropization is rapid enough with respect to advection, there arises a population of backstreaming ions that can overrun the shock barrier, which is “tuned” for preventing downstream thermal protons from returning upstream. The fraction of injected heavy ions is thus controlled by how rapid isotropization is, which depends on A/Z and on the strength of the magnetic turbulence in the shock layer.Fig. <ref> shows the x-p_x phase spaces for our benchmark run; we consider ions with Z=1 and A={1,2,4,8} at times t={200,700}ω_c^-1. We see that, while protons are promptly isotropized behind the shock (dashed vertical lines), ions with larger A tend to retain their anisotropy further in the downstream.At early times (left column), protons show the characteristic non-thermal, isotropic population of particles diffusing around the shock <cit.>; ions with A/Z=2 have also started being injected and accelerated. Injection may be quite “bursty” when patches of quasi-parallel magnetic field are advected through the shock <cit.>, resultingin coherent batches of particles protruding back into the upstream (as for A/Z=2 at t=200 ω_c^-1 in Fig. <ref>). Ions with A/Z=4 have just started overrunning the shock, but there are only few particles with p_x<0 in the upstream, implying that DSA has not yet been established. Finally, ions with A/Z=8 isotropize too far downstream to overrun the shock barrier and are not injected yet.At later times (t=700 ω_c^-1, right column in Fig. <ref>), instead, all of the species show the typical DSA spectrum comprising non-thermal particles that diffuse on both sides of the shock. From the color code it is also possible to see how the fraction of particles that leak back into the upstream is larger for heavier ions.Proton injection is controlled by the quasi-periodic reformation of the shock barrier <cit.>;instead, injection of heavier ions relies on rapid electromagnetic fluctuations larger than those induced by the local shock reformation and happens at later times for heavier species, and always after the onset of non-linear turbulence [The expected dependence E_max,i(t)∝ Z is achieved only when the acceleration time becomes much larger than the injection time, which may explain why in some heliospheric shocks Fe spectra roll over at significantly lower energies compared with O <cit.>.].A more quantitative characterization of the trajectories of the ions that get injected into DSA is beyond the scope of this Letter.Note, however, that the behavior reported here is not equivalent to the so-called thermal leakage scenario for particle injection <cit.>, in that the injected ions are not those in the tail of the Maxwellian (strictly speaking, they have not yet thermalized). The global shock structure is always controlled by species with the most inertia, so that density and fields jump within one gyroradius of thermal protons.The isotropization length for heavy ions is effectively larger than for protons, but injection is controlled by how rapidly they can be isotropized (i.e., reverse their p_x), which depends on the local electromagnetic fluctuations and not on the ion energy.We conclude that, while the injected protons are reflected by the shock barrier and need to be pre-energized via few cycles of shock-drift acceleration <cit.>, heavy ions reflect off post-shock magnetic irregularities.The enhancement in ions with A/Z≫1 is then due to the fact that they are not affected by the proton-regulated shock barrier (their kinetic energy being much larger than the barrier potential), so that they do not experience shock-drift acceleration but rather start diffusing right away.Ions with A/Z≳ 1 exhibit intermediate properties between protons and heavier ions because their probability of being reflected or transmitted at the shock barrier depends on the actual angle between their momentum and the shock normal: ions with velocity mainly along the shock surface are more proton-like because they can be reflected by the shock barrier <cit.>.Conclusions.— We have presented the first ab-initio calculation of ion DSA at non-relativistic shocks, finding that species with large A/Z show enhanced non-thermal tails with respect to protons, in quantitative agreement with the chemical abundances observed in GCRs. In forthcoming publications we will discuss the implications of these findings also for what concerns the discrepant hardening of non-H species in GCRs <cit.> and for the role of accelerated He in SNR shocks <cit.>.This research was supported by NASA (grant NNX17AG30G to DC), NSF (grant AST-1517638 to AS), and Simons Foundation (grant 267233 to AS). Simulations were performed on computational resources provided by the Princeton High-Performance Computing Center, the University of Chicago Research Computing Center, and XSEDE TACC (TG-–AST100035). | http://arxiv.org/abs/1704.08252v1 | {
"authors": [
"Damiano Caprioli",
"Dennis T. Yi",
"Anatoly Spitkovsky"
],
"categories": [
"astro-ph.HE",
"hep-th",
"physics.geo-ph",
"physics.plasm-ph"
],
"primary_category": "astro-ph.HE",
"published": "20170426180001",
"title": "Chemical Enhancements in Shock-accelerated Particles: Ab-initio Simulations"
} |
Fluctuations in an established transmission in the presence of a complex environment Fabrice Mortessagne December 30, 2023 ====================================================================================We develop a discrete version of paracontrolled distributions as a tool for deriving scaling limits of lattice systems, and we provide a formulation of paracontrolled distributions in weighted Besov spaces. Moreover, we develop a systematic martingale approach to control the moments of polynomials of i.i.d. random variables and to derive their scaling limits. As an application, we prove a weak universality result for the parabolic Anderson model: We study a nonlinear population model in a small random potential and show that under weak assumptions it scales to the linear parabolic Anderson model. MSC: 60H15, 60F05, 30H25 Keywords: paracontrolled distributions; scaling limits; weak universality; Bravais lattices; Besov spaces; parabolic Anderson model § INTRODUCTIONParacontrolled distributions were developed in <cit.> to solve singular SPDEs, stochastic partial differential equations that are ill-posed because of the interplay of very irregular noiseand nonlinearities. A typical example is the two-dimensional continuous parabolic Anderson model,∂_t u = Δ u + u ξ - u ∞,where u_+ ×^2 → and ξ is a space white noise, the centered Gaussian distribution whose covariance is formally given by 𝔼[ξ(x) ξ(y)] = δ(x-y). The irregularity of the white noise prevents the solution from being a smooth function, and therefore the product between u and the distribution ξ is not well defined. To make sense of it we need to eliminate some resonances between u and ξ by performing an infinite renormalization that replaces u ξ by u ξ - u ∞. The motivation for studying singular SPDEs comes from mathematical physics, because they arise in the large scale description of natural microscopic dynamics. For example, if for the parabolic Anderson model we replace the white noise ξ by its periodization over a given box [-L,L]^2, then it was recently shown in <cit.> that the solution u is the limit of u^(t,x) = e^-c^ t v^( t/^2,x/), where v^_+ ×{-L/, …, L/}^2 → solves the lattice equation∂_t v^ = Δ^ v^ +v^η,where Δ^ is the periodic discrete Laplacian and (η(x))_x ∈{-L/, …, L/}^2 is an i.i.d. family of centered random variables with unit variance and sufficiently many moments.Results of this type can be shown by relying more or less directly on paracontrolled distributions as they were developed in <cit.> for functions of a continuous space parameter. But that approach comes at a cost because it requires us to control a certain random operator, which is highly technical and a difficulty that is not inherent to the studied problem. Moreover, it just applies to lattice models with polynomial nonlinearities. See the discussion below for details. Here we formulate a version of paracontrolled distributions that applies directly to functions on Bravais lattices and therefore provides a much simpler way to derive scaling limits and never requires us to bound random operators. Apart from simplifying the arguments, our new approach also allows us to study systems on infinite lattices that converge to equations on ^d, while the formulation of the Fourier extension procedure we sketch below seems much more subtle in the case of an unbounded lattice. Moreover, we can now deal with non-polynomial nonlinearities which is crucial for our main application, a weak universality result for the parabolic Anderson model. Besides extending paracontrolled distributions to Bravais lattices we also develop paracontrolled distributions in weighted function spaces, which allows us to deal with paracontrolled equations on unbounded spaces that involve a spatially homogeneous noise. And finally we develop a general machinery for the use of discrete Wick contractions in the renormalization of discrete, singular SPDEs with i.i.d. noise which is completely analogous to the continuous Gaussian setting, and we build on the techniques of <cit.> to provide a criterion that identifies the scaling limits of discrete Wick products as multiple Wiener-Itô integrals.Our main application is a weak universality result for the two-dimensional parabolic Anderson model. We consider a nonlinear population model v^_+ ×^2 →,∂_t v^(t,x) = Δ^(d) v^(t,x) + F(v^(t,x)) η^(x),where Δ^(d) is the discrete Laplacian, F ∈ C^2 has a bounded second derivative and satisfies F(0) = 0, and (η^(x))_x ∈^2 is an i.i.d. family of random variables with Var(η^(0)) = ^2 and 𝔼[η^(0)] = - F'(0) ^2 c^ for a suitable sequence of diverging constants c^∼ |log|. The variable v^(t,x) describes the population density at time t in the site x. The classical example would be F(u) = u, which corresponds to the discrete parabolic Anderson model in a small potential η^. In that case v^ describes the evolution of a population where every individual performs an independent random walk and finds at every site x either favorable conditions if η^(x) > 0 that allow the individual to reproduce at rate η^(x), or non-favorable conditions if η^(x)<0 that kill the individual at rate -η^(x). We can include some interaction between the individuals by choosing a nonlinear function F. For example, F(u) = u(C - u) models a saturation effect which limits the overall population size in one site to C because of limited resources. In Section <ref> we will prove the following result: [see Theorem <ref>] Assume that F and (η^(x)) satisfy the conditions described above and also that the p-th moment of η^(0) is uniformly bounded infor some p>14. Then there exists a unique solution v^ to (<ref>) with initial condition v^(0,x) = 1_· = 0, up to a possibly finite explosion time T^ with T^→∞ for → 0, and u^(t,x) = ^-2 v^ε(^-2 t, ^-1 x) converges in law to the unique solution u _+ ×^2 → of the linear continuous parabolic Anderson model ∂_t u = Δ u + F'(0) u ξ - F'(0)^2 u ∞,u(0) = δ, where δ denotes the Dirac delta. It may appear more natural to assume that η^(0) is centered. However, we need the small shift of the expectation away from zero in order to create the renormalization -F'(0)^2 u ∞ in the continuous equation. Making the mean of the variables η^(x) slightly negative (assume F|_[0,∞)≥ 0 so that F'(0) ≥ 0) gives us a slightly higher chance for a site to be non-favorable than favorable. Without this, the population size would explode in the scale in which we look at it. A similar effect can also be observed in the Kac-Ising/Kac-Blume-Capel model, where the renormalization appears as a shift of the critical temperature away from its mean field value <cit.>. Note that in the linear case F(u)=u we can always replace η^ by η^ + c if we consider e^c t v^(t) instead. So in that case it is not necessary to assume anything about the expectation of η^, we only have to adapt our reference frame to its mean. The condition p>14 might seem rather arbitrary. Roughly speaking this requirement is needed to apply a form of Kolmogorov's continuity criterion, see Remark <ref> for details.Structure of the paperBelow we provide further references and explain in more details where to place our results in the current research in singular SPDEs and we fix some conventions and notations. In Sections <ref>- <ref> we develop the theory of paracontrolled distributions on unbounded Bravais lattices, and in particular we derive Schauder estimates for quite general random walk semigroups. Section <ref> contains the weak universality result for the parabolic Anderson model, and here we present our general methodology for dealing with multilinear functionals of independent random variables. The appendix contains several proofs that we outsourced. Finally, there is a list of important symbols at the end of the paper. Related worksAs mentioned above, we can also use paracontrolled distributions for functions of a continuous space parameter to deal with lattice systems. The trick, which goes back at least to <cit.> and was inspired by <cit.>, is to consider for a lattice function u^ on say {k: -L/≤ k ≤ L/}^2 the unique periodic function Ext(u^) on ( / (2L))^2 whose Fourier transform is supported in [-1/,1/]^2 and that agrees with u^ in all the lattice points. If the equation for u^ involves only polynomial nonlinearities, we can write down a closed equation for Ext(u^) which looks similar to the equation for u^ but involves a certain “Fourier shuffle” operator that is not continuous on the function spaces in which we would like to control Ext(u^). But by introducing a suitable random operator that has to be controlled with stochastic arguments one can proceed to study the limiting behavior of Ext(u^) and thus of u^. This argument has been applied to show the convergence of lattice systems to the KPZ equation <cit.>, the Φ^4_3 equation <cit.>, and to the parabolic Anderson model <cit.>, and the most technical part of the proof was always the analysis of the random operator. The same argument was also applied to prove the convergence of the Kac-Ising / Kac-Blume-Capel model <cit.> to the Φ^4_2 / Φ^6_2 equation. This case can be handled without paracontrolled distributions, but also here some work is necessary to control the Fourier shuffle operator. This difficulty is of a technical nature and not inherent to the studied problems, and the line of argumentation we present here avoids that problem by analysing directly the lattice equation rather than trying to interpret it as a continuous equation.Other intrinsic approaches to singular SPDEs on lattices have been developed in the context of regularity structures by Hairer, Matetski and Erhard <cit.> and in the context of the semigroup approach to paracontrolled distributions by Bailleul and Bernicot <cit.>, and we expect that both of these works could be combined with our martingale arguments of Section <ref> to give an alternative proof of our weak universality result.We call the convergence of the nonlinear population model to the linear parabolic Anderson model a “weak universality” result in analogy to the weak universality conjecture for the KPZ equation. The (strong) KPZ universality conjecture states that a wide class of (1+1)-dimensional interface growth models scale to the same universal limit, the so called KPZ fixed point <cit.>, while the weak KPZ universality conjecture says that if we change some “asymmetry parameter” in the growth model to vanish at the right rate as we scale out, then the limit of this family of models is the KPZ equation. Similarly, here the influence of the random potential on the population model must vanish at the right rate as we pass to the limit, so the parabolic Anderson model arises as scaling limit of a family of models. Similar weak universality results have recently been shown for other singular SPDEs such as the KPZ equation <cit.> (this list is far from complete), the Φ^2n_d equations <cit.>, or the (stochastic) nonlinear wave equation <cit.>.A key task in singular stochastic PDEs is to renormalize and to construct certain a priori ill-defined products between explicit stochastic processes. This problem already arises in rough paths <cit.> but there it is typically not necessary to perform any renormalizations and general construction and approximation results for Gaussian rough paths were developed in <cit.>. For singular SPDEs the constructions become much more involved and a general construction of regularity structures for equations driven by Gaussian noise was found only recently and is highly nontrivial <cit.>. For Gaussian noise it is natural to regroup polynomials of the noise in terms of Wick products, which goes back at least to <cit.> and is essentially always used in singular SPDEs, see <cit.> and many more. Moreover, in the Gaussian case all moments of polynomials of the noise are equivalent, and therefore it suffices to control variances. In the non-Gaussian case we can still regroup in terms of Wick polynomials <cit.>, but a priori the moments are no longer comparable and new methods are necessary. In <cit.> the authors used martingale inequalities to bound higher order moments in terms of variances.In our case it may look as if there are no martingales around because the noise is constant in time. But if we enumerate the lattice points and sum up our i.i.d. variables along this enumeration, then we generate a martingale. This observation was used in <cit.> to show that for certain polynomial functionals of the noise (“discrete multiple stochastic integrals”) the moments are still comparable, but the approach was somewhat ad-hoc and only applied directly tothe product of two variables in “the first chaos”. Here we develop a general machinery for the use of discrete Wick contractions in the renormalization of discrete, singular SPDEs with i.i.d. noise which is completely analogous to the continuous Gaussian setting. Moreover, we build on the techniques of <cit.> to provide a criterion that identifies the scaling limits of discrete Wick products as multiple Wiener-Itô integrals. Although these techniques are only applied to the discrete 2d parabolic Anderson model, the approach extends in principle to any discrete formulation of popular singular SPDEs such as the KPZ equation or the Φ^4_d models.§.§ Conventions and NotationWe use the common notation ≲, ≳ in estimates to denote ≤, ≥ up to a positive constant. The symbol ≈ means that both ≲ and ≳ hold true. For discrete indices we mean by ij that there is a N≥ 0 (independent of i,j) such that i≤ j+N, i.e. that 2^i2^j, and similarly for ji; the notation i ∼ j is shorthand for i j and j i.We denote partial derivatives by ∂^α for α∈ℕ^d := {0,1,2, …}^d and for α=(_i = j)_j we write ∂^i = ∂^α. Our Fourier transform follows the convention that for f∈ L^1(^d)f(y) :=∫_^d f(x) e^-2π xyx ,^-1 f (x):=∫_^d f(y) e^2π xyy , where xy denotes the usual inner product on ^d. The most relevant notations are listed in a glossary at theend of this article. § WEIGHTED BESOV SPACES ON BRAVAIS LATTICES §.§ Fourier transform on Bravais lattices A Bravais-lattice in d dimensions consists of the integer combinations of d linearly independent vectors a_1,…,a_d ∈^d, that is GG:=a_1 + … +a_d .Given a Bravais lattice we define the basis a_1,…,a_dof the reciprocal lattice by the requirement a_ia_j= δ_ij ,and we set ℛ :=a_1 +… +a_d. CR However, we will mostly work with the (centered) parallelotope which is spanned by the basis vectors a_1,…,a_d: widehatGG:=[0,1) a_1+… +[0,1) a_d-1/2(a_1 + … +a_d)=[-1/2,1/2) a_1+… +[-1/2,1/2) a_d . We callthe bandwidth or Fourier-cell ofto indicate that the Fourier transform of a map onlives on , as we will see below. We also identify ≃^d / ℛ and turninto an additive group which is invariant under translations by elements in ℛ.If we choose the canonical basis vectors a_1=e_1,…,a_d=e_d, we have simply=^d,ℛ = ^d, =𝕋^d=[-1/2,1/2)^d.Compare also the left lattice in Figure <ref>. In Figure <ref> we sketched some Bravais latticestogether with their Fourier cells . Note that the dashed linesbetween the points of the lattice are at this point a purely artistic supplement. However, they will become meaningful later on: If we imagine a particle performing a random walk on the lattice , then the dashed lines could be interpreted as the jumps it is allowed to undertake. From this point of view the lines will be drawn by the diffusion operators we introduce in Section <ref>.Given a Bravais latticeas defined in (<ref>) we write ^:=for the sequence of Bravais lattice we obtain by dyadic rescaling with =2^-N,N≥ 0.Whenever we say a statement (or an estimate) holds for ^ we mean that it holds (uniformly) for all =2^-N, N≥ 0.The restriction to dyadic lattices fits well with the use of Littlewood-Paley theory which is traditionally built from dyadic decompositions. However, it turns out that we do not lose much generality by this. Indeed, all the estimates below will hold uniformly as soon as we know that the scale of our lattice is contained in some interval (c_1,c_2)⊂⊂ (0,∞). Therefore it is sufficient to group the members of any positive null-sequence (_n)_n ≥ 0 in dyadic intervals [2^-(N+1),2^-N) to deduce the general statement.Given φ∈ℓ^1() we define its Fourier transform as FFgφ(x):=|| ∑_k∈φ(k) e^- 2π kx, x ∈,where we introduced a “normalization constant” ||:=|(a_1,…,a_d)|that ensures that we obtain the usual Fourier transform on ^d as || tends to 0. We will also write || for the Lebesgue measure of the Fourier cell .If we consider φ as a map on ^d, then it is periodic under translations in ℛ. By the dominated convergence theorem φ is continuous, so sinceis compact it is in L^1():= L^1(,x), where x denotes integration with respect to the Lebesgue measure. For any ψ∈ L^1() we define its inverse Fourier transform as^-1ψ(k) :=∫_ψ(x) e^2π kx x, k ∈.Note that || = 1/|| and therefore we get at least for φ with finite support ^-1φ = φ. The Schwartz functions onare() := {φ→ : sup_k ∈ (1+|k|)^m |φ(k)| < ∞ for allm ∈},and we have φ∈ C^∞() (with periodic boundary conditions) for all φ∈(), because for any multi-index α∈^d the dominated convergence theorem gives∂^αφ(x) = || ∑_k∈φ (k) (-2π k)^α e^- 2π kx.By the same argument we have ^-1ψ∈() for all ψ∈ C^∞(), and as in the classical case = ^d one can show thatis an isomorphism from () to C^∞() with inverse ^-1. Many relations known from the ^d-case carry over readily to Bravais lattices, e.g. Parseval's identity∑_k∈ ||· |φ(k)|^2=∫_ |φ(x)|^2x(to see this check for example with the Stone-Weierstrass theorem that (||^1/2e^2 π k ·)_k ∈ forms an orthonormal basis of L^2(,x)) and the relation between convolution and multiplication(φ_1 ∗_φ_2)(x) :=(∑_k∈ || φ_1(k) φ_2(·-k) )(x)= φ_1(x) ·φ_2(x),^-1(ψ_2 ∗_ψ_2)(k) :=^-1( ∫_ψ_1(x) ψ_2([-x]_) x )(k)=^-1ψ_1(k) ·^-1ψ_2(k).where [z]_ is for z∈^d the unique element insuch that z-[z]_∈. nwGGSince () consists of functions decaying faster than any polynomial, the Schwartz distributions onare the functions that grow at most polynomially,'() := { f → : sup_k ∈ (1+|k|)^-m |f(k)| < ∞ for somem ∈},and f(φ) := || ∑_k ∈ f(k) φ(k) is well defined for φ∈(). We extend the Fourier transform to '() by setting( f)(ψ):= f( ^-1ψ) = || ∑_k ∈ f(k) ^-1ψ(k), ψ∈ C^∞(),where (…) denotes the complex conjugate. This should be read as ( f)(ψ) = f(ψ), which however does not make any sense because for ψ∈ C^∞() we did not define the Fourier transform ψ but only ^-1ψ. The Fourier transform ( f)(ψ) agrees with ∫_ f(x)·ψ(x) x in case f ∈(). It is possible to show that f∈'(), where'():= {u C^∞() →: uis linear and ∃C>0, m∈ s.t.|u(ψ)| ≤ C ψ_C^m_b()}for ψ_C^m_b() := ∑_|α| ≤ m∂^αψ_L^∞(), and thatis an isomorphism from '() to '() with inverse(^-1 u)(φ) := || ∑_k ∈ u(e^2π k()) φ(k). As in the classical case =ℤ it is easy to see that we can identify every f∈'() with a “Dirac comb” distribution f_dir∈'(^d) by settingf_dir=||∑_k∈ f(k) δ(·-k),where δ(·-k)∈'(^d) denotes a shifted Dirac delta distribution. We can identify any element g∈'() of the frequency space with an ℛ-periodic distribution g_ext∈'(^d) by setting g_ext(φ):=g(∑_k∈ℛφ(·-k)),φ∈(^d) .If g∈'() coincides with a function onone sees thatg_ext(x)=g([x]_)where [x]_ is, as above, the (unique) element [x]_∈ such that [x]_-x∈a_1+…+a_d=. Conversely, every ℛ-periodic distribution g∈'(^d) can be seen as a restricted element g_res∈'(), e.g. by considering g_res(φ):= (ψ· g)(φ_ext)=g(ψ·φ_ext),φ∈ C^∞()where ψ∈ C^∞_c(^d) is chosen such that ∑_k∈ℛψ(·-k)=1 and where we used in the second equality the definition of the product between a smooth function and a distribution. To construct such a ψ it suffices to convolve _ with a smooth, compactly supported mollifier, and it is easy to check that (g_ext)_res = g for all g ∈'() and that g_res does not depend on the choice of ψ. This motivates our definition of the extension operatorbelow in Lemma <ref>.With these identifications in mind we can interpret the concepts introduced above as a sub-theory of the classical Fourier analysis of tempered distributions. We will sometimes use the following identity for f∈'()f=(f_dir) ,which is easily checked using the definitions above. Next, we want to introduce Besov spaces on . Recall that one way of constructing Besov spaces on ^d is by making use of a dyadic partition of unity.A dyadic partition of unity is a family (φ_j)_j≥ -1⊆ C^∞_c(^d) of nonnegative radial functions such that * φ_-1 is contained in a ball around 0, φ_j is contained in an annulus around 0 for j≥ 0, * φ_j=φ_0(2^-j·) for j≥ 0, * ∑_j≥ -1φ_j (x) =1 for any x∈^d, * If |j-j'|>1 we have φ_j ∩φ_j'=∅, Using such a dyadic partition as a family of Fourier multipliers leads to the Littlewood-Paley blocks of a distribution f ∈'(^d),_j f:=^-1 (φ_j · f).Each of these blocks is a smooth function and it represents a “spectral chunk” of the distribution. By choice of the (φ_j)_j≥ -1 we have f=∑_j≥ -1_j f in '(^d), and measuring the explosion/decay of the Littlewood-Paley blocks gives rise to the Besov spacesℬ^α_p,q (^d)={ f∈'(^d) :(2^jα_j f_L^p)_j≥ -1_ℓ^q<∞}. In our case all the information about the Fourier transform of f∈'(), that is f∈'(), is stored in a finite bandwidth . Therefore, it is more natural to decompose the compact set , so that we consider only finitely many blocks. However, there is a small but delicate problem: We should decomposein a smooth periodic way, but if j is such that the support of φ_j touches the boundary of , the function φ_j will not necessarily be smooth in a periodic sense. We therefore redefine the dyadic partition of unity for x∈ asvarphijφ^_j(x)= {[ φ_j(x), j<j_,; 1-∑_j<j_φ_j(x),j=j_ , ]. where jGG j≤ j_:=inf{j : φ_j∩∂≠∅}.Now we set for f∈'()varDeltaGGj^_j f := ^-1(φ^_j · f) ,which is now a function defined on . As in the continuous case we will also use the notation S_j^ f=∑_i<j_i^ f. Of course, for a fixedit may happen that ^_-1 = Id, but if we rescale the latticeto , the Fourier cellchanges to ^-1 and so for → 0 the following definition becomes meaningful. Given α∈ and p,q∈[1,∞] we define ℬ^α_p,q():={ f∈'()| f_ℬ^α_p,q() =(2^jα_j^ f_L^p())_j=-1,…,j__ℓ^q<∞},where we define the L^p() norm by f_L^p():=(|| ∑_k∈ |f(k)|^p )^1/p =||^1/p f _ℓ^p.We write furthermore 𝒞^α_p():=ℬ^α_p,∞(). The reader may have noticed that since we only consider finitely many j=-1,…,j_ (and since _j L^p()→ L^p() is a bounded operator, uniformly in j, as we will see below), the two spacesℬ^α_p,q() and L^p() are in fact identical with equivalent norms! However, since we are interested in uniform bounds onfor → 0, we are of course not allowed to switch between these spaces. Whenever we consider sequences ^ of lattices we construct all dyadic partitions of unity (φ_j^^)_j=-1,…,j_^ from the same partition of unity (φ_j)_j≥ -1 on ^d. With the above constructions at hand it is easy to develop a theory of paracontrolled distributions on a Bravais latticewhich is completely analogous to the one on ^d. For the transition from the rescaled lattice models on ^ to models on the Euclidean space ^d we need to compare discrete and continuous distributions, so we should extend the lattice model to a distribution in '(^d). One way of doing so is to simply consider the identification with a Dirac comb, already mentioned in (<ref>), but this has the disadvantage that the extension can only be controlled in spaces of quite low regularity because the Dirac delta is quite irregular. We find the following extension convenient: Let ψ∈ C^∞_c(^d) be a positive function with ∑_k ∈ℛψ(· - k) ≡ 1 and set f := ^-1(ψ· f),f ∈'(),where the periodic extension ·'()→'(^d) is defined as in (<ref>). Then f ∈ C^∞(^d) ∩'(^d) and f(k) = f(k) for all k ∈. We have f ∈'(^d) becausef is in '(^d), and therefore also f=^-1 ( ψ· f) ∈'(^d).Knowing that f is in '(^d), it must be in C^∞(^d) as well because it has compact spectral support by definition. Moreover,we can write for k ∈ f(k) =f(ψ·e^2 π k (·))=f(∑_r∈ℛψ(·-r) e^2π k(·-r))=ℱ_f(e^2π k (·))=f(k) , where we used the definition of · from (<ref>) and that kr ∈ for all k ∈ and r ∈. It is possible to show that if ^ denotes the extension operator on ^, then the family (^)_ > 0 is uniformly bounded in L(ℬ^α_p,q(^), ℬ_p,q^α(^d)), and this can be used to obtain uniform regularity bounds for the extensions of a given family of lattice models.However, since we are interested in equations with spatially homogeneous noise, we cannot expect the solution to be in ℬ^α_p,q() for any α, p, q and instead we have to consider weighted spaces. In the case of the parabolic Anderson model it turns out to be convenient to even allow for subexponential growth of the form e^|·|^σ for σ∈ (0,1), which means that we have to work on a larger space than '(), where only polynomial growth is allowed. So before we proceed let us first recall the basics of the so called ultra-distributions on ^d. §.§ Ultra-distributions on Euclidean space A drawback of Schwartz's theory of tempered distributions is the restriction to polynomial growth. As we will see later, it is convenient to allow our solution to have subexponential growth of the form e^λ |·|^σ for σ∈ (0,1) and λ > 0. It is therefore necessary to work in a larger space _ω'(^d)⊇'(^d), the space of so called (tempered) ultra-distributions, which has less restrictive growth conditions but on which one still has a Fourier transform. Similar techniques already appear in the context of singular SPDEs in <cit.>, where the authors use Gevrey functions that are characterized by a condition similar to the one in Definition <ref> below. Here, we will follow a slightly different approach that goes back to Beurling and Björck <cit.>, and which mimics essentially the definition of tempered distribution via Schwartz functions. For a broader introduction to ultra-distributions see for example <cit.> or <cit.>. Let us fix, once and for all, the following weight functions which we will use throughout this article.We denote by (x):=log(1+|x|), (x):=|x|^σ, σ∈ (0,1) .where x∈^d, σ∈ (0,1) For ω∈ω:={}∪{ | σ∈ (0,1)} we denote by ρ(ω) the set of measurable, strictly positive ρ:^d→ (0,∞) such thatρ(x)≲ρ(y) e^λω(x-y)for some λ=λ(ρ)>0. We also introduce the notation ():=⋃_ω∈(ω). The objects ρ∈() will be called weights. ww rromegaNote that the sets ρ(ω) are stable under addition and multiplication for a fixed ω∈. The indices “pol” and “exp” of the elements inindicate the fact that elements in ρ∈ρ() are polynomially growing or decaying while elements in ρ() are allowed to have subexponential behavior. Note that()⊆()and that(1+|x|)^λ∈() and e^λ|x|^σ∈() for λ∈, σ∈ (0,1). The reason why we only allow for σ<1 will be explained in Remark <ref> below. We are now ready to define the space of ultra-distributions.We define for ω∈ the locally convex space _ω(^d):={f∈(^d) | ∀λ>0, α∈^d p_α,λ^ω (f)+π_α,λ^ω(f)<∞} ,which is equipped with the seminorms p_α,λ^ω(f):=sup_x∈^d e^λω(x) |∂^α f(x)| , π_α,λ^ω(f):=sup_x∈^d e^λω(x) |∂^α f(x)|.Its topological dual '(^d) is called the space of tempered ultra-distributions. Sww We here follow <cit.> and equip the dual '(^d) with the strong topology. The choice of the weak-* topology is however also common in the literature <cit.>. The reason why we excluded the case σ≥ 1 forin Definition <ref> is that we wantto contain functions with compact support, which then allows for localization and thus for a Littlewood-Paley theory. But if ω= with σ≥ 1 and f∈(^d) the requirement π^ω_0,λ(f)<∞ implies that f can be bounded by e^-c|x|, c>0 , which means that f is analytic and the only compactly supported f∈(^d) is the zero-function f=0.In the case ω=,σ∈ (0,1) the space _ω' is strictly larger than '. Indeed: e^c|·|^σ'∈_ω'(^d)\'(^d) for σ'∈ (0,σ]. In the case ω= we simply have_ω(^d)=(^d)with a topology that can also be generated by only using the seminorms p^ω_α,λ so that the dual of _ω(^d)=(^d) is given by'_ω(^d)='(^d) .The theory of “classical” tempered distributions is therefore contained in the framework above.The role of the triple (^d):=C^∞_c(^d)⊆(^d)⊆ C^∞(^d) in this theory will be substituted by spaces (^d), C^∞_ω(^d) such that (^d)⊆_ω(^d)⊆(^d) . Let U⊆^d be an open set and ω∈={}∪{ | σ∈ (0,1)}. We define for ω= the set (U) to be the space of f∈ C^∞(U) such that for every >0 and compact K⊆ U there exists C_,K>0 such that for all α∈^dsup_K |∂^α f|≤ C_,K ^|α| (α!)^1/σ . For ω= we set C^∞_ω(U)=C^∞(U). We also define(U)=C^∞_ω(U)∩ C^∞_c(U) . The elements of C^∞_ω(U) are called ultra-differentiable functions and the elements of the dual space '(^d) are called ultra-distributions. EwwThe space '(^d) is equipped with a suitable topology <cit.> which we did not specify since this space will not be used in this article and is just mentioned for the sake of completeness.The factor α! in (<ref>) can be replaced by |α|! or |α|^|α| <cit.> as can be easily seen from α!≤ |α|!≤ d^|α|α! and Stirlings formula.The relation between ,, and their properties are specified by the following lemma.Let ω∈. i) We have (^d)⊆(^d) and (^d)=(^d)∩ C^∞_c(^d) .In particular (^d)⊆(^d)⊆C^∞_c(^d). ii) The space (^d) is stable under addition, multiplication and convolution. iii) The space (^d)is stable under addition, multiplication and division in the sense that f/g ·1_ f∈(^d) for f,g ∈(^d),f ⊆∘ g.We only have to prove the statements for ω∈{ | σ∈ (0,1)}. Take f∈(^d) and >0. We then have for α∈^d∂^α f(x)=(2π)^|α|∫_^de^2π x ξ ξ^αf(ξ) ξUsing further that for λ>0 (we here follow <cit.>)∫ |ξ|^|α| e^-λ |ξ|^σξ≲∫_0^∞ r^|α|+d-1 e^-λ r^σ r ≲λ^-|α|/σΓ((|α|+d)/σ)≲λ^-|α|/σ C^|α| |α|^|α|/σ,we obtain for x∈^d|∂^α f (x)|≲ C_λλ^-|α|/σ C^|α| |α|^|α|/σ·π^ω_0,λ(f) .Choosing λ>0 big enough shows that f satisfies the estimate in (<ref>) (with global bounds) and thus f∈(^d) and (^d)⊆(^d). In particular we get (^d)∩ C^∞_c(^d)⊆(^d). To show the inverse inclusion consider f∈(^d). We only have to show that π_α,λ^ω(f)<∞ for any λ>0 and α∈^d. And indeed for x∈^d with |x|≥ 1 (without loss of generality)[We here follow ideas from<cit.>.]| e^λ |x|^σ f(x) |≤∑_k=0^∞λ^k/k! |x|^σ k | f(x)| ≤∑_k=0^∞λ^k C^k/k! |x|^⌈σ k ⌉ | f(x) |≤∑_i=1^d∑_k=0^∞λ^k C^k/k! |x_i|^⌈σ k ⌉ | f(x) | =∑_i=1^d∑_k=0^∞λ^k C^k/k!| ∫ e^2πξ∂^⌈σ k ⌉ e_i f(ξ) ξ|(<ref>) & ≤C_∑_k=0^∞λ^k C^k ^k<∞where C, C_>0 denote as usual constants that may change from line to line and where in the last step we chose >0 small enough to make the series converge; note that the bound (<ref>) holds on all of ^d because f is compactly supported by assumption. The stability of _ω(^d) under addition, multiplication and convolution are quite easy to check, see <cit.>. It is straightforward to check that f· g∈(U) for f,g∈(U) using Leibniz's rule. For the stability under composition see e.g. <cit.>, from which the stability under division can be easily derived.Many linear operations such as addition or derivation that can be defined on distributions can be translated immediately to the space of ultra-distributions ((^d))'. We see with (<ref>) that (^d) should be interpreted as the set of smooth multipliers for ultra-distributions in '(^d) and in particular for tempered ultra-distributions '(^d)⊆'(^d).The space _ω'(^d) is small enough to allow for a Fourier transform.For f∈_ω'(^d) and φ∈_ω(^d) we set f (φ) :=f(φ),^-1 f(φ) :=f(^-1φ).By definition of _ω(^d) we have thatand ^-1 are isomorphisms on _ω(^d) which implies thatand ^-1 are isomorphisms on _ω'(^d).The following lemma proves that the set of compactly supported ultra-differentiable functions (^d) is rich enough to localize ultra-distributions, which gets the Littlewood-Paley theory started and allows us to introduce Besov spaces based on ultra-distributions in the next section.Let ω∈. For every pair of compact sets K⊊ K'⊆^d there is a φ∈(^d) such that φ|_K =1 , φ⊆ K' . §.§ Ultra-distributions on Bravais lattices For the discrete setup we essentially proceed as in Subsection <ref> and define spaces𝒮_ω ()={ f:→ |sup_k∈ e^λω(k) |f(k)|<∞λ>0}, and their duals (when equipped with the natural topology)𝒮_ω'()={ f:→ |sup_k∈ e^-λω(k) |f(k)|<∞λ>0},with the pairing f(φ)=||∑_k∈ f(k) φ(k), φ∈𝒮_ω(). As in Subsection <ref> we can then define a Fourier transformon 𝒮_ω'() which maps the discrete space 𝒮_ω() into the space of ultra-differentiable functions S_ω():=C^∞_ω() with periodic boundary conditions. The dual space _ω'() can be equipped with a Fourier transform ^-1 as in (<ref>) such that ,^-1 become isomorphisms between 𝒮_ω '() and_ω'() that are inverse to each other. For a proof of these statements we refer to Lemma <ref>. Performing identifications as in the case of '(^d) we can interpret these concepts as a sub-theory of the Fourier analysis on _ω'(^d) with the only difference that we have to choose the function ψ, satisfying ∑_k∈ℛψ(· -k)=1, on page IdentificationsFourierTheory as an element of _ω(^d), see page subsec:ExtensionOperator below for details.§.§ Discrete weighted Besov spacesWe can now give our definition of a discrete, weighted Besov space, where we essentially proceed as in Subsection <ref> with the only difference that ρ∈ρ(ω) is included in the definition and that the partition of unity (φ_j)_j≥ -1, from which (φ_j^)_j≥ -1 is constructed as on page DiscreteDyadicPartition, must now be chosen in (^d). Given a Bravais lattice , parameters α∈, p,q∈[1,∞] and a weight ρ∈(ω) for ω∈ we defineℬ^α_p,q(,ρ):={ f∈_ω'()| f_ℬ^α_p,q(,ρ): = ( 2^jαρ·_j^ f_L^p())_j=-1,…,j__ℓ^q<∞} ,where the Littlewood-Paley blocks (_j^)_j=-1,…,j_ are built from a dyadic partition of unity (φ_j^)_j=-1,…,j_⊆() onconstructed from some dyadic partition of unity (φ_j)_j≥ -1⊆(^d) on ^d as on page DiscreteDyadicPartition. If we consider a sequence ^ as in Definition <ref> we take the same (φ_j)_j≥ -1⊆(^d) to construct for allthe partitions (φ_j^)_j=-1,…, j_ on .We write furthermore 𝒞^α_p(,ρ)=ℬ^α_p,∞(,ρ) and defineL^p(,ρ):={f∈_ω()| f_L^p(,ρ):=ρ f_L^p()<∞} ,i.e. f_ℬ^α_p,q(,ρ) =(2^jα_j^ f_L^p(,ρ))_j_ℓ^q. Balphapq mathcalCalphapWhen we introduce the weight we have a choice where to put it. Here we set f_L^p(,ρ)=ρ f_L^p(), which is analogous to <cit.> or <cit.>, but different from <cit.> who instead take the L^p norm under the measure ρ(x)x. For p=1 both definitions coincide, but for p=∞ the weighted L^∞ space of Mourrat and Weber does not feel the weight at all and it coincides with its unweighted counterpart.The formulation of this definition for continuous spaces ℬ^α_p,q(^d,ρ), 𝒞^α_p(^d,ρ) and L^p(^d,ρ) is analogous.We can write the Littlewood-Paley blocks as convolutions (on ):_j^ f(x)=∗_ f(x)= || ∑_k∈(x -k) f(k) ,x ∈ ,where PsiGGj:=^-1φ^_j. We also introduce the notation PsiGGleqjΨ^,<j:=∑_i<jΨ^,j .Due to our convention to only consider dyadic scalings we always have the useful property=2^jdϕ_(2^j·)for a lattice sequence ^ as in Definition <ref>, where = -1, j=-1, 0, -1<j<j_^,∞, j=j_^,and where ϕ_-1, ϕ_0, ϕ_∞∈(^d) are Schwartz functions on ^d with ϕ_∈(^d). The functions ϕ_-1, ϕ_0, ϕ_∞ depend on the latticeused to construct ^= but are independent of . In a way, this is a discrete substitute for the scaling one finds on ^d for Ψ^j:=^-1φ_j =2^jd (^-1φ_0)(2^j·) (for j≥ 0) due to the choice of the dyadic partition of unity in Definition <ref>. We prove the identity (<ref>), together with a similar result for Ψ^,<j, in Lemma <ref> below. It turns out that (<ref>) is helpful in translating arguments from the continuous theory into our discrete framework. Let us once more stress the fact that ϕ_ is defined on all of ^d, and therefore (<ref>) actually makes sense for all x∈^d. With the ϕ_ from Lemma <ref> this “extension” coincides with ^ (Δ^_j f), where the extension operator ^ is defined as in Lemma <ref> below. The following Lemma, a discrete weighted Young inequality, allows us to handle convolutions such as (<ref>). Given ^ as in Definition <ref> and Φ∈_ω(^d) for ω∈ we have for any δ∈ (0,1] with δ≳ and p∈ [1,∞], λ>0 for Φ^δ:=δ^-dΦ(δ^-1·) the boundsup_x∈^dΦ^δ(·+x)_L^p(^,e^λω(·+x))≲δ^-d(1-1/p) .where the implicit constant is independent of >0. In particular, Φ^δ_L^p(^,e^λω)≲δ^-d(1-1/p) and for ρ∈ρ(ω)Φ^δ f_L^p(^,ρ)≲f_L^p(^,ρ),Φ^δ f_L^p(^d,ρ)≲f_L^p(^,ρ) , where we used in the second estimate that x↦ (Φ^δ f)(x)= |^|∑_k∈^Φ^δ(x-k) f(k)can be canonically extended to ^d. Using δ=2^-j for j∈{-1,…,j_^} this covers in particular the functions Ψ^^,j=^-1φ_j^^ via (<ref>). The case p = ∞ follows from the definition of _ω(^d) and e^λω(k)≤ e^λω(δ^-1 k), so that we only have to show the statement for p<∞.And indeed we obtain Φ^δ^p_L^p(^,e^λω) =∑_k∈^ |^| |Φ^δ(k)|^p e^ pλω(k) = δ^- d p^d∑_k∈ || |Φ(δ^-1 k)|^p e^ pλω( k)≤δ^- d p^d ∑_k∈ || |Φ(δ^-1 k)|^p e^pλω(δ^-1 k)≲δ^-d(p-1)∑_k∈ || δ^ -d^d1/1+|δ^-1 k|^d+1≲δ^-d(p-1)∫_^d z(δ^-1)^d 1/1+|δ^-1 z|^d+1≲δ^-d(p-1) ,where we used that Φ∈_ω(^d) and in the application of Lemma <ref> that for |x-y|≲ 1 the quotient 1+|δ^-1 x|/1+|δ^-1 y| is uniformly bounded. Inequality (<ref>) can be proved in the same way since it suffices to take the supremum over |x|≲. The estimates for Φ^δ f then follow by Young's inequality on ^ and a mixed Young inequality, Lemma <ref> below, applied to the right hand side ofρ(x) |Φ^δ f (x)|≤∑_k∈ || ρ(x)|Φ^δ(x - k)|· |f(k)| (⋆)≲∑_k∈ ||e^λω(x-k) |Φ^δ(x-k)|·ρ(k)|f(k)|= |e^λωΦ||ρ f| (x) .In the step (⋆) we used that ρ(x)≲ e^λω(x-k)ρ(k) for some λ>0 due to (<ref>).From Lemma <ref> ( and Remark <ref>) we see in particular that the blocks _j^^ map the space L^p(^,ρ) into itself for any p∈ [1,∞]:_j^ f_L^p(,ρ)=Ψ^,j f_L^p(,ρ)≲f_L^p(,ρ) ,where the involved constant is independent ofand j=-1,…,j_. This is the discrete analogue of the continuous version_j f_L^p(^d,ρ)≲f_L^p(^d,ρ)for j≥ -1 (which can be proved in essentially the same manner).As in the continuous case we can state an embedding theorem for discrete Besov spaces. Since it can be shown exactly as its continuous (and unweighted) cousin (<cit.> or <cit.>) we will not give its proof here. Given ^ as in Definition <ref> for any α_1 ∈, 1≤ p_1 ≤ p_2 ≤∞ ,1≤ q_1≤ q_2 ≤∞ and weights ρ_1,ρ_2 with ρ_2≲ρ_1 we have the continuous embedding (with norm of the embedding operator independent of ∈ (0,1]) ℬ^α_1_p_1,q_1(^,ρ_1) ⊆ℬ^α_2_p_2,q_2(^,ρ_2)for α_2-d/p_2≤α_1-d/p_1. If α_2< α_1-d(1/p_1-1/p_2) and lim_|x|→∞ρ_2(x)/ρ_1(x)=0 the embedding is compact. For later purposes we also recall the continuous version of this embedding. For any α_1 ∈, 1≤ p_1 ≤ p_2 ≤∞ ,1≤ q_1≤ q_2 ≤∞ and weights ρ_1,ρ_2 with ρ_2≲ρ_1 we have the continuous embedding (with norm independent of ε∈ (0,1]) ℬ^α_1_p_1,q_1(^d,ρ_1) ⊆ℬ^α_2_p_2,q_2(^d,ρ_2)for α_2≤α_1-d(1/p_1-1/p_2). If α_2< α_1-d(1/p_1-1/p_2) and lim_|x|→∞ρ_2(x)/ρ_1(x)=0 the embedding is compact.§.§.§ The extension operator Given a Bravais latticeand a dyadic partition of unity (φ_j)_j≥ -1 on ^d such that j_, as defined on page DiscreteDyadicPartition, is strictly greater than 0 we construct a discrete dyadic partition of unity (φ_j^)_-1,…,j_ from (φ_j)_j≥ -1 as on page DiscreteDyadicPartition.We choose a symmetric function ψ∈(^d) which we refer to as the smear function and which satisfies the following properties: * ∑_k ∈ℛψ(·-k)=1,* ψ=1 on φ_j for j<j_,* (ψ∩φ^_j)\≠∅ ⇒ j=j_ .psi The last property looks slightly technical, but actually only states that the support of ψ is small enough such that it only touches the support of the periodically extended φ_j^ with j<j_ inside . Using (∂,⋃_j<j_ (φ^_j)_ext)>0 it is not hard to construct a function ψ as above: Indeed choose via Lemma <ref> some ψ̃∈(^d) that satisfies property 3 and ψ̃|_=1 and set ψ:=ψ̃/∑_k∈ψ̃(·-k).The rescaled ψ^ := ψ(·) satisfies the same properties on ^ (remember that by convention we construct the sequence (φ_j^^)_j=-1,…,j_^ from the same (φ_j)_j≥ -1). This allows us to define an extension operator ^ in the spirit of Lemma <ref> as ^ f:= ℱ^-1_^d(ψ^· f), f∈_ω'(^),mathscrEeps and as in Lemma <ref> we can show that ^ f∈(^d)∩_ω'(^d) and ^ f|_^=f. Using (<ref>) we can give a useful, alternative formulation of ^ f^f=^-1ψ^^-1 f =^-1ψ^ f_dir=^-1ψ^ f=|^|∑_z∈^^-1ψ^(·-z) f(z) ,where as in (<ref>) we readthe convolution in the second line as a function on ^d using that ^-1ψ^∈(^d) is defined on ^d.By property 3 of ψ we also have for j<j_^_j ^ f=^_j^^ fFinally, let us study the interplay of ^ with Besov spaces.For any α∈, p,q∈ [1,∞] and ρ∈()the family of operators^ℬ^α_p,q(^, ρ )⟶ℬ^α_p,q(^d,ρ) , defined above, is uniformly bounded in . We have to estimate _j ^ f for j≥ -1. For j<j_^ we can apply (<ref>) and (<ref>) together with Lemma<ref> to bound _j ^ f_L^p(^d,ρ)=^-d(ψ)(^-1·)_j^^ f_L^p(^d,ρ)≲_j^^ f_L^p(^,ρ)≲ 2^-jαf_ℬ^α_p,q(^, ρ )For j≥ j_^ only j∼j_^ contributes due to the compact support of ψ^. By spectral support properties we have_j ^ f=_j (^∑_i∼ j_^_i^^ f)From (<ref>) we know that _j maps L^p(^d,ρ) into itself and we thus obtain_j ^ f_L^p(^d,ρ)≲^∑_i∼ j_^_i^^ f_L^p(^,ρ)≲ 2^-j_^αf_ℬ^α_p,q(^,ρ) , where we applied once more (<ref>) and Lemma <ref> in the second step.Below, we will often be given some functional F(f_1,…,f_n) on discrete Besov functions taking values in a discrete Besov space X (or some space constructed from it) that satisfies a bound of the typeF(f_1,…,f_n)_X ≤ c(f_1,…,f_n).We then say that the estimate (<ref>) has the property () (on X) if there is a “continuous version” F of F and a continuous version X of X and a sequence of constants o_→ 0such that^ F(f_1,…,f_n)-F(^ f_1,…,^ f_n )_X≤ o_· c(f_1,…,f_n) . In other words we can pull the operator ^ inside F without paying anything in the limit.With the smear function ψ introduced above when can now also give the proof of the announced scaling property (<ref>) of the functions Ψ^^,j. Let ^ be as in Definition <ref> and let ω∈. Let (φ^^_j)_j=-1,…,j_^⊆() be a partition of unity of ^ as defined on page DiscreteDyadicPartition and take Ψ^^,j=^-1φ^^_j and Ψ^^,<j:=∑_i<jΨ^^,i. The extensionsΨ̃^,j :=^Ψ^^,j=^-1_^d (ψ^·φ^^_j)Ψ̃^,<j :=^Ψ^^,<j=^-1_^d(ψ^·(∑_i<jφ^^_i)_ext)are elements of (^d). Moreover there are ϕ̌_-1,ϕ̌_0,ϕ̌_∞, ϕ̌_Σ∈(^d), independent of , such that for for j=-1,…,j_^ and j'=0,…,j_^ withas in (<ref>)ψ^·φ^^_j=ϕ̌_(2^-j·),ψ^·(∑_i<j'φ^^_i)_ext=ϕ̌_Σ(2^-j'·) .The functions ϕ̌_0 and ϕ̌_∞ have support in an annulus ⊆^d.In particular we have for j=-1,…,j_^ andj'=0,…,j_^. Ψ̃^,j=2^jd·ϕ_(2^j ·) ,Ψ̃^,<j'=2^j'd·ϕ_Σ(2^j'·)where ϕ_i:=^-1_^dϕ̌_i for i∈{-1,0,∞, Σ}.Denote by (φ_j)_j ≥ -1⊆(^d) the partition of unity on ^d from which the partitions (φ_j^)_j=-1,…,j_ are constructed. Let us recall the following facts about (φ_j)_j≥ -1φ_j=φ_0(2^-j·)∑_i<j'φ_i= φ_-1(2^-j'·)The second property can be seen by rewriting ∑_i<j'φ_i =1-∑_l≥ j'φ_0(2^-l·)=1-∑_l'≥ 0φ_0( 2^-(j'+l')·)=( 1-∑_l'≥ 0φ_l')(2^-j'· )=φ_-1(2^-j'·) .Recall further that φ_0 has support in an annulus around 0. To prove the claim we only have to show (<ref>) and (<ref>). For j<j_^ and 0≤ j'≤ j_^ we use that by construction of φ^^_j out of(φ_j)_j≥ -1 we have inside φ^^_j=φ_j ,∑_i<j'φ^^_i=∑_i<j'φ_iso that due to property 2 and 3 of the smear function ψ^ and (<ref>) it is enough to take ϕ̌_Σ=φ_-1and for j<j_^ by the scaling property of φ_j from (<ref>)ϕ̌_:=φ_j(2^j·)∈{φ_-1(·/2), φ_0} .For the construction of ϕ_∞ a bit more work is required.Recall that by definition of our lattice sequence ^ we took a dyadic scaling =2^-N which implies in particular 2^-j_^=· 2^kfor some fixed k∈. Using once more (<ref>) and relation (<ref>) we can write for x∈φ^^_j_^(x)=1-∑_j<j_^φ_j(x)=1-φ_-1(2^-j_^x)=χ( x)for some symmetric function χ∈(^d). As in (<ref>) let us denote for x∈^d by [x]_∈^ε the unique element of ^ε for which x-[x]_∈^. One then easily checks[x]_= [ x]_ .Applying (<ref>) and (<ref>) we obtain for x∈^d that the periodic extensionφ^^_j_^(x)=φ^^_j_^([x]_)=χ( [x]_)=χ( [ x]_)is thescaled version of the smooth, -periodic function χ([·]_)∈() (to see that the composition with [·]_ does not change the smoothness, note that χ equals 1 on a neighborhood of ∂). Consequentlyψ(·) φ^^_j_^=(ψχ([·]_))(·) ,so that setting ϕ̌_∞=(ψχ([·]_))(2^-k·) with k as in (<ref>) finishes the proof.§ DISCRETE DIFFUSION OPERATORSOur aim is to analyze differential equations on Bravais lattice that are in a certain sense semilinear and “parabolic”, i.e. there is a leading order linear difference operator, which here we will always take as the infinitesimal generator of a random walk on our Bravais lattice. In the following we analyze the regularization properties of the corresponding “heat kernel”.§.§ Definitions Let us construct a symmetric random walk on a Bravais lattice ^ with mesh sizewhich can reach every point (our construction follows <cit.>). First we choose a subset of “jump directions” {g_1,…,g_l}⊆\{0} such that g_1 + … +g_l = and a map κ{ g_1, …,g_l }→ (0,∞). We then take as a rate for the jump from z∈^ to z± g_i∈^ the value κ(g_i)/2^2. In other words the generator of the random walk isL^ u (y)= ^-2∑_e∈{± g_i}κ(e)/2 (u(y+ e)-u(y)),which converges (for u ∈ C^2(^d)) pointwise to L u =1/2∑_i=1^lκ(g_i)g_i∇^2 u g_i astends to 0. In the case =^d and κ(e_i)=1/d we obtain the simple random walk with limiting generator L=1/2dΔ.We can reformulate (<ref>) by introducing a signed measure μ = κ(g_1) (1/2δ_g_1+1/2δ_-g_1)+… + κ(g_l)(1/2δ_g_l+1/2δ_-g_l)- ∑_i=1^l κ(g_i) δ_0, which allows us to write L^ u= ^-2∫_^d u(x+ y)μ (y) and L u= 1/2∫_^d y∇^2 uyμ(y). In fact we will also allow the random walk to have infinite range.We write μ∈(ω)=(ω,) for ω∈ if μ is a finite, signed measure on a Bravais latticesuch that * ⟨μ⟩ =, * μ|_{0}^c≥ 0, * for any λ>0 we have ∫_ e^λω(x)|μ| (x)<∞, where |μ| is the total variation of μ, * μ(A)=μ(-A) for A⊆ and μ()=0,where ⟨·⟩ denotes thesubgroup generated by · in (,+). We associate a norm on ^d to μ∈(ω) which is given by x_μ^2 =1/2∫_ | x y |^2 μ(y) .We also write ():=⋃_ω∈(ω). mmomega The function ‖·‖_μ of Definition <ref> is indeed a norm. The homogeneity is obvious and the triangle inequality follows from Minkowski's inequality. If x_μ=0 we have x g=0 for all g∈μ. Since ⟨μ⟩ = we also have x a_i=0 for the linearly independent vectors a_1,…,a_d from (<ref>), which implies x=0.Given μ∈() as in Definition <ref> we can then generalize the formulas we found above.Forω∈, μ∈(ω) as in Definition <ref> and ^ as in Definition <ref> we set L^_μ u (x)= ^-2∫_ u(x+ y) μ(y)for u∈_ω'(^) and (L_μ u) (φ):= 1/2∫_ y ∇^2 uy μ(y) (φ):= 1/2∫_ y ∇^2 u(φ)yμ (y)for u∈'_ω(^d) and φ∈_ω(^d). We write further ℒ^_μ,ℒ_μ for the parabolic operators ℒ^_μ=∂_t - L^_μ and ℒ_μ =∂_t-L_μ.mathscrLepsmu L^_μ is nothing but the infinitesimal generator of a random walk with sub-exponential moments (Lemma <ref>). By direct computation it can be checked that for =^d and with the extra condition ∫ y_i y_j μ(y)=2 δ_ij we have the identities ‖·‖_μ=|·| and L_μ=Δ_^d. In general L_μ is an elliptic operator with constant coefficients,L_μu =1/2∫_ y ∇^2 uyμ(y)= 1/2∑_i,j∫_y_i y_jμ(y) ·∂^ij u =:1/2∑_i,ja^μ_ij·∂^iju,where (a^μ_ij) is a symmetric matrix. The ellipticity condition follows from the relation x(a^ μ_ij)x = 2 x_μ^2 and the equivalence of norms on ^d. In terms of regularity we expect therefore that L^_μ behaves like the Laplacian when we work on discrete spaces.We have for α∈, p∈ [1,∞], ω∈ and μ∈(ω), ρ∈ (ω) L^_μ u _𝒞^α-2_p(^,ρ)≲u_𝒞^α_p(^,ρ) ,where 𝒞^α_p(^,ρ)=^α_p,∞(^,ρ) is as in Definition <ref>, and where the implicit constant is independent of . For δ∈ [0,1] we further have(L^_μ-L_μ )u_𝒞^α-2-δ_p(^d,ρ)≲^δu_𝒞^α_p(^d,ρ) ,where the action of L^_μ on u∈'(^d) should be read as (L^_μ u)(φ)= u(^-2∫_φ(·+ y)μ(y) ) = u(^-2∫_φ(·- y)μ(y) ) = u(L^ε_μφ)for φ∈(^d), where we used the symmetry of μ in the second step. We start with the first inequality. With Ψ^^,j:=∑_-1≤ i ≤ j_^:|i-j|≤ 1Ψ^^,i∈_ω(^) we have by spectral support properties _j^^ u=Ψ^^, j_j^^u. Via (<ref>) we can read Ψ^^,j and thus Ψ^j,^ as a smooth function in (^d) defined on all of ^d. In this sense we read_j^^ u=|^|∑_z∈ |^|Ψ^^, j(·-z) _j^^ u(z) ,as a smooth function on ^d in the following. Since μ integrates affine functions to zero we can rewrite_j^^ L_μ^ u(x) =^-2∫_ μ (y)[_j^^ u(x+ y)-_j^^ u(x)-∇(_j^^ u)(x)· y] =∫_ μ(y) ∫_0^1 ζ_1 ∫_0^1 ζ_2 y∇^2(_j^^ u)(x+ζ_1 ζ_2 y) y.Using (<ref>) and the Minkowski inequality on the support of μ we then obtainρ_j^^ L_μ^ u _L^p(^)≲∫_ μ(y) ∫_0^1 ζ_1∫_0^1 ζ_2 e^λω(ζ_1 ζ_2 y) |y|^2 ρ(·+ζ_1 ζ_2 y) |∇^2(_j^^ u)(· +ζ_1 ζ_2 y)| _L^p(^) ,where λ is as in (<ref>). By definition of (ω) and monotonicity of ω∈ we have∫_0^1 ζ_1 ∫_0^1 ζ_2 ∫_μ(y)|y|^2 e^λω(ζ_1 ζ_2 y)≤∫_0^1 ζ_1 ∫_0^1 ζ_2 ∫_μ(y)|y|^2 e^λω( y)<∞so that we are left with the task of estimatingρ(·+ζ_1 ζ_2 y) |∇^2(_j^^ u)(· +ζ_1 ζ_2 y)| _L^p(^)≲∇^2Ψ^^,j(·+ζ_1ζ_2)_L^1(^,e^λω(·+ζ_1ζ_2)) _j^^ u_L^p(^,ρ) ,where we applied (<ref>) and Young's convolution inequality on ^. Due to (<ref>) and Lemma <ref> we can estimate the first factor by 2^j2 so that we obtain the total estimate_j^^ L_μ^ u _L^p(^,ρ)≲ 2^-j(α-2)u_𝒞^α_p(^,ρ)and the first estimate follows. To show the second inequality we proceed essentially the same but use instead Ψ^j=∑_i: |i-j|≤ 1Ψ^i, where Ψ^j=^-1_^dφ_j now really denotes the inverse transform of the partition (φ_j)_j≥ -1 on all of ^d. We then have _j=Ψ^j ∗_j, so that _j(L^_μ -L_μ)u=∫_0^1 ζ_1 ∫_0^1 ζ_2∫_μ(y) ∫_^d z y(∇^2Ψ^j(· +ζ_1ζ_2 y-z)-∇^2Ψ^j(· -z)) y_j u(z) .As above we can then either get 2^-j(α-2)u_𝒞^α_p(^,ρ), by bounding each of the two second derivatives separately, or2^-j(α-3)u_𝒞^α_p(^,ρ), by exploiting the difference to introduce the third derivative. We obtain the second estimate by interpolation. §.§ Semigroup estimatesIn Fourier space L^_μ can be represented by a Fourier multiplier l^_μ: →:(L_μ^ u)=-l^_μ· u, lepsmu for u∈'(^). The multiplier l^_μ is given by l^_μ(x)=-∫_e^ 2π xy/^2 μ(y)=∫_1-cos( 2π xy)/^2 μ(y)=2∫_sin^2(π xy)/^2 μ(y) ,where we used that μ is symmetric with μ()=0 and the trigonometric identity 1-cos=2sin^2.The following lemma shows that l^_μ is well defined as a multiplier (i.e. l^_μ∈ C^∞_ω(^)). It is moreover the backbone of the semigroup estimates shown below. Let ω∈ and μ∈(ω). The function l^_μ defined in (<ref>) is an element of _ω(^)=C^∞_ω(^) and * if ω= with σ∈ (0,1) it satisfies |∂^k l_μ^(x)|≲_δ^(|k|-2)∨ 0 (1+|x|^2) δ^|k| (k!)^1/σ for any δ>0, k ∈^d, * for every compact set K ⊆^d with K∩ℛ={0}, where ℛ is the reciprocal lattice of the unscaled lattice , we have l^_μ(x) ≳_K | x|^2 for all x ∈^-1 K.The implicit constants are independent of .We start by showing |∂^k l^_μ(x)|≲_δ^(|k|-2)∨ 0 (1+|x|^2) δ^|k| (k!)^1/σ if ω =,which implies in particular l^_μ∈_ω(^) in that case. The proof that l^_μ∈_ω(^) for μ∈() is again similar but easier and therefore omitted. We study derivatives with |k|=0,1 first. We have |l^_μ(x)|=2|∫_sin^2 (π x y)/^2μ(y) |≲|∫_sin^2 (π x y)/|π x y|^2 |x y|^2μ(y) | ≲∫_ |y|^2|μ| (y)·|x|^2 ≲ |x|^2,and for i=1,…,d|∂^i l_μ^(x)|| ≲∫_|sin(π x y)|/|π x y| |x||y|^2|μ| (y)≲ |x| .For higher derivatives we use that ∂_x^k e^ 2π x y=( 2π)^|k| y^k e^ 2π x y which gives (where C>0 denotes as usual a changing constant) |∂^k l_μ^(x)|≤^|k|-2 C^|k|∫_ |y|^|k| |μ|(y)≤^|k|-2 C^|k|max_t ≥ 0 (t^|k| e^-λ t^σ) ∫_ e^λ |y|^σμ(y)for any λ>0.Using max_t≥ 0 t^a e^-λ t^σ= λ^-a/σ (a/σ)^a/σ e^-a/σ for a>0 we end up with |∂^k l_μ^(x)|≲^|k|-21/λ^|k|/σ C^|k| |k|^|k|/σ≲^|k|-21/λ^|k|/σC^|k|(k!)^1/σ ,and our first claim follows by choosing λ^1/σ := C/δ. It remains to show that l_μ^/|·|^2≳ 1 on ^-1 K, which is equivalent to l_μ^1/|·|^2≳ 1 on K. We start by finding the zeros of l^1_μ which, by periodicity can be reduced to finding all x∈ with l^1_μ(x)=0. But if l^1_μ(x)=0, then y x∈ for any y∈μ, which yields with ⟨μ⟩= that we must have a_ix∈ for a_i as in (<ref>). But since x ∈ we have x=x_1 â_1+… +x_d â_d with x_i∈ [-1/2,1/2) and â_i as in (<ref>). Consequently x_i=x a_i∈∩ [-1/2,1/2)={0} , and hence x = 0. Since l^1_μ is periodic under translations in the reciprocal lattice ℛ, its zero set is thus precisely ℛ. By assumption K∩ℛ={0} and it remains therefore to verify l^1_μ(x) ≳ |x|^2 in an environment of 0 to finish the proof.Note that there is a finite subset V⊆μ such that 0∈ V and ⟨ V ⟩ =, sinceonly finitely many y∈μ are needed to generate a_1,…,a_d. We restrict ourselves to V:l^1_μ(x) =2 ∫_sin^2 (π x y) μ(y) ≥ 2 ∫_Vsin^2 (π x y) μ(y)For x∈\{0} small enough we can now bound ∫_Vsin^2 (π x y) μ(y) ≳∫_V |x y|^2 μ(y). The term on the right hand side defines (the square of) a norm by the same arguments as in Lemma <ref>, and since it must be equivalent to |·|^2 the proof is complete. Using that _ω(^)=C^∞_ω(^) is stable under composition with functions in (^d) we see that e^-t l^_μ∈ C^∞_ω(^) for t≥ 0 and can thus define the Fourier multipliere^t L^_μ f:= ^-1 (e^-t l^_μ f)for t≥ 0 and f∈_ω'(^), which gives the (weak) solution to the problem ℒ_μ^ g=0, g(0)=f. The regularizing effect of the semigroup is described in the following proposition.We have for α∈,β≥ 0, p∈ [1,∞], ω∈, μ∈(ω) and ρ∈(ω) e^t L^_μ f _𝒞_p^α+β(^,ρ) ≲ t^-β/2 f _𝒞^α_p(^,ρ) , e^t L^_μ f_𝒞^β_p(^,ρ) ≲ t^-β/2f_L^p(^,ρ) ,and for α∈ (0,2)(e^t L^_μ-Id) f_L^p(^,ρ)≲ t^α/2f_𝒞^α_p(^,ρ) ,uniformly on compact intervals t∈ [0,T]. The involved constants are independent of .We show the claim for ω==|x|^σ, σ∈ (0,1) , the arguments for ω= are similar but easier.Using spectral support properties we can rewrite for j= -1,…,j_Δ_j^ e^tL^_μ f =^-1(∑_i: |i-j|≤ 1φ^^_i e^-t l^_μ·_j^^ f )=𝒦_j(t,·)_j^^ f ,where we set for z∈^𝒦_j(t,z):=∫_ y e^2π z y∑_i: |i-j|≤ 1φ^^_i(y) e^-t l^_μ(y). Using the smear function ψ^=ψ(·) from Subsection <ref> we can rewrite this as an expression that is well-defined for all x∈^d𝒦_j(t,x):=∫_^d y e^2π xyψ^(y)∑_i: |i-j|≤ 1φ^^_i(y) · e^-t l^_μ(y) ,where · is given as in (<ref>) and where we extended l^_μ (periodically) to all of ^d by relation (<ref>).Consequently, we can apply Lemma <ref> to give an expression for the scaled kernel𝒦_(j)(t,x):=2^-jd𝒦_j(t,2^-jx)=∫_^d y e^2π xy φ_(j)(y) · e^-t l^_μ(2^j y) ,where we wrote φ_(j)=∑_i: |i-j|≤ 1ϕ̌_⟨ i ⟩_(2^-(i-j)·) with ϕ̌_⟨ i ⟩_ as in Lemma <ref>. Suppose we already know that for any λ>0 and x∈^ the estimate|𝒦_(j)(t,x)|≲_λe^-λ |x|^σ 2^-j β t^-β/2=: 2^-jβ t^-β/2Φ(x)holds. We then obtain from (<ref>) with Φ^2^-j(x):=2^jdΦ(2^j x)=2^jde^-λ |2^j x|^σ the boundΔ_j^ e^tL^_μ f_L^p(^,ρ)≲ 2^-jβ t^-β/2Φ^2^-j |Δ_j^ e^tL^_μ f|_L^p(^,ρ)and an application of Lemma <ref> shows (<ref>) and (<ref>) (for (<ref>) we also need (<ref>)). Note that we cheated a little bit as Lemma <ref> actually requires Φ∈(^d) which is not true, inspecting however the proof of Lemma <ref> we see that all we used was a suitable decay behavior which is still given. We will now show (<ref>).Using Lemma <ref> below we can reduce this task to the simpler problem of proving the polynomial bound for i=1,…,d and n∈t^β/2 |x_i|^n |𝒦_(j)(t,x)|≲_δδ^n C^n (n!)^1/σ 2^-jβ,δ > 0,with a constant C>0 that does not depend on δ. To show (<ref>) we assume that 2^j≤ 1. Otherwise we are dealing with the scale 2^j≈^-1 and the arguments below can be easily modified. Integration by parts gives|x_i|^n |𝒦_(j)(t,x)|= C^n | ∫_^d y e^2π xy ∂^n· e_i(φ_(j)e^-t l^_μ(2^j· ))(y) | ≤ C^n ∫_^d y | ∂^n· e_i(φ_(j) e^-t 2^2j l_μ^2^j)(y) | ,where we used that l^_μ(2^j y )=2^2j l_μ^2^j(y) by (<ref>). Now we have the following estimates for k∈|∂^k· e_iφ_(j)(y)| ≲_δδ^k (k!)^1/σ , |∂^k· e_i l^ 2^j_μ(y)| ≲_δδ^k (k!)^1/σ,|(2^2j t)^β/2∂^k (e^t2^2j·)(l^2^j _μ(y))| ≲_δ k^k/σδ^k ,where we used that φ_(j)∈(^d) (with bounds that can be chosen independently of j by definition) and we applied Lemma <ref> with the assumption 2^j≤ 1 (which we need because we only defined l^'_μ for ' ≤ 1). Together with Leibniz's and Faà-di Bruno's formula and a lengthy but elementary calculation (<ref>) follows, which finishes the proof of (<ref>) and (<ref>).The last estimate (<ref>) can be obtained as in the proof of Lemma <cit.> by using Lemma <ref> below.Let g^d→, σ>0 and B>0. Suppose for any δ>0 there is a C_δ>0 such that for all z∈^d, l≥ 0 and i=1,…,d |z_i^lg(z)|≲_δδ^l C_δ^l (l!)^1/σB .It then holds for any λ>0 and z∈^d |g(z)|≲_λ Be^-λ |z|^σ .This follows ideas from <cit.>. Without loss of generality we can assume |z|>1 (otherwise we get the required estimate by taking l=0). Recall that we have |z|^l ≤ C^l ∑_i=1^d |z_i|^l, where C>0 denotes a constant that changes from line to line and is independent of l. Consequently,Stirling's formula gives |e^λ |z|^σ g(z)|=|∑_k=0^∞λ^k/k! |z|^σ k g(z) | ≲∑_k=0^∞λ^k C^k/k^k |z|^⌈ kσ⌉|g(z)| ≲∑_k=0^∞λ^k C^k/k^k∑_i=1^d |z_i^⌈ kσ⌉ g(z)| ≲ B ∑_k=0^∞λ^k C^k δ^kσ/k^k⌈ kσ⌉^⌈ kσ⌉ /σ≲ B ∑_k=0^∞λ^k C^k δ^kσ/k^kk^k = B ∑_k=0^∞λ^k C^k δ^kσ≲_λ B ,where we used ⌈ kσ⌉≤ k⌈σ⌉ so that ⌈ kσ⌉^⌈ kσ⌉ /σ≤ (⌈σ⌉ k)^kσ+1/σ≲ C^k k^k and where we chose δ<(C λ)^-1/σ in the last step.§.§ Schauder estimatesWe will follow here closely <cit.> and introduce time-weighted parabolic spaces ℒ^γ,α_p,T that interplay nicely with the semigroup e^t L^_μ. Given γ≥0, T>0 and an increasing family of normed spaces X=(X(s))_s∈ [0,T] we define the spaceℳ^γ_T X:={ f[0,T]→ X(T)| f_ℳ^γ_T X=sup_t∈ [0,T]t^γ f(t)_X(t)<∞} , and for α>0C^α_T X:={f∈ C([0,T],X(T)) | f_C^α_T X<∞} ,wheref_C^α_T X :=sup_t∈ [0,T]f(t)_X(t) + sup_0≤ s≤ t≤ Tf(s)-f(t)_X(t)/|s-t|^α .For a lattice , parameters γ≥ 0,T>0,α≥0, p∈ [1,∞] and a pointwise decreasing map ρ[0,T]∋ t ↦ρ(t) ∈ρ(ω) we set ℒ^γ,α_p,T(, ρ) := {f[0,T] →𝒮_ω'() | f_ℒ^γ,α_p,T(,ρ)<∞} ,wheref_ℒ^γ,α_p,T(,ρ) := t↦ t^γ f(t)_C^α/2_T L^p(,ρ)+f_ℳ^γ_T 𝒞^α_p(,ρ) .mathscrLgammaalphapT mathcalMgammaTX As in Remark <ref> the definition of the continuous version ℒ^γ,α_p,T(^d, ρ) is analogous. Standard arguments show that if X is a sequence of increasing Banach spaces with decreasing norms, all the spaces in the previous definition are in fact complete in their (semi-)norms.The Schauder estimates for the operatorI^_μ f(t)=∫_0^t e^(t-s) L^_μf(s) sand the semigroup (e^tL^_μ) in the time-weighted setup aresummarized in the following lemma, for which we introduce the weights pkappaesigmalp^κ(x)=(1+|x|)^-κe^σ_l+t(x) =e^-(l+t)(1+|x|)^σwith κ>0 and l,t∈. The parameter t should be thought of as time. The notation ℒ^γ,α_p,T(,e^σ_l) means therefore that we take the time-dependent weight (e^σ_l+t)_t ∈ [0,T], while e^σ_l p^κ stands for the time-dependent weight (e^σ_l+t p^κ)_t ∈ [0,T].Let ^ be as in Definition <ref>, α∈ (0,2),γ∈ [0,1), p∈ [1,∞], σ∈ (0,1) and T>0. If β∈ is such that (α+β)/2∈ [0,1), then we have uniformly in s↦ e^sL^_μf_0_ℒ^(α+β)/2,α_p,T(^,e^σ_l)≲f_0_𝒞^-β_p(^,e^σ_l) ,and if κ≥ 0 is such that γ+κ/σ∈ [0,1), α+2κ/σ∈ (0,2) alsoI^_μ f_ℒ^γ,α_p,T(^,e^σ_l)≲f_ℳ^γ_T𝒞_p^α+2κ/σ-2(^,e^σ_l p^κ) . The proof is along the lines of Lemma 6.6 in <cit.> with the use of the simple estimatep^κ e^σ_l+s≲e^σ_l+t/|t-s|^κ/σ , t ≥ s, which is similar to an inequality from the proof of Proposition 4.2 in <cit.> and the reason for the appearance of the term 2κ/σ in (<ref>) (the factor 2 comes from parabolic scaling). We need γ+κ/σ∈ [0,1) so that the singularity |t-s|^-γ - κ/σ is integrable on [0,t]. For the comparison of the parabolic spaces ℒ^γ,α_p,T the following lemma will be convenient. Let ^ be as in Definition <ref>. For α∈ (0,2), γ∈ (0,1), ∈ [0,α∧ 2γ), p∈ [1,∞], T>0 and a pointwise decreasing _+∋s ↦ρ(s)∈ρ(ω) we have f_ℒ^γ-/2,α-_p,T(^,ρ)≲f_ℒ^γ,α_p,T(^,ρ) ,and for γ∈ [0,1) and ∈ (0,α)f_ℒ^γ,α-_p,T(^,ρ)≲1_γ=0f(0)_𝒞^α-_p(^,ρ)+ T^/2f_ℒ^γ,α_p,T(^,ρ) . All involved constants are independent of . The first estimate is proved as in <cit.>. For γ=0 the proof of the second inequality works as in Lemma 2.11 of <cit.>. The general case follows from the fact that f∈ℒ^γ,α_p,T if and only if t↦ t^γ f∈ℒ^0,α_p,T.§ PARACONTROLLED ANALYSIS ON BRAVAIS LATTICES §.§ Discrete Paracontrolled Calculus Given two distributions f_1,f_2∈'(^d), Bony <cit.> defines their paraproduct asf_1 f_2 := ∑_1≤ j_2∑_-1≤ j_1<j_2-1_j_1 f_1 ·_j_2 f_2 = ∑_1≤ j_2 S_j_2-1f_1 ·_j_2 f_2 ,which turns out to always be a well-defined expression. However, to make sense of the product f_1 f_2 it is not sufficient to consider f_1f_2 and f_1f_2 := f_2f_1, we also have to take into account the resonant term <cit.>f_1 f_2 := ∑_-1≤ j_1, j_2:|j_1-j_2|≤ 1_j_1 f_1 ·_j_2 f_2 ,which can in general only be defined under compatible regularity conditions such as f_1∈𝒞^α_∞(^d), f_2∈𝒞_∞^β(^d) with α+β>0 (see e.g. <cit.> or <cit.>). If these conditions are satisfied we decompose f_1 f_2=f_1f_2+ f_1f_2 +f_1f_2. Bony's construction can easily be adapted to a discrete and weighted setup, where of course we have no problem in making sense of pointwise products but we are interested in uniform estimates.Letbe a Bravais lattice, ω∈ and f_1,f_2 ∈_ω'(^d). We define the discrete paraproductf_1 ^ f_2 := ∑_1≤ j_2≤ j_∑_-1≤ j_1<j_2-1_j_1^ f_1 ·_j_2^ f_2 =∑_1≤ j ≤ j_ S_j-1^ f_1 ·_j f_2,where the discrete Littlewood-Paley blocks _j^ are constructed as in Section <ref>.We also write f_1 ^ f_2 : = f_2 ^ f_1. The discrete resonant term is given byf_1 ^ f_2 := ∑_1≤ j_1,j_2 ≤ j_, |j_1-j_2|≤ 1_j_1^ f_1 ·_j_2^ f_2.If there is no risk for confusion we may drop the indexon , , and . para reso In contrast to the continuous theory ^ is well defined without any further restrictions since it only involves a finite sum. All the estimates that are known from the continuous theory carry over.Given ^ as in Definition <ref>, ρ_1,ρ_2∈ρ() and p∈ [1,∞] we have the bounds: * For any α_2∈ f_1f_2_𝒞_p^α_2(^,ρ_1·ρ_2)≲f_1_L^∞(^,ρ_1) f_2_𝒞_p^α_2(^,ρ_2)∧f_1_L^p(^,ρ_1) f_2_𝒞^α_2_∞(^,ρ_2) , * for any α_1<0, α_2∈f_1f_2_𝒞_p^α_1+α_2(^,ρ_1·ρ_2)≲f_1_𝒞_p^α_1(^,ρ_1) f_2_𝒞_∞^α_2(^,ρ_2)∧f_1_𝒞_∞^α_1(^,ρ_1) f_2_𝒞_p^α_2(^,ρ_2) , * for any α_1,α_2∈ with α_1+α_2>0f_1f_2_𝒞_p^α_1+α_2(^,ρ_1·ρ_2)≲f_1_𝒞_p^α_1(^,ρ_1) f_2_𝒞_∞^α_2(^,ρ_2)∧f_1_𝒞_p^α_1(^,ρ_1) f_2_𝒞_∞^α_2(^,ρ_2) ,where all involved constants only depend onbut not on . All estimates have the property (<ref>) if the regularity on the left hand side is lowered by an arbitrary κ>0.Similarly as in the continuous case S_j-1^ f_1 ·_j^ f_2 is spectrally supported on a set of the form 2^j∩, whereis an annulus around 0. Similarly, we have for i,j with i∼ j that the function _i^f_1·^_j f_2 is spectrally supported in a set of the form 2^j ∩, whereis a ball around 0. We give a proof of these two facts in the appendix (Lemma <ref>). Using these two observations the proof of the estimates in (i.)-(iii.) follows along the lines of <cit.>) (which in turn is taken from <cit.>).We are left with the task of proving the property (<ref>). We show in Lemma <ref> below that there is an N∈ (independent ofand j) such that for -1≤ i≤ j ≤ j_-N^(_i^ f_1 ·_j^ f_2) = _i ^ f_1·_j ^ f_2 . Consequently we can write^(f_1 ^ f_2)=∑_1≤ j≤ j_^( S_j-1^^ f_1·_j^ f_2 )= ∑_1≤ j≤ j_-N S_j-1^ f_1 ·_j ^ f_2+ ∑_j_-N< j≤ j_^( S_j-1^^ f_1·_j^ f_2 ) ,where we used (<ref>) and S_j-1^=∑_-1≤ i <j-1_i^, S_j-1=∑_-1≤ i< j-1_i. On the other hand we can write^ f_1 ^ f_2=∑_ 1≤ j S_j-1^ f_1 ·_j^ f=∑_1≤ j≤ j_-N S_j-1^ f_1 ·_j ^ f_2+ ∑_j∼ j_ S_j-1^ f_1 ·_j ^ f_2 ,where we used in the second step that ^ f_2=(ψ(·) ( f_2)_ext) is spectrally supported in a ball of size ^-1≈ 2^j_^ to drop all j with j j_. In total we obtain^(f_1 ^ f_2)-^ f_1 ^ f_2=∑_j∼ j_^( S_j-1^^ f_1·_j^ f_2 ) - ∑_j∼ j_ S_j-1^ f_1 ·_j ^ f_2 . Note that the two sums on the right hand side are spectrally supported in an annulus of size 2^j_. Using Lemma <ref>, the fact _i L^p(^d,ρ)→ L^p(^d,ρ) (by (<ref>)) and that ^ L^p(^,ρ)→ L^p(^d,ρ) (due to (<ref>) and Lemma <ref>), with uniform bounds, we can thus estimate_i(^(f_1 ^^ f_2)-^ f_1 ^ f_2)_L^p(^d,ρ) ≲1_i∼ j_^( ∑_j∼ j_ S_j-1^^ f_1 ·_j^^ f_2_L^p(^,ρ)+ ∑_j∼ j_S_j-1^ f_1 ·_j ^f_2_L^p(^d,ρ)). Assume without loss of generality that the right hand side of estimate (i.) is bounded by 1. We then have using S_j-1^ L^q(,ρ)→ L^q(,ρ) (by Lemma <ref> and Lemma <ref>) and S_j-1 L^q(^d,ρ)→ L^q(^d,ρ) (by (<ref>) and Young's inequality) for q∈ [1,∞], both with uniform bounds,_i(^(f_1 ^^ f_2)-^ f_1 ^ f_2)_L^p(^d,ρ)≲1_i∼ j_^∑_j∼ j_ 2^-jα_2≲1_i∼ j_^ 2^-j_α_2≲ 2^-i(α_2-κ)^κ .In the last step we used that 2^-j_≈ by definition of j_. This shows the property (<ref>) for estimate (i.). If the right hand side of estimate (ii.) is uniformly bounded by 1 we obtain the bound_i(^(f_1 ^^ f_2)-^ f_1 ^ f_2)_L^p(^d,ρ) ≲1_i∼ j_^∑_j∼ j_∑_-1≤ j'< j-1 2^-j'α_1 2^-jα_2≲1_i∼ j_^ 2^-j_^(α_1+α_2)≲ 2^-i(α_1+α_2-κ)^κ and the property (<ref>) for (ii.) follows. Considering case (iii.) assume once more that the right hand side is bounded by 1. We get, by once more applying (<ref>), ^ (f_1 ^^ f_2)-^ f_1 ^ f_2 = ∑_j, j'∼ j_^: |j-j'|≤ 1^(_j^^ f_1 ·_j'^^ f_2) - ∑_j, j' j_^: |j-j'|≤ 1_j ^ f_1 ·_j'^ f_2 = ∑_j, j'∼ j_^: |j-j'|≤ 1( ^(_j^^ f_1 ·_j'^^ f_2) -_j ^ f_1 ·_j'^ f_2 ) , where we used in the second line that the spectral support of ^ f_1 and of ^ f_2 is contained in a ball of size ^-1≈ 2^j_ to reduce the sum in the second term to j, j'∼ j_^. Using then that the terms on the right hand side are spectrally supported in a ball of size 2^j we get for i≥ -1_i( ^ (f_1 ^^ f_2)-^ f_1 ^ f_2)= ∑_j, j'∼ j_^: |j-j'|≤ 11_i j( ^(_j^^ f_1 ·_j'^^ f_2) -_j ^ f_1 ·_j'^ f_2 ) ,so that we obtain, using once more ^ L^p(^,ρ)→ L^p(^d,ρ) and _iL^p(^d,ρ)→ L^p(^d,ρ),_i( ^ (f_1 ^^ f_2)-^ f_1 ^ f_2)_L^p(^d,ρ) ≲∑_j,j'∼ j_^: |j-j'|≤ 11_i j·2^-(j α_1+j'α_2)≲1_i j_·2^-j_ (α_1+α_2-κ)^κ≲2^-i(α_1+α_2-κ)^κ ,where we chose κ>0 in the second line small enough so that α_1+α_2-κ>0.Let ^ be as in Definition <ref>, ω∈ and construct Littlewood-Paley blocks as in Subsection <ref>. Let ψ, ψ^ and ^ be as in Subsection <ref>. There is a N=N(,ψ)∈ such that for alland -1≤ i≤j≤ j_-N and f_1, f_2∈'(^)^(_i^ f_1 ·_j^ f_2) = _i ^ f_1·_j ^ f_2 . Let us fix r_:=(∂,0) so that B(0,r_)⊆. From Lemma <ref> and the construction of our discrete partition of unity on page DiscreteDyadicPartition we know that the spectral support of _i^ f_1 ·_j^ f_2 and the support of φ_i^^· f_1 and φ_j^^· f_2 are contained in a set of the form 2^j ∩, whereis a ball around 0. Choose N∈ such that for j with -1≤ j ≤ j_-N (if any) we have 2^j ⊆ 2^j_-N⊆ B(0,r_/4) (note that N is independent ofsince r_=c· 2^j_ by the dyadic scaling of our lattice). In particular we have 2^j ⊆, 2^j ∩=2^j. Choose N further so big that we have for the smear function ψ^εψ^|_2^j =ψ(· )|_2^j =1 ,ψ^∩ (2^j +^\{0}) =∅for -1≤ j≤ j_-N (independently of ). Choose a χ∈(^d) such that χ|_B(0,r_/4)=1 and χ=0 outside B(0,r_/2). We can then reshape^(_i^ f_1 ·_j^ f_2)=ψ^·(φ_i^ f_1 ∗_φ_j^ f_2 )_ext = χ·(φ_i^ f_1 ∗_φ_j^ f_2 )_ext ,where we used the support properties above to replace ψ^ by χ. Now, note that (using formal notation to clarify the argument)χ(x) ·(φ_i^ f_1 ∗_φ_j^ f_2 )_ext(x)= χ(x) ·∫_ (φ_i^ f_1)(z)· (φ_j^ f_2)([x -z]) z .Since only x∈ B(0,r_/2) and z∈ B(0,r_/4) contribute we have x-z∈ B(0,3/4 r)⊆ so that [x-z]=x-z in (<ref>). Using that φ_i^∪φ^_j⊆ we can replace φ_i^ and φ_j^ in (<ref>) by φ_i, φ_j (the dyadic partition of unity on ^d from which φ_j^ is constructed as on page DiscreteDyadicPartition), replace f_1,f_2 by their periodic extension and extend the integral to ^d so that in total^(_i^ f_1·_j^ f_2)(x) =χ(x) ·∫_^d (φ_i ( f_1)_ext)(z)· (φ_j ( f_2)_ext)(x -z) z=∫_^d (φ_i ψ^ ( f_1)_ext)(z)· (φ_j ψ^ ( f_2)_ext)(x -z) z =(_i^ f_1 _j ^ f_2)(x),where we used in the second line that the support of the convolution is once more contained in B(0,r_/4) to drop χ and that ψ^|_2^j =1 to introduce smear functions in the integral. The claim follows.The main observation of <cit.> is that if the regularity condition α_1 + α_2 > 0 is not satisfied, then it may still be possible to make sense of f_1f_2 as long as f_1 can be written as a paraproduct plus a smoother remainder. The main lemma which makes this possible is an estimate for a certain “commutator”. The discrete version of the commutator is defined as C^(f_1,f_2,f_3):=(f_1 ^ f_2) ^ f_3 - f_1 (f_2^ f_3) .If there is no risk for confusion we may drop the indexon C.(<cit.>)Given ρ_1, ρ_2, ρ_3∈ρ(ω), p∈ [1,∞] and α_1,α_2,α_3∈ with α_1+α_2+α_3>0 and α_2 + α_3 ≠ 0 we have C^(f_1,f_2,f_3)_𝒞_p^α_2+α_3(^,ρ_1 ρ_2 ρ_3)≲f_1_𝒞^α_1_p(^,ρ_1)f_2_𝒞^α_2_∞(^,ρ_2)f_3_𝒞_∞^α_3(^,ρ_3) .Further, property (<ref>) holds for C if the regularity on the left hand side is reduced by an arbitrary κ>0. The proof of the estimates works line-by-line as in <cit.> and the (<ref>)-property follows as in Lemma <ref> via a modification of Lemma <ref> to three factors.§.§ The Modified Paraproduct It will be useful to define a lattice version of the modified paraproductthat was introduced in <cit.> and also used in <cit.>. Fix a function φ∈ C_c^∞((0,∞);_+) such that ∫_φ(s)s=1 and defineQ_i f(t):=∫_-∞^t 2^2idφ(2^2i(t-s)) f(s∨ 0)s, i≥ -1 .We then set f_1 ^ f_2:=∑_-1≤ j_1,j_2≤ j_:j_1< j_2-1 Q_j_2_j_1^ f_1 ·_j_2^ f_2for f_1,f_2_+ →_ω'() where this is well defined. We may drop the indexif there is no risk for confusion. mpara As in <cit.> we silently identify f_1 in f_1f_2 with t↦ f(t) 1_t>0 if f_1∈ℳ^γ_T 𝒞^α_p(,ρ) with γ>0. Once more the translation to the continuous case f_1,f_2 _+→_ω'(^d) is analogous. The modified paraproduct allows for similar estimatesas in Lemma <ref>. Let β∈,p∈ [1,∞], γ∈ [0,1), t>0,α<0 and let ρ_1,ρ_2_+ →ρ(ω) with ρ_1 pointwise decreasing. Thent^γfg (t)_𝒞^α+β_p(^, ρ_1(t) ρ_2(t))≲f_ℳ^γ_t 𝒞^α_p (^,ρ_1)g(t)_𝒞^β_∞(^,ρ_2(t))∧f_ℳ^γ_t 𝒞^α_∞(^,ρ_1)g(t)_𝒞^β_p(^,ρ_2(t))and t^γfg (t)_𝒞^β_p(^,ρ_1(t) ρ_2(t))≲f_ℳ^γ_t L^p (^,ρ_1)g(t)_𝒞^β_∞(^,ρ)∧f_ℳ^γ_t L^∞(^,ρ_1)g(t)_𝒞^β_p(^,ρ_2(t)) .Both estimates have the property (<ref>) if the regularity on the left hand side is decreased by an arbitrary κ>0. The proof is the same as for <cit.>. Property (<ref>) is shown as in Lemma <ref>. We further have an estimate in terms of the parabolic spaces ℒ^γ,α_p,T(,ρ) that were introduced in Definition <ref>.We have for α∈ (0,2),p∈ [1,∞], γ∈ [0,1) and ρ_1,ρ_2_+ →ρ(ω), pointwise decreasing in s, the estimatef g_ℒ^γ,α_p,T(^,ρ_1ρ_2)≲f_ℒ^γ,δ_p,T(^,ρ_1) (g_C_T𝒞_∞^α(^,ρ_2)+ℒ^ g_C_T𝒞^α-2_∞(^,ρ_2))for any δ>0 and any diffusion operator ℒ_μ^ as in Definition <ref>. This estimate has the property (<ref>) if the regularity α on the left hand side is lowered by an arbitrary κ>0. The proof is as in <cit.> and uses Lemma <ref> below. The proof of the property (<ref>) is as in Lemma <ref>.The main advantage of the modified paraproducton ^d is its commutation property with the heat kernel ∂_t -Δ (or ℒ_μ=∂_t -L_μ) which is essential for the Schauder estimates for paracontrolled distributions, compare also Subsection <ref> below. In the following we state the corresponding results for Bravais lattices. For α∈ (0,2), β∈, p∈ [1,∞], γ∈ [0,1) and ρ_1,ρ_2_+ →ρ(ω), with ρ_1 pointwise decreasing, we have for t>0t^γ(f g - f g)(t) _𝒞^α+β_p(^,ρ_1(t)ρ_2(t))≲f_ℒ^γ,α_p,t(^,ρ_1)g(t)_𝒞^β_∞(^,ρ_2(t))and t^γ(ℒ_μ^(f g) - fℒ_μ^ g)(t) _𝒞^α+β-2_p(^,ρ_1(t)ρ_2(t))≲f_ℒ^γ,α_p,t(^,ρ_1)g(t)_𝒞^β_∞(^,ρ_2(t)) .where ℒ_μ^=∂_t-L^_μ is a discrete diffusion operator as inDefinition <ref>. These estimates have the property (<ref>) if the regularity on the left hand side is lowered by an arbitrary κ>0. Again we can almost follow along the lines of the proof in <cit.> with the only difference that in the derivation of the second estimate the application of the “product rule” of ℒ^ε_μ does not yield a term -2∇ f ∇ g but a more complex object, namely ∫_^dμ(y)/ε^2 D^_y fD^_y g,where D^_y f(t,x)=f(t,x+ε y)-f(t,x) and similarly for g. The bound for (<ref>) follows from Lemma <ref> once we showD^_y φ_𝒞^γ-1_p(^ε,ρ_1)≲φ_𝒞^γ_p(^ε,ρ_1)|y|·εfor any γ∈. Note that due to Lemma <ref> we can write_j D^y_φ=(Ψ̃^,j(·+ε y) - Ψ̃^,j)φ ,where Ψ̃^,j=ℰ^Ψ^^,j=2^jdϕ_(2^j· ) with ϕ_∈_ω(^d). WithΨ̃^,j(x+ε y) - Ψ̃^,j(x)=2^j∫_0^12^jdϕ_(2^j (x+ζε y))ζ· y εwe get (<ref>) by applying Lemma <ref>. The proof of the property (<ref>) is as in Lemma <ref> and it uses Lemma <ref>.§ WEAK UNIVERSALITY OF PAM ON ^2With the theory from the previous sections at hand we can analyze stochastic models on unbounded lattices using paracontrolled techniques. As an example, we prove the weak universality result for the linear parabolic Anderson model that we discussed in the introduction.For F∈ C^2(;) with F(0)=0 and bounded second derivative we consider the equationℒ^1_μ v^=F(v^)·η^, v^(0)=||^-11_·=0on _+ ×, where ⊆^2 is a two-dimensional Bravais lattice, ℒ^1_μ=∂_t-L^1_μ is a discrete diffusion operator on the latticeas described in Definition <ref>, induced by μ∈(ω) with ω= for σ∈(0,1). The upper index “1” indicates that we did not scale the latticeyet. The family (η^(z))_z∈∈_ω'() consists of independent (not necessarily identically distributed) random variables satisfying for z∈𝔼[η^(z)]=-F'(0) c^_μ^2 ,Var(η^(z))=1/|^|= 1/|| ^2 ,where c^_μ>0 is a constant of order O(|log|) which we will fix in equation (<ref>) below. We further assume that for everyand z∈ the variable η^(z) has moments of order p_ξ>14 such thatsup_z ∈^ε𝔼[|η^(z)-𝔼[η^(z)]|^p_ξ]≲^p_ξ .The lower bound 14 for p_ξ might seem quite arbitrary at the moment, we will explain this choice in Remark <ref> below. Note that η^ is of order O() while its expectation is of order O(^2 |log|), so we are considering a small shift away from the “critical” expectation 0. We are interested in the behavior of (<ref>) for large scales in time and space. Setting u^(t,x):=^-2 v^(^-2 t,^-1 x) and ξ^(x):=^-2 (η^(^-1 x )+F'(0) c^_μ^2 ) modifies the problem toℒ^_μ u^=F^(u^)(ξ^ -F'(0) c^_μ), u^(0)=|^|^-11_·=0 ,where u^_+ ×^→ is defined on refining lattices ^ in d=2 as in Definition <ref> and where F^:=^-2 F(^2· ). The potential (ξ^(x))_x∈^ is scaled so that it satisfies for z∈^ * 𝔼[ξ^(z)]=0, * 𝔼[ |ξ^(z)|^2]= |^|^-1 = ||^-1^-2, * sup_z ∈^𝔼[ |ξ^(z)|^p_ξ]≲^- p_ξ for some p_ξ > 14.We consider ξ^ as a discrete approximation to white noise in dimension 2.In particular, we expect ^ξ^ to converge in distribution to white noise on ^2, and we will see in Lemma <ref> below that this is indeed the case. In Theorem <ref> we show that ^ u^ converges in distribution to the solution u of the linear parabolic Anderson model on ^2,ℒ_μ u=F'(0) u (ξ - F'(0)∞), u(0) = δ,where ξ is white noise on ^2, δ is the Dirac delta distribution, “-∞” denotes a renormalization and ℒ_μ is the limiting operator from Definition <ref>. The existence and uniqueness of a solution to (<ref>) were first established in <cit.> (for more regular initial conditions) by using a “partial Cole-Hopf transformation” which turns the equation into a well-posed PDE. Using the continuous versions of the objects defined in the Sections above we can modify the arguments of <cit.> to give an alternative proof of their result, see Corollary <ref> below. The limit of (<ref>) only sees F'(0) and forgets the structure of the non-linearity F, so in that sense the linear parabolic Anderson model arises as a universal scaling limit.Let us illustrate this result with a (far too simple) model: Suppose F is of the form F(v)=v(1-v) and let us first consider the following ordinary differential equation on [0,T]:∂_t v=η· F(v), v(0)∈ (0,1) ,for some η∈. If η>0, then v describes the evolution of the concentration of a growing population in a pleasant environment, which however shows some saturation effects represented by the factor (1-v) in the definition of F. For η<0 the individuals live in unfavorable conditions,say in competition with a rival species. From this perspective equation (<ref>) describes the dynamics of a population that migrates between diverse habitats. The meaning of our universality result is that if we tune down the random potential η^ and counterbalance the growth of the population with some renormalization (think of a death rate), then from far away we can still observe its growth (or extinction) without feeling any saturation effects.The analysis of (<ref>) and the study of its convergence are based on the lattice version of paracontrolled distributions that we developed in the previous sections and it will be given in Subsection <ref> below. In that analysis it will be important to understand the limit of ^ξ^ and a certain bilinear functional built from it, and we will also need uniform bounds in suitable Besov spaces for these objects. In the following subsection we discuss this convergence.§.§ Discrete Wick calculus and convergence of the enhanced noiseWe develop here a general machinery for the use of discrete Wick contractions in the renormalization of discrete, singular SPDEs with i.i.d. noise which is completely analogous to the continuous Gaussian setting. Moreover, we build on the techniques of <cit.> to provide a criterion that identifies the scaling limits of discrete Wick products as multiple Wiener-Itô integrals. Our results are summarized in Lemma <ref> and Lemma <ref> below and although the use of these results is illustrated only on the discrete parabolic Anderson model, the approach extends in principle to any discrete formulation of popular singular SPDEs such as the KPZ equation or the Φ^4_d models. In order to underline the general applicability of these methods we work in this subsection in a general dimension d.Take a sequence of scaled Bravais lattices ^ in dimension d as in Definition <ref>. As a discrete approximation to white noise we take independent (but not necessarily identically distributed) random variables (ξ^(z))_z∈^ that satisfy * 𝔼[ξ^(x)]=0, * 𝔼[ |ξ^(x)|^2]= |^|^-1 = ||^-1^-d, * sup_z ∈^𝔼[ |ξ^(z)|^p_ξ]≲^- d/2·p_ξ for some p_ξ≥ 2. Note that the family (ξ^(z))_z∈^ we defined in the introduction of this Sectionfits into this framework (with d=2 and p_ξ>14). Let us fix a symmetric χ∈_ω(^d), independent of , which is 0 on 1/4· and 1 outside of 1/2· and define X_μ^ := χ/l^_μ(D_^)ξ^:=^-1( χ/l^_μ·ξ^) . Let us point out that the χ used in the construction of X_μ^ does not depend onand only serves to erase the “zero-modes” of ξ^ to avoid integrability issues. Note that ℒ_μ^ X_μ^ = -L^_μ X_μ^=χ(D_^)ξ^=^-1(χ·ξ^) so that X_μ^ is a time independent solution to the heat equation on ^ driven by χ(D_^)ξ^. Our first task will be to measure the regularity of the sequences (ξ^), (X_μ^) in terms of the discrete Besov spaces introduced in Subsection <ref>. For that purpose we need to estimate moments of sufficiently high order. Fordiscrete multiple stochastic integrals with respect to the variables (ξ^(z))_z ∈^, that is for sums ∑_z_1,…,z_n ∈^f(z_1,…,z_n) ξ^(z_1)…ξ^(z_n) with f(z_1,…, z_n) = 0 whenever z_i = z_j for some i ≠ j it was shown in <cit.> that all moments can be bounded in terms of the ℓ^2 norm of f and the corresponding moments of the (ξ^(z))_z∈^. However, typically we will have to bound such expressions for more general f (which do not vanish on the diagonals) and in that case we first have to arrange our random variable into a finite sum of discrete multiple stochastic integrals, so that then we can apply <cit.> for each of them. This arrangement can be done in several ways, here we follow <cit.> and regroup in terms of Wick polynomials. Given random variables (Y(j))_j ∈ J over some index set J and I = (j_1,…,j_n) ∈ J^n we setY^I = Y(j_1) … Y(j_n) = ∏_k=1^n Y(j_k)as well as Y^∅=1. According to Definition 3.1 and Proposition 3.4 of <cit.>, the Wick product Y^♢ I can be defined recursively by Y^♢∅ := 1 andY^♢ I := Y^I - ∑_∅≠ E ⊂ I𝔼[Y^E] ·Y^♢ I ∖ E .For I = (j_1,…,j_n) ∈ J^n we also write Y(j_1)♢…♢ Y(j_n):= Y^♢ I .By induction one easily sees that this product is commutative. In the case j_1=…=j_n we may write instead Y(j_1)^♢ n . diamond Letbe as in Definition <ref> and let (ξ^(z))_z∈^ be a discrete approximation to white noise as above, n≥ 1 and assume p_ξ≥ 2 n. For f ∈ L^2((^ε)^n) define the discrete multiple stochastic integral w.r.t (ξ^(z)) byℐ_nf:=∑_z_1,…,z_n ∈^ε |^ε|^n f(z_1,…,z_n)ξ^ε(z_1)♢…♢ξ^ε(z_n) .It then holds for 2≤ p≤ p_ξ/n ℐ_n f_L^p(ℙ)≲f_L^2((^ε)^n) . In the following we identify ^ε with an enumeration byso that we can writeℐ_n f =∑_1≤ r≤ n, a∈ A_r^n r! ∑_z_1<…<z_r|^ε|^n f̃_a(z_1,…,z_r)·ξ^ε(z_1)^♢a_1×…×ξ^ε(z_r)^♢a_r ,where A_r^n:={a∈ℕ^r |∑_i a_i=n}, f̃_a denotes the symmetrized version of f_a(z_1,…,z_r):=f(z_1,…,z_1^a_1 ×,…,z_r,…,z_r^a_r ×)·1_z_i≠ z_j∀ i≠ j ,and where we used the independence of ξ^ε(z_1),…,ξ^ε(z_r) to decompose the Wick product (we did not show this property, but it is not hard to derive it from the definition of ♢ we gave above). The independence and the zero mean of the Wick products allow us to see this as a sum of nested martingale transforms so that an iterated application of the Burkholder-Davis-Gundy inequality and Minkowski's inequality as in <cit.> gives the desired estimateℐ_n f ^2_L^p(ℙ) ≲∑_1≤ r≤ n, a∈ A_r^n∑_z_1<…<z_r |^ε|^n ·f̃_a(z_1,…,z_r)·ξ^ε(z_1)^♢a_1×…×ξ^ε(z_r)^♢a_r_L^p(ℙ)^2 ≲∑_1≤ r≤ n, a∈ A_r^n∑_z_1<…<z_r|^ε|^2n· |f̃_a(z_1,…,z_r)|^2 ·∏_j=1^r ξ^ε(z_j)^♢a_j_L^p(ℙ)^2 ≲∑_1≤ r≤ n, a∈ A_r^n∑_z_1,…,z_r |^ε|^n |f̃_a(z_1,…,z_r)|^2 ≤f_L^2((^ε)^n)^2 ,where we used the bound ξ^ε(z_r)^♢a_j_L^p(ℙ)^2≲ |^ε|^-a_j which follows from (<ref>) and our assumption on ξ^. As a direct application we can bound the moments of ξ^ and X_μ^ in Besov spaces. We also need to control the resonant term X_μ^ξ^, for which we introduce the renormalization constantc^_μ:=∫_^χ(x)/l^_μ(x)x ,which is finite for all > 0 because ^ is compact and χ is supported away from 0. We define a renormalized resonant product by bulletX^_μ∙ξ^ε:=X^_μξ^ε-c^ε_μ .Since l^ε_μ≈|·|^2 (Lemma <ref> together with the easy estimate l^ε_μ≲|·|^2) we have c^ε_μ≈ -logε in dimension 2. Using Lemma <ref> we can derive the following bounds. Let ξ^, X^ and X^_μ∙ξ^ε be defined on ^ as above with p_ξ≥ 4 (where p^ξ is as on page ApproximationToWhiteNoise) and let d<4. For μ∈(), ζ<2-d/2-d/p_ξ and κ>d/p_ξ we have𝔼[ξ^ε^p_ξ_𝒞^ζ-2(^ε,^κ)] + 𝔼[X^_μ^p_ξ_𝒞^ζ(^ε,^κ)] + 𝔼[X^_μ∙ξ^ε^p_ξ/2_𝒞^2ζ -2(^ε,^2κ)]≲ 1.The implicit constant is independent of . Let us bound the regularity of X^_μ. Recall that by Lemma <ref> we have the continuous embedding (with norm uniformly bounded in ε) ℬ^ζ+d/p_ξ_p_ξ,p_ξ(^ε,^κ)⊆𝒞^ζ(^ε,^κ). To show (<ref>) it is therefore sufficient to bound for β<2-d/2𝔼[ X^_μ^p_ξ_ℬ^β_p_ξ,p_ξ(^ε, ^κ)]=∑_-1≤ j ≤ j_^ε 2^jp_ξβ∑_z ∈^ε |^ε|𝔼[|_j^^ε X^_μ(z)|^p_ε ] 1/(1+|z|)^κ p_ξ .By assumption we have κ p_ξ > d and can bound ∑_z ∈^ε |^ε| (1+|z|)^-κ p_ξ≲ 1 uniformly in ε (for example by Lemma <ref>). It thus suffices to derive a bound for 𝔼[|_j^^ε X^_μ(x)|^p_ε ], uniformly in ε and x. Note that by (<ref>) _j^^ε X^_μ(x)=∑_z∈^ε |^ε| 𝒦^ε_j(x-z) ξ^ε(z) with 𝒦^ε_j=^-1 (φ^^ε_j χ/l^ε_μ) so that Lemma <ref>, Parseval's identity(<ref>) and l^ε_μ≳|·|^2 on(from Lemma <ref>) imply𝔼[ |_j^^ X^_μ (x)|^p_ξ] ≲𝒦_j^ε(x - ·) _L^2(^ε)^p_ξ≲ 2^jp_ξ (d/2-2) ,which proves the bound for X^_μ. The bound for ξ^ε follows from the same arguments or with Lemma <ref>.Now let us turn to X^_μ∙ξ^ε. A short computation shows that𝔼[(X^_μξ^ε)(x)] = 𝔼[(X^_μ·ξ^ε)(x)] = c^ε_μ,x ∈^ε ,and, by a similar argument as above, it suffices to bound X^_μ∙ξ^ε in ℬ^β_p_ξ/2,p_ξ/2(^d,^2κ) for β<2-d. We are therefore left with the task of bounding the p_ξ/2-th moment of ^^ε_k ( ∑_|i-j|≤ 1^^_i X^_μ^^_jξ^ε -𝔼[^^_i X_μ^ε^^_jξ^ε] )(x)=∑_z_1,z_2,y |^ε|^3 ∑_|i-j|≤ 1Ψ^^ε, k(x-y) 𝒦_i^ε(y-z_1) Ψ^^ε,j (y-z_2) (ξ^ε(z_1)ξ^ε(z_2)-𝔼[ξ^ε(z_1)ξ^ε(z_2)])= ∑_z_1,z_2 |^ε|^2 (∑_|i-j|≤1∑_y |^ε| Ψ^^ε, k(x-y) 𝒦_i^ε(x-z_1) Ψ^^ε,j(x-z_2) )ξ^ε(z_1)♢ξ^ε(z_2),which with Lemma <ref> and Parseval's identity (<ref>) can be estimated by𝔼[|∑_z_1,z_2 |^ε|^2 ( ∑_|i-j|≤1 |^ε| Ψ^^ε,k(x-y) 𝒦_i^ε(x-z_1) Ψ^^ε,j(x-z_2) )ξ^ε(z_1)♢ξ^ε(z_2)|^p_ξ/2]^2/p_ξ≲∑_|i-j|≤1∑_y|^ε| Ψ^^ε,k(x-y) 𝒦_i^ε(x-z_1) Ψ^^ε,j(x-z_2) _L^2_z_1,z_2((^ε)^2) = ∑_|i-j|≤1∑_y |^ε| Ψ^^ε,k(x-y)_(^ε)^2( 𝒦_i^ε(x- ·)⊗Ψ^^ε,j(x-·) )(ℓ_1, ℓ_2) _L^2_ℓ_1,ℓ_2((^ε)^2) =e^-2π (ℓ_1 + ℓ_2) · x∑_|i-j|≤1_^εΨ^^ε,k(-(ℓ_1+ℓ_2)) _^ε𝒦_i^ε(-ℓ_1) _^εΨ^^ε,j(-ℓ_2) _L^2_ℓ_1,ℓ_2((^ε)^2) = ∑_|i-j|≤1φ^^ε_k (ℓ_1+ℓ_2) φ^^ε_i(ℓ_1) χ(ℓ_1)/l^ε_μ(ℓ_1)φ^^ε_j(ℓ_2) _L^2_ℓ_1,ℓ_2((^ε)^2) ,where in the last step we used that all considered functions are even. Since φ^^ε_k (ℓ_1+ℓ_2) = 0 unless |ℓ_m| ≳ 2^k for m=1 or m=2 and since φ^^ε_m_L^2(^ε)≲ 2^m d/2, we get∑_|i-j|≤1φ^^ε_k (ℓ_1+ℓ_2) φ^^ε_i(ℓ_1) χ(ℓ_1)/l^ε_μ(ℓ_1)φ^^ε_j(ℓ_2) _L^2_ℓ_1,ℓ_2((^ε)^2) ≲∑_|i-j|≥ 1, j ≳ k 2^-2iφ^^ε_k (ℓ_1+ℓ_2)φ^^ε_j(ℓ_2) _L^2_ℓ_1,ℓ_2((^ε)^2)≲∑_|i-j|≥ 1, j ≳ k 2^-2i 2^kd/2 2^jd/2≲ 2^k (d-2),using d/2-2 < 0 in the last step. By the compact embedding result in Lemma <ref> together with Prohorov's theorem we see that the sequences (^εξ^ε), (^ε X^_μ), and (^ε(X^_μ∙ξ^ε)) have convergent subsequences in distribution – note that while the Hölder space 𝒞^ζ(^d, p^κ) is not separable, all the processes above are supported on the closure of 𝒞^ζ'(^d, p^κ') for ζ' > ζ and κ' < κ, which is a separable subspace and therefore we can indeed apply Prohorov's theorem. We will see in Lemma <ref> below that ^εξ^ε converges to the white noise ξ on ^d. Consequently, the solution X^_μ to -L^ε_μ X^_μ=χ(D_^)ξ^ε should approach the solution of -L_μ X_μ=χ(D_^d)ξ :=^-1( χ ξ), i.e.X_μ=χ(D_^d)/(2π)^2D_^d_μ^2ξ=^-1(χ/(2π)^2·_μ^2ξ) =𝒦^0_μ∗ξ,𝒦^0_μ:=^-1_^dχ/(2π)^2‖·‖_μ^2 .where ·_μ is defined as in Definition <ref>. The limit of ^ε(X^_μ∙ξ^ε) will turn out to be the distributionX_μ∙ξ(φ):=∫_^d∫_^2𝒦^0_μ(z_1-z_2) φ(z_1)ξ( z_1)♢ξ( z_2) -(X_μξ +ξ X_μ)(φ) for φ∈_ω(^d), where the right hand side denotes the second order Wiener-Itô integral with respect to the Gaussian stochastic measure ξ( z) induced by the white noise ξ, compare <cit.>. Note that X_μ∙ξ is not a continuous functional of ξ, so the last convergence is not a trivial consequence of the convergence for ^ξ^. To identify the limit of ^(X_μ^∙ξ^) we could use a diagonal sequence argument that first approximates the bilinear functional by a continuous bilinear functional as in <cit.>. Here prefer to go another route and instead we follow <cit.> who provide a general criterion for the convergence of discrete multiple stochastic integrals to multiple Wiener-Itô integrals, and we adapt their results to the Wick product setting of Lemma <ref>. Let ^,n∈ and (ξ^(z))_z∈^ be as in Lemma <ref>. For k = 0, …, n let f_k^ε∈ L^2((^ε)^k). We identify (^ε)^k with a Bravais lattice in k · d dimensions via the orthogonal sum (^ε)^k=⊕_i=1^k ^ε⊆⊕_i=1^k^d= (^d)^kto define the Fourier transform ℱ_(^ε)^kf^ε_k∈ L^2((^ε)^k) of f_k^ε. Assume that there exist g_k ∈ L^2((^d)^k) with |_(^ε)^kℱ_(^ε)^kf^ε_k|≤ g_k for all ε and f_k∈ L^2((^d)^k) such that lim_ε→ 0_(^ε)^kℱ_(^ε)^kf^ε_k- _(^d)^k f_k_L^2((R^d)^k) = 0 for all k ≤ n. Then the following convergence holds in distributionlim_ε→ 0∑_k=0^n ℐ_k f_k^ε = ∑_k=0^n ∫_(^d)^k f_k(z_1, …, z_k)ξ( z_1)♢…♢ξ( z_k) ,where ξ( z_1)♢…♢ξ( z_k) denotes the Wiener-Itô integral against the Gaussian stochastic measure induced by the white noise ξ on ^d.The proof is contained in the appendix.The identification of the limits of the extensions of ξ^,X_μ^ and X_μ^∙ξ^ is then an application of Lemma <ref>. In the setup of Lemma <ref>with ξ, X_μ and X_μ∙ξ defined as above and with ζ,κ as in Lemma <ref> we have for d<4(^εξ^ε,^ε X^_μ,^ε (X^_μ∙ξ^ε))ε→ 0⟶(ξ,X_μ,X_μ∙ξ)in distribution in 𝒞^ζ-2(^d,^κ)×𝒞^ζ(^d,^κ)×𝒞^2ζ-2(^d,^2κ ).Recall that the extension operator ^ is constructed from ψ^=ψ(· ) where the smear function ψ∈(^d) is symmetric and satisfies ψ=1 on some ball around 0. Since from Lemma <ref> we already know that the sequence (^εξ^ε,^ε X^_μ,^ε (X^_μ∙ξ^ε)) is tight in 𝒞^ζ-2(^d,^κ)×𝒞^ζ(^d,^κ)×𝒞^2ζ-2(^d,^2κ ), it suffices to prove the convergence after testing against φ∈_ω(^d):(^εξ^ε(φ_1),…, ^εξ^ε(φ_n),^ε X^_μ(ψ_1),…,^ε X^_μ(ψ_n) ,^ε (X^_μ∙ξ^ε)(f_1), …,^ε (X^_μ∙ξ^ε)(f_n) ) ε→0→ (ξ(φ_1), …, ξ(φ_n), X_μ(ψ_1),…, X_μ(ψ_n),X_μ∙ξ(f_1),…, X_μ∙ξ(f_n)) ,and by taking linear combinations and applying Lemma <ref> we see that it suffices to establish each of the following convergences:^εξ^ε(φ) ε→ 0⟶ξ(φ),^ε X^_μ(φ) ε→0⟶ X_μ(φ),^ε (X^_μ∙ξ^ε)(φ)) ε→ 0→ X_μ∙ξ(φ)for all φ∈_ω(^d). We can even restrict ourselves to those φ∈_ω(^d) with φ∈(^d), which impliesφ⊆ and ^-1(ψ^φ )=φ for ε small enough, which we will assume from now on. Note that φ⊆ implies(φ|_) =(φ)|_ since by definition of ^-1^-1 ((φ)|_)=(^-1φ ) |_^= φ|_^ .To show the convergence of ^εξ^ε(φ) to ξ(φ) note that we have from (<ref>)^εξ^ε(φ)=∑_z∈^ε |^ε| (^-1ψ^∗φ)(z)ξ^ε(z)= ∑_z∈^ε |^ε|^-1(ψ^ φ)(z)ξ^ε(z)=∑_z∈^ε |^ε| φ(z) ξ^ε(z)where we used in the first step that ψ^ is symmetric and in the last step that ^-1(ψ^φ )=φ by our choice of φ and . Using Lemma <ref> and relation (<ref>) the convergence of ^εξ^ε(φ) to ξ(φ) follows. For the limit of ^ε X^_μ we use the following formula, which is derived by the same argument as above:^ε X^_μ(φ)=∑_z_1, z_2∈^ε |^ε|^2 φ(z_1) 𝒦^ε_μ(z_2-z_1) ξ^ε(z_2)with 𝒦_μ^ε=^-1 (χ/l^ε_μ). In view of Lemma <ref> it then suffices to note thatf̂^:=(φ𝒦_μ^ε)=φ·χ/l^ε_μ(<ref>)=φ·χ/l^_μis dominated by a multiple of χ/|·|^2 on ^ε due to Lemma <ref>, and it converges to φ·χ/(2π)^2‖·‖_μ^2by the explicit formula for l^_μ in (<ref>). We are left with the convergence of the third component. Since ^εξ^ε→ξ and ^ε X^_μ→ X_μ we obtain via the (<ref>)-Property of the paraproductlim_ε→0^ε (X^_μ^ξ^ε) = lim_ε→0^ε X^_μ^εξ^ε = X_μξand similarly one gets ^ε (ξ^ε^ X^_μ) →ξ X_μ. We can therefore show instead ^ε(X^_μξ^ε-𝔼[X^_μξ^ε])(φ)→ (X_μ∙ξ +ξ X_μ+ X_μξ)(φ) .Note that we have the representations ^ε(X^_μξ^ε-𝔼[X^_μξ^ε])(φ) =∑_z_1,z_2∈^ε|^ε|^2 φ(z_1) 𝒦^ε_μ(z_1-z_2)ξ^ε(z_1)♢ξ^ε(z_2), (X_μ∙ξ +ξ X_μ+ X_μξ)(φ) =∫_^2∫_^2φ(z_1) 𝒦^0_μ(z_1-z_2) ξ( z_1)♢ξ( z_2)with 𝒦^_μ as above and 𝒦^0_μ as in (<ref>). The (^ε)^2-Fourier transform of φ(z_1) 𝒦^ε_μ(z_1-z_2) is φ̂_ext(x_1-x_2) χ(x_2)/l^ε_μ(x_2) for x_1,x_2∈^ε, where φ̂_ext denotes the periodic extension from (<ref>) for φ|_∈() (recall again that φ⊆). We can therefore apply Lemma <ref> since for d<4 the function (χ(x_2)/l^ε(x_2))^2≲1_|x|≳ 1 /|x|^4 is integrable on ^ε and thus we obtain (<ref>).We have shown the convergence in distribution of all the components in (<ref>). By Lemma <ref> we can take any linear combination of these components and still get the convergence from the same estimates, so that (<ref>) follows from the Cramér-Wold Theorem. §.§ Convergence of the lattice model We are now ready to prove the convergence of ^ u^ announced at the beginning of this section. The key statement will be the a priori estimate in Lemma <ref>. The convergence of ^ u^ to the continuous solution on ^2, constructed in Corollary <ref>, will be proven in Theorem <ref>. We first fix the relevant parameters. §.§.§ Preliminaries Throughout this subsection we use the same p∈ [1,∞], σ∈ (0,1), μ∈(), a polynomial weight p^κ for some κ>2/p_ξ > 1/7 and a time dependent sub-exponential weight (e^σ_l+t)_t∈ [0,T]. We further fix an arbitrarily large time horizon T>0 and require l≤ -T for the parameter in the weight e^σ_l. Then we have 1≤ e^σ_l+t≤ (e^σ_l+t)^2 for any t≤ T, which will be used to control a quadratic term that comes from the Taylor expansion of the non-linearity F^. We take ξ^ as in the beginning of this section with p_ξ>14 (see Remark <ref> below) and construct X^_μ as in Subsection <ref>. We further fix a parameter α∈ (2/3-2/3·κ/σ, 1- 2 /p_ξ-2κ/σ)with κ/σ∈ (2/p_ξ,1) small enough such that the interval is non-empty, which (as we will discuss in the following remark) is possible since 2/p_ξ<1/7. Let us sketch where the boundaries of the interval (<ref>) come from. The parameter α will measure the regularity of u^ below. The upper boundary, that is1- 2 /p_ξ-2κ/σ, arises due to the fact that we cannot expect u^ to be better than X^, which has regularity below 1-2/p_ξ due to Lemma <ref>. The correction -2κ/σ is just the price one pays in the Schauder estimate in Lemma <ref> for the “weight change”. The lower bound 2/3-2/3·κ/σ is a criterion for our paracontrolled approach below to work. We increase below the regularity α of our solutions, by subtraction of a paraproduct, to 2α. By Lemma <ref> this allows us to uniformly control products with ξ^ provided2α+(α+2κ/σ-2)>0 ,because ξ^∈𝒞^α+2κ/σ-2_p^κ. This condition can be reshaped to α>2/3-2/3·κ/σ, explaining the lower bound. The interval (<ref>) can only be non-empty if2/3-2/3·κ/σ<1- 2/p_ξ-2κ/σ ⇔ 2/3<1- 2/p_ξ-4/3·κ/σLemma <ref> forces us to take κ/σ>2/p_ξ so that the the right hand side can only be true provided 2/3<1- 2/p_ξ-4/3 · 2/p_ξ, which is equivalent top_ξ>14 .Let us mention the simple facts 2α+2κ/σ,2α+4κ/σ∈ (0,2), α+κ/σ,α+2κ/σ∈ (0,1) and 3α+2κ/σ-2>0 which we will use frequently below. We will assume that the initial conditions u_0^ are uniformly bounded in 𝒞^0_p(^,e^σ_l) and are chosen such that ^ u_0^ converges in _ω'(^2) to some u_0. For u^_0 = |^|^-11_· = 0 it is easily verified that this is indeed the case and the limit is the Dirac delta, u_0 = δ.Recall that we aim at showing that (the extension of) the solution u^ to ℒ_μ^ u^=F(u^)(ξ^-F'(0)c^_μ), u^(0)=u^_0=|^|^-11_·=0converges to the solution of ℒ_μ u =F'(0) u ξ, u(0)=u_0=δ ,where uξ is a suitably renormalized product defined in Corollary <ref> below. Our solutions will be objects in the parabolic space ℒ^α,α_p,T which does not require continuity at t=0. A priori there is thus no obvious meaning for the Cauchy problems (<ref>), (<ref>) (although of course for (<ref>) we could use the pointwise interpretation). We use the common interpretation of (<ref>, <ref>) as equations for distributions u^,u∈_ω'(^1+2) (compare for example <cit.>) by requiring u^,u ⊆_+×^2 andℒ_μ^ u^ =F(u^)(ξ^-F'(0)c^_μ)+δ⊗ u^_0 ,ℒ_μ u=F'(0) uξ +δ⊗ u_0 ,in the distributional sense on (-∞,T) ×^2, where ⊗ denotes the tensor product between distributions. Since we mostly work with the mild formulation of these equations the distributional interpretation will not play a crucial role. Some care is needed to check that the only distributional solutions are mild solutions, since the distributional Cauchy problem for the heat equation is not uniquely solvable <cit.>. However, under generous growth conditions for u,u^ for x→∞ (compare <cit.>) there is a unique solution. In our case this fact can be checked by considering the Fourier transform of u,u^ in space. §.§.§ A priori estimatesWe will work with the following space of paracontrolled distributions. We identify a pair(u^,X,u^,♯) [0,T]→_ω'(^)^2with u^:=u^,X X_μ^+u^,♯ and introduce a normu^_𝒟^γ,δ_p,T(^,e^σ̃_l):=(u^,X,u^,♯)_𝒟^γ,δ_p,T(^,e^σ̃_l):=u^,X_ℒ^γ/2,δ_p,T(^,e^σ̃_l)+u^,♯_ℒ^γ,δ+α_p,T(^,e^σ̃_l)for α as above, σ̃∈ (0,1) and γ≥ 0, δ∈ (0, 2-α). We denote the corresponding space by 𝒟^γ,δ_p,T(^,e^σ̃_l). If the norm (<ref>) is bounded for a sequence (u^=u^,X X_μ^+u^,♯)_ε we say that u^ is paracontrolled by X_μ^. In view of Remark <ref> we can also define a continuous version 𝒟^γ,δ_p,T(^d,e^σ̃_l) of the space above. mathscrDgammaalphapT As in <cit.> it will be useful to have a common bound on the stochastic data: LetM_:=ξ^_𝒞^α+2κ/σ-2_∞(^,p^κ)∨X_μ^_𝒞^α+2κ/σ_∞(^,p^κ)∨X_μ^∙ξ^_𝒞^2α+4κ/σ-2_∞(^,p^2κ)(compared to Lemma <ref> we have ζ=α+2κ/σ). The following a priori estimates will allow us to set up a Picard iteration below.In the setup above consider γ∈{0, α} and u_0∈^0_p(). If γ=0 we require further that u_0∈^α_p(^,ρ) and u_0^♯:=u_0-F'(0)u_0 X_μ^∈^2α_p(^,e^σ_l). Define a mapℳ^_γ,u_0 𝒟_p,T^γ,α(^,e^σ_l) ∋ (u^,X, u^,♯)⟼ (v^,X, v^,♯) ∈𝒟_p,T^γ,α(^,e^σ_l)for u^=u^,X X_μ^+ u^,♯ with u^(0)=u_0 via v^,X:=F'(0)u^ and v^,♯ := v^-v^,X X_μ^, where v^ is the solution to the problemℒ^_μ v^ :=F^(u^)ξ^ - F^(u^,X/F'(0)) F'(0) c^_μ,v^(0)=u_0 .The map ℳ^_γ,u_0 is well defined for γ∈{0,α} and we have the bound (v^,X, v^,♯)_𝒟_p,T^γ,α(^,e^σ_l) ≤ C_u_0+C_M_· T^(α-δ)/2 (u^_𝒟_p,T^γ,α(^,e^σ_l)+^νu^_𝒟_p,T^γ,α(^,e^σ_l)^2 )for δ∈(2-2α-2κ/σ,α) and some ν>0, where C_M_=c_0 (1+M_^2) and C_u_0= 1_γ=α c_0 u_0_𝒞_p^0(^,e^σ_l) +1_γ=0c_0 (u_0^♯_𝒞_p^2α(^,e^σ_l)+ u^,X(0)_𝒞_p^α(^,e^σ_l)+ u^,♯(0)_𝒞_p^2α(^,e^σ_l)) ,for some c_0>0 that does not depend on ξ^ε, c^ε_μ or u_0. The complicated formulation of (<ref>) is necessary because when we expand the singular product on the right hand side we getF^(u^)ξ^ = F'(0) (C(u^,X,X_μ^,ξ^) + u^,X (X_μ^ξ^)) + … ,so to obtain the right renormalization we need to subtract F'(0) u^,X c^_μ, which is exactly what we get if we Taylor expand the second addend on the right hand side of (<ref>). If u^ε = v^ε = ℳ^ε_γ,u_0 u^ε is a fixed point,then u^,X = v^,X = F'(0) u^ and the “renormalization term” is just F^(u^) F'(0) c^_μ. Moreover we have in this caseℒ^_μ u^ = F^(u^)(ξ^ - F'(0) c^_μ) , u^(0)=u_0 . We assume for the sake of shorter formulas (1+M_^2)≲ 1, the general case can be easily included in the reasoning below.The solution to (<ref>) can be constructed using the Green's function ^-1 e^-t l^_μ and Duhamel's principle. To uncluster the notation a bit, we will drop the upper indexon u, v, X_μ, ℒ_μ,…in this proof.We show both estimates at once by denoting by γ either 0 or α. Throughout the proof we will use the fact that u_ℒ^γ/2,α_p,T(^,e^σ_l)=u^X X_μ +u^♯_ℒ^γ/2,α_p,T(^,e^σ_l)≲u_𝒟_p,T^γ,β(^,e^σ_l)for all β∈ (0,α] which follows from Lemma <ref>. In particular (with β=δ) we havev^X_ℒ^γ/2,α_p,T(^,e^σ_l) = F'(0) u _ℒ^γ/2,α_p,T(^,e^σ_l)(<ref>)≲u_𝒟^γ,δ_p,T(^,e^σ_l)≲1_γ=0 (u^X(0)_𝒞_p^α(^,e^σ_l)+u^♯(0)_𝒞_p^2α(^,e^σ_l))+ T^α-δ/2u_𝒟^γ,α_p,T(^,e^σ_l) . This leaves us with the task of estimating v^♯_ℒ^γ,2α_p,T(^,e^σ_l). We split ℒ_μ v^♯ =ℒ_μ(v-F'(0)u X_μ) =F^(u)ξ-F^ε(u^Y/F'(0)) F'(0)c_μ-F'(0)ℒ_μ(u Y )= F'(0)uξ- F'(0)u^X c_μ -F'(0)ℒ_μ(u X_μ)+R(u)u^2 ξ - R(u^X/F'(0)) (u^X)^2 /F'(0) c_μ=F'(0)[ u (ξ-ξ̅)+uξ̅ -uξ̅ +u ξ̅- ℒ_μ (uX_μ)+ u ξ +C(u^X,X_μ,ξ)+u^X(X_μ∙ξ) + u^♯ξ ] ♯ +R(u)· u^2 ξR_u -R(u^X/F'(0))(u^X)^2/F'(0) c_μR_u^X ,where ξ=χ(D)ξ so that ℒ_μ X_μ=ξ̅ with ξ-ξ̅∈⋂_β∈𝒞^β_∞(^,p^κ) and where R(x)=^2∫_0^1 (1-λ) F”(λ^2 x)λ.We have by Lemmas <ref>, <ref>(<ref>)_ℳ^γ_T𝒞^2α+2κ/σ-2_p(^,e^σ_l p^κ)≲u_ℒ^γ/2,α_p,T(^,e^σ_l)≲u_𝒟^γ,δ_p,T(^,e^σ_l)and further with Lemma <ref> and Lemma <ref>(<ref>)_ℳ^γ_T 𝒞^2α+4κ/σ-2(^,e^σ_l p^2κ)≲u_𝒟^γ,δ_p,T(^,e^σ_l) ,while the term (<ref>) can be bounded with Lemma <ref> by u^♯ξ_ℳ^γ_T𝒞^2α+2κ/σ-2_p(^,e^σ_l p^κ)≲u^♯_ℒ^γ,α+δ_p,T(^,e^σ_l)≤u_𝒟^γ,δ_p,T(^,e^σ_l) .To estimate (<ref>) weuse the simple bounds ^β' f_𝒞_q^β+β'(^,ρ)≲f_𝒞^β_q(^,ρ) for β∈, β'>0, q∈ [1,∞], ρ∈ρ(ω) and^-β f_L^q(^,ρ)≲^-β∑_j≲ j_^ 2^-jβf_𝒞^β_q(^,ρ)≲f_𝒞^β_q(^,ρ)for β<0, q∈ [1,∞], ρ∈ρ(ω), together with the assumption F”∈ L^∞, and obtain for ν' > 0(<ref>)_ℳ^γ_T𝒞^2α+2κ/σ-2_p(^,e^σ_l p^κ) ≲F”_∞^α+2κ/σ u^2_ℳ^γ L^p(^,e^σ_l) ^2-(α+2κ/σ)ξ_L^∞(^,p^κ)≲^α+2κ/σ u^2_ℳ^γ_T L^p(^,(e^σ_l)^2) ξ_𝒞^α+2κ/σ-2_∞(^, p^κ)≲^α/2+κ/σ u_ℳ^γ/2_T L^2p(^,e^σ_l)^2 ≲^α/2+κ/σ u_ℳ^γ/2_T 𝒞_p^d/2p+ν'(^,e^σ_l)^2≤^α/2+κ/σ u_ℳ^γ/2_T 𝒞_p^1+ν'(^,e^σ_l)^2 ≲^α/2+κ/σ-(1+ν'-α) u^2_ℳ^γ/2_T 𝒞^α_p(^,e^σ_l)≲^3α+2κ/σ-2(1+ν')u_𝒟^γ,δ_p,T(^,e^σ_l)^2 ≤ε^νu_𝒟^γ,δ_p,T(^,e^σ_l)^2 for all ν∈ (0,3α+2κ/σ-2(1+ν')] (which is nonempty if ν' is sufficiently small). Similarly we get for ν'∈ (0,δ)(<ref>)_ℳ^γ_T𝒞^2α+2κ/σ-2_p(^,e^σ_l p^κ) ≲F”_L^∞()· c_μ u^X_ℳ^γ/2_T L^2p(^,e^σ_l)^2 ≲ c_μ u^X_ℳ^γ/2_T 𝒞^1+ν'_p(^,e^σ_l)^2≲^2(δ-ν') |log()| u^X_ℳ^γ/2_T 𝒞^δ_p(^,e^σ_l)^2 ≲^νu_𝒟^γ,δ_p,T(^,e^σ_l)^2for all ν∈ (0,δ-ν'].In total we have ℒ_μ v^♯_ℳ^γ_T𝒞^2α+2κ/σ-2_p(^,e^σ_l p^κ)≲u_𝒟^γ,δ_p,T(^,e^σ_l) + ^νu_𝒟^γ,δ_p,T(^,e^σ_l)^2 , v^,♯(0)=1_γ=0 u_0^♯+1_γ=α u_0 ,where we used for the initial condition that by Definition <ref> and Convention <ref> we have (F'(0)u X_μ)(0)=F'(0)u_0 X for γ=0 and (F'(0)u X_μ)(0)=0 for γ=α>0.The Schauder estimates of Lemma <ref> yield on these groundsv^♯_ℒ^γ,2α_p,T(^,e^σ_l) ≲1_γ=αu_0_𝒞^0_p(^,e^σ_l) + 1_γ=0u_0^♯_𝒞^2α_p(^,e^σ_l)+ u_𝒟^γ,δ_p,T(^,e^σ_l) + ^νu_𝒟^γ,δ_p,T(^,e^σ_l)^2 ≲1_γ=αu_0_𝒞^0_p(^,e^σ_l)+1_γ=0(u_0^♯_𝒞^2α_p(^,e^σ_l) +u^♯(0)_𝒞_p^2α(^,e^σ_l) +u^X(0)_𝒞^α_p(^,e^σ_l)) + T^(α-δ)/2 (u_𝒟^γ,α_p,T(^,e^σ_l) + ^νu_𝒟^γ,α_p,T(^,e^σ_l)^2) ,where in the last step we used Lemma <ref>. Together with (<ref>) the claim follows.As we mentioned in Remark <ref> we aim at finding fixed points of ℳ^_γ,a_0 which is achieved by the following Corollary.With the notation of Lemma <ref> choose T_^loc:=1/2(C_M_+C_M_^ν r( u_0))^-2/(α-δ) for a sufficiently large r(u_0)>0, depending on u_0. Then the map ℳ^_γ,u_0 from Lemma <ref> has a unique fixed point u^=u^,X X_μ^+u^,♯ on 𝒟^γ,α_p,T_^loc(,e^σ_l). This fixed point solvesℒ^_μ u^ = F^(u^)(ξ^ - F'(0) c^_μ) , u^(0)=u_0 ,and u^ε,X = F'(0) u^ε. Moreover, we haveu^_𝒟^γ,α_p,T_^loc(,e^σ_l)≤ r(u_0) .We construct the fixed point u^ by a Picard type iteration. To avoid notational clashes with the initial condition u_0, we start the iteration with n=-1 for which we define u_-1^:=F'(0) u_0 X_μ^+u^♯_0=u_0 X_μ^+u^♯_0=u_0 for γ=0 and u_-1^:=0 X_μ^+e^tL^_μu_0 for γ=α (which is in 𝒟_p,T^γ,α(^,e^σ_l) due to Lemma <ref>).Define recursively for n≥ 0 the sequence u_n^:=ℳ^_γ,u_0 u^_n-1 (with u^_n=u^,X_n X_μ^+u_n^,♯ to be read as a pair as in Definition <ref>). Choose now r(u_0) so big that u_-1^_𝒟_p,1^γ,α(^,e^σ_l)≤ r(u_0) and such thatC_u_0≤1/2 r(u_0)with C_u_0 as in Lemma <ref>. Note that for u_n+1^ε the constant C_u_0 in principle depends on u^_n(0), but in fact we can choose it independently of n since u_n^,X(0)=F'(0)u_0 for all n ≥ -1 (by definition of ℳ^_γ,u_0) and u_n^,♯(0)=1_γ=0 u_0^♯+1_γ=α u_0 (by Definition <ref> and Convention <ref>) in the second term of (<ref>).Since T_^loc≤ 1 we already know for n=-1 thatu_n^_𝒟_p,T_^loc^γ,α(^,e^σ_l)≤ r(u_0).We show recursively that (<ref>) is in fact true for any n≥ -1. Suppose we have already shown the statement for n-1, we then obtain by Lemma <ref>u_n^_𝒟^γ,α_p,T_^loc(^,e^σ_l) ≤ C_u_0+(T_^loc)^α-δ/2· C_M_ (r(u_0)+^ν (r(u_0))^2)≤r(u_0)/2+ (T_^loc)^α-δ/2 (C_M_+C_M_^ν r(u_0)) · r(u_0) =r(u_0)/2+r(u_0)/2=r(u_0) .By Lemma <ref> in the appendix inequality (<ref>) implies that for α'∈ (0,α) and σ'∈ (0,σ) there is a subsequence (u_n_k^ε)_k≥ 0, convergent in 𝒟^γ,α'_p,T_^loc(^,e^σ'_l) to some u^ε∈𝒟^γ,α_p,T_^loc(^,e^σ_l), and u^_𝒟^γ,α_p,T_^loc(^,e^σ_l)≤lim inf_k→∞u^_n_k_𝒟^γ,α_p,T_^loc(^,e^σ_l)≤ r(u_0) .In particular u^ε is a fixed point of ℳ^_γ,u_0 that satisfies (<ref>). It remains to check uniqueness. Choose two fixed points u^, v^, which then satisfyℒ_μ^(u^-v^) =(F^(u^)-F^(v^))(ξ^-c^_μ F'(0)) =∫_0^1 F'(u^+λ(v^-u^))λ_=:ℱ· (v^-u^)(ξ^-c^_μ F'(0)) . We will use that for ρ∈() and ζ, ζ'∈ with ζ'≥ζf_^ζ'_p(^,ρ)≲^-(ζ'-ζ)f_^ζ_p(^,ρ) ,which is an easy consequence of Definition <ref> and which we essentially already used in the proof of Lemma <ref>. In other words, we can consider our objects as arbitrarily “smooth” if we are ready to accept negative powers of . In particular, we can consider the initial condition u_0 as paracontrolled, that is u_0∈^α_p(^,e^σ_l), u_0^♯∈^2α_p(^,e^σ_l) (and thus u^,X(0)=v^,X(0)=F'(0)u_0∈_p^α(^,e^σ_l)), so that with Lemma <ref>we obtain u^,v^∈𝒟^0,α_p,T_^loc(^,e^σ_l). Consequently, since also e^σ_l ≥ 1, we get u^, v^∈ C_T_^loc L^∞(^) which implies that the integral term ℱ is in C_T_^loc L^∞(^) and, by using once more (<ref>), we can consider it as an element of C_T_^loc𝒞^β_∞(^) for any β∈. The product (v^-u^)(ξ^-c^_μ F'(0)) can then be estimated as in the proof of Lemma <ref>. Since multiplication by ℱ only contributes an (-dependent) factor we obtain for T'≤ T_^loc a bound of the formu^-v^_𝒟^0,α_p,T'(^,e^σ_l)≲_ (T')^α-δ/2u^-v^_𝒟^0,α_p,T'(^,e^σ_l) ,which shows u^-v^_𝒟^0,α_p,T'(^,e^σ_l)=0 for T' small enough. Iterating this argument gives u^=v^ on all of [0,T_^loc]. §.§.§ Convergence to the continuumIt is straightforward to redo our computations in the continuous linear case (i.e. F(x) = c x), which leads to the existence of a solution to the continuous linear parabolic Anderson model on ^2, a result which was already established in <cit.>. Since the continuous analogue of our approach is a one-to-one translation of the discrete statements and definitions above from ^ε to ^d we do not provide the details.Let u_0∈𝒞^0_p(^d,e^σ_l). Let ξ be a white noise on ^2, and let ℒ_μ be defined as in Section <ref>. Then there is a unique solution u=F'(0)u X_μ+u^♯∈𝒟^α,α_p,T(^d,e^σ_l) to ℒ_μ u=F'(0)uξ, u(0)=u_0 , blacklozenge on [0,T], whereuξ:=ξ u+ uξ + F'(0) C(u,X_μ,ξ)+ F'(0)u (X_μ∙ξ)+ u^♯ξwith X_μ, X_μ∙ξ as in (<ref>), (<ref>). As in Lemma <ref> we can build a map ℳ_α,u_0 𝒟^α,α_p,T(^d,e^σ_l)→𝒟^α,α_p,T(^d,e^σ_l): u=u^X_μ X_μ + u^♯↦ v= F'(0)u X_μ + v^♯ viaℒ_μ v:= F'(0) u ξ , v(0)=u_0 .As in Corollary <ref> there is a time T^loc such that ℳ_α,u_0 has a (unique) fixed point u^(0)=F'(0)u^(0) X_μ+u^(0),♯ in 𝒟^α,α_p,T^loc(^d,e^σ_l) that solves ℒ_μ u^(0) = F'(0) u^(0)ξ , u^(0)(0)=u_0.on [0,T^loc]. Since the right hand side of (<ref>) is linear, this time can be chosen of the form T^loc=1/2K^-2/(α-δ), where K>0 is a (random) constant that only depends on ξ, X_μ, X_μ∙ξ, but not on the initial condition. Proceeding as above but starting in u^(0)(T^loc) we can construct a map ℳ_0,u^(0)(T^loc) 𝒟^0,α_p,T^loc(^d,e^σ_l) →𝒟^0,α_p,T^loc(^d,e^σ_l) by (the continuous version of) Lemma <ref> and Lemma <ref>. The map ℳ_0,u^(0)(T^loc) has again a fixed point on [0,T^loc] which we call u^(1). Starting now in u^(1)(T^loc) we can construct u^(2) as the fixed point of ℳ_0,u^(1)(T^loc) on [0,T^loc] and so on. As in <cit.>) the sequence of local solutions u^(0), u^(1),u^(2),… can be concatenated to a paracontrolled solution u=F'(0)u X_μ+u^♯∈𝒟^α,α_p,T(^d,e^σ_l) on [0,T]. To see uniqueness take two solutions u, v in 𝒟^α,α_p,T(^d,e^σ_l) and consider h=u-v. Using that h(0)=0 and ℒ_μ h = hξ one derives as in Lemma <ref>h_𝒟_p,T^α,α(^d,e^σ_l)≤C · T^(α-δ)/2 h_𝒟_p,T^α,α(^d,e^σ_l)so that choosing T first small enough and then proceeding iteratively yields h=0.We can now deduce the main theorem of this section. The parameters are as on page eq:ParameterCondition1. Let u^_0 be a uniformly bounded sequence in 𝒞^0_p(^,e^σ_l) such that ^ u^_0 converges to some u_0 in _ω'(^2). Then there are unique solutions u^∈𝒟^α,α_p,T^(^,e^σ_l) to ℒ_μ^ u^=F^(u^)(ξ^ - c^_μ F'(0)), u^(0)=u^_0,on [0,T^] with random times T^∈ (0,T] that satisfy ℙ(T^=T)→ 0⟶ 1. The sequence u^=F'(0)u^ X_μ+u^,♯∈𝒟^α,α_p,T^(^,e^σ_l) is uniformly bounded and the extensions ^ u^ converge in distribution in 𝒟^α,α'_p,T(^d,e^σ'_l), α'<α, σ'<σ, to the solution u of the linear equation in Corollary <ref>.Since T^ is a random time for which it might be true that P(T^<T)>0 the convergence in distribution has to be defined with some care: We mean by ^ u^→ u in distribution that for any f∈ C_b (𝒟^α,α'_p,T(^,e^σ_l);), we have 𝔼[f(^ u^)1_T^ = T]→𝔼[f(u)] and further ℙ(T^ < T)→ 0. The local existence of a solution to (<ref>) is provided by Corollary <ref>. Proceeding as in the proof of Corollary <ref> we can in fact construct a sequence of local solutions (u^,(n))_n≥ 0 on intervals [0,T_^loc,(n)] with u^,(n)(0)=u^,(n-1)(T_^loc,(n-1)), where we set T_^loc,(-1):=0 and u^,(-1):=u_0. Due to Corollary <ref> the time T_^loc,(n) is given by T_^loc,(n):=1/2 (C_M_+C_M_^ν r( u^,(n-1)(T_^loc,(n-1))))^-2/(α-δ) .Note that, in contrast to the proof of Corollary <ref>, T_^loc,(n) now really depends on n and we might have ∑_n≥ 0 T_^loc,(n)<∞. As in <cit.> we can concatenate the sequence u^,(0),u^,(1),… to a solution u^ to (<ref>) which is defined up to its “blow-up” timeT_^blow-up=∑_n≥ 0 T_^loc,(n) (which might be larger than T or even infinite). Let us setT^:= T∧T_^blow-up/2 .To show ℙ(T^=T)→ 0⟶ 1 we prove that for any t>0 we have ℙ(T_^blow-up<t)→ 0. By inspecting the definition of r(…) in the proof of Corollary <ref> we see that given the (bounded) sequence of initial condition u_0^ the size of T_^blow-up can be controlled by the quantity M^. More precisely there is a deterministic, decreasing function T_^det: _+→_+ such thatT_^blow-up≥ T_^det(M^)and such that for any K>0 (due to the presence of the factor ^ν in (<ref>))T_^det(K)→ 0⟶∞ .Let t>0 and K^_t:=sup{K>0 | T_^det(K) ≥t}. Note that we must have K^_t→ 0⟶∞ since otherwise we contradict (<ref>). But this already implies the desired convergence:ℙ(T_^blow-up< t)≤ℙ(T_^det(M^)< t)≤ℙ(M^≥ K^_t)K^_t→∞⟶ 0 ,where we used in the last step the boundedness of the moments of M^ due to Lemma <ref>. It remains to show that the extensions ^ u^ converge to u. By Skohorod representation we know that ^ξ^, ^ X_μ^, ^ (X_μ^∙ξ^) in Lemma <ref> converge almost surely on a suitable probability space. We will work on this space from now on. The application of the Skohorod representation theorem is indeed allowed since the limiting measure of these objects has support in the closure of smooth compactly supported functions and thus in a separable space. We can further assume by Skohorod representation that (a.s.) T_^blow-up→∞ so that almost surely we have T^=T for all ε≤ε_0 with some (random) ε_0. Having proved that the sequence u^ is uniformly bounded in 𝒟^α,α_p,T^(^,e^σ_l) we know, by Lemma <ref>, that ^ u^ is uniformly bounded in 𝒟^α,α_p,T^(^d,e^σ_l). Due to (the continuous version of) Lemma <ref> there is at least a subsequence of ^_n u^_n that converges to some u∈𝒟^α,α_p,T(^d,e^σ_l)(^d) in the topology of 𝒟^α,α'_p,T(^d,e^σ'_l). If we can show that this limit solves (<ref>) we can argue by uniqueness that (the full sequence) ^ u^ converges to u. We haveℒ^_n_μ^_n u^_n=^_nℒ^_n_μ u^_n =^_n(F^_n(u^_n)(ξ^_n -c^_n_μ F'(0))) ,where ℒ^_μ^ u^ should be read as in (<ref>). Note that the left hand side of (<ref>) converges asℒ^_μ^_n u^_n = (ℒ^_n_μ - ℒ_μ)^_n u^_n+ℒ_μ^_n u^_n_n→ 0⟶ 0+ℒ_μ u=ℒ_μ udue to Lemma <ref>. For the right hand side of (<ref>) we apply the same decomposition as in (<ref>)=(<ref>)+(<ref>)+(<ref>)+(<ref>)+(<ref>). While (the extensions of) the terms (<ref>),(<ref>) of (<ref>) vanish astends to 0, we can use the property (<ref>) of the operators acting in the terms (<ref>), (<ref>), (<ref>) to identify their limits. Consider for example the product u^,X^_μ(X_μ^∙ξ^)= F'(0) u^ (X_μ^∙ξ^) in (<ref>) whose extension we can rewrite as ^_n( F'(0) u^_n (X_μ^_n∙ξ^_n) ) = F'(0) ^_n(u^_n (X_μ^_n∙ξ^_n) +u^_n (X_μ^_n∙ξ^_n) +u^_n (X_μ^_n∙ξ^_n) ) (<ref>)=F'(0)[^_n u^_n^_n(X_μ^_n∙ξ^_n) +^_n u^_n^_n(X_μ^_n∙ξ^_n) + ^_n u^_n^_n(X_μ^_n∙ξ^_n)]+o__n(1) ,where we applied the property (<ref>) of ,, (Lemma <ref>) in the second step. By continuity of the involved operators and Lemma <ref> we thus obtain lim__n→ 0^_n( F'(0) u^_n (X_μ^_n∙ξ^_n) ) = F'(0) [ u (X∙ξ)+u (X∙ξ)+ u (X∙ξ)]=F'(0)u (X∙ξ) .Proceeding similarly for all terms in the decomposition of the right hand side of (<ref>) one arrives at ℒ_μ u=lim__n→ 0^_nℒ^_n_μ u^_n = lim__n→ 0^_n(F^_n(u^_n)(ξ^_n -c^_n_μ F'(0)))=F'(0) uξ ,which finishes the proof.Since the weights we are working with are increasing, the solutions u^ and the limit u are actually classical tempered distributions. However, since we need the _ω spaces to handle convolutions in e^σ_l weighted spaces it is natural to allow for solutions in '_ω.In the linear case, F=Id, we can allow for sub-exponentially growing initial conditions u_0 since the only reason for choosing the parameter l in the weight e_l+t^σ smaller than -T was to be able to estimate e^σ_l+t≤ (e^σ_l+t)^2 to handle the quadratic term. In this case the solution will be a genuine ultra-distribution. § APPENDIX§.§ Results related to Section <ref> The mappings (_,^-1_) defined in Subsection <ref> map the spaces (_ω(), _ω()) and ('_ω(), '_ω()) to each other.We only consider the non-standard case ω=|·|^σ. Givenf ∈_ω() the sequence_ f(x)=||∑_k∈ f(k) e^2π k xobviously converges to a smooth function that is periodic on . We estimate on(and thus by periodicity uniformly on ^d)|∂^α∑_k∈ || f(k) e^2π k x|≲_λ∑_k∈ |||k|^|α| e^-λ |k|^σWe can use Lemma <ref> for |·|^|α| e^-λ |·|^σ with Ω= and c>0 of the form c=C(λ) · C^|α| (C denoting a positive constant that may change from line to line) which yields |∂^α∑_k∈ || f(k) e^2π k x|≲_λ C^|α|∫_^d |x|^|α| e^-λ|x|^σ xWe now proceed as in <cit.> and estimate the integral by the Γ-function ∫_^d |x|^|α| e^-λ|x|^σ x ≲∫_0^∞ r^|α|+d-1 e^-λ r^σ r ≲_λλ^- |α|/σ∫_0^∞ r^|α|+d-1 e^-r^σ r ≲λ^- |α|/σΓ((|α|+d-1)/σ) ≲λ^-|α|/σ C^|α| |α|^|α|/σ .Since we can choose λ>0 arbitrarily large we see that indeed f∈ C^∞_ω().For the opposite direction, f∈_ω(), we use that by integration by parts |z_i^l ·^-1_ f(z) | ≲ C^l sup_ (∂^i)^l f ≲ C^l ^l l^l/σ for all z∈, l≥ 0, i=1,…,d. With Stirling's formula and Lemma <ref> we then obtain |_^-1 f(z)|≲e^λ |z|^σ. This shows the statement for the pair (_ω(), _ω()). The estimates above show that _,^-1_ are in fact continuous w.r.t to the corresponding topologies so that the statement for the dual spaces ('_ω(), '_ω()) immediately follows.Given a latticeas in (<ref>) we denote the translations of the closed parallelotope G:=[0,1]a_1 +… + [0,1]a_d by 𝔾:={g+G |g∈}. Let Ω⊆ and set Ω:=⋃_G'∈𝔾,G'∩Ω≠∅ G' . If for a measurable function f:Ω→_+ there exists c≥ 1 such that for any g∈Ω there is a G'(g)∈𝔾,g∈ G'(g) with f(g)≤ c ·essinf _x∈ G' f(x)then it also holds∑_g∈Ω || f(g) ≤ c· 2^d∫_Ω f(x)x .Indeed ∑_g∈Ω || f(g)≤ c ∑_g∈Ω∫_G'(g) f(x)x ≤ c ∑_g∈Ω∑_G'∈𝔾: g∈ G'∫_G'(g) f(x)x ≤c ∑_G' ⊆Ω∑_g∈Ω: g∈ G'∫_G' f(x)x ()= 2^d c ∑_G' ∈Ω∫_G' f(x)x = 2^d c ∫_Ω f(x)x ,where we used in () that the d-dimensional parallelotope has 2^d vertices.For f^d→ℂ and g→ℂ we set for x∈^df∗_ g(x):=∑_k∈ || f(x-k) g(k)Then for r,p,q∈ [1,∞] with 1+1/r=1/p+1/q f∗_ g_L^r(^d)≤sup_x∈^d f(x-·) ^1-p/r_L^p()·f_L^p(^d)^p/rg_L^q()(with the convention 1/∞=0, ∞/∞=1).We assume p,q,r∈ (1,∞). The remaining cases are easy to check. The proof is based on Hölder's inequality onwith 1/r+1/rp/r-p+1/rq/r-q=1|f∗_ g(x)|≤∑_k∈ || ( |f(x-k)|^p |g(k)|^q )^1/r· |f(x-k)|^r-p/r |g(k)|^r-q/rHölder≤( |f(x-·)|^p |g(·)|^q)^1/r_L^r()· |f(x-·)|^r-p/r_L^rp/r-p()·|g(·)|^r-q/r_L^rq/r-q()≤( ∑_k∈ || (|f(x-k)|^p |g(k)|^q )^1/rsup_x'∈^df(x'-·)^r-p/r_L^p()g^r-q/r_L^q() .Raising this expression to the rth power and integrating it shows the claim. §.§ Results related to Section <ref> For T≥ 0, p∈ [1,∞], ρ∈ρ(ω) we have uniformly in t ∈ [0,T] and ε∈ (0,1]e^tL^_μ f _L^p(^,ρ)≲f_L^p(^,ρ) ,and for β>0e^t L^_μ f_L^p(^,ρ)≲ t^-β/2f_𝒞^-β_p(^,ρ) .With the random walk (X_t^)_t∈_+ which is generated by L^_μ on ^ we can express the semigroup as e^t L^_μ f(x)=𝔼[f(x+X_t^)], so thatρ e^t L^_μf _L^p(^ε)= 𝔼[ρ(·)/ρ(·+X^ε_t)ρ(· + X^ε_t) f(· +X_t^)] _L^p(^ε)≤𝔼[ sup_x ∈^ερ(x)/ρ(x+X^ε_t) f _L^p(^ε, ρ)]≲𝔼[e^λω(X^ε_t)]f _L^p(^ε, ρ)An application of the next lemma finishes the proof of the first estimate.The second estimate follows as in Lemma 6.6. of <cit.>.The random walk generated by L^_μ on ^ satisfies for any λ >0 and t∈ [0,T] 𝔼[e^λω(X_t^)]≲_λ,T 1 .We assume ω=, if ω is of the polynomial form the proof follows by similar, but simpler arguments.In this proof we write shorthand s=1/σ.By the Lévy-Khintchine-formula we have 𝔼[e^θ X_t^]=e^-t/^2 ∫_ (1-e^θ x)μ(x) =e^-t l^_μ(θ) for all θ∈.We want to bound first for k≥ 1𝔼[|X_t,1^|^k+…+|X_t,d^|^k]=∑_j=1^d |∂_θ_j^k |_θ=0𝔼[e^θ X_t^]| .To this end we apply Faá-di-Brunos formula with u(v)=e^-tv, v(θ)=l^_μ(θ). Note that with Lemma <ref> for m ∈ and j=1,…,du^(m)(0)=(-t)^m |∂^m_θ_jv(0)|≲_δδ^m (m!)^s.Thus with A_m,k={(α_1,…,α_m)∈^m | ∑_i=1^m α_i ·i =k} we get for any δ∈ (0,1]|∂_θ_j^k |_θ=0𝔼[e^θ X_t^]| =| ∑_1≤ m≤ k, α∈ A_m,kk!/α! u^(m)(0) ∏_i=1^m (1/i!∂^i_θ_j v(0))^α_i| ≲_δ∑_1≤ m≤ k , α∈ A_m,kk!/α! t^m ∏_i=1^m (i!)^α_i(s-1)δ^i·α_iStirling≤δ^k C^k ∑_1≤ m≤ k, α∈ A_m,kk!/α! t^m∏_i=1^m C^iα_i i^iα_i(s-1)i≤ m≤ k≤δ^k C^k ∑_1≤ m≤ k, α∈ A_m,kk!/α! t^m k^k(s-1)Stirling≤δ^k C^k ∑_1≤ m≤ k, α∈ A_m,k(k!)^s/α! t^m (α!)^-1≤ 1≤δ^k C^k (k!)^s∑_1≤ m≤ k |A_m,k|t^m = δ^k C^k (k!)^s∑_1≤ m≤ kk-1m-1 t^m = δ^k C^k (k!)^s t (1+t)^k-1≤δ^k C^k (k!)^s (1+t)^k,where C>0 denotes as usual a generic constant that changes from line to line. With |x|_k^k:=|x_1|^k+…+|x_d|^k we get𝔼[|X^_t|_k^k] ≲δ^k C^k (k!)^s (1+t)^kand therefore, using once more Stirling's formula and |x|^k ≲ C^k · |x|_k^k,𝔼[e^λ |X_t^|^σ]≲ 1+ 𝔼[e^λ|X_t^|^σ1_|X_t^|≥ 1] ≤ 1+∑_k=0^∞λ^k/k!𝔼[|X^_t|^⌈ k σ⌉] ≲ 1+∑_k=0^∞C^k (1+t)^⌈ k σ⌉/k^kδ^⌈ k σ⌉⌈ k σ⌉^⌈ k σ⌉ s≲ 1+ (1+t)∑_k=0^∞C^k δ^k σ (1+t)^k σ/k^k k^k ≲ 1 ,where in the last step we chose δ>0 small enough to make the series converge.§.§ Results related to Section <ref> Let ^ as in Definition <ref>, let ω∈, and let (φ_j^^)_j=-1,…,j_ be a partition of unity as on page DiscreteDyadicPartition. For -1≤ i≤ j≤ j_ the function_i^ f_1 ·_j^ f_2 ∈'(^)is spectrally supported in a set of the form 2^j ∩, whereis a ball around 0 that can be chosen independently of i, j and . For f_1, f_2∈'() and 0<j≤ j_^ the functionS_j-1^ f_1 ·_j^ f_2 ∈'(^) ,is spectrally supported in a set of the form 2^j ∩, whereis an annulus around 0 that can be chosen independently of j and .We can rewrite ( ^_i f_1 ·_j^ f_2 )= (φ_i^ f)∗_(φ_j^ f_2) = ∫_ (φ_i^ f)(z) ·(φ_j^ f_2)([· -z]_)z,where we used formal notation in the last step and [·]_ as in (<ref>). From this one sees that the spectral support of ^_i f_1 ·_j^ f_2 is contained in(φ_i^+ φ_j^^ + ^) ∩ ,where we recall that φ^_i={x∈ | φ_i^(x)≠ 0 } is a subset of (the closure of) ⊆^d, while the sum of sets in the parentheses should be read as a subset of ^d. Now, by the dyadic scaling of φ_j^ we have for all i ≤ jφ_i^+ φ_j^^⊆ B(0,2^j b)for some b>0, independent ofand j. Set: _1:=B(0,b) and consider first the case 2^j ℬ_1=B(0,2^j b)⊆. In this case we have (φ_i^+ φ_j^^ + ^) ∩⊆ (2^j _1 + ^) ∩ = 2^j_1 ∩^ = 2^j _1 .On the other hand, if 2^j_1=B(0,2^j b)⊊ we are in the regime j∼ j_ and take a ball _2 around 0 such that 2^j_2 ⊇ and hence 2^j_2∩ = for all j∼ j_ (by the dyadic scaling offrom Definition <ref> we have 2^j_=c·^-1 so that we can choose _2 independently of ). Choosing then =_1∪_2 shows the first part of the claim.Let us now consider S_j-1^ f_1 ·_j^ f_2. With φ_<j-1^^:= ∑_j'<j-1φ_j'^ we see as above that the spectral support of S_j^ f_1 ·_j^ f_2 is contained in(φ_<j-1^+ φ_j^^ + ^) ∩ ,We already know from above that this set is contained in a ball of size 2^j so that is enough to show that (<ref>) is bounded away from 0. Since φ_<j-1^ and φ_j^ are symmetric and disjoint, we have due to the scaling from (<ref>) and (<ref>), which we observed in the proof of Lemma <ref>, that (φ_<j-1^ + φ_j^,0) ≥ 2^j afor some a>0 andφ_<j-1^ + φ_j^⊆ B(0,2^j · b') , for some b'>0. Note, that we can choose b'>0 small enough such that B(0,2^j_ b')∩^={ 0 }. Indeed, otherwise there are x_1∈φ_<j_-1^, x_2∈φ_j_^ such that x_1+x_2=r for some r∈^\{0}. But from |x_1|<(∂,0) one sees that |x_2|=|r-x_1|> ()/2 which contradicts x_2∈φ_j^⊆. This choice of the parameter b' can be done independently ofdue to the dyadic scaling of our lattice (Definition <ref>).Consequently, there exists r>0 such that (B(0,2^j b')+^\{0},0)= 2^j r (to see that r>0 is independent of , use once more the dyadic scaling of the sequence ^). But then we have ((φ_<j-1^+ φ_j^^ + ^) ∩, 0)≥ (a ∧ r) · 2^j ,which closes the proof. §.§ Results related to Section <ref>We will write shorthand f^ε_k:=_(^ε)^kf^ε_k and f_k:=_(^d)^k f_k. The claimed convergence is a consequence of the results in <cit.>. For z∈^ε let G^ε(z)=z+[-ε/2,ε/2)a_1+…+[-ε/2,ε/2)a_d, where a_1,…,a_d denote the vectors that span . For x∈^d let [x]_ be the (unique) element in ^ε such that x∈ G^ε([x]_) and for x ∈ (^d)^k set [x]_ = ([x_1]_, …, [x_k]_).We will start by showinglim_ε→ 0 f^ε_k([ ·]_) - f_k_L^2((^d)^k) = 0for all k.By Parseval's identity we have f^ε_k([ ·]_) - f_k_L^2((^d)^k) = ℱ_(^d)^k(f^ε_k([ ·]_)) - f_k_L^2((^d)^k), where ℱ_(^d)^k denotes the Fourier transform on (^d)^k for which one easily checks thatℱ_(^d)^k(f^ε_k([ ·]_)) = (f^ε_k)_ext· p^ε_k,where we recall that (f^ε_k)_ext is the periodic extension of the discrete Fourier transform of f^ε_k (on (^d)^k) as in (<ref>) and wherep^ε_k(y_1, …, y_k) = ∫_G^1(0)^k z_1 … z_k/|^1|^k e^-2πε (y_1z_1 + … + y_kz_k).The function p^ε_k is uniformly bounded and tends to 1 as ε goes to 0. Now we apply Parseval's identity, once on (^d)^k and once on (^ε)^k, and obtain∫_(^d)^k x_1 … x_k| ((f^ε_k)_ext p^ε)(x_1, …, x_k) |^2 = ∑_z_1, …, z_k ∈^ε |^ε|^k |f^ε_k(z_1, …, z_k)|^2= ∫_(^ε)^k x_1 … x_k | f^ε_k(x_1, …, x_k)|^2and thus∫_((^ε)^k)^c x_1 … x_k | ( (f^ε_k)_extp^ε)(x_1, …, x_k) |^2 = ∫_(^ε)^k x_1 … x_k (| f^ε_k|^2 (1 - |p^ε|^2 )(x_1, …, x_k).Since _(^ε)^kf^ε_k is uniformly in ε bounded by g_k ∈ L^2((^d)^k) and since 1 - |p^ε|^2 converges pointwise to zero, it follows from the dominated convergence theorem that _((^ε)^k)^c (f^ε_k)_ext p^ε_k converges to zero in L^2((^d)^k). Thus, we getlim_ε→ 0 (f^ε_k)_extp^ε_k - f_k_L^2((^d)^k)= lim_ε→ 0_(^ε)^kf^ε_k p^ε_k- f_k_L^2((^d)^k)≤lim_ε→ 0 (_(^ε)^kf^ε_k- f_k) p^ε_k_L^2((^d)^k) + lim_ε→ 0f_k (1 - p^ε_k ) _L^2((^d)^k) = 0,where for the first term we used that p^ε_k is uniformly bounded in ε and that by assumption _(^ε)^kf^ε_k converges to f_k in L^2((^d)^k) and for the second term we combined the fact that p^ε_k converges pointwise to 1 with the dominated convergence theorem. We have therefore shown (<ref>). Note that this impliesf^ε_k([·]_) 1_∀ i≠ j[z_i]_≠ [z_j]_-f_k_L^2(^d)→ 0 &f^ε_k([·]_)1_∃ i≠ j [z_i]_=[z_j]__L^2(^d)→ 0 .As in the proof of Lemma <ref> we identify ^ε with an enumeration →^ε and use the set A^k_r={a∈ℕ^r| ∑_i a_i=k} so that we can writeℐ_k f^ε_k=∑_1≤ r≤ k, a∈ A^k_r r! ∑_z_1<…<z_r |^ε|^k f̃^k_ε,a(z_1,…,z_r)·∏_j=1^r ξ^(z_j)^♢a_j ,where we denote as in the proof of Lemma <ref> by f̃^k_ε,a the symmetrized restriction of f^k_ε to (^d)^r. By Theorem 2.3 of <cit.> we see that due to (<ref>) the r=k term of ℐ_k f^ε_kconverges in distribution to the desired limit, so that we only have to show that the remaining terms vanish as ε tends to 0. The idea is to redefine for fixed a ∈ A^k_r the noise as ξ^ε_j(z)=ξ^(z)^♢a_j/r_j^ε(z) where r^ε_j(z):=√(Var(ξ^(z)^♢a_j)· |^ε|)≲ |^ε|^(1-a_j)/2, so that in view of <cit.> it suffices to show that ∑_z_1<…<z_r |^ε|^r ∏_j=1^r r^ε_j(z_j)^2 · |f̃^ε_k,a(z_1,…,z_r)|^2≲∑_z_1<…<z_r |^ε|^k · |f̃^ε_k,a(z_1,…,z_r)|^2 → 0 ,but this follows from (<ref>).Let (f_n)_n≥ 0 be a sequence which is bounded in the space ℒ^γ,α_p,T(,e^σ_l) and let α'∈(0,α) and σ'∈ (0,σ). There is a subsequence (f_n_k)_k≥ 0, convergent in ℒ^γ,α'_p,T(,e^σ'_l), with limit f such thatf_ℒ^γ,α_p,T(,e^σ_l)≤lim inf_k→∞f_n_k_ℒ^γ,α_p,T(,e^σ_l)Take in the following α̃=α+α'/2 and σ̃=σ+σ'/2. By Definition of ℒ^γ,α_p,T(,e^σ_l) we know that (g_n)_n≥ 0:=((t,x)↦ t^γ f_n(t,x) )_n≥ 0 is bounded in C_T^α/2L^p(,e^σ_l)∩ C_T_p^α(,e^σ_l). Interpolation then shows that (g_n)_n≥ 0 is bounded in C_T^α̃/2_p^δ_x(,e^σ_l)∩ C_T^δ_t_p^α̃(,e^σ_l) for some δ_x, δ_t>0. We obtain by compact embedding (Lemma <ref>) for δ_x'∈ (0,δ_x), δ_t'∈ (0,δ_t) the existence of a convergent subsequence (g_n_k)_k≥ 0 in C_T^α'_p^δ_x'(,e^σ'_l)∩ C_T^δ_t'_p^α'(,e^σ'_l) with some limit g. From the convergence of g_n_k→ g in C_T^α'_p^δ_x'(,e^σ'_l)∩ C_T^δ_t'_p^α'(,e^σ'_l) it follows that for f:=t^-γ g we have f_n_k→ f in ℒ^γ,α'_p,T(,e^σ'_l). The estimate (<ref>) is then just an iterative application of Fatou like arguments for the norms from which ·_ℒ^γ,α_p,T(,ρ) is constructed. 10Bahouri H. Bahouri, J.-Y. Chemin, and R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equations. Springer-Verlag, 2011.Bailleul2016 I. Bailleul and F. Bernicot. Heat semigroup and singular PDEs. J. Funct. Anal., 270(9):3344–3452, 2016.Bjoerck G. Björck. Linear partial differential operators and generalized distributions. 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Lattice approximation to the dynamical ϕ_3^4 model. Ann. Probab., 46(1):397–455, 2018. | http://arxiv.org/abs/1704.08653v2 | {
"authors": [
"Jörg Martin",
"Nicolas Perkowski"
],
"categories": [
"math.PR"
],
"primary_category": "math.PR",
"published": "20170427165633",
"title": "Paracontrolled distributions on Bravais lattices and weak universality of the 2d parabolic Anderson model"
} |
Source File Set Search for Clone-and-Own Reuse Analysis Takashi Ishio12, Yusuke Sakaguchi1, Kaoru Ito1, Katsuro Inoue1 1 Graduate School of Information Science and Technology, Osaka University, Osaka, Japan 2 Graduate School of Information Science, Nara Institute of Science and Technology, Nara, Japan Email: {ishio, s-yusuke, ito-k, inoue}@ist.osaka-u.ac.jp December 30, 2023 ======================================================================================================================================================================================================================================================================================================================= Swarms of robots will revolutionize many industrial applications, from targeted material delivery to precision farming. Controlling the motion and behavior of these swarms presents unique challenges for human operators, who cannot yet effectively convey their high-level intentions to a group of robots in application. This work proposes a new human-swarm interface based on novel wearable gesture-control and haptic-feedback devices. This work seeks to combine a wearable gesture recognition device that can detect high-level intentions, a portable device that can detect Cartesian information and finger movements, and a wearable advanced haptic device that can provide real-time feedback. This project is the first to envisage a wearable Human-Swarm Interaction (HSI) interface that separates the input and feedback components of the classical control loop (input, output, feedback), as well as being the first of its kind suitable for both indoor and outdoor environments. § HUMAN-SWARM INTERACTION: THE EMERGENT FIELDWith a strong initial influence from nature and bio-inspired models <cit.>, swarm systems are known for their adaptability to different environments <cit.> and tasks <cit.>. As a result, swarm robotics research has recently been gaining popularity –- Fig. <ref>[Information retrieved from http://www.scopus.comScopus research database.] –-. As the cost of robotic platforms continues to decrease, the number of applications involving multiple robots is increasing. These include targeted material transportation <cit.>, where groups of small robots are used to carry tall, and potentially heavy, objects; precision farming <cit.>, where a fleet of autonomous agents shifts operator activities in agricultural tasks; and even entertainment systems <cit.>, where multiple robots come together to form interactive displays.The efficiency of performing tasks with robotic teams depends on two main factors: the level of robot autonomy, and the ability of human operators to command and control the team of robots. Regarding the latter, the transition from current application scenarios where several human operators control a single robot <cit.> to environments where a single human control multiple robots, has been identified as one of the main challenges in robotics research <cit.>.One of the clearest examples of this necessity is when the task conducted by the team of robots becomes extremely complex and begins to require high-level, cognitive-based decisions inline (e.g., exploration of dynamic, unstructured, and unpredictable environments for search and rescue applications). When a robot swarm needs to react to or quickly respond to an abrupt event (e.g., a fast stop), the absence of human intervention can even lead to complete mission failure. In these situations, full autonomy is still far from being reached by robot units alone, and human intervention is necessary for adequate performance. However, the ability to command a swarm of robots requires a significant cognitive effort from human operators. Previous works <cit.>, have emphasized the complexity of these tasks and have compared them to computational complexity (O). Likewise, swarm operators traditionally perform a repetitive sequence of steps to enable the system (i.e., the robot swarm) to fulfill an objective, or reach some desired goal state <cit.>. Normally, these sequences of steps become more complex as the operator has to share his/her cognitive resources among a higher number of robots <cit.>.Under this framework, different command and control operations involving robot swarms can have different levels of complexity (Fig. <ref>). For instance, control modes, such as the leader-follower approach <cit.> where the number of possible actions (n) is independent of the number of robots, can represent a relatively low-level of complexity (ideally O_(1)) for human operators under their cognitive limit. In contrast, if several robots are performing independent tasks, the complexity level might increase linearly as new robots and tasks are included into the swarm (O_(n)), eventually surpassing the cognitive abilities of the operator and making the operation of the swarm unsustainable. Moreover, task scenarios where robots need to tightly coordinate (e.g., transporting objects with deformable <cit.> shapes) are considered to have an exponential complexity level (O_(>n)) due to the inter-dependencies between robots, making the operation of such group of robots even harder.The primary purpose of this cognitive complexity framework was to emphasize the effort of human operators required to control a swarm robotics system, and the basic need of creating tools and techniques that allow operators to control higher number of robots without reaching their cognitive limits.Human-Swarm Interaction (HSI) is a prominent research field –- Fig. <ref>[1] –- that aims to allow a human operator to be aware of certain swarm-level information that he/she can use to make decisions regarding the swarm behavior. However, this is a complicated process since some kind of mechanism is needed to bridge the information gap between the human operator and the robot swarm. Human supervision normally relies on global goals such as mission statements or route planning. In contrast, simple robot units are usually hardware-limited and can access only to local information.The design of interfaces that allow operators to control a swarm of robots is receiving increasing research attention <cit.>. Several well-known technologies – including vision-based systems, haptic devices and electromyographical (EMG) receptors – have been proposed. However, seamless interaction between operators and robot swarms has not yet been achieved, not only due to the complexities of translating numerous local information streams (i.e., the robot swarm) to a unified global input scheme (i.e., the human operator), but also due to the complex infrastructure settings of existing interfaces such as vision-based sensors or global positioning systems, which only work in controlled environments, and a lack of appropriate feedback that can guide the operator and provide accurate information about the swarm’s state. These obstacles notwithstanding, a general-purpose human-swarm interface is required to tackle the next wave of challenges facing industry and advance the technology to a new state of the art.In the following, I will discuss two promising technologies that, if combined, could support the development of a general purpose HSI interface to control robotic swarms in an efficient and natural manner.§.§ Gesture Recognition: a versatile high-level input mechanismGestures and body movements are a natural way to communicate intentions and strengthen messages. Gestures are part of our social communication skill set <cit.>, which humans can use, understand and analyze. Hand gestures were very early adopted in research on human-robot interaction <cit.>. However, it took more than 20 years to utilize them in HSI to convey an operator’s intentions to small swarms of robots <cit.>.Recently, gesture-based HSI has evolved with the development of rich gesture taxonomies –- e.g., Fig. <ref> –-, which operators employ to control a group of robots. These taxonomies have mainly focused on remote interaction (i.e., tele-operation) applications conducted in controlled indoor environments <cit.>. Despite the high correct classification rates (CCRs) and solid conceptual foundation for future research achieved by these works, their models are difficult to use in other experimental settings as they require complex infrastructure such as vision-based sensors and global positioning systems –- Fig. <ref> –- or specialized hardware –- Fig. <ref> –-.A different approach was proposed in <cit.>, in which robots had to distinguish in a distributed fashion the orders and commands provided by the operator. This method was designed to enable the operator to interact with the swarm in a proximity environment –- Fig. <ref> –-, making the human operator a ‘special’ swarm member. However, the robots required a direct line of sight to the operator in order to detect and classify the operator’s gestures. A consensus mechanism was then used within the robot swarm to reach an agreement about the operator’s intentions. Its lack of complex infrastructure makes this method suitable for a wider range of scenarios, such as on-the-spot progress checks of a swarm’s operation, as well as in outdoor environments. However, the approach is nonetheless hamstrung by the limited sensing and computational power of individual robots, and so it is unclear if or how it could be applied to large swarms <cit.>.Even though the above-described remote and proximity interaction approaches are promising steps towards achieving suitable gesture-based control methods for specific applications, they have key limitations. Methods proposed for remote interaction rely on complex infrastructure, while proximity interaction methods suffer from scalability issues. Despite these problems, the aforementioned works prove that gestures can be a feasible way to control a swarm of robots in both remote and proximity interaction scenarios. Given enough flexibility, they may be used in a more general interface that could be suitable in both application scenarios in the future.§.§ Haptic Feedback: augmented assistance for the operatorAnother popular approach to combine robot swarms with human input has been to explore the haptic channel. Haptic technology provides a way in which information related to swarm status can be transferred back to the operator via tactile or force feedback. Haptic devices such as http://www.dentsable.com/haptic-phantom-omni.htmPhantom Omni – Fig. <ref> – or http://www.forcedimension.com/downloads/specs/specsheet-omega.3.pdfOmega3 – Fig. <ref> – have been extensively used to orchestrate the movements of whole swarms of robots <cit.>, certain subgroups of the swarm <cit.> or robot teams using a leader-follower approach <cit.>.In previous research, haptic feedback has been used in combination with existing methods such as continuous visual input to assist a human operator <cit.>. Haptic information has proven useful in guiding the operator in situations where a robot swarm is operating in obstacle-populated environments <cit.> or unstructured areas <cit.>. In such scenarios, attraction/repulsion forces are calculated according to environmental obstacles or swarm members’ positions, and transferred back to the operator to assist his/her decisions.Even though current haptic devices provide a way to receive feedback, they suffer from several limitations. Such devices rely only on Cartesian input information (trajectories, vectors, etc.) but cannot process high-level commands such as those provided using gesture-based interaction. Further, they unify input and feedback components, which could confuse the operator in situations where input and elaborate haptic feedback signals occur simultaneously. For instance, the operator might need feedback about the energy status of a robot at the same time as he/she is trying to accurately guide it through an obstacle-free path. Another drawback of devices such as the Omega3 and Phantom Omni is that they represent a single point of failure in case of malfunction, which might be a problem in terms of building a general-purpose interface. Finally, one of their most significant constraints is that they are limited to indoor environments equipped with a computer terminal, thus excluding missions in outdoor or other environments.§ A WEARABLE GESTURE-HAPTIC INTERFACE FOR HUMAN-SWARM INTERACTIONThe gaming and wearable technology industries have been a good source of breakthroughs and disruptive devices not only for commerce, but for academic research as well. Devices such as the http://dev.windows.com/en-us/kinectMicrosoft Kinect or http://www.oculus.com/en-usOculus Rift, initially designed as interactive controllers for common console platforms and game engines, have been extensively used within the robotics community to assist key research activities. In the last few years, we have observed how the second generation of wearable and gaming devices has reached the market with a special focus on haptic and monitoring capabilities. As the cost of these new devices decreases and their technology starts to provide enhanced features, novel application domains open up for their use in robotics research. Gesture recognition devices such as the http://www.thalmic.com/en/myo/Myo armband – Fig. <ref> – or http://www.leapmotion.comLeap motion – Fig. <ref> – demonstrate the feasibility of transferring gestures and simple commands to a computer system. Their low cost and suitability to both indoor and outdoor environments make them interesting candidates for the gesture recognition component of a general-purpose interface.Myo is a wearable armband device that can recognize a rich set of gestures –- Fig. <ref> -– by using 8 EMG muscle sensors installed in its frame. Its built-in accelerometers and gyroscopes also allow Myo to detect arm motion accurately. Several vibration motors give the additional possibility of providing haptic feedback in the form of short, medium and long vibrations. Myo is equipped with a rechargeable lithium-ion battery which is designed to last a day of continuous use. Finally, its Bluetooth connectivity allows Myo to transmit data to a computer or external device.Leap Motion is a portable 3D motion sensor that is able to detect hand movements as well as finger positions in a large interaction space (eight cubic feet) – Fig. <ref> –. The core of the device consists of two cameras that track the light reflected by three built-in infrared LEDs. Leap Motion has been used in academic research focused on fingers and their movements, such as on sign language recognition <cit.>.Leap Motion is a great complement to Myo since they each detect different types of information. Myo can recognize high-level intentions (gestures) whereas Leap Motion can recognize Cartesian information such as paths or routes, as well as deictic information like numbers through finger patterns. The possibility of combining these types of information allows for the creation of a wide range of control modes. For instance, high-level gestures could trigger ‘actions’ such as “follow path” or “change parameter value”, while deictic patterns could provide detailed information about the shape of the path to follow or the value of the parameter to change. Advanced haptic devices have been recently introduced into the gaming industry to increase the immersion experience while playing video games. Gaming vests such as http://www.korfx.com/startKOR-FX – Fig. <ref> – or http://tngames.com/3^rd Space – Fig. <ref> – allow video game players to deeply engage with a game’s current events by producing vibrations (KOR-FX) or pneumatic pulses (3^rd Space) according to the in-game situation. Users of these devices obtain rich information such as direction, intensity, and rhythm. Transforming swarm-centric data such as force fields and obstacle/landmark positions into these modalities would give the human operator a better understanding of the state of a robot swarm in the field. Brooks changed the course of artificial intelligence (AI) by arguing that the world is its own best model <cit.>. In a similar fashion, we now argue that the best-placed entity to obtain feedback about a swarm’s ‘body’ is the operator’s body.Table <ref> outlines the main benefits, limitations, and mitigation strategies for all devices proposed before. First, Myo provides wireless gesture recognition without complex infrastructure settings. However, its gesture range is limited to the ones depicted in Fig. <ref>. Even though this does not represent a problem at the current, proof–of-concept stage, advanced users could extend the repertoire of recognizable gestures as their applications require. In that case, custom gestures could be played, recorded, and stored by using the device’s built-in EMG monitoring tool.Second, Leap Motion offers high precision joint tracking for both hands. Even though Leap Motion has a small size and large interaction space, it requires a USB connection to operate. The inclusion of a Single Board Computer (SBC) such as Raspberry Pi[<http://www.raspberrypi.org/products/raspberry-pi-3-model-b/>] in the interface configuration could solve this problem, as well as provide additional computing and communication capabilities to the whole interface. In addition, the Leap Motion sensor needs to be located such that it can clearly sense the operator’s hands. To this end, a support frame could be attached to the haptic vest to place it in a suitable location. 3D-printing this support frame would ensure precise conformance with the vest’s dimensions.Third, the 3^rd Space gaming vest provides a large haptic interaction body area (both frontal and back parts) and a solid software development kit (SDK) with which to build applications. However, device operation requires a USB connection as well as a portable air compressor. The former requirement could be solved by the inclusion of a mini PC (in the same way as described before) in the wearable interface; the latter could be solved by the inclusion of a small pocket in the vest to carry the air compressor, as well as a Li-ion rechargeable battery that could power the whole system. In a nutshell, the combination of a wearable gesture recognition device that can detect high-level intentions, a portable device that can detect Cartesian information and finger movements, and a wearable advanced haptic device that can provide real-time feedback is a promising scheme for a general-purpose wearable interface for HSI applications. As far as this author knows, this work is the first to envision a wearable HSI interface that separates the input and feedback components of the classical control loop (input, output, feedback). Moreover, the proposed interface is suitable for both indoor and outdoor environments. In addition, such an enhanced interface might be able to provide other advanced interaction and interesting control capabilities: some are described below. §.§ Enhanced haptic feedbackPrevious swarm robotics research involving haptic feedback has only explored the use of traditional static devices normally attached to a desktop computer terminal <cit.>. This work is the first to suggest the use of wearable technology in a wider range of scenarios to allow the operator to obtain richer feedback. For instance, classical force feedback could be conveyed to the operator using this wearable technology and, therefore, increase the immersion experienced by the operator. Moreover, the operator might obtain additional information using this technology by utilizing different pulse or vibration patterns (e.g., heartbeat-like pulse patterns could serve to communicate the battery status of swarm robots). Finally, by decoupling the feedback and input components of the interface, a more robust and fault-tolerant interface can be achieved. §.§ Timing based inputRecent research <cit.> has demonstrates that improper timing of control input by operators can lead to problems when commanding a swarm of robots, such as group fragmentation (i.e. unintentional division of the swarm). Group fragmentation causes delays in coordination, as well as motion and sensing errors that hurt the performance of swarm tasks. Operators who issue commands frequently showed higher levels of swarm fragmentation than those who allowed the swarm to adjust between new commands. Optimal timing studies are just beginning to emerge, and they are an interesting area for future research.An interface that could determine optimal human timing as well as provide guidance and assistance could be crucial to achieving effective interaction between human operators and robot swarms. The proposed interface outlined in this work offers a suitable platform to conduct research on optimal human timing algorithms since it has all the necessary components to develop effective models. First, an embedded computing unit (e.g., Raspberry Pi) gathers input data from the robotic swarm and calculates proper timing threshold parameters. Second, an advanced haptic component (e.g., 3^rd Space Gaming Vest or KOR-FX) provides intuitive patterns based on previously calculated timing parameters to the operator to assist with his/her decision-making. Finally, high-level gesture (e.g., Myo) and deictic recognition (e.g., Leap Motion) components send commands to the robot swarm after the feedback is taken into account. §.§ Hierarchical controlSeveral recent surveys <cit.> have pointed out that one of the main problems in the swarm robotics field is that robotic swarms cannot switch between different behaviors during the same mission at present.Due to the loosely-coupled settings of an interface composed of different wearable parts, it may be possible to create a taxonomy of commands suitable for a wide range of robot behaviors. For instance, high-level commands such as gestures could serve as a switch mechanism between different robot behaviors, while deictic movements could command the swarm within that specific mode of operation§.§ Simple interfaceEarly studies <cit.> in the field of swarm visualization and representation indicated that simplifying the large state of a swarm to a lower-dimensional representation can be beneficial when controlling a group of robots. Reducing the amount of noise as well as fusing information to simplify the problem of determining a swarm’s state are further promising topics in the HSI field.The proposed interface outlined in this paper allows the possibility of mapping the state of the robot swarm to an operator’s body through haptic feedback. This capability could dramatically increase the amount of status information available without increasing the complexity of its representation.§ CONCLUSIONSRobotic swarms are expected to become an integral part of emerging technologies and open the door to future economic possibilities. However, the lack of a general purpose human-swarm interface that provides seamless interaction between a human operator and a group of robots confines the field to academic laboratories. Recent advances in gaming technology have brought sophisticated devices that, if combined, could further advance the HSI discipline. Wearable gesture recognition devices recognize and interpret gestures in a wide range of scenarios. In addition, they provide the mobility required for an operator to command a group of robots in both remote and proximity interaction scenarios. Also, new haptic devices such as gaming vests provide a means for an operator to receive haptic feedback without interfering with his/her input signal. At the same time, they constitute a novel platform to obtain rich feedback without increasing the complexity of the swarm’s status information.The aim of this work is to incorporate the underlying principles of these two novel technologies into a general-purpose HSI interface that is able to control adaptive robotic swarms, which can be controlled in a natural and seamless manner by human operators in order to tackle complex tasks. Potential applications, such as remote and proximity interaction with swarms of unmanned aerial vehicles (UAVs), could be achieved without complex calibration and infrastructure settings. Finally, outdoor applications (e.g. agricultural tasks) could greatly benefit from the proposed interface.plain | http://arxiv.org/abs/1704.08393v2 | {
"authors": [
"Eduardo Castelló Ferrer"
],
"categories": [
"cs.RO",
"cs.HC",
"cs.MA"
],
"primary_category": "cs.RO",
"published": "20170427005750",
"title": "A wearable general-purpose solution for Human-Swarm Interaction"
} |
http://arxiv.org/abs/1704.08174v1 | {
"authors": [
"Maxime Oliva",
"Ole Steuernagel"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170426155833",
"title": "Structures far below sub-Planck scale in quantum phase-space through superoscillations"
} |
|
APS/[email protected] Department of Nuclear Physics, Research School of Physics and Engineering, The Australian National University, Canberra, ACT 2601, Australia [email protected] Department of Nuclear Physics, Research School of Physics and Engineering, The Australian National University, Canberra, ACT 2601, Australia Department of Nuclear Physics, Research School of Physics and Engineering, The Australian National University, Canberra, ACT 2601, Australia ARC Centre of Excellence for Particle Physics at the Terascale and CSSM, Department of Physics, The University of Adelaide, SA 5005, Australia BackgroundThe Skyrme energy density functional is widely used in mean-field calculations of nuclear structure and dynamics. However, its reliance on phenomenology may compromise its isovector properties and its performance for exotic nuclear systems. PurposeThis work investigates the possibility of removing some phenomenology from the density functional by drawing on the high-energy degrees-of-freedom of the quark-meson coupling (QMC) model. The QMC model has microscopically derived isovector properties and far fewer adjustable parameters. MethodThe parameters of the Skyrme interaction are fixed using the energy density functional of the QMC model, to give the Skyrme-QMC (SQMC) parameterisation. ResultsHartree-Fock-Bogoliubov calculations of the Sn, Pb and N=126 chains are reported, in which SQMC performs with an accuracy comparable to modern phenomenological functionals. ConclusionsThe isovector terms of the energy density functional are significant for the binding energies of neutron-rich nuclei. The isovector dependence of the nuclear spin-orbit interaction must be taken into account for calculations of r-process nucleosynthesis abundances.Isovector properties of the nuclear energy density functional from the quark-meson coupling model A. W. Thomas December 30, 2023 =================================================================================================§ INTRODUCTIONInterest in exotic nuclear systems is growing as new experimental facilities and techniques constantly expand the limits of known nuclei. Astrophysical models, of r-process nucleosynthesis <cit.> and neutron stars for example, rely heavily on the properties of very exotic nuclei, many of which are unknown. To guide these searches predictive theories of nuclear physics are required. However, one must face the significant challenges posed by the quantum many-body problem. This is further complicated by the interaction between nucleons, which is not well understood in-medium. For very neutron-rich nuclei the isovector properties of the interaction, which are far less known than the isoscalar channels, will become increasingly important. One possible approach is to consider systematic applications of chiral effective field theory <cit.> to nuclear interactions and hence to nuclear structure.It is certainly appealing to develop such an approach based on the symmetries of quantum chromodynamics.Nevertheless, chiral perturbation theory does not yet allow systematic applications to structure and dynamics across the nuclear chart. Self-consistent mean-field approaches basedonnuclear energy density functionals (EDF) make the problem computationally tractable. The EDF approach relies on the Hohenberg and Kohn theorem <cit.>, according to which the nucleon density distributions uniquely determine the ground-state energy of the nuclear system. An energy density functional can be derived from an effective nucleon-nucleon force, for example the Skyrme interaction, and used at the mean-field level in a range of approaches based on Hartree-Fock approximations, which offer a unified description of nuclear structure and dynamics <cit.>. However, this requires the introduction of some amount of phenomenology. For instance, the form of the density dependence of the coupling constants is essentially guessed. As a result, the Skyrme EDF contains a large number of parameters without clear physical meanings, compromising its predictive power when applied to very exotic systems. Another difficulty in the Skyrme approach isthat the strength of the spin-orbit coupling must be fitted to experimental data, such as the energy splitting between spin-orbit single-particle level partners.Such data are often not known away from stability. This leads to difficulties in determining the isovector dependence of the spin-orbit force. It is thus desirable to develop approaches which would account for thespin-orbit interaction without requiring additional parameters.Indeed, including as little phenomenology as possible should improve the reliability of predictions for very exotic nuclei.In fact, few models can predict the nuclear spin-orbit interaction, a relativistic effect which, unlike the electronic spin-orbit force, is not a small correction. As shown by Goeppert-Mayer and Jensen <cit.>, the inclusion of the nuclear spin-orbit coupling is essential for correctly predicting all magic numbers above 20. The strength of the spin-orbit interaction plays a role in determining the location of the superheavy island of stability <cit.>, as well as providing a dissipation mechanism which is crucial in heavy-ion collisions <cit.>. Covariant energy density functionalsoffer another interesting approach to the nuclear many-body problem <cit.>. Being fully relativistic, they naturally generate a spin-orbit interaction and, if exchange terms are properly included, some isovector dependence <cit.>.However, as in the Skyrme EDF, the density dependence of the coupling constants requires some phenomenology, leading to a relatively large number of parameters needed to fit experimental data or pseudodata.One promising avenue is to obtain the nuclear EDF by drawing on the high-energy degrees-of-freedom of the nuclear system through a self-consistent mean-field model of the in-medium modification of the quark structure of the bound nucleons: the quark-meson coupling (QMC) model <cit.>. The QMC model predicts both the central and spin-orbit channels of the effective nucleon-nucleon interaction, along with their isovector behaviour. This is achieved with a much smaller number of parameters than in standard Skyrme or covariant EDF approaches. There exist many codes based on Hartree-Fock approximations and their generalisations that can be used to compute the properties of atomic nuclei and their reactions (see, e.g., <cit.>). The majority of these codes are based on Skyrme functionals. To take advantage of the Skyrme framework, this work looks at the process of using the QMC model to fix the parameters of a Skyrme functional. With applications to exotic nuclei in mind, we focus on the importance of the isovector behaviour of the energy density functional and, in particular, the isovector dependence of the spin-orbit term as derived within the QMC model. The philosophy of the QMC model is briefly outlined in Sec. <ref>. Section <ref> introduces the Skyrme energy density functional. A method for fixing the Skyrme parameters from the QMC model is then detailed in Sec. <ref>. Hartree-Fock-Bogoliubov (HFB) calculations are presented in Sec. <ref>, along with an analysis of the level of success of this method. Finally, the spin-orbit functional and its importance for exotic nuclei is investigated in Sec. <ref>, including a comparison to the approach of standard relativistic mean-field (RMF) theory. § THE QUARK-MESON COUPLING MODELThe quark-meson coupling model is a relativistic mean-field approach to the quantum many-body problem. Proposed by P. A. M. Guichon in Ref. <cit.> and since developed in Refs. <cit.>, the QMC model posits that the nucleon is not immutable and that modification of the internal quark structure by the nuclear environment is an important factor in modelling nuclear systems. In its simplest form, the model considers three valence quarks confined to an MIT bag <cit.>, which interact with the quarks of the surrounding nucleons via the exchange of σ, ω and ρ mesons. The ω and ρ fields are identified with the real vector-isoscalar and vector-isovector particles, respectively. The σ field is an effective representation of scalar-isoscalar exchange, including two correlated pions. Thanks to its inclusion of quark degrees-of-freedom, the QMC model can be used to describe effects associated with the modification of the structure of hadrons in the nuclear medium.As an example, it proposes a possible explanation <cit.> to the famous “European Muon Collaboration” (EMC) effect, which involves the loss of momentum from valence quarks in a nucleus compared with a free nucleon. (Note that an alternate development by Bentz and Thomas has applied the same concept to the Nambu-Jona-Lasinio (NJL) model <cit.>, which allows a covariant formulation of the calculation of nuclear structure functions, with similar results <cit.>.) The QMC model was also applied to the masses and properties of other hadrons immersed in a nuclear medium <cit.>, which amongst other things led to a remarkably successful description of hypernuclei <cit.>. In particular, the QMC model naturally explains the virtual absence of a spin-orbit force in Λ hypernuclei, repulsion for Σ hyperons in nuclei and it predicted binding at the level of a few MeV for Ξ hypernuclei, for which the first evidence has recently been reported atJPARC<cit.>. Beginning from a Lorentz-invariant Lagrangian density <cit.> of quark and meson fields, the system is solved self-consistently to give a non-relativistic energy density functional <cit.> which can be applied to the study of finite nuclei. The EDF can be written as=ρ M_N+τ/2M_N+_0+3+_eff+_fin+_SO,separated based on the occurrence of the total (neutron, proton) particle density, ρ (ρ_n, ρ_p); kinetic density, τ (τ_n, τ_p); and spin-orbit density, J (J_n, J_p). The QMC EDF of Ref. <cit.> gives _0+3^QMC=(-3G_ρ/32+G_σ/8(1+dρ G_σ)^3-G_σ/2(1+dρ G_σ)+3G_ω/8)ρ^2 +(5G_ρ/32+G_σ/8(1+dρ G_σ)^3-G_ω/8)(ρ_n-ρ_p)^2,_eff^QMC=(G_ρ/4m_ρ^2+G_σ/2M_N^2)ρτ+(-G_ρ/8m_ρ^2-G_σ/2m_σ^2+G_ω/2m_ω^2-G_σ/4M_N^2)∑_q=n,pρ_qτ_q,_fin^QMC=(-3G_ρ/16m_ρ^2-G_σ/2m_σ^2+G_ω/2m_ω^2-G_σ/4M_N^2)ρ∇^2ρ +(9G_ρ/32m_ρ^2+G_σ/8m_σ^2-G_ω/8m_ω^2+G_σ/8M_N^2)∑_q=n,pρ_q∇^2ρ_q, and _SO^QMC=-1/4M_N^2[(G_σ+G_ω(2μ_s-1))ρ∇·J +(G_σ/2+G_ω/2(2μ_s-1)+3G_ρ/8(2μ_v-1))∑_q=n,pρ_q∇·J_q]. One assumes experimental values for the free nucleon mass, M_N, the ω and ρ meson masses, m_ω and m_ρ, and the isoscalar and isovector nucleon magnetic moments, μ_s and μ_v. This leaves only the σ meson mass, m_σ, and a coupling constant for each of the three meson fields, G_σ, G_ω and G_ρ, as free parameters in the model.The `scalar polarisability', d, is a constant which describes the self-consistent response of the confined quarks to the applied scalar mean-field. It gives rise very naturally to the many-body interactions within the system of nucleons, as we see from the appearance of d in the denominators of Eq. (<ref>). In this work, as in Ref. <cit.>, for an appropriate choice of m_σ (e.g. 700 MeV), the meson coupling constants are fixed using infinite nuclear matter pseudodata: the nucleon density, energy per nucleon, and symmetry energy at saturation (ρ_0=0.16 fm^-3, e_∞=-15.85 MeV, and a_s=30 MeV, respectively). As the parameters are not determined by a fit to experimental masses and radii, the functional can be used consistently in both the intrinsic frame of a nucleus and the centre-of-mass frame of a collision between two nuclei <cit.>. Reference <cit.> on the other hand, fixes the free parameters of the QMC model through a fit to ground-state properties of nuclei, while ensuring consistency with nuclear matter properties within reasonable uncertainties, obtaining similar values. § THE SKYRME FUNCTIONALThe Skyrme interaction began as a general form dictated by the symmetries that are required of the force between nucleons <cit.>. It has since been generalised to an energy density functional of local densities and currents up to second-order in derivatives. Separated into the terms of Eq. (<ref>), the Skyrme EDF takes the form:_0+3^Skyrme=1/12{[6t_0(1+x_0/2) + t_3(1+x_3/2)ρ^α] ρ^2 -[6t_0(x_0+1/2)+t_3(x_3+1/2)ρ^α] ∑_q=n,pρ_q^2},_eff^Skyrme=1/4{[t_1(1+x_1/2)+t_2(1+x_2/2)]ρτ - [t_1(x_1+1/2)-t_2(x_2+1/2)]∑_q=n,pρ_qτ_q},_fin^Skyrme=-1/16{[3t_1(1+x_1/2)-t_2(1+x_2/2)]ρ∇^2ρ - [3t_1(x_1+1/2)+t_2(x_2+1/2)]∑_q=n,pρ_q∇^2ρ_q}, and _SO^Skyrme=-1/2(W_0ρ∇·J + W_0'∑_q=n,pρ_q∇·J_q) ,with elevenparameters x_0-3, t_0-3, α, W_0, and W_0'.These parameters are typically determined by a fit to nuclear equation of state properties and to the experimental masses and radii of a selection of atomic nuclei <cit.>.Most modern functionals also include uncertainties on their parameters <cit.>. While Skyrme functionals remain a widely successful approach, the phenomenological nature of the traditional Skyrme functional can call into question its predictions for very exotic systems. Because of the correlations between parameters and the many choices of fitting procedure, which place anemphasis on different experimental data, many Skyrme parameterisations exist. They give conflicting predictions for the properties of very exotic nuclei <cit.>, with no clear winner. § THE SKYRME-QMC FUNCTIONALDespite its limitations, the Skyrme functional is very convenient and the vast majority of Hartree-Fock codes are based on it. In an attempt to improve the predictive power of the Skyrme-Hartree-Fock approach, this paper determines the parameters of the Skyrme EDF using the QMC model.In essence, this is achieved by equating the terms of the QMC density functional [Eqs. (<ref> – <ref>)] to those of the Skyrme functional [Eqs. (<ref> – <ref>)]. §.§ QMC700 A similar approach was adopted by Guichon et al. <cit.> to produce the QMC700 Skyrme parameterisation (“700” refers to the choice of m_σ=700 MeV). QMC700 was obtained by approximating the QMC functional [Eqs. (<ref> – <ref>)] with a simplified Skyrme EDF analogous to the SkM* parameterisation <cit.>. As a result,QMC700 had only 6 parameters (t_0-3, x_0, and W_0), setting x_1-3=0, W_0=W_0', and α=1/6 (see Table <ref>). Because ofthe reduced number of parameters, the _eff, _fin and _SO functionals of Skyrme and QMC were equated for only N=Z (assuming ρ_n=ρ_p), to yield t_1, t_2 and W_0. The microscopically derived density dependence of the _0+3 term of the QMC functional [Eq. (<ref>)] is significantly more complicated than that of the Skyrme functional [Eq. (<ref>)] because of the terms involving the scalar polarisability d. Unlike the other terms, _0+3^Skyrme cannot be equated to _0+3^QMC. Instead it is fitted numericallyover a large range of nucleon density, ρ∈[0, 0.2 fm^-3] with ρ_n=ρ_p, to give t_0 and t_3, and with Z/A=82/208 (corresponding to ^208Pb) to give x_0 <cit.>. Out of 240 Skyrme parameterisations tested in Ref. <cit.>, QMC700 and QMC650 (a parameterisation derived identically but with m_σ=650 MeV) were two of only 16 parameter sets shown to satisfy all constraints (a range of nuclear matter properties derived from experiment).§.§ QMC700* The complicated density dependence of the QMC model [Eq. (<ref>)] cannot be well reproduced by a Skyrme functional with a single ρ^α dependence as in Eq. (<ref>).This is illustrated in Fig. <ref>, which shows the equations of state for symmetric nuclear matter for various EDF parameterisations. By restricting the range of density used in the _0+3 fit to ρ∈[0.12, 0.2 fm^-3], the reproduction of the QMC functional is improved around saturation density where this bulk term will be most important (see also Table <ref>). Hartree-Fock calculations of nuclear masses and radii show a corresponding improvement in performance as compared to QMC700. This refitting alters t_0, t_3 and x_0, and we label this new parameter set QMC700*. The other parameters remain unchanged, so that QMC700* has the same isovector properties as QMC700. §.§ SQMC Given that our aim is to study the isovector dependence of the nuclear density functional and its importance for exotic systems, we create one final QMC-motivated Skyrme force by extending QMC700* to include additional isovector parameters, x_1-3 and W_0'. This yields a Skyrme-QMC functional which perfectly reproduces _eff^QMC, _fin^QMC and _SO^QMC for all densities and nucleon asymmetries, and improves the fit for _0+3^QMC. This parameterisation is labelled “SQMC” and is the primary focus of this paper. It is consistent with the parameterisation of Ref. <cit.> (labelled SQMC^ in the following) for the central terms.However, SQMC^ does not include the second spin-orbit parameter, fixing W_0=W_0', unlike the present version of SQMC. §.§ IncompressibilityThe incompressibility of symmetric nuclear matter has long been the subject of debate. The QMC model parameterisation in Ref. <cit.> has an incompressibility of 346 MeV, in line with most other relativistic approaches. For all Skyrme parameterisations discussed thus far (QMC700, QMC700* and SQMC), the density dependence is fixed with α=1/6. This restricts the functionals to a lower incompressibility of around 220 MeV, consistent with many common Skyrme functionals, such as SLy4 <cit.> and UNEDF1 <cit.>. To generate a QMC functional with an incompressibility closer to the QMC model, we use the spherical Hartree-Fock-Bogoliubov code hfbrad <cit.>, which allows for two fractional density dependences in _0+3. This makes it possible to reproduce the full QMC functional very accurately over a wide range of densities, using α_1=1/3 and α_2=2/3, as shown in Fig. <ref>. The resulting Skyrme parameterisation, labelled SQMC^α_1,α_2 hereafter, shares the high incompressibility of the QMC functional. However, this parameterisation does not perform well, yielding binding energies approximately 60 MeV away from experiment for tin nuclei, as shown in Fig. <ref>, due to the high incompressibility.The incompressibility of the QMC is expected to be reduced by the inclusion of Fock terms involving pion exchange, reducing the incompressibility to be within the bounds recently deduced from an extensive review of experimental data in Ref. <cit.> of 250 – 315 MeV.Indeed, the relativistic version <cit.> of the QMC EDF, including pion exchange, lowers the incompressibility from 340 MeV to ∼300 MeV. As a non-relativistic version of the QMC model with finite range terms coming from pion exchange remains to be developed, and the high-incompressibility functional SQMC^α_1,α_2 performs poorly, we will focus on QMC700, QMC700* and SQMC, where the restricted density dependence gives a low incompressibility and substantially more accurate results. § ISOVECTOR EFFECTSIn the following, the isovector dependence of the SQMC functional will be investigated by comparing its behaviour to that of QMC700*, as the latter employs the same treatment of the density dependence, only differing in its exclusion of the four parameters {x_1-3,W_0'} in the fitting procedure. Hartree-Fock-Bogoliubov codes are a class of microscopic mean-field approach which are well-suited to the study of exotic nuclear structure, as the treatment of pairing remains robust for weakly bound nuclei.All of the results that follow were calculated using the hfbthov2.00d code <cit.>, which allows for axial deformation. hfbtho also has all the properties required to properly implement the modern Universal Nuclear Energy Density Functional 1 (UNEDF1) <cit.> Skyrme parameterisation. UNEDF1 is used below as a representative of the current generation of phenomenological functionals and their level of accuracy. Long isotopic and isotonic chains provide a large range of differences between proton and neutron number, useful for revealing trends in the isovector properties of the functionals. HFB calculations for the tin (Z=50) and lead (Z=82) isotopic chains, and the N=126 isotonic chain give a systematic picture of each functional's performance. We compare our various theoretical calculations to the experimental binding energies reported in the 2012 Atomic Mass Evaluation (AME2012 <cit.>), labelled E_exp in what follows.All calculations were performed without any centre-of-mass corrections. Indeed the parameters of UNEDF1 were fitted without these corrections so that it can be used both in structure calculations and in time-dependent Hartree-Fock simulations of heavy ion collisions <cit.>. The mixed (surface and volume) pairing effective interaction takes the form V_pair(r,r') = V_0(1-1/2ρ(r)/ρ_0)δ(r-r'), in the ^1S_0 channel with the pairing strength V_0 adjusted for each parameterisation to give a neutron pairing gap of 1.245 MeV for ^120Sn. UNEDF1 calculations use the proton and neutron pairing strengths prescribed in Ref. <cit.>. All calculations include the Lipkin-Nogami (LN) prescription for particle-number projection <cit.>. Figure <ref> shows the ground-state binding energies for the experimentally measured even-even tin nuclei as calculated using the UNEDF1, SQMC and QMC700* parameterisations. Figures <ref> and <ref> repeat this analysis for the lead isotopes and for the N=126 isotonic chain, respectively.Comparing the results of QMC700* to SQMC it becomes apparent that while a single parameter (x_0) controlling the isovector dependence is adequate for nuclei close to symmetry, the extra terms included in SQMC quickly become significant for neutron-rich systems. Except for a handful of nuclei, the binding energies given by SQMC are significantly closer to experiment than those of QMC700*, supporting the isovector dependence microscopically derived within the QMC model. The accuracy of the SQMC results is comparable to that of UNEDF1, especially away from stability. This is particularly remarkable when one recalls that no experimental masses were directly used in the production of the SQMC, only nuclear matter pseudodata and the QMC model. UNEDF1 on the other hand, included the masses of ^108, 112-124Sn and ^198-214Pb in its fitting procedure <cit.>. Also shown in Figs. <ref> and <ref> are the two-neutron separation energies, S_2n, for the same nuclei, where S_2n, cal(N, Z) = E_cal(N-2, Z) - E_cal(N, Z). Similarly, Fig. <ref> shows the two-proton separation energies. While the trends are less clear for these separation energies than for the total binding energies, all three parameterisations perform well, generally lying within 2 MeV of the experimental data. § THE SPIN-ORBIT FUNCTIONALThe nuclear spin-orbit interaction is a manifestation of relativity. Non-relativistic models, such as the Skyrme interaction, must add it by hand. The isovector properties of the spin-orbit interaction have remained largely unknown, with little theoretical guidance available and significant difficulties faced extracting them from experimental data <cit.>. §.§ Isovector dependenceIn the case of the Skyrme functional, the isovector dependence of the spin-orbit functional is controlled by the relationship between the two terms of Eq. (<ref>), i.e. the ratio of the parameters W_0' to W_0. This ratio offers a convenient way to compare the predictions of different approaches, as done in Table <ref>. When only Hartree terms are considered the zero-range, two-body spin-orbit force of the Skyrme interaction necessitates that the resulting functional have only an isoscalar term, with W_0'=0. The inclusion of the Fock (exchange) term introduces an isovector dependence, however there remains only one spin-orbit parameter, as W_0' is fixed equal to W_0. While this is sufficient for a good fit near stability, a single degree-of-freedom allows for no control over the isovector dependence and no guarantee of reliability when extrapolating to exotic nuclei. By relaxing the connection to the two-body Skyrme interaction, it is possible to introduce a second free spin-orbit parameter at the level of the energy density functional. Though it was first proposed by Reinhard and Flocard in 1995 <cit.>, it was not widely utilised until the recent UNEDF parameterisations <cit.>. An extensive fit to medium and heavy mass nuclei yielded a significantly stronger isovector dependence for UNEDF1 (W_0'/W_0=1.86) than standard Skyrme forces. In the QMC model the nucleons are assumed to move slowly and non-relativistic limits are taken to obtain the energy density functional. However, as the model is based upon a relativistic model of quarks and mesons, important relativistic corrections remain, chief among them the spin-orbit interaction. The spin-orbit interaction derived within the QMC model emerges from a combination of the variation of the vector meson fields (ω and ρ) across the finite nucleon, and Thomas precession, a purely relativistic effect induced when changing frames of reference. The resulting density functional [Eq. (<ref>)] has no extra free parameters, unlike non-relativistic approaches. The microscopically derived isovector dependence of the (S)QMC functional is remarkably similar to that of the phenomenological UNEDF1 functional (see Tables <ref> and <ref>). The significance of the similarity between the spin-orbit functionals of the QMC model and UNEDF1 may be determined by examining whether the isovector dependence of the spin-orbit term has any impact on Hartree-Fock calculations of nuclei. The spin-orbit terms of the SQMC parameterisation and the QMC model are identical, making SQMC a convenient tool to test the isovector dependence of the original QMC functional. It is possible to extract the impact of only the isovector dependence of the spin-orbit term by comparing the SQMC parameterisation (with W_0'/W_0=1.78) against a baseline which has only one spin-orbit parameter (W_0'=W_0=104.3872^5) but is otherwise identical. hfbtho calculations, using the same pairing strength for both parameterisations, were performed for all bound nuclei of the Sn and Pb isotopic chains and the N=126 isotonic chain, and are reported in Figures <ref>, <ref> and <ref>, respectively.The figures show a relatively strong isovector dependence, varying rapidly with N and Z.The isovector dependence of the spin-orbit term induces a binding energy change of approximately 1 MeV in the regions of the r-process (^134 - 152Sn, ^238 - 266Pb and ^180 - 184Z_126) as predicted by SQMC. A change of only 500 keV for nuclei in the region of ^140Sn is already expected to have a significant impact on r-process abundances <cit.>. This illustrates that it is crucial to properly account for the isovector properties of the spin-orbit functional, as included in UNEDF1 and predicted in the QMC model. §.§ Impact of deformation The lower panels of Figs. <ref> and <ref> show the deformation of the mean-field ground-states in tin and lead isotopes.We observe that the effect of the isovector contribution to the spin-orbit functional is clearly enhanced by the presence of prolate deformation in neutron-rich nuclei (^146 - 162Sn and ^224 - 252Pb). A more detailed investigation of the interplay between deformation and the isovector contribution to the spin-orbit functional will be the focus of a future study.We should point out that the deformations reported in Figs. <ref> and <ref> are those of the lowest energy mean-field state in the potential energy curve. In reality, the ground-state of soft nuclei is expected to be made of a superposition of mean-field states with various deformations.This is illustrated in Figs. <ref> and <ref> showing the potential energy versus the quadrupole deformation parameter β_2 in ^196Pb and ^198Pb nuclei, respectively, for UNEDF1, QMC700* and SQMC parameterisations.Interestingly, QMC700* and SQMC both predict an oblate minimum in ^196Pb while the UNEDF1 minimum is spherical.In ^198Pb, all functionals predict a spherical minimum.However, all parameterisations predict a soft potential with respect to quadrupole deformation.An improved description of the ground-states in such nuclei would thus require beyond mean-field approaches such as the generator coordinate method (GCM) <cit.>. §.§ Comparison to relativistic mean-field The term “relativistic mean-field” (RMF) encompasses a constantly expanding set of theories. The version discussed here is based on a hadronic Lagrangian of inert nucleons exchanging σ, ω and ρ mesons <cit.>. Like the QMC model, RMF is able to predict a nuclear spin-orbit interaction. However, it is often argued that, in RMF, Fock terms are negligible <cit.> and Dirac nucleon magnetic moments, μ_p=1 and μ_n=0, are used. Once again we use the ratio W_0'/W_0 to compare the isovector dependence of the spin-orbit energy density functionals in Table <ref>. RMF gives a substantially weaker isovector dependence than the QMC model, UNEDF1 and even standard Skyrme. RMF is very close to the Hartree value of W_0'=0, only deviating due to a small contribution from the vector-isovector ρ meson. By keeping only the direct (Hartree) terms of the QMC spin-orbit functional and setting the nucleon magnetic moments to their Dirac rather than the experimental (μ_s=μ_p+μ_n=0.88, μ_v=μ_p-μ_n=4.7) values, one would obtain a similarly weak isovector dependence (Tab. <ref>). This implies that the weak isovector dependence of RMF is primarily due to these two approximations: neglect of exchange terms and use of Dirac magnetic moments. These are approximations which are not required in the QMC model. § CONCLUSIONThe predictions of ground-state masses from the SQMC parameterisation are of a comparable level of accuracy to UNEDF1 for the studied systems, showing the promise of replacing some level of phenomenology with input from a more microscopic theory based on quark degrees-of-freedom. The isovector behaviour of the functional can be derived from the QMC model and is significant for exotic nuclei. It is remarkably similar to that of UNEDF1 and much stronger than RMF. This is significant because this dependence is an important factor, which must be taken into account in r-process calculations. The SQMC functional may be used in any existing Skyrme-Hartree-Fock codes to study further exotic structure effects such as shell evolution, driplines or superheavy magic numbers. Also, using a time-dependent Hartree-Fock code, dynamic processes can be investigated, including reactions with exotic nuclei, fusion barriers, transfer and fission, revealing how they are affected by the high-energy physics upon which the QMC model is based. It will also be possible to perform similar investigations using the original QMC functional, rather than the SQMC parameterisation, by modifying Skyrme-based Hartree-Fock codes to accept the more complicated density dependence of the central terms. Stone et al. <cit.> modified the static skyax code <cit.> in this way and used nuclear data to fix the few free parameters of the QMC model. They found its performance across the nuclear chart to be on a level comparable to a traditional Skyrme functional with many more parameters and particularly impressive for the masses of superheavy nuclei not included in the fit. A similar implementation in a time-dependent code will allow for a study of the giant monopole resonance to illuminate the issue of nuclear matter incompressibility and the importance of single pion exchange, among many other dynamic processes.CS thanks G. Lane for useful discussions.This research was undertaken with the assistance of resources from the National Computational Infrastructure (NCI), which is supported by the Australian Government.This research was supported by an Australian Government Research Training Program (RTP) Scholarship and by the Australian Research Council through ARC Grants FT120100760 (CS), DP160101254 (CS), FL110100098 (ECS), DP170102423 (ECS), DP150103164 (AWT) and by the ARC Centre of Excellence for Particle Physics at the Terascale, CE110001104 (AWT). | http://arxiv.org/abs/1704.07991v1 | {
"authors": [
"E. McRae",
"C. Simenel",
"E. C. Simpson",
"A. W. Thomas"
],
"categories": [
"nucl-th"
],
"primary_category": "nucl-th",
"published": "20170426071233",
"title": "Isovector properties of the nuclear energy density functional from the quark-meson coupling model"
} |
http://arxiv.org/abs/1705.00707v1 | {
"authors": [
"V. Ambethkar",
"Durgesh Kushawaha"
],
"categories": [
"physics.flu-dyn",
"math.DS",
"physics.comp-ph"
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"primary_category": "physics.flu-dyn",
"published": "20170426203119",
"title": "Numerical simulations of fluid flow and heat transfer in a four-sided, lid-driven rectangular domain"
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|
1Institute of Astronomy, National Central University, Jhongli,Taiwan 2Space Science Institute, Macau University of Science and Technology, Macau 3Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA 4Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, U.S.A. 5Department of Astronomy and Space Science, Chungbuk National University, 1, Chungdae-ro, Seowon-Gu, Cheongju, Chungbuk, 28644 Korea 6Korea Astronomy and Space Science Institute, 776, Daedeok-daero, Yuseong gu, Daejeon, 305-348 South [email protected] April 4 2017 April 27 2017 (144977) 2005 EC_127 is an V-/A-type inner-main-belt asteroid with a diameter of 0.6 ± 0.1 km. Asteroids of this size are believed to have rubble-pile structure, and, therefore, cannot have a rotation period shorter than 2.2 hours. However, our measurements show that asteroid 2005 EC_127 completes one rotation in 1.65 ± 0.01 hours with a peak-to-peak light-curve variation of ∼ 0.5 mag. Therefore, this asteroid is identified as a large super-fast rotator. Either a rubble-pile asteroid with a bulk density of ρ∼ 6 g cm^-3 or an asteroid with an internal cohesion of 47 ± 30 Pa can explain 2005 EC_127. However, the scenario of high bulk density is very unlikely for asteroids. To date, only six large super-fast rotators, including 2005 EC_127, have been reported, and this number is very small when compared with the much more numerous fast rotators. We also note that none of the six reported large SFRs are classified as C-type asteroids. § INTRODUCTIONThe large (i.e., a diameter of a few hundreds of meters) super-fast rotators (hereafter, SFRs) are of interest for understanding asteroid interior structure. Because asteroids of sub-kilometer size are believed to have rubble-pile structure (i.e., gravitationally bounded aggregations) and cannot have super-fast rotation, defined as a rotation period shorter than 2.2 hours <cit.>[The 2.2-hour spin barrier was calculated for an asteroid with a bulk density of ρ = 3 g cm^-3.]. However, the first large SFR, 2001 OE84, a near-Earth asteroid of ∼ 0.7 km in size and completing one rotation in 29.19 minutes <cit.>, cannot be explained by rubble-pile structure, and, consequently, internal cohesion was proposed to be a possible solution <cit.>. Although several attempts were made to discover large SFRs with extensive-sky surveys <cit.>, this asteroid group was not confirmed until another large SFR, 2005 UW163, was found by <cit.>. Up to now, five large SFRs have been reported, additionally including 1950 DA <cit.>, 2000 GD65 <cit.>, and 1999 RE88 <cit.>. However, the population size of large SFRs is still not clear. Compared with the 738 large fast rotators (i.e., diameters between 0.5–10 km and rotation periods between 2-3 hours) in the up-to-date Asteroid Light Curve Database <cit.>, large SFRs are rare. Either the difficulty of discovering them due to their sub-kilometer sizes (i.e., relatively faint) or the intrinsically small population size of this group could lead to this rarity in detection. Therefore, a more comprehensive survey of asteroid rotation period with a wider sky coverage and a deeper limiting magnitude, such as the ZTF[Zwicky Transient Facility; http://ptf.caltech.edu/ztf], could help in finding more large SFRs. With more SFR samples, a thorough study of their physical properties could be conducted, and, therefore, further insights about asteroid interior structure are possible. To this objective, the TANGO project[Taiwan New Generation OIR Astronomy] has been conducting asteroid rotation-period surveys since 2013 using the iPTF[intermediate Palomar Transient Factory; http://ptf.caltech.edu/iptf] <cit.>. From these surveys, two large SFRs and 27 candidates were discovered. Here we report the confirmation of asteroid (144977) 2005 EC_127 as a new large SFR. The super-fast rotation of (144977) 2005 EC_127 was initially and tentatively identified in the asteroid rotation-period survey using the iPTF in Feb 2015 <cit.>, and then later confirmed in this work by follow-up observations using the Lulin One-meter Telescope in Taiwan <cit.>.This article is organized as follows. The observations and measurements are given in Section 2, the rotation period analysis is described in Section 3, the results and discussion are presented in Section 4, and a summary and conclusions can be found in Section 5. § OBSERVATIONS The iPTF, LOT, and spectroscopic observations that support the findings in this work are described in this section.The details of each of these observation runs are summarized in Table <ref>.§.§ iPTF ObservationsThe iPTF is a follow-on project of the PTF, a project whose aim is to explore the transient and variable sky synoptically.The iPTF/PTF employ the Palomar 48-inch Oschin Schmidt Telescope and an 11-chip mosaic CCD camera with a field of view of ∼7.26 deg^2 <cit.>. This wide field of view is extremely useful in collecting a large number of asteroid light curves within a short period of time. Four filters are currently available, including a Mould-R, Gunn-g', and two different H α bands. The exposure time of the PTF is fixed at 60 seconds, which routinely reaches a limiting magnitude of R ∼21 mag at the 5σ level <cit.>. All iPTF exposures are processed by the IPAC-PTF photometric pipeline <cit.>, and the Sloan Digital Sky Survey fields <cit.> are used in the magnitude calibration. Typically, an accuracy of ∼ 0.02 mag can be reached for photometric nights <cit.>. Since the magnitude calibration is done on a per-night, per-filter, per-chip basis, small photometric zero-point variations are present in PTF catalogs for different nights, fields, filters and chips.In the asteroid rotation-period survey conducted on Feb 25–26, 2015, we repeatedly observed six consecutive PTF fields near the ecliptic plane, in the R-band with a cadence of ∼10 minutes. Asteroid 2005 EC_127 was observed in the PTF field centered at R.A. = 154.04^∘ and Dec. = 10.12^∘ when it was approaching its opposition at a low phase angle of α∼ 1.3^∘. After all stationary sources were removed from the source catalogs, the light curves for known asteroids were extracted using a radius of 2 to match with the ephemerides obtained from the JPL/HORIZONS system. The light curve of 2005 EC_127 contains 42 clean detections from this observation run (i.e., the detections flagged as defective by the IPAC-PTF photometric pipeline were not included in the light curve). §.§ LOT ObservationsThe follow-up observations to confirm the rotation period of 2005 EC_127 were carried out on Sept 24, 2016 using the LOT when 2005 EC_127 had a magnitude of r' ∼ 19.2 at its low phase angle of α∼ 2.6^∘. The average seeing during the observations was ∼1.3. All images were taken in the r'-band with a fixed exposure time of 300 seconds using the Apogee U42 camera, a 2K×2K charge-coupled device with a pixel scale of 0.35. We acquired a total of 84 exposures over a time span of ∼ 440 minutes, and the time difference between consecutive exposures was ∼ 5 minutes. The image processing and reduction included standard procedures of bias and flat-field corrections, astrometric calibration using astrometry.net[http://astrometry.net], and aperture photometry using SExtractor <cit.>. The photometric calibration was done against Pan-STARRS1 point sources of r' ∼ 14 to 22 mag <cit.> using linear least-squares fitting, which typically achieved a fitting residual ∼ 0.01 mag. We improved the photometric accuracy by employing the trail-fitting method <cit.> to accommodate the streaked image of 2005 EC_127 as a result of asteroid motion over the 300-second exposure time. §.§ Spectroscopic ObservationsTo determine the taxonomic type for 2005 EC_127, its optical spectra were obtained using the Palomar 200-inch Hale Telescope (hereafter, P200) and the Double-Beam Spectrograph <cit.> in low-resolution mode (R∼1500). Three consecutive exposures were taken on Oct 4, 2016 with an exposure time of 300 seconds each. An average bias frame was made out of 10 individual bias frames and a normalized flat-field frame was constructed out of 10 individual lamp flat-field exposures. For the blue and red arms, respectively, FeAr and HeNeAr arc exposures were taken at the beginning of the night. Both arms of the spectrograph were reduced using a custom -based pipeline[https://github.com/ebellm/pyraf-dbsp] <cit.>. The pipeline performs standard image processing and spectral reduction procedures, including bias subtraction, flat-field correction, wavelength calibration, optimal spectral extraction, and flux calibration. The average spectrum of 2005 EC_127 was constructed by combining all individual exposures, and then it was divided by the solar spectrum[The solar spectrum was obtained from <cit.>, and was then convolved with a Gaussian function to match the resolution of the spectrum of 2005 EC_127.] to obtain the reflectance spectrum of 2005 EC_127 (Fig. <ref>). The trend of the reflectance spectrum suggests an V-/A-type asteroid for 2005 EC_127, according to the Bus-DeMeo classification scheme <cit.>. § ROTATION-PERIOD ANALYSISBefore measuring the synodic rotation period for 2005 EC_127, the light-curve data points were corrected for light-travel time, and were reduced to both heliocentric (r) and geocentric (Δ) distances at 1 AU by M = m + 5log(rΔ), where M and m are reduced and apparent magnitudes, respectively. A second-order Fourier series <cit.> was then applied to search for the rotation periods:M_i,j = ∑_k=1,2^N_k B_ksin[2π k/P (t_j-t_0)] + C_kcos[2π k/P (t_j-t_0)] + Z_i,where M_i,j is the reduced magnitude measured at the light-travel-time-corrected epoch, t_j; B_k and C_k are the Fourier coefficients; P is the rotation period; t_0 is an arbitrary epoch; and Z_i is the zero point. For the PTF light curve, the fitting of Z_i also includes a correction for the small photometric zero-point variations mentioned in Section <ref> <cit.>. To obtain the other free parameters for a given P, we used least-squares minimization to solve Eq. (<ref>). The frequency range was explored between 0.25–50 rev/day with a step of 0.001 rev/day. To estimate the uncertainty of the derived rotation periods, we calculated the range of periods with χ^2 smaller than χ_best^2+χ^2, where χ_best^2 is the chi-squared value of the picked-out period and χ^2 is obtained from the inverse chi-squared distribution, assuming 1 + 2N_k + N_i degrees of freedom.The rotation period of 1.64 ± 0.01 hours (i.e., 14.6 rev/day) of 2005 EC_127 was first identified using the PTF light curve <cit.>. Although the derived frequency of 14.6 rev/day is significant in the periodogram calculated from the PTF light curve, the corresponding folded light curve is relatively scattered (see upper panels of Fig. <ref>). Therefore, we triggered the follow-up observations using the LOT. The rotation periods of 2005 EC_127 derived from the LOT light curve is 1.65 ± 0.01 hours (i.e., 14.52 rev/day), which is in good agreement with the PTF result (see lower panels of Fig. <ref>). Both folded light curves show a clear double-peak/valley feature for asteroidrotation (i.e., two periodic cycles). The peak-to-peak variations of the PTF and LOT light curves are ∼ 0.6 and ∼ 0.5 mag, respectively. This indicates that 2005 EC_127 is a moderately elongated asteroid and rules out the possibility of an octahedronal shape for 2005 EC_127, which would lead to a light curve with four peaks and an amplitude of Δ m < 0.4 mag <cit.>. Moreover, we cannot morphologically distinguish between the even and odd cycles in the LOT light curve. Therefore, we believe that 1.65 hours is the true rotation period for 2005 EC_127.§ RESULTS AND DISCUSSIONTo estimate the diameter, D, of 2005 EC_127, we use:D = 1329 √(p_v) 10^-H/5<cit.>. Since the phase angle of the asteroid had a small change during our relatively short observation time span, the absolute magnitude of 2005 EC_127 is simply calculated using a fixed G slope of 0.15 in the H–G system <cit.>. We obtain H_R' = 17.27 ± 0.22 and H_r'17.30 ± 0.02 mag from the PTF and LOT observations, respectively[A G slope of 0.24 for S-type asteroids <cit.> would make the H magnitude ∼ 0.03 mag fainter, which is equivalent to a ∼ 0.01 km diameter difference, and within the uncertainty of our estimation.]. Because the absolute magnitude derived from the LOT observation has a smaller dispersion, we finally adopt H_r' = 17.30 mag for 2005 EC_127. We use (V-R) = 0.516 in the conversion of H_r' to H_V <cit.>, and then obtain H_V = 17.82 for 2005 EC_127. Assuming an albedo value of p_v = 0.36 ± 0.10 for V-type and p_v = 0.19 ± 0.03 for A-type asteroids <cit.>, diameters of D ∼ 0.6 ± 0.1 and ∼ 0.8 ± 0.1 km, respectively, are estimated for 2005 EC_127, where the uncertainty includes the residuals in light-curve fitting and the range of assumed albedos. Even when an extreme albedo value of p_v = 1.0 is applied, a diameter of 0.4 km is still obtained for 2005 EC_127. Since A-type asteroids are relatively uncommon in the inner main belt, we therefore assume an V-type asteroid for 2005 EC_127 in the following discussion. As shown in Fig. <ref>, 2005 EC_127 lies in the rubble-pile asteroid region and has a rotation period shorter than 2 hours. Therefore, we conclude that 2005 EC_127 is a large SFR.If 2005 EC_127 is a rubble-pile asteroid, a bulk density of ρ∼ 6 g cm^-3 would be required to withstand its super-fast rotation (see Fig. <ref>). This would suggest that 2005 EC_127 is a very compact object, i.e., comprised mostly of metal. However, such high bulk density is very unusual among asteroids. Moreover, 2005 EC_127 is probably an V-type asteroid. Therefore, this is a very unlikely scenario indeed.Another possible explanation for the super-fast rotation of 2005 EC_127 is that it has substantial internal cohesion <cit.>. Using the Drucker-Prager yield criterion[The detailed calculation is given in <cit.>. This method has been widely used, e.g., in <cit.>.], we can estimate the internal cohesion for asteroids. Assuming an average ρ = 1.93 g/cm^3 for V-type asteroids <cit.>, a cohesion of 47 ± 20 Pa results for 2005 EC_127[For an A-type asteroid with average density ρ = 3.73 g/cm^3 <cit.>, the cohesion would be 52 Pa.]. This modest value is comparable with that of the other large SFRs (see Table <ref>), and also nearly in the cohesion range of lunar regolith, i.e., 100-1000 Pa <cit.>.As shown by <cit.>, the size-dependent cohesion would allow large SFRs to be present in the transition zone between monolithic and rubble-pile asteroids. However, only six large SFRs have been reported to date (including this work). This number is very small when compared with the number of large fast rotators (i.e., 738 objects in the LCDB). The reason for the rarity in detecting large SFRs from previous studies (i.e., the sparse number of large SFRs in the transition zone in Fig. <ref>) could be that: (a) The rotation periods are difficult to obtain for large SFRs due to their small diameters (i.e., faint brightness); or (b) The population size of large SFRs is intrinsically small. Therefore, a survey of asteroid rotation period with a larger sky coverage and deeper limiting magnitude can help to resolve the aforementioned question. If it is the latter case, these large SFRs might be monoliths, which have relatively large diameters and unusual collision histories.We also note that none of the six reported large SFRs are classified as C-type asteroids. Therefore, any discovery of a large C-type SFR would fill out this taxonomic vacancy and help to understand the formation of large SFRs. In addition, the determination of the upper limit of SFR diameter is also important for understanding asteroid interior structure, since this can constrain the upper limit of internal cohesion of asteroids.§ SUMMARY AND CONCLUSIONS(144977) 2005 EC_127 is consistent with an V-/A-type inner-main-belt asteroid, based on our follow-up spectroscopic observations, with a diameter estimated to be 0.6 ± 0.1 km from the standard brightness/albedo relation. Its rotation period was first determined to be 1.64 ± 0.01 hours from our iPTF asteroid rotation-period survey, and then confirmed as 1.65 ± 0.01 hours by the follow-up observations reported here using the LOT. We categorize 2005 EC_127 as a large SFR, given its size and since its rotation period is less than the 2.2-hour spin-barrier.Considering its 0.6 km diameter, 2005 EC_127 is most likely a rubble-pile asteroid. For 2005 EC_127 to survive under its super-fast rotation, either an internal cohesion of 47 ± 20 Pa or an unusually high bulk density of ρ∼ 6 g/cm^3 is required. However, the latter case is very unlikely for large asteroids, and more so for V-/A-type asteroids, as 2005 EC_127 has been classified. Only six large SFRs have been reported in the literature, including 2005 EC_127, the subject of this work. This number is very small compared with the number of existing large fast rotators. Therefore, future surveys will help to reveal whether this rarity in detection is due to the intrinsically small population size of large SFRs. Moreover, none of the known super-fast rotators have been classified as C-type asteroids, and the discovery of a large super-fast rotator of this type in future work would be an interesting development to further our understanding of the formation of large super-fast rotators.This work is supported in part by the Ministry of Science and Technology of Taiwan under grants MOST 104-2112-M-008-014-MY3, MOST 104-2119-M-008-024 and MOST 105-2112-M-008-002-MY3, and also by Macau Science and Technology Fund No. 017/2014/A1 of MSAR. We are thankful for the indispensable support provided by the staff of the Lulin Observatory and the staff of the Palomar Observatory. We thank the anonymous referee for his useful suggestions and comments.llccccccccccObservational details. 0pt Telescope Date Filter RA (^∘) Dec. (^∘) N_exp Δ t (hours) α (^∘) r (au) Δ (au) m (mag) H (mag) PTF Feb 25–26 2015 R'154.0410.12 43 28.3 1.3 2.45 1.46 20.3 17.3LOT Sept 24 2016 r' 23.81 2.81 847.3 2.5 2.03 1.03 19.2 17.3 P200 Oct 4 2016 Spec.: 0.4-–0.9 μm 23.65 2.2130.5 7.7 2.05 1.07 19.5 Δ t is observation time span and N_exp is the total number of exposures. rllcccccrcrrrlConfirmed large SFRs to date. 0ptAsteroid Tax. Per. Δ m Dia. H Coh. a e i Ω ω Ref. (hours) (mag) (km) (mag) (Pa) (au)(^∘) (^∘) (^∘) (144977) 2005 EC_127 V/A 1.65 ± 0.010.5 0.6 ± 0.1 17.8 ± 0.1 47 ± 30 2.210.174.75 336.9 312.8 This work (455213) 2001 OE_84S 0.49 ± 0.000.5 0.7 ± 0.1 18.3 ± 0.2 ∼1500^b2.280.479.3432.2 2.8 <cit.> (335433) 2005 UW_163 V 1.29 ± 0.010.8 0.6 ± 0.3 17.7 ± 0.3 ∼200^b 2.390.151.62 224.6 183.6 <cit.>(29075) 1950 DA M 2.12 ± 0.000.2^a 1.3 ± 0.1 16.8 ± 0.2 64 ± 20 1.700.51 12.17 356.7 312.8 <cit.> (60716) 2000 GD_65S 1.95 ± 0.000.3 2.0 ± 0.6 15.6 ± 0.5 150–4502.420.103.1742.1 162.4 <cit.>(40511) 1999 RE_88S 1.96 ± 0.011.0 1.9 ± 0.3 16.4 ± 0.3 780 ± 500 2.380.172.04 341.6 279.8 <cit.> The orbital elements were obtained from MPC website, http://www.minorplanetcenter.net/iau/mpc.html. aΔ m is adopted from <cit.>. bThe cohesion is adopted from <cit.>. [Bellm & Sesar(2016)]Bellm2016 Bellm, E. 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"Chan-Kao Chang",
"Hsing-Wen Lin",
"Wing-Huen Ip",
"Zhong-Yi Lin",
"Thomas Kupfer",
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"Quan-Zhi Ye",
"Russ R. Laher",
"Hee-Jae Lee",
"Hong-Kyu Moon"
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"published": "20170427070728",
"title": "Confirmation of Large Super-Fast Rotator (144977) 2005 EC127"
} |
Figures/ | http://arxiv.org/abs/1704.08604v2 | {
"authors": [
"Gregory Faye",
"Matt Holzer"
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"published": "20170427144301",
"title": "Bifurcation to locked fronts in two component reaction-diffusion systems"
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A.J. Holanda*, M. Matias, S.M.S.P. Ferreira,G.M.L. Benevides & O. Kinouchi Character Networks and Book Genre Classification Departamento de Computação e Matemática – FFCLRPUniversidade de São PauloAv. Bandeirantes 3900, CEP 14040-901,Ribeirão Preto, SP, [email protected] Departamento de Física – FFCLRPUniversidade de São PauloAv. Bandeirantes 3900, CEP 14040-901,Ribeirão Preto, SP, Brazil Departamento de Educação, Informação eComunicação – FFCLRPUniversidade de São PauloAv. Bandeirantes 3900, CEP 14040-901,Ribeirão Preto, SP, Brazil Prefeitura do Campus USP de Ribeirão PretoUniversidade de São PauloAv. Bandeirantes 3900, CEP 14040-901,Ribeirão Preto, SP, Brazil Departamento de Física - FFCLRPUniversidade de São PauloAv. Bandeirantes 3900, CEP 14040-901,Ribeirão Preto, SP, [email protected] Networks and Book Genre Classification O. Kinouchi December 30, 2023 ================================================ Day Month Year Day Month YearWe compare the social character networks of biographical, legendary and fictional texts, in search for marks of genre differentiation. We examine the degree distribution of character appearance and find a power law that does not depend on the literary genre or historical content.We also analyze local and global complex networks measures, in particular, correlation plots between the recently introduced Lobby (or Hirsh H(1)) index and Degree, Betweenness and Closeness centralities.Assortativity plots, which previous literature claims to separate fictional from real social networks, were also studied.We've found no relevant differences in the books for these network measures and we give a plausible explanation why the previous assortativity result is not correct.PACS Nos.: ****§ INTRODUCTION Social networks gathered from literary texts have been studied from some years now. Most of the analyses characterized the networks of pure fictional texts with different indexes <cit.>. Others proposed or tested automatic social network extraction algorithms <cit.>.We examined a different aspect of character networks, comparing social networks extracted from texts with pure fictional, legendary and biographical types, called “genres”. The aim of the study is to find a measure or method that is able to separate the literary social networks into genres.We apply a recent node centrality index, the Lobby index <cit.>, also called Hirsh index <cit.>, to literary networks, analyzing the correlation between it and Degree, Betweenness and Closeness centralities. Indeed, we study the degree distribution of character appearances and a simple but meaningful index, in such context, that we've called Happax Legomena (HL) whose meaning we borrow from corpus linguistics.Previous literature claimed that some measures (degree, clustering coefficient, assortativity) can distinguish character networks from real social networks <cit.>. We argue that this claim is probably incorrect because the examined corpus (Marvel Universe) has a biographical-like nature similar to our corpus (where such indexes are non discriminative), which differs from real social (e.g., Facebook) networks that have no central character.§ MATERIALS AND METHODSWe use the following definition of fictional, legendary and biographical works: Biographical works are those recognized as such by modern standards describing details of a person's life. The biographies are the books: * James Gleick's Isaac Newton <cit.> (Newton);* Anthony Peake'sA Life of Philip K. Dick <cit.> (Dick);* Humphrey Carpenter's Tolkien: a Biography <cit.> (Tolkien);* Jane Hawking's Travelling to Infinity: The True Story Behind TheTheory of Everything <cit.> (Hawking). Legendary texts are those that, in the view of modern scholars, contain fictional narratives mixed with possible biographical traces. In this genre are the books: * Luke Gospel <cit.> (Luke);* Acts of the Apostles <cit.> (Acts);* Philostratus's Life ofApollonius of Tyana <cit.> (Appolonius);* Iamblicus's Life of Pytaghoras <cit.> (Pytaghoras). Fiction is denoted as texts that are recognized as such by the author of the book. The books classified as such are: * Charles Dickens's David Copperfield <cit.>(David);* Mark Twain's Huckleberry Finn <cit.> (Huck);* J. R. R. Tolkien's The Hobbit <cit.>(Hobbit);* Bernard Cornwell's The Winter King: a novel of Arthur <cit.> (Arthur).All networks were generated from the books using characters as nodes and characters' encounters represented as undirected links without the existence of self-loops.We gathered all data, with exception of David Copperfield and Huckleberry Finn that were obtained from Stanford GraphBase project <cit.>. The data files for each book contain the characters represented by two-letter, for example, the label GA in hobbit.dat file represents the character Gandalf of Hobbit book. Sometimes a group of people is considered like acting as a character, for example, the Thessalians (TH) in Apollonius of Tyana (apollonius.dat). The links are represented as “cliques of encounters”, for example, the entry AP,DM,KB in Apollonius of Tyana represents the encounter among Apollonius, Damis and the king of Babylon. The nodes are separated by comma and the cliques by semicolon.We calculated the following measures using graph-tool <cit.> library: density D, average clustering coefficient C_c, Degree K_i, node Betweenness B_i and Closeness C_i. We also wrote Python scripts to evaluate the Lobby index for node centrality <cit.> and Assortativity plots <cit.>.Additional information about project's data and source code can be found at Github page called charnet[<https://ajholanda.github.io/charnet/>].The density D of a network is the ratio of the number of links and the possible number of linksD = 2M/N(N-1)where M is the number of links and N is the number of nodes.The number of neighbors of node i is its degree K_i. The network average degree is ⟨ K_i ⟩ = 1/N ∑_i^N K_i.The clustering coefficient C_c is calculated as follows:C_c=1/N∑_i=1^N 2 l_i/K_i(K_i-1) ,wherel_i is the numberof links between the K_i neighbors of node i.For nodes in the social network, we can use the following measures of centrality: * The Degree normalized by the number of nodes not including i:K_i^N = K_i/(N-1);* The Betweenness centrality B_i^N, defined as the number of shortest paths that pass through a node i, normalized by the number of pair of nodes not including i, that is (N-1)(N-2)/2;* The Closeness centrality C_i, defined asthe sum of shortest distances between a node i andall other reachable nodes, normalized to a maximum value C^N_i=1;* The Lobby index, which is the maximumnumber L_i such that the node hasat least L_i neighbors with degree larger than or equal to L_i, normalized as L_i^N = L_i/(N-1), because the maximum degree of a node is N-1, when it is linked with all nodes but self-loop in the network.We've also analyzed the Assortativity mixing <cit.> by plotting the average degree ⟨ k_nn⟩ of neighbors of a node as a function of its degree k.An assortative mixing is found when the slope of the curve is positive, and a disassortative mixing is found when the slope is negative.We've studied the degree distribution k_i of a given character in the network fitting data using powerlaw <cit.> package. Finally, we've counted characters that appear only once which is called Hapax Legomena and twice which is called Dis Legomena.§ RESULTSGlobal indexes. <ref> indicates that Density values for fiction texts were larger (D>0.1) than other genres in our sample. The exception is Arthur that could also be considered legendary and has the actions concentrated on the main character, King Arthur, that is characteristic of a biography. Legendary and biographical texts are normally dedicated to describe the story of a few main characters with secondary characters orbiting around them and with few links among secondary characters. For example, in Apollonius of Tyana, Appolonius, Damis and Iarchas are the most proeminent characters with 151, 40 and 33 appearances, respectively. After them, king Phraotes, king of Babylon and Menippus appear only 13, 12 and 11 times respectively, with few interactions (degree), 5, 5 and 8, respectively. We do not find any clustering trend for these global measures in the plot of C_c vs D showed in <ref>. Node centrality indexes. The individual centrality indexes are Degree K_i, Betweenness B_i, Closeness C_i and Lobby L_i. As we have four quantities, one could examine six types of correlation plots for each of the books, that is, at first we should report 6 × 12 = 72 plots.Here we choose to concentrate the analysis on the least studied Lobby index versus the other classical indexes, so we report only L_i × K_i, L_i × B_i and L_i × C_i.We plot the normalized Lobby index L^N_i vs normalized degree K^N_i for all characters in <ref>[Some graphs, as Pythagoras, show few points because they have the same (L^N_i,K^N_i) coordinates.].We can see that there is an initial linear correlation between the Degree and Lobby indexes followed by a saturation in almost all graphs. This behavior can be explained by the fact that it is much difficult for Lobby index to continue increasing after a certain value of degree. For example, it is possible for the central character to have degree K^N_i=(N-1)/N ≈ 1 (he/she meets all the other characters) but to have L_i^N ≈ 1, the graph must be complete where not only the central nodes link to all other nodes, but any of their neighbors link to all other nodes too.By the comparison of the twelve plots, we noticed that Lobby and Degree are well correlated, with the exception of Pythogoras that suffers from finite size effect (N=31, M=41). Even though the measures have a good degree of correlation, the genres cannot be classified by applying Lobby vs Degree correlation.See, for example, the plots for David, Luke and Tolkien are almost indistinguishable.The Pearson correlation is low between Lobby vs Betweenness ( <ref>). We've noticed that the correlation is larger for biographies than for most of the fictional and legendary texts.However, the fictional book Arthur has a larger correlation than Tolkien, reinforcing the biographical-legendary nature of the text previously discussed.We've observed an interesting phenomenon in the Lobby vs Closeness plot ( <ref>). It shows clusters in the data, a feature found in a study of biological networks <cit.>.It seems that Lobby can detect communities that the other indexes couldn't. So, anew, these correlation plots cannot separate the book genres.The <ref> presents the Assortativity plots where each point is the degree k_i for a given character of degree k.The plot also shows the average k_nn =⟨ k_i(k) ⟩. We've observed that it doesn't matter the book genre, all plots are disassortative. Disassortativity means that characters with high degree interact preferentially with characters with low degree. An explanation is that all books have been selected as fictional or not biographies of central characters and there is no coexistence of several strong characters, perhaps with the exception of Peter and Paul in Acts. Degree distribution. We plot the degree distribution k_i so that each character now has a degree k and a cumulative probability P(k). The <ref> presents the P(k)× k for all books.Hapax Legomena. From literary criticism, we have words that appear a single time in a text named Hapax Legomena.Here we consider only Hapax Legomena (HL) for character labels, that is, labels with frequency f_i=1. They are presented in <ref>, with the books ranked from the largest to the lowest Hapax Legomena ratio HL^N= HL/N (number HL of labels with f_i = 1 divided by total number of characters N). The reasoning for using the Hapax Legomena to separate the books is the following: for a fictional text, it seems unusual the author to have the effort to create a character but use it only once.But for biographies, this seems to pose no problem. So the conclusion would be that fictional texts have less Hapax Legomena than the other genres. Surprisingly, this trend does not appear in our <ref>.The fact that the legendary texts have the larger Hapax Legomena fraction seems to be more related to the fact that they are small texts compared to the other books, so that there is less space to cite the same character several times.§ DISCUSSION The separation of book genres based on complex networks indexes is a hard task. But we've concluded that even negative results are very interesting because they refute, in a Popperian way, the conjecture that network indexes could separate literary social networks. For example, Alberich et al. <cit.> noticed differences between the average degree and clustering coefficients of the Marvel Universe (MU) network and non-literary social networks.In the MU, there is a predominance of a few characters (for example Captain America and Spider Man) with very large degree. Also, Gleiser <cit.> pointed out that the MU is very different from real social networks because it is disassortative.However, low average degree, low clustering coefficient and disassortative behavior also occurred in our character networks, because they are based in biographical-like texts which imply very central characters (e.g, Arthur, Jesus or Stephen Hawking).That is, our data suggests that Alberich et al. and Gleiser findings can be alternatively explained considering that Marvel books are a “biographical” texts of a few central heroes that should not be compared with usual (e.g., Facebook) social networks.Indeed, the hard task to distinguish real from purely fictional social networks becomes harder when we add legendary texts, which we define as text that cannot be trusted as historical biographies but could have some historical traces due to oral traditions.We have no certainty that the social network described is fictional or some information refers to true historical social relations. This is the case of the narratives about Pythagoras, Jesus of Nazareth, the first apostles and Apollonius of Tyana.The degree distributionfollowed a power law that does not depend on the literary genre studied(see <ref>). Even though this statement needs to be confirmedwith a larger corpora, it suggests that α̂ is not a good measureto distinguish historical from fictional texts, which is our primary objective.In the case of global measures average degree, density and average clustering coefficient ( <ref>), we've observed no trend that splits the genres.This result suggests that they aren't good metrics to classify the texts because they are linked with size and length of the network and don't take into account the weight of the links, for example, to highlight the importance of frequent interactions that could help in the discrimination of biographical or legendary texts.A legendary or biographical text normally has few characters with high degree and some links with high weight; the same arrangement normally doesn't occur with fictional texts. In our study, for example, in Apollonius of Tyana book (N=93, M=138), the highest weighted link has 35 interactions (27% of the encounters) between Apollonius (k=72) and Damis (k= 12); while in Huckleberry Finn (N=74, M=301), Huckleberry (k=53) is tied with the highest weighted link with 28 interactions (5.2% of the encounters) between him and Jim (k=16) . In the biography of Stephen Hawking (N=248, M=444), Hawking (k=99) meets Jane (k=152) 108 times (24.2%); while in David Copperfield, David (k=82) meets Betsey (k=31) 54 times (13.3%). Using the same reasoning as in Hapax Legomena, this is not an universal law but it can help to figure out the genre a book is most likely to fit in.Recently, Ronqui and Travieso <cit.> proposed that the analysis of correlations between centrality indexes is interesting to characterize and distinguish between natural and artificial networks. In these plots, each point refers to one character.We examined the correlation plots for the Lobby index vs Degree ( <ref>), Betweenness ( <ref>) and Closeness ( <ref>). Such comparisons revealed that social networks, fictional and legendary or historical are very similar and they cannot be distinguished.Although these are negative results, we think that they are important ones. After all, with such small sample, we cannot aim to have corroboration by induction (a large number of results suggesting clear clustering).Indeed, even with perhaps a sample of one thousand books, there's no guarantee that in the next one studied conclusions will be refuted.On the other hand, negative results refute conjectures.And, indeed, our small sample refutes a lot of a priori conjectures concerning the capacity of traditional network indexes or Hapax Legomena to separate the genres.§ CONCLUSION AND PERSPECTIVES In this paper we examined three questions: first, is there some difference among pure fictional social networks (centered in a main character), legendary social networks and networks extracted from a historical biography?Second, are there complex network indexes with potential to separate these genres? Third, what is the behavior of the recently introduced Lobby index in this respect?This preliminary study is important by proposing the problem and exploring its possible solutions. Even with a small sample, our findings seems to refute some ideas such as comparing degree distributions. By examining local node centrality indexes like Degree, Closeness, Betweenness and Lobby, what we obtain is that to separate the genres by using only the social networks is a hard and non trivial task.Although negative, these results are important as guide for future research.To overcome the limitations of this paper, we foresee only a methodological advance: to have a good Natural Language Processing algorithm that extracts automatically social networks from raw texts. Since this methodology is yet under development <cit.>, our study can be thought as both preliminary and as a benchmark for further studies.§ ACKNOWLEDGMENTSThis paper results from research activity on theFAPESP Center for Neuromathematics (FAPESP grant 2013/07699-0).OK acknowledges support from CNPq andNúcleo de Apoio á PesquisaCNAIPS-USP. MM received support from PUB-USP.§ AUTHOR CONTRIBUTIONS STATEMENT GMLB, SMSPF, MM and AJH extracted the books character networks andcharacter frequency data. AJH organized the public database, performed the complex network analyses and analyzed the data. OK proposed the original problem and analyzed the data. AJH and OK wrote the paper. All authors reviewed the manuscript.§ COMPETING FINANCIAL INTERESTSThe authorsdeclare no competing financial interests.elsarticle-num | http://arxiv.org/abs/1704.08197v2 | {
"authors": [
"Adriano J. Holanda",
"Mariane Matias",
"Sueli M. S. P. Ferreira",
"Gisele M. L. Benevides",
"Osame Kinouchi"
],
"categories": [
"cs.SI",
"physics.soc-ph"
],
"primary_category": "cs.SI",
"published": "20170426163619",
"title": "Character Networks and Book Genre Classification"
} |
The effect of hardware-computed travel-time on localization accuracy in the inversion of experimental (acoustic) waveform data Mika Takala,Timo D. Hämäläinen, Sampsa Pursiainen M. Takala and S. Pursiainen (corresponding author) arewith the Laboratory of Mathematics, Tampere University of Technology, Finland. M. Takala and T. D. Hämäläinen arewith Laboratory of Pervasive Computing, Tampere University of Technology, Finland.Copyrightc 2017 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. This paper has supplementary downloadable material available at http://ieeexplore.ieee.org., provided by the author. The material includes a set of audio wave measurement data files, a readme file and a Matlab plot scrip. Contact [email protected] for further questions about this work.December 30, 2023 =================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== This study aims to advance hardware-level computations for travel-time tomography applications in which the wavelength is close to the diameter of the information that has to be recovered. Such can be the case, for example, in the imaging applications of(1) biomedicalphysics,(2) astro-geophysics and(3) civil engineering. Our aim is to shed light on the effect of that preprocessing the digital waveform signal has on the inversion results and to find computational solutions that guarantee robust inversion when there are incomplete and/or noisy measurements. We describe a hardware-level implementation for integrated and thresholded travel-time computation (ITT and TTT). We compare the ITT and TTT approaches in inversion analysis with experimental acoustic travel-time data recorded using a ring geometry for the transmission and measurement points. The results obtained suggest that ITT is essential for maintaining the robustness of the inversion with imperfect signal digitization and sparsity. In order to ensure the relevance of the results,the specifications of the test setup were related to those of applications(1)–(3). Inverse Imaging, Waveform Tomography, Travel-Time Measurements, Field Programmable Gate Array (FPGA), High-Level Synthesis.§ INTRODUCTION This paper concerns waveform tomography in which either an acoustic or electromagnetic wave travels through a target object and, based on the external measurements of the wave,the task is to estimate the distribution(s) of a given parameter within the target <cit.> such asthe velocity of the wave or the absorbtion parameter. Tomographic imaging based on wave propagation requires computationally heavy mathematical inversion of the data in order to retrieve the relevant result set, such as an image or three-dimensional model of the test subject. In this paper, we explore the problem of determining the travel-time of the signal<cit.> and its relation to accuracy of the the inverse localization. In particular, we focus on the effect of hardware-level computations in applications where the signal wavelength may be expected to be close to the diameter of the details that are to be recovered. Such can be the case, for example, in biomedical microwave or ultrasonic tomography <cit.>, in astro-geophysical subsurface imaging<cit.> and in non-destructive testing of civil engineering materials and structures <cit.>.Our aim is determine what effect the preprocessing of digital waveform signal has on the inversion results and in that context to find computational solutions that guarantee robust inversion even with incomplete and noisy measurements. We use travel-time data, as it is known to yield robust information of the unknown parameter <cit.>, and also because it requires minimal data transfer between the hardware and the inversion routines. Data preprocessing performed on embedded hardware has its limitations, and therefore, we aim to find out how the hardware-level evaluation of the travel-time is related to the tomography results. To perform thepreprocessing operations, a field programmable gate array (FPGA) <cit.> chip on a development board is used as aplatform for the design. In order to make the implementation fast and flexible, the hardware on the FPGA was implemented by adapting Matlab scripts <cit.> to C code <cit.> and then utilizing Catapult C hardware synthesis <cit.>to generate the hardware from the adapted C code.FPGA-based processing of tomographic travel-time data is utilized in all above-mentioned application fields. As specific examples we consider (1) the microwave and ultrasonic computedtomography (MCT and UCT) of the breast <cit.>, (2)tomography of small solar system bodies (SSSBs), e.g., the COmet Nucleus Sounding Experiment by Radiowave Transmission (CONSERT)<cit.>, and (3) the acoustic/electromagnetic imaging and testing of concrete structures <cit.>. An M/UCT breast scan (Figure <ref>) can be performed by utilizinga 2D sensor ring<cit.> which records data slices sharing the direction ofthe plane normal with the ring. MCT and UCT have recently been shown to have the potential to detect and classify breast lesionsat least as reliably as other imaging methods, such as computed X-ray tomography (CT) and magnetic resonance imaging (MRI)<cit.>. Other recently studied methods for breast cancer diagnosis include, e.g., optical tomography <cit.>. The CONSERT experiment took place as a part of the European Space Agency's Rosetta mission. The objective of CONSERT was to recover the internal structure of the nucleus of the comet 67P/Churyumov-Gerasimenko based on sparse lander-to-orbiter signal transferbetween the Rosetta spacecraft and a single comet lander Philae (Figure <ref>). Space technology also involves also the challenging space environment as a limitation <cit.>, leading, e.g.,to sparse measurements. In concrete testing <cit.>, the task can be for example to detect interior defects within a concrete element (Figure <ref>) in a similar way to, e.g. the way voids are localized in geoscientific applications. In the numerical experiments, we compared integrated and thresholded travel-time (ITT and TTT)approaches via inversion analysis utilizing experimental acoustic waveform data. A 16-bit and 8-bit analog-to-digital (A/D) conversions were tested together with two different threshold criteria and normalization levels. The results obtained suggest that ITT is essential for maintaining the robustness of the inversion if the A/D conversion is incomplete. In order to ensure the relevance of the results,the specifications of the test setup were related to those of applications(1)–(3). This paper is organized as follows. Section 2 describes the materials and methods, including the mathematicalforward modeling and inversion techniques, the test setup, the equipment and the FPGA implementation.Section 3 presents the inversion results. Section 4 sums up the findings of this work and discusses the results and the potential direction of future work.§ MATERIALS AND METHODS§.§ Forward modeling§.§.§ Signal wave In this study, awaveform signal is modeled as a scalar field u representing an electromagnetic or acoustic wave. The computational domain Ω is assumed to contain the target object of the tomography together with its immediate surroundings. During the measurements, that is, when t ∈ [t_1, t_2], the transmitters and receivers can be either fixed or moving or touching the surface or remote from it. In (t, x⃗)∈[t_1, t_2] ×Ω, the scalar field u is assumed to obey the following second-order wave equation system: 1/𝚌^2∂^2 u/∂ t^2 - Δ_x⃗ u= f(t)δ(x⃗ - x⃗^(0)),u(0, x⃗) =∂ u/∂ t(0, x⃗) =0,where 𝚌 denotes the signal velocityin Ω, f(t) is the time dependence of the signal transmission,x⃗^(0) is the point (spatial location) of the transmission, and δ(x⃗ - x⃗^(0)) is a Dirichlet delta function, i.e., it is zero everywhere except at x⃗ = x⃗^(0) and satisfiesthe integral identity ∫_Ωδ(x⃗ - x⃗^(0)) d Ω = 1. The left-hand side of Equation (<ref>) is the standard hyperbolic operator of the wave equation <cit.>, and the right-hand side represents a point source transmitting an isotropic signal pulse.The isotropic radiation pattern is altered, if the source is placed in front of a reflector (with 𝚌 = 0). In this study, a loudspeaker profile is used as a reflector.Additionally, it is assumed that only the set of points belonging to Ω cantransmit a signal and that there are no other signal sources present. Defining a new variableg⃗(t, x⃗) = ∫_0^t ∇ u(τ, x⃗) d τ and h(t) = ∫_0^t f(τ) d τ, the resulting system is of the following first-order form: 1/𝚌^2∂ u/∂ t -∇·g⃗= h(t)δ(x⃗ - x⃗^(0)), ∂g⃗/∂ t - ∇ u=0.This system can be discretized spatially using the finite element method (FEM) <cit.> and temporally via the finite-difference time-domain (FDTD) method <cit.> leading to so-called leap-frog formulae which enable the simulation of the complete wave<cit.>.§.§.§ Measurement modelIn this study, we aim at localizing perturbations in the inverse of the velocity distribution 𝚗 = 1/𝚌 based on the travel-time T of the signal, as this is known to provide robust information about the velocity<cit.>. The wave u_m(t, x⃗) measured at x⃗ is a sum of the propagating wave u(t, x⃗) and a random noise term ε(t, x⃗), i.e.,u_m(t, x⃗) = u(t, x⃗) + ε(t, x⃗).The noise can include both modeling and measurement errors. In order to minimize the effect of the noise,a priori information should be used, for example, to determine the most relevant time interval for each measurement point. However, deriving a complete statistical model for ε(t, x⃗) can be difficult due to potential but unknown error sources, such as reflections, refractions and absorption. There is also no unique way to obtain T based on the waveform measurement u_m <cit.>.§.§.§ Integrated and thresholded travel-time estimatesIn this paper, we study integration and thresholding as two alternative techniques for estimating T. Given the measured wave u_m, the integrated travel-time (ITT) estimate for a signal received atx⃗ within the time interval [t_1, t_2] can be defined via the formula<cit.>T(x⃗) = ∫_t_1^t_2 tu_m(t, x⃗)^2 dt/∫_t_1^t_2 u_m(t, x⃗)^2 dt.To interface the mathematic model and real signals, some decisions have to be made. These include how to decide the time interval [t_1, t_2] and whether to use the ITT or an alternative strategy, e.g., the thresholded travel-time (TTT) estimate. ITT and TTT were calculated as follows:* Detect the time value t_0 (TTT) where the amplitude reaches a pre-selected threshold value. * Set t_1 = t_0 - τ_1 and t_2 = t_0 + τ_2, where τ_1 and τ_2 are auxiliary a prioriparameters ensuring that the essential part of the signal pulse will be contained in [t_1, t_2]. Then, obtain ITT through Equation (<ref>). Here,τ_1 extends the inspected time interval in the reverse direction, that is, (before) the signal detection point t_0. Parameter τ_2cuts the signal based on the a priori information of the pulse length in order to prevent noise due to reflections. Figure <ref> visualizes the calculation of ITT and TTT for measurement and simulated data. Figureshows that in the measurements, the travel-time can only be calculated if the starting point (control pulse) is also measured. The measured travel-time can be compared to the simulated case (Figure <ref>) where the starting point (zero-point) is defined exactly.Based on the measurements, exactly one travel-time value T is calculated for each transmitter-receiver position pair within the measurement configuration. The resulting set of values is referred to as the measurement data. The simulated signal refers to the wave that can be obtained via the FDTD method <cit.> usingan initial (constant) estimatefor the unknown parameter.§.§.§ Path integrals and ray-tracing In addition to FEM/FDTD computations, a forward simulation for the travel-time can be obtained via anintegral of the formT = ∫_𝒞𝚗 d s, where 𝒞 is the signal path <cit.>. In this approach, the prediction for 𝒞 has a central role. This can be done, e.g., based on Snell's law of reflection and reftraction <cit.>. We use the approximation(<ref>) in order to find a reconstruction of 𝚗. We assume that the signal paths 𝒞_1, 𝒞_2, …, 𝒞_M are straight line segments and that 𝚗 is of the form 𝚗 = 𝚗_p + 𝚗_0, where𝚗_0 is an a priori known constant background distribution and 𝚗_p is an unknown perturbation.A discretized version of (<ref>) can be obtained by subdividing the computational domain into pixels P_1, P_2, …, P_N and estimating 𝚗_p with a pixelwise constant distribution.Using x to denote the vector of pixel values, the equation (<ref>) can be written in thediscretized form y =Lx+y_0,where L_i, j = ∫_𝒞_i ∩ P_jd s and y_0 is a simulated vector estimated from the travel-time data corresponding to 𝚗_0. §.§ Inversion procedure To invert the data, we use a classical total variation based regularization technique, which aims at minimizing (Appendix) the regularized objective funcion Ψ( x) = y -y_0 -L x_2 + α D x_1 via the iterative recursion procedurex^(k) = (L^TL + α DΓ^(k) D)^-1 L^T ( y -y_0), where Γ^(k) is a diagonal weighting matrix defined as Γ^(k) = diag( D x^(k))^-1 and D is a regularization matrix whose entries satisfy D_i, j = βδ_i,j + ∫_P_i ∩ P_j( 2δ_i,j-1) ds/max_i,j(∫_P_i ∩ P_j ds)with δ_i,j = 1 if i=j, and zero if otherwise. In D, the first term penalizes the norm of x and the second one the total variation, i.e., the total sum ∑_i,j∫_P_i ∩ P_j | x_i - x_j| ds in which each non-zero term equals to the total jump of x between two adjacent pixels multiplied by the pixel's side-length <cit.>. This simple inversion approach usually converges sufficiently in a relatively low number of iteration steps, e.g., five. The value of the parameter β determines the balance between the regularization matrices. A small value for β leads to inverse estimates with low total variation and larger values can be expected to result in well-localized estimates <cit.>. §.§ Test setup and scenario §.§.§ Domain An acoustic setup with one speaker and two microphones was utilized to gather the experimental data. The computational domain Ω was a 59 cm diameter disk drawn on a 0.5 cm thick soft foam covering (Figure <ref>). In the experiment, the locations of three foam cylinders A, B and C with diameters 15, 10 and 10 cm, respectively, were to be detected. These were placed in the disk Ω in the upright position and apart from each other. The perimeter of Ω contained 64 equally spaced control points 1–64 for localizing the transmitter and the receiver of the signal. A top-down view of the experiment setup is included in Figure<ref>. The diameters and positions of Ω and A–C can be found in Table <ref>. §.§.§ Signal The signal pulsetransmitted (Figure <ref>) was of the form f(t) = sin(36 t) exp[-35(t - 0.60)^2]for t = [0, 1.2](milliseconds),and f(t) = 0, in all other times. The resulting signal with 5.8 kHz center frequency was transmitted from the points1, 24 and 43 using an active two-wayspeaker. Corresponding to each point of transmission, the signal was recorded for the 57 centermost control points opposite the transmitter. Based on the measurements, two travel-time data sets (I)and (II)(Figure <ref>) were formed. In (I) (dense set),the 47 centermost points were utilized in the final data set consisting of 3 × 47 = 141individual travel-time values, while only every other point of the set (I) was included in (II) (sparse set) resulting in 3 × 24 = 72 travel-time values. The measured waveform data can be found in Figure <ref>.The simulated data were obtained using the FDTD method using the constant 334 m/s as the signal velocity (the speed of sound in air at 20 ^∘C). Both the measured and simulated data have been included as supplementary material.§.§.§ Equipment The data recording device was an ordinary laptop computer equipped with an external sound card interface. Two-channel audio data was recorded using the waveform audio file format (WAV), 24 bitresolution and a 48 kHz sample rate. To eliminate any possible hardware or software based delays, a control pulse was recorded by one microphone placed 3.5 cm directly above the speaker. The actual data pulse was recorded with the other microphone positioned on the perimeter of the target area in the radial direction. Figure <ref> shows the test setup with both microphones near each other. The technical data of the hardware used in the setup can be found in Table <ref>.§.§.§ Noise Figure <ref> includes a comparison between the transmitted signal pulse and the onereceived at the perimeter of the (empty) test domain Ω without the foam cylinders placed inside. The tail of the signal received was observed to contain noise due to echoes andinaccuracies in transmission (ringing). Based on a comparison with the original signal the noise peaks were estimated to be mainly 10 dB (greyed area) below the main peak.§.§.§ Relevance The relevance of the test setup with respect to a real travel-time tomography application is the following. The transmitter sends a waveform signal pulse at a known position which is then recorded by the receiver in a known position, and the resulting signal recordings are then sent via a communication link. Further processing will consist of compression or some other processing technique, such as calculating travel-time values.§.§ Experimental FPGA hardwareThe travel-time calculation was implemented using a high-level synthesis of hardware on an FPGA development board (Altera DE2) with the typical performance of an embedded signal acquisition system. FPGAenables fastprocessing of data which is essential in waveform imaging because of the massive data input needed to record a complete wave. A high-level data flow chart of the implementation has been included in Figure <ref>. In this study, we investigated different bit resolution, threshold and normalization levels. These were set in a separate parameters file. A functioning Matlab script was first transformed directly into C code and the result was then further modified to accept parameters. The hardware was generated by using the Mentor Graphics Catapult High-Level Synthesis (HLS) tool <cit.>. HLS is a method where digital hardware is generated from a high level programming language, such as C <cit.>. The tool takes the modified, algorithmic C code and generates register transfer level (RTL) code, which is synthesized as digital logic on an Field Programmable Gate Array (FPGA) chip <cit.>.The basic hardware for signed 16-bit input (audio) data was implemented first for ITT and TTT, and then adapted separately for the signed 8-bit input data. The software on computer uses common datatypes <cit.>, but efficient hardware requires accurately defined bit resolutions for all inputs, outputs and intermediate variables <cit.>. The HLS tool does not offer a way to calculate the maximum bit resolution for an arbitary integer number. An example of this is the divisor in Equation (<ref>), which has the sum of the squares of the numbers for a part of the signal. Knowing the bit resolution of the element and thus the maximum values, the bit resolution for the square could be calculated.The 8-bit and 16-bit versions of the hardware required individual optimizations. §.§ Numerical experiments In the numerical experiments, the initial guessfor the signal velocity distribution was set to be 𝚌 = 334 m/s (the speed of sound in air at 20 ^∘ C).The number of inverse iteration steps was set at three and the regularization parameters α and β were both given the value 0.01 which, based on our preliminary tests, is a reasonable approximaitoion of the midpoint ofthe interval of the workable values.To evaluate the data processing artifacts, bit resolutions of the signed 16 and the signed 8 bits were used as in the A/D conversion units of practical applications <cit.>. The maximum amplitude in the measurement data set was normalized to a given level ν dB FS (full of the bit scale) and the other signals proportional thereto.The control signal was normalized to 0 dB FS for each individual measurement point. 16-bit preprocessing was evaluated at 100 % (ν = 0 dB FS) normalization and 8-bit at 100 % (ν = 0 dB FS) and 6 % (ν = -24 dB FS) normalization.The 6 % level represents an extreme case where the signal noise is large, thus simulating either a weak reception of transmitted signals or an insensitive receiver. Two different threshold levels 90 % and 70 % for initial signal detection were tested corresponding to around -1 dB FS and -3 dB FS of the maximum value of a normalized signal. The data vector y was obtained as the difference between the measurement and simulation based travel-time both of which were computed using either the ITT or TTT approach. The interval of the ITT was determined by τ_1 = 5 and τ_2 = 250 (samples at 48 kHz) resulting in a total length of 256 samples (5.3 ms).TTT was evaluated using two alternative strategies TTT 1 and TTT 2. In the former, ITT was applied to the control pulse and TTT to thedata pulse. In the latter, the travel-time of both the control and data pulse was computed via TTT. The motivation for investigating TTT 1, was the potential situation in which the simulated and measured travel-time are obtained via different techniques, e.g., due to different suppliers of computer software and measurement equipment. The reconstructions were analyzed by measuring the Relative Overlapping Area (ROA) and the minimum relative overlap ROA_ min between the foam cylinders 𝒮_ A, 𝒮_ B, 𝒮_ Cthe set 𝒮_recin which the value of the reconstruction wasless than a fixed limit such that Area(𝒮_ rec) = Area(𝒮_ A∪𝒮_ B∪𝒮_ C ). ROAand ROA_ min were calculated as given by the equationsROA=Area(𝒮_ rec∩ [𝒮_ A∪𝒮_ B∪𝒮_ C ])/Area(𝒮_ A∪𝒮_ B∪𝒮_ C ) ROA_ min=min( Area(𝒮_ rec∩𝒮_ A)/Area(𝒮_ A), Area(𝒮_ rec∩𝒮_ B)/Area(𝒮_ B),Area(𝒮_ rec∩𝒮_ C)/Area(𝒮_ C)).The reconstructions were computed using a laptop computer equipped with the 2.8 GHz Intel Core i7 processor 2640M and 8 GB of RAM. Computing a single reconstruction took 9 seconds of CPU time. § INVERSION RESULTS Figures <ref>-<ref> show the test area recovery (inversion) results. These figures consist of image pairs. The outlines of the three foam cylinders are superimposed on the top of the images. For each image the left side image is the actual result image. From that image, an area with strongest values is collected so that it has the same area as that of the foam cylinders. The right side image shows only the solid black areas. ITT was found to yield robust results with respect to the A/D conversion bit resolution, signal normalization, threshold, and sparsity of the measurements. The highest ROA(ROA_ min) obtained with ITT was 64 % (56 %) and the lowest one 54 % (38 %).TTT was observed to be advantageous under optimizedconditions with respect to the bit resolution, normalization, threshold, and density of the data. It was also significantly more sensitive than ITT to variations in any of these parameters. For TTT 1, the highest and lowest ROA (ROA_ min) were 70 (57 %) and 17 % (0 %), respectively.For TTT 2, these values were 71 (68 %) and 25 % (0 %), respectively. The results obtained with TTT 2 were slightly superior to those achieved using TTT 1. The 16-bit signed integer travel-time calculation module produced essentially the same outcome as64-bit floating point arithmetics. The HLS method was found to work appropriately in developing the FPGA hardware. A complete prototyping cycle took a few hours, most of which time was spent implementing a new feature in C. It was found that since the algorithmic C-like code is much more maintainable than traditional designs using VHDL, further changes and reuse of the code can be done more easily via HLS.A summary of the synthesis results for the 8-bit and 16-bit designs for the ITT and TTT is presented in Table <ref>. The hardware for ITT was almost as fast as that of TTT, and the difference in chip area between these two methods was not significant. § DISCUSSION This paper focused on developing the processing and inversion of waveform tomography datafor applications in whichthe signal wavelength is close to the diameter of the details that are to be recovered. In particular, FPGA-based hardware was used.We compared the integrated and thresholded travel-time (ITT and TTT) in numerical experiments in which three foam cylinders were to be localized based on an experimental 5.8 kHz acoustic signal. We tested a 16-bit and 8-bit analog-to-digital (A/D) conversion together with two different threshold criterions and normalization levels. As reference applications of this study, we considered (1) microwave and ultrasonic computed tomography (MCT and UCT), (2)tomography of small solar system bodies (SSSBs), in particular, the CONSERT experiment and (3) ultrasonic/microwave detection of concrete defects. §.§ Experiments The recovery of the test object locations on the target area by using the described inversion methods was found to work appropriately.Our results show that ITT is more stable than TTT if the signal quality decreases. This can been seen from the stability of the relative overlapping area (ROA) percentages (53 - 64 %). These percentages are comparable to our previous research on waveform inversion within a 2D domain. In <cit.> the best ROA percentage 71 % was obtained by using the full wave data. Using TTT the best recovery result was exactly the same 71%. ITT was 7 % less accurate, but more stable when data preprocessing was performed using source data with lesser bit resolution. The results show that ITT isinvariant with respect to source data bit resolution reduction in the time-domain and the level of thresholding used to locate the signal pulse from recorded audio data. ITT was also found to be more reliable than TTT with respect to the normalization of the signal and the sparsity of the measurements. We normalized the 8-bit signal to two different levels, 100 % (0 dB FS) and 6 % (-24 dB FS) amplitude, to simulate weak receivers, i.e., to decrease signal-to-noise ratio (SNR). This is relevant in applications where the signal quality is reduced. For example, in astro-geoscientific applications, the signals can be weak in some directions. This was evident from the CONSERT experiment <cit.> in which the power and quality of the received signal varied significantly depending on the direction of the measurement <cit.>. The result concerning the sparsity of the measurements is essential for CONSERT, and other applications in which not all the data can be gathered.Based on the results, we suggest that ITT can be superior to TTT with respect to the robustness of the inversion. It also seems obvious that TTT might achieve a higher ROA than ITT for a high-quality signal and a well-chosen threshold parameter. Namely,under optimal conditions TTT filters out the noisy tail part of the signal that is present in the computation of the ITT. This advantage isutilized, e.g., in the first-arrival sound speed inversion <cit.>. The present results suggest that the ITT method does not significantly diminish the inversion quality, and indeed, it can increase the reliability of the results with respect to the uncertainty factors and incompleteness of the data. §.§ Applications Tomography applications differ from each other by the speed and length of the electromagnetic or sound waves. The feature size d that can be detected is dependent on the wave length λ (e.g. λ/2). In Table <ref>, the significance of the test setup with respect to the present reference applications (1)–(3) has been summarized based on d, λ, the diameter of the target domain D and the ratios d/λ andD/λ. The values d, λ, andD utilized in Table <ref> can be reasoned as follows.(1) In MCT, signal frequencies 1–6 GHz are being usedand the diameter of the sensor ring can be, e.g., D = 15 cm <cit.>. At 5 GHz,the wavelength isλ≈ 1.9 cm corresponding to the relative permittivity of the breast ε_r ≈ 10<cit.>. Medical UCT utilizes frequencies in the range of 1 - 20 MHz <cit.>. Using the 1 MHz value and the speed of 1500 m/s for ultrasound propagation in human tissue <cit.>, the wavelength is λ = 1.5 mm. In UCT and MCT, the feature to be recovered can be, e.g., a small T1 or T1a tumor which can have a maximum diameter of d = 2 cm and d =0.5 cm, respectively. (2) A signal frequency of 10 MHz has been suggested for the tomography of SSSBs <cit.> matching roughly with the wavelength λ≈ 15 m (ε_r ≈ 4, e.g., for dunite and kaolinite <cit.>) which isthe estimated resolution of the CONSERT experiment <cit.>. The diameter of the SSSB can be for example D=150 m. (3) In ultrasonic material testing for concrete the velocity of the wave is 3500 m/s<cit.>. At a typical ultrasound frequency of 100 kHz the wavelength of the signal is λ = v/f = 3.5 m and the diameter of the concrete beam being tested could be, for example, D=30 cm. A fault or crack inside the beam could be d = 3 cm. Based on Table <ref>, the present experiment setup can be considered applicable to all application contexts (1)–(3). In MCT and UCT the best match to the test geometry is obtained with the T1 and T1a tumor size, respectively. Obviously, the difference between ITT and TTT can be less significant in applications, where the wave length is likely to be considerably smaller than the smallest detail to be detected. As an additional comparison, in the tomography of the ionosphere, the speed of the electromagnetic wave is around the speed of light and a typical frequency used in tomography is between 120 MHz and 400 MHz. Using a value of 200 MHz the wavelength is λ = 1.5 m. The size of the ionosphere is up to D=1000 km from the surface of the Earth and a typical vertical feature to be recovered can be d = 100 km, for example <cit.>. Consequently, d/λ and D/λ can be 100 and 670, respectively. This suggests that our test scenario might be too different fromthe tomography of the ionosphere to be able to draw inferences from the results.The noise peaks in the experimental data were estimated to be mainly 10 dB below the maximum peak. This can be considered as appropriate for the applications(1)–(3).For a MCT imaging system, the relative reconstruction error has been shown to stay under 10 % for amplitude errors down to 10 dB SNR<cit.>.The total noise peak level of around 20 dB was observed in the CONSERT experiment <cit.>. In concrete testing, structural noise <cit.>, e.g., echoes from walls, can be significant resulting in noise peaks that can be comparable to the main peak. §.§ Hardware The use of high-level synthesis (HLS) to develop the test hardware was found to be essential, as the specifications for the travel time calculation changed during the implementation phase and so changes to the hardware had to be made quickly. We aimed at a very direct workflow in implementing the hardware, and thus some optimization methods in the design partitioning and in the HLS tool were not utilized. In addition, having the different bit resolution implementations in separate files increased the work when changes had to be made. The resulting digital hardware was synthesized on an FPGA platform for demonstration and to facilitate further development of the data gathering system towards a laboratory instrument.Many operations in the computation scripts, such as the normalization of values, require divisions. In the ITT calculation formula(<ref>) there is a large-valued divisor that first sums the squares of the values and then divides the sum with that value. Catapult generates divisions as combinatorial logic if written directly as it is in typical C code <cit.>. Combinatorial logic <cit.>is not synchronized with the hardware clock, which makes it unreliable in use. This gives a false sense of flexibility in the HLS tool and the user has to know what is being generated. Catapult has a math library which has synthesizable, basic algorithmic division operators for integers. Changing all divisions to use division operators offered by this library improved the results, and resulted in aworking design for most division operations whereas combinatorial dividers did not. Travel-time calculation can be seen as an extreme form of compression in time domain. There are more compression methods such as filtering in the frequency domain. Our work used reduced the bit resolution for input audio data. Calculation of the travel times was also performed with integer arithmetics. These reduce the accuracy of the results. bit resolution reduction is relevant when using FPGAs in general. This is because vendor-provided hardware multipliers available for use on FPGA chips such as Altera's Cyclone II have limited bit resolutions <cit.>. For example, a FIR filter can use these hardware multipliers on FPGAs as in <cit.>. Because the operator is a multiplication that can increase the required bit resolution to store the intermediate results, the original bit resolution of the data has to be reduced. One system <cit.> has implemented Fast Fourier Transform (FFT) filtering in the frequency domain and inverse FFT (IFFT) signal reconstruction back to the time domain. In currently available FPGA DSP chips, the available bit resolutions for multipliers are much larger <cit.>, but reduction in bit resolution can still be required. This makes the invariance of the source data bit resolution of our ITT method an important point of interest. An example of decreased signal quality is found in Ground Penetrating Radar (GPR) applications. In <cit.> an instrumentation system with either 8-bit or 16-bit Analog-to-Digital conversion was employed. This shows that bit resolution limitations are also found in instrument hardware. §.§ Outlook Finally, the present results indicate various directions for future work. We will study hardware-level solutions regarding (i) processing and (ii) inverting waveform data.(i) Investigating harware constraints utilizing a more sophisticated statistical travel-time detection model, such as the akaike information criterion <cit.>,would be an interesting goal. The calculation of travel-time values is only one data preprocessing method that can be used, and mathematical methods to accomodate other types of filtering, such as compressing in the frequency domain could be developed. The goal in preprocessing depends on the application: in the space environment minimal data transfer between the sensors and the computation unit is important for the limited communication capacity available, whereas in biomedical and civil engineering the main objective can be to optimize the speed of the procedure in order to allow recordingas much data as possible. (ii) We aim to develop inversion approaches utilizing the FPGA environment, so extending the current study of hardware-level constraints to include inversion algorithms is essential. § ACKNOWLEDGEMENTS M.T. and S.P. were supported by the Academy of Finland Key Project 305055 and the Academy of Finland Centre of Excellence in Inverse Problems Research.§.§ AppendixIn the inversion procedure (<ref>), the matrix D is symmetric and can thus be diagonalized. When β>0, D is also positive definite and also invertible. Hence,one can define x̂ =D x and L̂=L D^-1. Substitutingx̂ and L̂ into (<ref>) leads to the following form <cit.>x̂_ℓ+1= (L̂^T L̂+ αΓ̂_ℓ )^-1L̂^Ty, Γ̂_ℓ = diag ( |x̂_ℓ|)^-1, Γ̂_0 =I .which can be associated with alternating conditionalminimization of the functionH(x̂, ẑ)=L̂x̂ -y^2_2 + α∑_j = 1^Mx̂^2_j/ẑ_j + α∑_j = 1^M ẑ_jin which z_j>0, for i = 1, 2, …, M. As H(x̂, ẑ) is quadraticwith respect to x̂,the conditional minimizer x̂'=min_x̂ H(x̂|ẑ) is given by the least-squares solution of the form x̂' =(L̂^T L̂+ αΓ̂_ẑ )^-1L̂^Ty, Γ̂_ẑ = diag ( ẑ)^-1 and Γ̂_0 =I. At the point of the conditional minimizer ẑ'=min_ẑ H(ẑ|x̂), the gradient of H( ẑ|x̂ ) vanishes with respect to ẑ, i.e., ∂ H( x̂, ẑ)/∂ẑ_j|_ẑ' =- αx̂^2_j/(ẑ'_j)^2 + 1 = 0, i.e.ẑ'_j = |x̂_j|√(α). Hence, the global minimizer can be estimated via the following alternating iterative algorithm.* Set ẑ_0 = (1, 1, …, 1) and ℓ = 1. For a desired number of iterations repeat the following two iteration steps.* Find the conditional minimizer x̂_ℓ = min_x̂ H(x̂, ẑ_ℓ-1). * Find the conditional minimizerẑ_ℓ = min_ẑ H(x̂_ℓ, ẑ).The sequence x̂_1, x̂_2, … produced by this algorithm is identical to that of (<ref>) and x_ℓ =D^-1x̂_ℓ equals to the ℓ-th iterate of (<ref>). If for some ℓ < ∞ the pair (x̂_ℓ, ẑ_ℓ) is a global minimizer ofH( x,z), then, since (ẑ_ℓ)_j =|(x̂_ℓ)_j|, j = 1, 2, …, M,then it is also the minimizer ofΨ̂(x̂) = H(x̂_1, x̂_2, … , x̂_M, |x̂_j|, |x̂_j|, …, |x̂_M|)= L̂x̂ -y^2_2 + α∑_j = 1^Mx̂_j^2/ẑ_j +∑_j = 1^M ẑ_j=L̂x̂ -y^2_2 + α∑_j = 1^Mx̂_j^2/|x̂_j| √(α) +∑_j = 1^M |x̂_j| √(α) =L̂x̂ -y^2_2 + 2√(α)x̂_1.Consequently, x_ℓ =D^-1x̂_ℓ is the minimizer of Ψ( x), asΨ( x) = Ψ̂(x̂). Furthermore, the minimizer of Ψ̂(x̂) is also the 1-norm regularized solution of the linearized inverse problem. IEEEtran[ < g r a p h i c s > ]Mika Takala (M’16) was born in Nurmo, Finland, in 1982. He received the B.Sc. degree in electrical engineering from the Tampere University of Technology (TUT), in 2015, and the M.Sc.(tech.) degree from TUT in 2016. His master’s thesis in the field of embedded systems concentrated on implementation of signal preprocessing modules with High-Level Synthesis for waveform inversion applications.In 2016 Mr. Takala started working at the Department of Mathematics, TUT, as a PhD student. He currently works on his PhD research related geophysical inversion strategies and embedded systems. He also works as a software architect in Granite Devices, Inc., Tampere, Finland. [ < g r a p h i c s > ]Timo D. Hämäläinen (M’95) received the M.Sc. and Ph.D. degrees from Tampere University of Technology (TUT), Tampere, Finland, in 1993 and 1997, respectively. He has been a Full Professor with TUT since 2001 and is currently theHead of the Laboratory of Pervasive Computing.He has authored over 70 journals and 210 conference publications. He holds several patents. His research interests include design methods and tools formultiprocessor systems-on-a-chip and parallel video codec implementations.[ < g r a p h i c s > ]Sampsa Pursiainen received his MSc(Eng)and PhD(Eng) degrees (Mathematics)in the Helsinki University of Technology(Aalto University since 2010), Espoo, Finland, in 2003 and 2009. He focuses on various forward and inversion techniques of applied mathematics. In 2010–11, he stayed at the Department of Mathematics, University of Genova, Italy collaborating also with the Institutefor Biomagnetism and Biosignalanalysis (IBB), University of Münster, Germany. In 2012–15, he worked at theDepartment of Mathematics andSystems Analysis, Aalto University,Finland and also at the Department of Mathematics, Tampere University of Technology, Finland, where he currently holds an Assistant Professor position. Received xxxx 20xx; revised xxxx 20xx. | http://arxiv.org/abs/1705.03087v1 | {
"authors": [
"Mika Takala",
"Timo D. Hämäläinen",
"Sampsa Pursiainen"
],
"categories": [
"astro-ph.IM",
"physics.geo-ph"
],
"primary_category": "astro-ph.IM",
"published": "20170427154745",
"title": "The effect of hardware-computed travel-time on localization accuracy in the inversion of experimental (acoustic) waveform data"
} |
Source File Set Search for Clone-and-Own Reuse Analysis Takashi Ishio12, Yusuke Sakaguchi1, Kaoru Ito1, Katsuro Inoue1 1 Graduate School of Information Science and Technology, Osaka University, Osaka, Japan 2 Graduate School of Information Science, Nara Institute of Science and Technology, Nara, Japan Email: {ishio, s-yusuke, ito-k, inoue}@ist.osaka-u.ac.jp December 30, 2023 =======================================================================================================================================================================================================================================================================================================================Clone-and-own approach is a natural way of source code reuse for software developers. To assess how known bugs and security vulnerabilities of a cloned component affect an application, developers and security analysts need to identify an original version of the component and understand how the cloned component is different from the original one. Although developers may record the original version information in a version control system and/or directory names, such information is often either unavailable or incomplete. In this research, we propose a code search method that takes as input a set of source files and extracts all the components including similar files from a software ecosystem (i.e., a collection of existing versions of software packages). Our method employs an efficient file similarity computation using b-bit minwise hashing technique. We use an aggregated file similarity for ranking components. To evaluate the effectiveness of this tool, we analyzed 75 cloned components in Firefox and Android source code. The tool took about two hours to report the original components from 10 million files in Debian GNU/Linux packages. Recall of the top-five components in the extracted lists is 0.907, while recall of a baseline using SHA-1 file hash is 0.773, according to the ground truth recorded in the source code repositories.Software reuse, origin analysis, source code search, file clone detection§ INTRODUCTIONSoftware developers often reuse source code of existing products to develop a new software product <cit.>.Mohagheghi et al. reported that reused components are more reliable than non-reused code <cit.>. While open source software projects reuse code from other OSS projects, industrial developers also use open source systems due to their reliability and cost benefits <cit.>.Clone-and-own approach is one of the popular approaches to source code reuse <cit.>. Dubinsky et al. <cit.> reported that cloning is perceived as a natural reuse approach by the majority of practitioners in the industry.Although many reusable components are available online in binary forms for various operating systems,developers copy source code of an existing component into their project's so that they can build and test their product using a particular version of a component or modify it for their own purpose. For example, Mozilla Firefox 45.0 includes a modified version of zlib 1.2.8 in itsdirectory; a developer added a header filein order to rename functions defined in the library. Koschke et al. <cit.> reported that copies of specific libraries are involved in a relatively large number of projects.Cloned components may introduce potential defects into an application. Sonatype reported that many applications include severe or critical flaws inherited from their components <cit.>.Hemel et al. <cit.> reported that each of Linux variants embedded in electronic devices has its own bug fixes. To investigate known bugs and security vulnerabilities of a cloned component, developers and security analysts need to identify an original version of the component and understand how the cloned component is different from the original one. However, in general identifying the original version is tedious and time-consuming. The main reason is that original component names and version numbers are often unrecorded <cit.>. Another reason is that a cloned component may be a derived version in a different project. For example, Firefox includes another copy of zlib indirectory; the version is a part of the NSS component. To identify an original version of a cloned component, an analyst has to compare its source code with all the existing versions of components in its software ecosystem.A baseline method to analyze a cloned component is code comparison using file hash values such as SHA-1 and MD5. Since the method cannot detect modified files, Kawamitsu et al. <cit.> proposed a code comparison technique that identifies the most similar file revision in a repository as an original version of a file. The experiment reported 20% of cloned files are modified in eight projects.Although the method is effective, its simple pairwise comparison of files is inefficient to analyze an entire software ecosystem, which includes millions of source files.In this research, we propose a code search method tailored for analyzing a cloned component. It takes as input a set of source files, and reports existing components including files that are similar to the input files.The method ranks components using aggregated file similarity, assuming a cloned component is the most similar to the original component. A reported list of components enables developers and security analysts to compare their cloned component with its original version.Our code search method employs the b-bit minwise hashing technique <cit.> that is an extension of Min-Hash technique <cit.>; in summary, the technique enables to estimate file similarity using hash signatures. Our method constructs a database of hash values for each file in a software ecosystem. Using the database, our method then extracts a subset of files likely similar to a query and then computes actual similarity for the subset. Although a database construction takes time, we can search similar files within a practical time. We also define a filtering method for components using file similarity. Since different versions of a library often include similar files,we select components having the most similar files as a representative set.We conducted an experiment to evaluate the effectiveness of the tool. As the ground truth, we manually identified original versions of 75 cloned components included in source code of Firefox 45.0 and Android 4.4.2_rc1. We then analyzed the components with a database of Debian GNU/Linux packages including 10 million source files. Recall of the top-five components in the extracted lists is 0.907, while recall of a baseline using SHA-1 file hash is 0.773. To obtain all the original components, a user needs to investigate 551 components in the lists. It is smaller than 931 components reported by the baseline method. The result shows that our method ranks the original components at higher positions and reduces manual effort of a user.The contributions of the paper are summarized as follows. * We defined a code search method to extract similar files from a huge amount of source files efficiently. * We defined a component filtering method to select likely original components. * We created the ground truth dataset of actual clone-and-own reuse instances for two major OSS projects, and evaluated our method using the dataset. Section <ref> shows related work of our approach.The approach itself is detailed in Section <ref>.Section <ref> presents the evaluation of our approach using OSS projects.Section <ref> describes the threats to validity of the work. Section <ref> describes the conclusion and future work. § RELATED WORK §.§ Code Clone Detection Code clone detection has been used to analyze source code reuse between projects. Kamiya et al. <cit.> proposed CCFinder to detect similar code fragments between files. German et al. <cit.> used CCFinder to detect code siblings reused across FreeBSD, OpenBSD and Linux kernels. They identify the original project of a code sibling by investigating the source code repositories of the projects. Hemel et al. <cit.> analyzed vendor-specific versions of Linux kernel using their own clone detection tool.Their analysis showed that each vendor created a variant of Linux kernel and customized many files in the variant.Koschke et al. <cit.> also used a clone detection technique to analyze code clone rates in 7,800 OSS projects. They found that a relatively large number of projects included copies of libraries. They excluded the copies from analysis, because the analysis did not focus on inter-project code reuse.Krinke et al. <cit.> proposed to distinguish copies from originals using the version information recorded in source code repositories.Krinke et al. <cit.> used the approach to analyze GNOME Desktop Suite projects. The result shows that there is a lot of code reuse between the projects. Although the version information is useful to select older files as candidates of reused files, our method does not use it because source code repositories are not always available.Sajnani et al. <cit.> proposed SourcererCC, a scalable code clone detection tool. They optimized comparison of two code fragments, based on an observation that most of files are different from one another. We employ b-bit minwise hashing technique to avoid unnecessary code comparison.Our approach can be combined with SourcererCC's optimization.Sasaki et al. <cit.> proposed a file clone detection tool named FCFinder. The tool normalizes source files by removing code comments and white space, and compare the resultant files using MD5 hash.This method is not directly applicable to our problem, because it cannot detect similar but modified files.Hummel et al. <cit.> proposed to use an index database for instant code clone detection. While it is similar to our approach, the clone index is designed to report source code locations. It is not suitable to compute source file similarity.Jiang et al. <cit.> proposed DECKARD, a code clone detection tool using a vector representation of an abstract syntax tree of source code. Nguyen et al. <cit.> extended a vector representation for a dependence graph and proposed a code clone detection tool named Exas. The tool has been used to detect common patterns in source code <cit.>, rather than identification of similar files.Detected instances of source code reuse are clues to extract the common functionalities in software products. Rubin et al. <cit.> reported that industrial developers extract reusable components as core assets from existing software products. Bauer et al. <cit.> proposed to extract code clones across products as a candidate of a new library. Ishihara et al. <cit.> proposed a function-level clone detection to identify common functions in a number of projects. Our method can be seen as a file-level detection of cloned components.While our method enables a user to select original components for comparison, it does not directly support a source code comparison activity itself. Duszynski <cit.> proposed a code comparison tool to analyze source code commonalities from a number of similar product variants. Sakaguchi et al. <cit.> also proposed a code comparison tool that visualizes a unified directory tree for source files of several products. Fischer et al. <cit.> proposed to extract common components from existing product variants and compose a new product.§.§ Origin Analysis Godfrey et al. <cit.> proposed origin analysis to identify merged and split functions between two versions of source code. The method compares identifiers used in functions to identify original functions. Steidl et al. <cit.> proposed to detect source code move, copy, and merge in a source code repository. The method identifies a similar file in a repository as a candidate of an original version. Kawamitsu et al. <cit.> proposed to identify an original version of source code in a library's source code repository. It is an extension of origin analysis across two source code repositories. A user of the method must know what library is included in the program. Our method does not need such knowledge, because it compares input files with all the existing components in a software ecosystem. Sojer et al. <cit.> pointed out that ad-hoc code reuse from the Internet has a risk of a license violation.Inoue et al. <cit.> proposed a tool named Ichi-tracker to identify the origin of ad-hoc reuse. It is a meta-search engine to obtain similar source files on the Internet and visualizes the similarities. While Ichi-tracker takes a single file as a query,our method enables a user to analyze a set of files as a component.Kanda et al. <cit.> proposed a method to recover an evolution history of a product and its variants from their source code archives without a version control. The approach also compares the full contents of source files, using a heuristic that developers tend to enhance a derived version and do not often remove code from the derived version. It might complement our approach, because it helps to understand an evolution history of components reported by our method.Antoniol et al. <cit.> proposed a method to recover the traceability links between design documents and source files. The method computes similarity of classes by aggregating similarity of their attribute names to identify an original class definition in design documents. Our method can be seen as a coarse-grained extension that identifies original components using aggregated file similarity.Hemel et al. <cit.> proposed a binary code clone detection to identify code reuse violating software license of a component. The method compares the contents of binary files between a target program and each of existing components. Sæbjørnsen et al. <cit.> proposed a clone detection for binary code. It uses a locality sensitive hashing to extract similar code fragments in binary files. Qiu et al. <cit.> proposed a code comparison method for a binary form to identify library functions included in an executable file. For Java software, Davies et al. <cit.> proposed a file signature to identify the origin of a jar file using classes and their methods in the file ignoring the details of code.German et al. <cit.> demonstrated the approach can detect OSS jar files included in proprietary applications. Mojica et al. <cit.> used the same approach to analyze code reuse among Android applications. Ishio et al. <cit.> extended the analysis to automatically identify libraries copied in a product. Differently from those approaches, our method directly compares source files because small changes in files might be important to understand differences between a cloned component and its original version.Luo et al. <cit.> proposed a code plagiarism detection applicable to obfuscated code. The detection method identifies semantically equivalent basic blocks in two functions.Chen et al. <cit.> proposed a technique to detect clones of Android applications. The analysis uses similarity between control-flow graphs of methods. Obfuscated code is out of scope of our method. Ragkhitwetsagul et al. <cit.> evaluated the performance of code clone detection and relevant techniques for source files modified by code obfuscators and optimizations. § OUR APPROACH Our method takes as input a set of source files and reports a list of components that likely include the original version of the files in a software ecosystem. In this paper, a software ecosystem is a collection of components { C_1, C_2, ⋯, C_n }.Each component comprises a set of files.Our implementation assumes that each component has a unique name such as “zlib-1.2.8” and “libpng-1.6.9”.We use source file similarity, because popular libraries written in C/C++ are used by many projects. While the main target of our method is C/C++, our implementation supports C/C++ and Java.Our method is language independent except for the lexical analysis step. The lexical analysis assumes that each source file has a correct file extension representing a programming language.Our method comprises two steps: component search and component-ranking. The first step extracts a set of components R including files similar to query files Q. The second step filters and ranks components according to aggregated file similarity. We use a simple assumption: A component in a database is likely original if it has the most similar file to a query file.Fig. <ref> shows an example input and output of our method. The example query Q includes five files: , , , , and . Our method compares each query file with files in a component database. In the example, our component search step detects similar files in three components , , and . Three edges connecting a product and components have labels indicating the sum of similarity values. Using file similarity and the aggregated similarity, our method ranksat the top andat the second.is the most likely original component, because it has four similar files.is the second, because its two filesandare more similar than files in .Those files are also likely original files. Our method filters out ,because it does not have a file whose similarity is higher than other components. §.§ Component Search Our component search uses a file-by-file comparison. Given a query set of source files Q, we extract candidates of original components R from a collection of components as follows.R = { C_i | ∃ q ∈ Q, f ∈ C_i. sim(q, f) ≥ th }where sim is a similarity function and th is a similarity threshold, respectively.It should be noted that the definition does not use release date of components, because there is a time lag between an official release of a project and its packaging for a software ecosystem. If accurate timestamps are available for both query files and components, our method can use a subset of components older than query files.Our similarity of source files is Jaccard index of token trigrams defined as follows.sim(f_1, f_2) = |trigrams(f_1) ∩ trigrams(f_2)|/|trigrams(f_1) ∪ trigrams(f_2)|where trigrams(f) is a multiset of trigrams extracted from a file f. We employ the Jaccard index because it approximates the edit distance <cit.>. A higher similarity indicates that a larger amount of source code could be reused. Compared with the longest common subsequence, it is less affected by moved code in a file. Furthermore, it can be efficiently estimated using the Min-Hash technique <cit.>.We use a token as a trigram element to ignore the length of identifier names. A lexer extracts a token sequence by removing comments and white space. The lexer keeps identifiers as they are, because identifiers are important clues to identify a version <cit.>. Our lexer also keeps preprocessor directives in C/C++ source files. Fig. <ref> shows a pair of example code fragments, their trigrams, and a similarity value obtained from the trigrams. In the figure, a trigram ABC indicates three consecutive tokens in a code fragment. A symbol “_” in a trigram indicates the beginning and end of a file.Algorithm <ref> shows an entire process of the component search step. The algorithm starts with a query Q and a file collection F. Since the same file may be included in multiple components,we use F to represent a set of existing unique files in a software ecosystem,and Owners(f) to represent a set of components including the file f. Our implementation uses SHA-1 file hash to detect files shared by components.The algorithm computes similarity values S(q, C) between query files and their most similar files in C defined as follows.S(q, C) = max { sim(q, f) | f ∈ C }The line 2 initializes S(q, C) to zero and the lines 10 and 11 update it. A file f ∈ C may update similarity values of multiple query files,because a developer could copy the file f to create the files. The whole process compares all the pairs of q ∈ Q and f ∈ F with two optimizations. First, it compares the size of trigram sets at the line 5. The statement uses the following property to avoid unnecessary comparison.min(|X|,|Y|)/max(|X|,|Y|) < th |X ∩ Y|/|X ∪ Y| < thThe property is derived from min(|X|,|Y|) ≥ |X ∩ Y| and max(|X|,|Y|) ≤ |X ∪ Y|. Secondly, the process computes sim_e(q, f) that is an estimated similarity computed by b-bit minwise hashing technique <cit.>. Since it may have a margin of error, line 6 uses th-m as a threshold, where m specifies allowable errors. Finally, the process computes an actual similarity metric sim(q, f) to compare trigrams. If it is higher than a threshold, components including f are added to R in the component-ranking step. The lines 10 and 11 record the highest similarity of a query file q and a component C_i. The recorded similarity values S(q, C_i) are used in the component-ranking step.The algorithm works efficiently because sim_e(q, f) avoids unnecessary actual similarity computation. In summary, b-bit minwise hashing technique approximates a similarity of files by comparing k pairs of b-bit signatures.Each signature represents a trigram sample in a file. In case of our implementation, we chose parameters b=1, k=2048; 2048 trigram samples in a source file are selected and then translated into 1-bit signatures.Consequently, a file is represented by a 2048-bit vector.To compute sim_e(q, f), we use k independent hash functions h_i(t) (1 ≤ i ≤ k). Each function translates a trigram t in a file into an integer. Our implementation uses 64-bit integers as described in the Appendix. Using the hash functions, min-hash signatures m_i(f) (1 ≤ i ≤ k) for a file f are computed as follows <cit.>.m_i(f) = min { h_i(t) | t ∈ trigrams(f) }A min-hash signature m_i(f) represents a trigram sample selected from a file. If two files f_1 and f_2 are more similar, more likely m_i(f_1) and m_i(f_2) select the same trigram and result in the same value.The probability of m_i(f_1)=m_i(f_2) is represented by the similarity of files <cit.>:P(0.9m_i(f_1)=m_i(f_2)) = sim(f_1, f_2)We use b-bit min-hash signatures b_i(f) (1 ≤ i ≤ k) extracted from min-hash signatures. In case of b=1, the signatures are computed as follows <cit.>.b_i(f) = LSB( m_i(f) )where LSB is the least significant bit; i.e., b_i(f) ∈{0, 1}. The probability of b_i(f_1)=b_i(f_2) is represented byP(0.9b_i(f_1)=b_i(f_2)) = sim(f_1, f_2) + 1 - sim(f_1, f_2)/2because the condition is satisfied when m_i(f_1)=m_i(f_2) or two different signatures have the same LSB by chance.We estimate a similarity of files q and f using their b-bit min-hash signatures b_i(q) and b_i(f) (1 ≤ i ≤ k).Since the probability of b_i(q)=b_i(f) is dependent on a similarity,we compute an estimated similarity sim_e(q, f) from an observed probability as follows.sim_e(q, f) = (P_o(q, f) - 1/2) × 2P_o(q, f) = 1 - 1/k∑_i=1^k XOR(b_i(q), b_i(f))where P_o(q, f) is an observed probability of b_i(q)=b_i(f) on k (2048) samples. The maximum value of sim_e(f_1, f_2) is 1.0. If two files are the same, their estimated similarity is always 1.0. Although the sim_e(f_1, f_2) could be negative, we simply regard them as zero. It should be noted that this is a simplified version for ease of implementation, compared with the original (strict) estimation that was conducted in <cit.>.Computation of sim_e is O(1), because it uses XOR bit-operations and bit counting. It is much efficient than Jaccard index computation that requires O(n) depending on file size n. Since a signature does not change, we compute b-bit min-hash signatures for each f ∈ F and store them in a database. The tool loads the entire database on memory because it is sufficiently compact; 1 GB memory can store signatures for four million files. For each query, we then compute signatures for q ∈ Q and compare them with signatures in the database. The line 6 in Algorithm <ref> uses th-m, where m specifies allowable errors. We use m=0.1 for our implementation. Fig. <ref> shows the distribution of errors of 10^8 randomly created P_o samples under the condition sim(f_1, f_2) = 0.6. In the figure, sim(f_1, f_2) - sim_e(f_1, f_2) is always less than 0.1.In other words, sim_e(f_1, f_2) > th-m. We confirmed that it was sufficient in the experiment.§.§ Component Ranking The second step filters and ranks extracted components R to enable a user to identify an original component easily. To filter components, we consider that C_1 is a better candidate than C_2 if C_1 provides more similar files than their corresponding source files in C_2. We define this relation C_1 ⊃_S C_2,C_1 ⊃_S C_2 (∀ q. S(q, C_1) ≥ S(q, C_2) ∧ ∃ q. S(q, C_1) > S(q, C_2) ) ∨(∀ q. S(q, C_1) = S(q, C_2)∧ |C_1|<|C_2|)where S(q, C) represents similarity values recorded in Algorithm <ref>. We then select a smaller component (in terms of the number of files) if two components have tied similarity, because it is likely a simpler version. Using the relation, we select a subset of components R_S from R:R_S = { C ∈ R | ∄ C_i ∈ R. C_i ⊃_S C } Table <ref> shows example similarity values S(q, C) for the example input in Fig. <ref>. Since the similarity values satisfy the condition of ⊃_S,we obtain R_S = {, } excluding .We assign a higher rank to a component that could provide a larger amount of code to the query files. To measure the degree of potential code reuse, we use the sum of file similarity S_Q(C) defined as follows.S_Q(C) =∑_q ∈ Q S(q, C)We rank components in the descending order of S_Q(C). Our method provides the following information to a user. * A list of components R_S sorted by S_Q(C). Each component is reported with attributes S_Q(C), |Q|, and |C|. The result for the example input is following. A pair of S_Q(C) and |Q| indicates the amount of reused code in Q. |C| is also important to analyze whether Q is a complete copy of C or not. * A full list of components R in the descending order of S_Q(C). We provide this list because our filtering may accidentally exclude an original component from R_S.A user can analyze all the components if necessary.* A table of similarity S(q, C). Although the component search step does not need file names,our implementation uses file names for this report. Table <ref> shows an excerpt of a similarity table fordirectory of Firefox 45.0. It shows that the analyzed directory is likely a clone of zlib 1.2.8 with some modification. It also shows that filein Firefox likely includes a similar change as MongoDB 3.2.8. These information enables a user to easily focus on candidates of original components and investigate actual source files in the components.§ EVALUATIONWe investigate two research questions to evaluate the effectiveness of our method. RQ1. Does our method accurately report an original component?RQ2. Is our method efficient?To answer the questions, we analyze actual clone-and-own instances in two products: Firefox 45.0 and Android 4.4.2_rc1. These projects reuse components in a well-organized manner. Firefox developers often record version numbers of reused components in commit messages in their source code repository. Android developers manage their own git repositories for cloned components and record Change-Id to identify changes of the original components. We manually analyzed directories whose names are likely cloned component names,and identified original versions using the commit-log messages.We then excluded components whose original versions are unidentifiable. We spent about one week for the analysis.Our database of components is the Snapshot Archive of Debian GNU/Linux <cit.>. We regard a version of a Debian package as a component. The archive includes all the existing source code packages released for Debian from 2005 until the present. We automatically downloaded files through its machine usable interface <cit.>. While Debian package maintainers sometimes apply their own patches, we included only original source tarballs whose names matched a pattern “”. The database includes 200,018 package files (868 GB in total).The resultant dataset includes 9,730,689 C/C++ files and 1,310,235 Java files.The total size is 5,733 MLOC (185 GB) including comments and white space.Our queries comprise 21 directories in Firefox and 54 directories in Android whose original versions are available as Debian packages. The directory names and corresponding Debian package names are included in the Appendix.Fig. <ref> plots the distributions of the numbers of files in each of directories. The queries include various size of components; the minimum one comprises two files, while the maximum one comprises 1,163 files. The medians of Firefox and Android queries are 76 and 79.5, respectively. Their total number is 13,720. Since the database includes copies of Firefox and Android themselves, we exclude their related packages from the search space.The baseline of evaluation is a simple file search using SHA-1 file hash instead of our similarity function (i.e., R = { C | Q ∩ C ≠ϕ}). We sort the extracted components in the descending order of |Q ∩ C| that is equivalent to S_Q(C).Our method uses five threshold for component search: th=0.6, 0.7, 0.8, 0.9, and 1.0 in order to evaluate the effect of threshold. The parameter th=1 ignores white space and comments, differently from the baseline.For each query, we obtain the rank of the original version in an extracted list. The rank approximates manual effort of a user. To evaluate the effect of our filtering method separately from search method,we use both our filtering result R_S and a full result without filtering R. In case of R_S, we assume that a user investigates all the elements in R_S,and then investigates R if R_S does not include an original component.§.§ Accuracy Fig. <ref> shows the distribution of the size of |R|, i.e. the number of reported components for each query.It shows that many similar files are included in various components. In case of the baseline method, the median of |R| is 60. Since our method detects similar files,a lower threshold results in a larger number of components. The medians of |R| are 145, 318, 367, 375, and 500 in cases of th=1.0 through th=0.6, respectively. It should be noted that the plots exclude queries that report no components. The baseline method reports no component for three queries. One of them is the smallest component whose files are modified. The other two components have differences in comments and white space.Fig. <ref> plots the number of selected components (|R_S|) for each query. As shown in this figure, the size is less than five components for most of queries. The median is 1 in cases of the baseline and th=1.0. The median is 2 in cases of th=0.9 through th=0.6.While a lower threshold results in a larger R_S, the difference is not statistically significant. Wilcoxon rank sum test results in p=0.244 for two cases th=0.9 and th=0.6. Our filtering approach successfully selects a small number of components. Table <ref> summarizes recall of components appeared in the top-k elements of resultant lists. For example, the baseline method ranks 64% of original components at the top of lists. The column “All” shows recall of components in the entire lists.The column “Rank Total” shows the sum of positions of the original components in the results.It approximates the effort to identify all the original components in the lists. Each row shows the result of a configuration. Each configuration uses a search method in the first column. The top six configurations simply reports a full list of R without our filtering. The bottom six configurations use a filtered list of R_S. The result shows that our method with th=0.9, R_S performs the best among the configurations. A user can identify all the original components in the top-5 components for 90.7% cases and 551 components in total.Our method reduces 40% of user's effort compared with the baseline reporting 931 components. The baseline method also requires additional effort to analyze three queries that resulted in no components.The result shows that modified files are important to identify original components. A simple exclusion of white space and comments does not improve the result;our similar file search with th=1 and R results in a longer list of components, although it slightly improves recall. Our method with th < 1 performs better than the baseline and th=1,because modified files provide a clue to sort components and identify an original version. A lower threshold does not improve a result, because it also detects various versions of the same components affecting the results.Our filtering method successfully improves results in all the cases. In case of th=0.9, it reduces 27% of manual effort to investigate the reported lists. This is because the most similar version of a library among several versions in R is actually the original version. In this experiment, only 882 out of 13720 query files (6.4%) are modified in Firefox and Android. Developers tend to reuse source files without modification.We do not regard other components in R_S as false positives, because some of them are also informative. For example, Android includes a modified version of libpng 1.2.46. Our method ranks the original version at the top. Our method also ranks libpng 1.2.49 at the second. Table <ref> shows an excerpt from the reported similarity values for them. It shows that filein Android is the same as a file in libpng 1.2.49. The file includes a security fix for CVE-2011-3048 <cit.> found in 1.2.x before 1.2.49. Our method successfully reports that the cloned component includes a part of the newer version. Our filtering method excluded some original components from R_S. An example is the Expat XML parser library in Android.Our method reports node-expat-2.3.12 that is a NodeJS binding component instead of the original version, because both Android and the component includes all the files in the original Expat library and an additional header filegenerated byscript. Another example is zlib 1.2.8 in Android. In this case, zlib 1.2.8 component in the database does not includefiles because of a license issue.Hence, our method accidentally reports another component including a full copy of the original version. This is a limitation of our repository-based approach. Our file search cannot identify any origins for 1240 files (9.04%) using th=0.6. They are likely added by the projects. We believe that this is also useful for a user to understand how a cloned component is modified. The baseline method does not provide this additional information.§.§ PerformanceOur experiment is performed on a workstation equipped withIntel Xeon E5-2690 v3 (2.6 GHz), 64 GB RAM, and 2 TB HDD. We use a single thread to execute a query.Figure <ref> shows the time spent for each query with different threshold. A lower threshold takes longer time, because it affects a size-based optimization of Algorithm <ref> (Line 5). The median is 77.7 seconds for th=0.6.The longest query takes 25 minutes. Table <ref> shows the median time required for each query and the total time for 75 queries for each parameter. The baseline method takes about 200 seconds to read files and compute SHA-1 file hash values. Comparison of SHA-1 file hash takes 70 seconds. While our method takes significantly longer time than the baseline,the time is still practical because a user can analyze a result of a query during execution of other queries. It should be noted that the query time is strongly dependent on component search. The component-ranking step takes less than 2 seconds in all the configurations. Our method compares each query file with a database of files. Since the total number of query files is 13,720,our method takes at most a few seconds to compare a file with 10 million files. This performance is achieved by b-bit minwise hashing technique. To analyze the effect of similarity estimation, we count the numbers of computed estimated and actual similarity values.They are shown in the columns #sim_e(q, f) and #sim(q, f) of Table <ref>.The table clearly shows that sim_e(q, f) enables us to avoid the computation of sim(q, f). The ratio of #sim(q, f) is reported to be less than 0.03% in all cases.The experiment uses m=0.1 in Algorithm <ref>.We executed the same experiment with m=0.2 and confirmed the same result. Hence, m=0.1 is sufficiently large. Fig. <ref> shows the distribution of errors between actual and estimated similarity. The lowest estimated similarity satisfying the sim(q, f) ≥ 0.6 is sim_e(q, f) = 0.513.Our implementation constructs a database of components in prior to search. We took about 5 days to extract all the files from package archives, remove duplicated files, and compute file signatures. The archive extraction step is the bottleneck, because the step has to process 2 TB of files in archives. Our database uses 4 GB for file signatures and 20 GB for component names and file names. Our implementation keeps file signatures on memory. Our database can be incrementally updated, by simply adding file signatures. A larger database can be hosted by multiple servers,because we can search multiple component sets independently and later merge the final result.§ THREATS TO VALIDITYWe analyzed Firefox and Android source code repositories, because developers in the projects keep their record of code reuse. Our result might be dependent on code reuse strategies of the projects.We manually analyzed commit messages recorded in the repositories. Since the version information is verified by two authors but no developers of the projects,the result has a risk of human error. The result is dependent on components included in a database. Our database represents a single software ecosystem: Debian GNU/Linux source code packages. Since we used original source code provided by each project, we believe the packages reflect activities of various projects. On the other hand, our analysis does not reflect source code modified by package maintainers of the operating system.The collection of software packages may miss important packages. For example, libpng project maintains a number of branches: libpng 0.x, 1.0.x, 1.2.x, 1.4.x, 1.5.x, 1.6.x, and 1.7.x. Since package maintainers selected major branches to create Debian packages,our dataset includes a subset of official versions.It may affect the analysis of variants in the experiment.While the Debian Snapshot Archive is publicly available,we found several errors (i.e., 404 not found) and corrupted archive files during our analysis. Since the entire dataset is large, the result might be affected by this accidental data corruption.The performance of b-bit minwise hashing signature is dependent on underlying hash functions. It may miss a similar file with a very low probability. We confirmed that we did not miss any files during the experiment. For replicability, our strategy to define 2048 hash functions is included in Appendix. § CONCLUSIONThis paper proposed a code search method to extract original components from a software ecosystem. In the experiment, our method successfully reported original components compared with the baseline method. Our implementation also reports the computed similarity values to enable further analysis.To implement an efficient code search, we used b-bit minwise hashing technique. It enabled us to extract less than 0.03% of likely similar files from a database in a second. Our method also introduced a component filtering method using aggregated file similarity. It reduces manual effort to analyze reported components.In future work, we would like to apply our method to analyze clone-and-own reuse activity in various projects including industrial organizations. We are also interested in a systematic method to analyze known issues and vulnerabilities caused by cloned components in a software product.§ ACKNOWLEDGMENT This work was supported by JSPS KAKENHI Grant Numbers JP25220003, JP26280021, and JP15H02683.We are grateful to Naohiro Kawamitsu for the implementation of hash functions and Raula Gaikovina Kula for his comments to improve the manuscript. Hash Function.Our implementation uses 2048 hash functions to translate a file into a file signature. We use the following hash functions on 64-bit integer:h_i(t) = a_i × base(t) + b_iwhere 1 ≤ i ≤ 2048, a_i and b_i are randomly generated 64-bit integers. The base(t) function translates a trigram into a 64-bit integer.Since a trigram in a multiset is identified by four elements , , , and i (i-th occurrence of a trigram ), the base function is defined as follows.base(, , , i) = (((i × 65537) + ) × 65537 + )× 65537 +whereismethod in Java. Component List. 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"authors": [
"Takashi Ishio",
"Yusuke Sakaguchi",
"Kaoru Ito",
"Katsuro Inoue"
],
"categories": [
"cs.SE"
],
"primary_category": "cs.SE",
"published": "20170427010612",
"title": "Source File Set Search for Clone-and-Own Reuse Analysis"
} |
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